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Faraday Discuss. Chem. Soc., 1982, 73, 153-172 \ Energetics and Dynamics of Large Van der W-aals Molecules BY UZI EVEN, AVIV AMIRAV, SAMUEL LEUTWYLER," MARY JO ONDRECHEN,? ZIVA BERKOVITCH-YELLIN $ AND JOSHUA JORTNER Department of Chemistry, Tel-Aviv University, Tel Aviv 69978, Israel Received 2nd December, 198 1 We report on the synthesis, identification, excited-state energetics, interstate electronic relaxation and intrastate nuclear dynamics in electronically-vibrationally excited states of Van der Waals molecules, consisting of a large aromatic molecule bound t o inert-gas atoms. INTRODUCTION A Van der Waals (VDW) complex, consisting of an organic aromatic molecule (M) bound to inert-gas (R) atoms '-lo can be viewed as a " guest " molecule embedded in a well-characterized local solvent configuration.Studies of excited-state energetics and dynamics of such large VDW complexes are expected to provide basic information on solvent perturbations, as explored from the microscopic point of view. With this goal in mind, we have undertaken to explore the spectroscopy and intramolecular dynamics of electronically-vibrationally excited states of a variety of MR, complexes, where M constitutes a substituted benzene (aniline), a medium-sized aromatic (fluorene), a large linear polyacene (anthracene, tetracene and pentacene) or a very large aromatic molecule (ovalene). In this paper, we shall focus on some of the features of these large VDW complexes, which can be classified in the following manner: A. Chemistry A.l Formation kinetics of MR, molecules.A.2 Identification and characterization of the chemical composition of MR, com- A.3 Specification of chemical isomers. A.4 Estimates of VDW binding energies. plexes. B. Spectroscopy B.l Microscopic spectral shifts of the electronic origin of M in MR,. B.2 Solvent effects on intramolecular vibrations of M in MR,. B.3 Intermolecular vibrations of low-frequency, large-amplitude M-R motion. C . Intramolecular Dynamics C.l Microscopic solvent effects on electronic relaxation of M in MR,. * Present address : Physikalisch Chemisches Institut der Universitat Basel, Klingelbergstrasse 80, 4056 Basel, Switzerland. 1- Present address: Department of Chemistry, Northeastern University, Boston, Massachusetts 021 15, U.S.A. $ Permanent adress: Department of Structural Chemistry, The Weizmann Institute of Science, Rehovot 76100, Israel.154 LARGE VAN D E R WAALS MOLECULES C.2 Non-reaction intramolecular vibrational energy flow in MR, complexes, involving C.3 Reactive vibrational predissociation of large MR, molecules.both intermolecular M-R modes and intramolecular modes of M. The understanding of the structure, energetics and dynamics of large VDW molecules requires a different conceptual framework than that applied, so successfully, to small VDW complexes. In the latter case, detailed spectroscopic information on realistic potential energy surfaces can be ~btained,""~ serving as central input data for the description of energy levels and relaxation phenomena. This approach is prohibitively difficult for large VDW complexes in view of the huge dimensionality of the potential surface.The elucidation of the features of such large VDW complexes currently rests on experimental spectroscopic information for the energetics, which is supplemented by crude model calculations of the potential surfaces for the elucidation of the structure and energetics, toget her with coarse-graining procedures for the description of intramolecular dynamics. EXPERIMENTAL The large VDW complexes were synthesized in seeded supersonic expansion^,^^^^^ where by an appropriate choice of the experimental conditions, i.e. stagnation pressure and the nozzle diameter, one can accomplish selective, gradual and controlled solvation of the arom- atic molecule. The experimental interrogation techniques are based on energy-resolved and time-resolved laser spectroscopy in seeded supersonic expansions.Supersonic expansions of three types were produced : (a) continuous axial-symmetric jets from a circular n~zzle,'~*'~ (b) pulsed axial-symmetric jets,* (c) pulsed planar supersonic jets from a nozzle slit.I6 The rare-gas diluent at the stagnation pressure p -- 10-1 O4 Torr was mixed with the organic vapour (at pressures of 10-2-1 Torr) in the heated sample chamber and sent through the nozzle. The following interrogation methods were utilized : (1) fluorescence excitation spectra,', corresponding to the intensity of the total fluorescence versus the laser wavelength, (2) energy-resolved emission ~pectra,'~ which correspond to the energy-resolved fluorescence resulting from excitation at a fixed laser wavelength, (3) time-resolved emission,lS correspond- ing to time-resolved fluorescence excited at a fixed laser wavelength, (4) energy-resolved and mass-resolved spectroscopy, * based on resonant two-photon ionization in conjunction with time-of-flight mass spectrometric detection of the large VDW ions.The fluorescence studies (1)-(3) were conducted in free supersonic jets, while ion spectroscopy method (4) was carried out in skimmed jets. RESULTS AND DISCUSSION ELECTRONIC EXCITATIONS OF V A N DER WAALS COMPLEXES Fig. 1 shows small portions of the fluorescence excitation spectra of several large molecules in supersonic expansions of Ar. On the basis of spectroscopic studies of these m o l e c ~ l e s , ~ * ~ * ~ ~ ~ ' ~ we can assign the electronic origin of the lowest spin-allowed S,+S, electronic transition of the bare ultracold molecule, which is marked 0-0 in fig.1 . The satellite bands, appearing in the low-energy side of the 0-0 transition of the bare molecule, are characterized by the following features: (1) Their energies depend on the identity of the diluent rare gas (fig. 2). (2) Their intensities exhibit a strong dependence on the stagnation pressure (fig. 3). (3) At higher pressures the spectral feature at lower energies become more prominent. All the spectral features appearing on the low-energy side of the electronic origin of the bare molecules (fig. 1 and 2) are assigned to S,+S, electronic excitations of MR, molecules. Such an inter-EVEN et a l . 155 0- 0 NH2 0-0 n= 1 n= 2 II I I I pu- I I I +Ar I I I - 4460 4 4 8 5 4510 5360 5380 5 4 0 0 wavelength/ A FIG.1 .-Fluorescence excitation spectra of the So+Sl electronic origin of several bare molecules, M, and MAr, complexes in supersonic expansions of Ar. The spectra were obtained under the follow- ing conditions: (1) aniline + Ar: continuous expansion from a 50 ym nozzle at pressure p = 1060 Torr and temperature T = 70 "C; (2) fluorene + Ar: pulsed expansion from a 35 mmx 0.2 mm nozzle slit at p = 150 Torr and T = 105 "C; (3) tetracene 4- Ar: continuous expansion from a 150 pm nozzle at p = 609 Torr and T = 200 "C; (4) pentacene + Ar: continuous expansion from a 200 ym nozzle at p = 295 Torr and T = 280 "C. The electronic origin of the bare molecule is marked as 0-0 and the coordination numbers of the MR, complexes are marked, pretation is borne out by the spectroscopic diagnostic methods, which will now be considered.SPECTROSCOPIC DIAGNOSIS The following spectroscopic diagnostic methods were utilized for the identification and characterization of the individual spectral features, which could unambiguously be assigned to distinct MR, VDW molecules. (A) COMPLEXATION OF BARE MOLECULES BY RARE GASES We have followed the reduction of the intensity [MI of the 0-0 transition of a bare As is evident from fig. 4 and 5, [MI molecule by increasing the stagnation pressure. obeys the relationship Thus, the attachment of the first R atom to a large aromatic molecule proceeds via a three-body collision M -+- R + R 2 MR1 + R. [MI = [MI0 exP(--lp2).156 LARGE VAN DER WAALS MOLECULES n = 2 n= 1 I I I I I 4460 4485 4510 4535 4560 wavelength/A FIG.2.-The fluorescence excitation spectra of tetracene-R, (R = Ar, Kr and Xe) in continuous supersonic expansion of rare gases. Tetracene at 200 "C (vapour pressure 0.1 Torr) was mixed with Ar (p = 710 Torr), Kr ( p = 465 Torr) or Xe ( p = 181 Torr) and expanded through a 150 p m nozzle. The electronic origin of the bare molecule is marked as 0-0 and the coordination numbers of the tetracene-R, complexes are marked. It is interesting to note that even for the huge pentacene molecule, which is char- acterized by 102 vibrational degrees of freedom, '' sticky " two-body collisions are ineffective in the stabilization of a longlived complex, and three-body collisions are necessary.The three-body mechanism is universal, applying to complexation of diatomics l8 and to huge molecules. (B) PRESSURE DEPENDENCE OF THE INTENSITIES The relative intensities of the spectral features attributed to individual MR, VDW molecules were found to exhibit a pressure dependence of the formp'" at low values of the stagnation p r e s s ~ r e . ~ ~ ~ Such p2" power law l8 reflects a three-body recombin- ation mechanism for the formation of higher MR, complexes MR,-, + R 4- R % MR, + R where K, is the three-body rate constant. Provided that these rates K,, are weakly dependent on n, we can invoke a " democratic assumption", setting K,, = K for all n. The distribution of the MR, complexes is then Poissonian, with the distribution para-EVEN e t a l .157 0-0 4460 4510 4560 wavelength/A FIG. 3.-Fluorescence excitation spectra of the So-+S, electronic origin of tetracene and of tetracene- Ar, complexes in a continuous supersonic expansion through a 160 pm nozzle. The backing pressures of Ar are (a) 180, (b) 478, Cc) 609, (d) 710 and ( e ) 853 Torr. The vibrationless 0-0 of the bare tetracene molecule is marked.158 LARGE V A N DER WAALS MOLECULES r------ 0 1 2 3 4 5 6 7 8 p21105 TO+ FIG. 4.-Dependence of the intensity [TI of the 0-0 transition of the bare tetracene molecule on the stagnation pressure,^, of the rare gas in a continuous expansionjn Ar.(O), Kr (0) and Xe (0) through a 150 p m nozzle. The nozzle temperature was 220 "C. The tetracene pressure was maintained con- stant in all experiments [TI = [To] exp (-lip;).pi/105 Torr2 FIG. 5.-Dependence of the intensity of the 0-0 transition of the bare aniline, tetracene and pentacene molecules in continuous supersonic expansions of Ar on the stagnation pressure : A, aniline-Ar, T = 70 "C, D = 50 ,urn; 0, tetracene-Ar, T = 220 "C, D = 150 pm; #, pentacene-Ar, T = 280 "C, D = 200pm.EVEN e t a l . 159 meter being Kp2. The mean value of the complex size is ( n ) = Kp2, while the variance of the distribution is An = K3p. The relative spectral intensity [MR,] of MR, is given by This relation was utilized for the analysis of the spectral features of tetracene-Ar, and of tetracene-Kr, complexes, which are portrayed in fig. 6 and 7. Seven distinct [MRflI/[Ml = (KP2)”. I - 60 - 40 - 20 -10 - 8 - 6 - 4 -2 100 200 400 600 1000 p/Torr FIG.6.-Dependence on the Ar pressure of the normalized intensities [tetracene-Ar,]/[tetracene] = [tetracene-Ar,] exp(Klp2) of the spectral features of the tetracene-Ar, molecules, normalized by the intensity of the bare molecule, for supersonic expansions of tetracene in Ar. The upper insert shows the fluorescence excitation spectrum of tetracene seeded in Ar expanded at p = 710 Torr. The individual spectral features are labelled by the indices j = 1,2 . . . 7 in the order of increasing wave- length. The pressure dependence of the normalized intensities of these spectral features is of the form p2”. The values of n obtained from this analysis are shown in the spectrum (upper insert). spectral features in the tetracene-Ar system were assigned to excitations of tetracene- Ar, complexes with coordination numbers n = 1 .. . 7, while seven spectral features in the tetracene-Kr system could be attributed to excitations of tetracene-Kr, com- plexes with n = I . . . 4 , with several distinct features corresponding to each of the chemical compositions, tetracene-Kr, and tetracene-Kr,. (c) ORDER OF APPEARANCE OF THE SPECTRAL FEATURES The order of appearance of the excitations of the MR, complexes with increasingp is expected to be sequential, starting with the spectral features due to excitations of1 60 LARGE V A N DER WAALS MOLECULES 7 n z i . 2 4460 4510 4 560 X I A ,,- -/o ___L_ I I I 150 200 300 4 0 0 p/Torr FIG. 7.-Dependence on the Kr pressure of the normalized intensities [tetracene-Kr,]/[tetracene] = [tetracene-Kr,,] exp( -K& of the spectral features of tetracene-Kr,, molecules, normalized by the intensity of the bare molecule, for supersonic expansions of tetracene in Kr.The upper insert shows the fluorescence excitation spectrum of tetracene seeded in Kr expanded at p = 465 Torr. The individual spectral features are labelled by the indices j = 1 . . . 10 in the order of increasing wave- length. The pressure dependence of the normalized intensities of these spectral features is of the form pz". The values of n obtained from this analysis are shown on the spectrum (upper insert). the complexes with n = 1, while, at higher values of p , complexes characterized by higher coordination numbers will be exhibited. This qualitative criterion has been extremely useful in the identification of MR, complexes with low n (= 1-3) values.(D) RESONANT TWO-PHOTON IONIZATION OF VDW MOLECULES The laser-induced fluorescence techniques (A)-(C) are applicable only to prominent absorption bands of fluorescent complexes. The utilization of tuneable laser two- photon ionization (2PI) method 19-21 to VDW molecules, combined with time of flight mass spectroscopy of the resulting MR: ions: provides an unambiguous identification of the detailed spectral features of MR, complexes. We applied the resonant 2PI method for the spectroscopy of fluorene-Ar, (n = 1-3) complexes. Fig. 8 shows the mass-resolved ion signal spectra of fluorene and of fluorene-Ar: (n = 1-3) ions. These energy-resolved and mass-resolved spectra monitor the spectral features of theEVEN e t a l .161 energy lcrn-' 33 800 33750 33700 I I I I I 2958 2962 2966 2970 wavelength/A FIG. 8.-Ion current plotted against laser wavelength of fluorene+ (a, m/e = 166), fluorene-Ar+ (b, m/e = 206) and fluorene-Ar2+ (c, m/e = 246). Fluorene at 1 0 0 "C (vapour pressure 1.4 Torr) was seeded into Ar at p = 1200 Torr and expanded through a 300 pm pulsed nozzle. The jet was skimmed and photoionized by the laser in the ion source of a time-of-flight mass spectrometer. intermediate resonant states, which involve the vibrationless S,+S, excitations of individual fluorene-Ar,, molecules. As is evident from table 1 , there is an excellent agreement between the prominent features in the fluorescence excitation spectra and the ion current spectra of fluorene-Ar, complexes, providing conclusive evidence for the assignment of the spectral features of these large VDW molecules.In addition, the resonant 2PI method unveils some additional weak spectral features of the com- plexes which are obscured in the fluorescence spectra, providing information on the multiplet structure of these electronic excitations. Finally, it is worthwhile to note162 LARGE V A N D E R W A A L S MOLECULES TABLE 1 .-EXCITATION ENERGIES OF FLUORENE-R, COMPLEXES (IN cm-l) a - d mass-resolved resonant 1 aser-i nduced molecule two-photon ionization fluorescence fluorene 0 0 fluorene-Ar, 41 fluorene-Ar2 - 76 - 75 fluorene-Ar3 - 118 - 120 fluorene-Krl - 33(W) - 63 - 33(W) - 63 a Energies (in cm-') represent spectral shifts from the electronic origin of the bare molecule.Negative signs correspond to red shifts. bThe origin of the bare molecule determined by mass- resolved resonant 2PI is at 2960.2 f 0.1 A, which is in agreement with the laser-induced fluorescence method and which gives 2960.0 f 0.1 A. The absolute accuracy of wavelength scale is *0.5 A. The weak satellite bands are denoted by (W). Accuracy of spectral shifts is f1.5 cm-'. TABLE 2.-EXCITATION ENERGIES 6V OF MR, COMPLEXES RELATIVE TO THE 0-0 TRANSITION OF M (ENERGIES GIVEN IN cm-') 4 - c Van der Waals interrogation complex n = l n == 2 n = 3 n = 4 method aniline-Ne,, -4.8 -9.6 fluorene-Ar,, - - 76 fluorene-Kr, - 63 - 33(W) -113 fluorene-Xe , - 73(W) - 63 tetracene-Ar , - 35 - 88 - 109 - 155 (A), (C) t etr acene-Kr - 70 -110 - 186 - 248 (A), (C) - 150(W) -223(W) - 200(W) t e t r acene-Xe ,, - 101 - 189 -83(W) - 147(W) - pen tacene-Ar, - 29 - 25 - pen tacene-Kr , - 62 - 55 (I Negative signs of 6v denote red spectral shifts.The accuracy of the 6v values is &2 cm- for aniline and fluorene complexes and &4 cm-' for tetracene and pentacene complexes. When multiple spectra, corresponding to MR, complexes with a fixed value of n, are exhibited, all the peak The weak satellite bands are labelled as (W). energies are listed.EVEN e t a l . 163 that resonant 2PI via the electronic origin of these VDW molecules results in the photo- selective production of large VDW ions, which are of considerable interest. Table 2 summarizes the energies of the lowest excitation of MR, molecules, each being given relative to the 0-0 transition of the corresponding bare M molecule.Only those features which were definitely identified using the spectroscopic diagnostic methods were included. These spectral features provide a rather comprehensive picture of the electronic excitations of MR,, which do not change the internal vib- rational state of M. These spectroscopic data provide a characterization of the composition of large VDW molecules. These " chemical type" data will now be supplemented with some crude information regarding potential surfaces to elucidate some features of the structure and intermolecular nuclear motion in these complexes. MODEL POTENTIALS One approach towards the elucidation of the structure and nuclear vibrational motion of large VDW molecules rests on model calculations of the intermolecular interactions. The potential surfaces of aromatic-rare-gas complexes were construc- ted as a superposition of pairwise atom-atom potentials, the R-carbon atom pair potentials being taken from the heats of adsorption of rare-gas atoms on graphite,22 the R-hydrogen atom pair potentials were estimated using empirical combination rules 23 and the R-R interaction pair potentials were represented in the conventional 6-12 form.23 Fig.9 and 10 provide a view of the potential surfaces of tetracene-Ar, FIG. 9.-Contour map for the tetracene-Ar, potential in the plane parallel to the aromatic plane at the distance z = 3.45 A from it. The innermost contour is at E = - 1.5 kcal mol-' and subsequent contour intervals are separated by 0.2 kcal mol-'.and of ovalene-Ar, parallel to the molecular xy plane at a distance of z = 3.45 8, away from it. Details of the potential surface of MR, complexes are portrayed in fig. 11 for tetracene-R, and in fig. 12 for pentacene-R, complexes. The following notable features of the potential surfaces of MR, complexes, consisting of a rare-gas atom bound to a linear polyacene, should be noted. First, the motion parallel to the short molecular axis x is characterized by a single potential minimum for all polyacenes. Secondly, the potential for motion parallel to the long molecular axis is characterized by several minima or inflection points, whose number is equal to the number of aro- matic rings. Thirdly, for complexes containing an even number of aromatic rings, there are two equivalent points of minimum energy on each side of the molecule, which give rise to " tunnelling type " motion of the R atom. Fourthly, for complexes containing164 LARGE V A N DER WAALS MOLECULES X Y FIG.10.-Contour map for the ovalene-Ar' potential in the plane parallel to the aromatic plane at the distance of z = 3.45 A from it. The innermost contour is at E = -1.80 kcal mo1-' and sub- sequent contour intervals are separated by 0.10 kcal mol-'. an odd number of rings, i.e. benzene, anthracene and pentacene, there is a single absolute minimum energy on each side. Fifthly, for the large odd-ringed pentacene, there are two energetically close but distinct potential minima on each side, differing by ca. 10 cm-l. Sixthly, the R atom parallel to the molecular plane exhibits large amplitude non-harmonic motion.Seventh, the potential energy for the motion of the R atom away from the molecular plane for all these polyacene complexes exhibits a universal form, which is close to the familiar 6-12 potential. Most of these gross features of the potential surfaces are determined essentially by the topology of the interaction. We now turn to somewhat less reliable numerical results pertaining to the ground- state energetics and structure. Table 3 summarizes the energies of the potential minima, which provide an estimate of the ground-state binding energies for a variety of complexes. The intermolecular equilibrium separation ro and the frequency v for the motion of R with respect to the molecular plane are summarized in table 4 for tetracene-R, molecules, being characteristic for all of these large complexes.Similar model calculations provided information on the equilibrium structure of MR, complexes containing several rare-gas atoms, which will briefly be considered in relation to the cardinal question concerning the existence of isomers. Regarding MR, complexes of large polyacenes, one can argue, without alluding to any numerical calculations, that the binding of two R atoms to a large aromatic molecule will result in an energetically favoured configuration, with the two R atoms being located on the same side of the aromatic ring. This single-sided structure is stabilized by the attractive interaction between the pair of the R atoms. This expectation is borne out by numerical calculations for tetracene-R, molecules (R = Ar, Kr and Xe), where the stabilization energy of the one-sided structure relative to the two-sided configurationEVEN e t al.165 - 6.0 - 2.0 2.0 6.0 Y -1.6 I I I I I I -4.8 0.0 4 . 8 Y / A FIG. 11.-Tetracene-Ari potential along the x, y and z axes. -1.30 -1.40 -1.50 -1.601 I 1 1 I I -4.0 -2.0 0 2 .o 4 .O Y / A FIG. 12.-Pentacene-Art potential along the x, y and z axes.166 LARGE V A N DER WAALS MOLECULES TABLE 3.-DISSOCIATION ENERGIES FROM THE MINIMA OF MODEL POTENTIALS FOR MR1 COMPLEXES (ENERGIES GIVEN IN cm-') a-c Ar Kr Xe \ R M \ benzene 393 [2] 458 [2] 545 [2] ant hracene 519 [2] 612 [2] 753 [2] te tracene 531 [4] 630 [4] 777 [4] 644 [2] 798 [2j pent acene a The contribution of the zero-point energy of M-R motion to the dissociation energy is neglected.For benzene, anthracene, tetracene and ovalene, a single value of the minimum energy is exhibited. In the case of pentacene, two minima appear in the potential surfaces, the energy of the second minimum being given in round brackets. 'Number of points of minimum energy are given in square brackets. is ca. 40-60 cm- l. Accordingly, at low temperatures a single equilibrium configuration is expected to be energetically favoured, and no isomers are expected for tetracene-Ri, complexes. The situation is more complicated and interesting for MR, complexes. Model calculations for tetracene-R, complexes raise the distinct possibility of the TABLE 4.-ESTIMATES OF FREQUENCY (V), DISSOCIATION ENERGY (D), NUMBER OF BOUND VIBRATIONAL LEVELS (2) AND EQUILIBRlUM DISTANCES (Yo) FOR THE PERPENDICULAR VIBRATIONAL MOTION OF A RARE-GAS ATOM IN TETRACENE-R1 COMPLEXES rare gas v/cm - D/cm- .x rolA Ar 57 53 1 18 3 .4 5 Kr 45 630 28 3.45 Xe 41 777 38 3 . 7 0 peaceful coexistence of two isomers, a two-sided structure and a triangular same-sided configuration, whose energies are very close.9 Such a pattern of nearly isoenergetic isomers is expected to be exhibited for large MR, complexes, which are characterized by a high coordination number n > 3. DISSOCIATION ENERGIES The model calculations of the preceding section indicate that the binding energies, D, of R atoms to aromatic molecules are relatively large. Pertinent experimental information regarding these binding energies was obtained from three sources : (1) Resonant 2PI of a MR1 complex, proceeding via an intermediate excitation of an intramolecular vibration in S1, will result in the M+ ion when the vibrational predissociation (VP) channel is open in the intermediate state while, when the VP channel is closed, extensive production of the MRC ion is expected. This approach was utilized to set upper and lower bounds on D for fluorene-R, complexes.(2) The lack of VP from an electronic-vibrational excitation of a MR, VDW com- plex, as interrogated by energy-resolved emission, indicates that the reactive channelEVEN e t a l . 167 is presumably closed. This approach was used to set a lower limit for D of tetra- cene-Ar,. (3) The heavy-atom effect on the decay lifetime of the S, state of VDW complexes was adopted to search for the onset of VP in tetracene-Kr, complexes, establishing both a lower and an upper limit for D .Table 5 summarizes the experimental findings for the binding energies of R atoms to TABLE S.-EXPERIMENTAL ESTIMATES OF BINDING ENERGIES OF LARGE VAN DER WAALS COMPLEXES complex dissociation energy/cm- method fluorene-Ar I 408 < D < 728 (1) t etracene-Ar D > 314 (2) tetracene-Krl 314 < D < 1200 (3) ' (1) Resonant 2P1, (2) energy-resolved emission, (31 vibrational predissociation interrogated by the external heavy-atom effect. large aromatics in the S, state. To obtain estimates of the ground-state D values, these values should be corrected by the (small) difference in the binding energy between S , and So.These experimental estimates of D concur with the results of our model calculations. MULTIPLE SPECTRA The spectral features of an intramolecular vibrationless excitation of a MR, complex, which corresponds to a single chemical composition (table 2), fall into two categories : (1) A single broadened spectral feature is exhibited for the MR, complex. This is the case for some large complexes, i.e. tetracene-Ar, (n = 1 and 2) and tetracene-Kr, (n = 1 and 2). Our model calculations predict that for tetracene-R, and tetracene- R2 complexes only a single chemical isomer is energetically stable, which is in accord with the experimental data. (2) A multiple spectrum is exhibited with several distinct spectral features for a MR, complex(es) with a fixed n.This state of affairs prevails for tetracene-Xe, (n = 1 and 2), tetracene-Kr, and tetracene-Kr,, fluorene-R, (R = Ar, Ke and Xe) and pentacene-R, (R = Ar, Kr and n = 1,2). The broadening of the spectral features and the appearance of multiple spectra can originate from the following causes: (A) The existence of nearly isoenergetic chemical isomers. The two separate spectral features in the spectrum of tetracene-Kr, are assigned to the existence of two isomers. These are characterized by a two-sided configuration and by a one-sided triangular configuration. The three distinct spectral features of tetracene-Kr, are also tentatively assigned to distinct chemical isomers, as implied by the model cal- culations presented earlier in this paper. (€3) Thermal population of intermolecular M-R vibrational states in So, resulting in hot bands and in sequence bands involving low-frequency modes.The broadening of the spectral features of tetracene-R, and tetracene-R, (R = Ar and Kr) may be due to this effect. (C) The existence of several conformers. When the potential surface of the com- plex is characterized by two or more close lying but non-degenerate minima, the168 LARGE VAN DER WAALS MOLECULES vibrational structure will consist of several close lying levels. Thermal populations of such vibrational states, which prevail even at the low temperatures attained in super- sonic expansions, are expected to result in several distinct vibrational excitations. Vibrational conformers differ from isomers [case (A)], as in the latter case the potential barrier separating the different structures is very large. Conformer splitting is expected to be exhibited by the odd-ringed pentacene-R, complex (fig.12). The multiple doublet spectra of pentacene-Ar, and of pentacene-Kr, (table 2) are tentatively attributed to conformers, whose minimum energies are separated by ca. 10 crn-'. (D) Vibrational structure due to M-R intermolecular motion in the S, state. This will be amenable to observation, provided that the relevant intermolecular vibrational frequency is sufficiently high and that the Franck-Condon vibrational overlap is favourable. Such vibrational structure is expected to be exhibited towards higher energies from the main vibrationless peak. The vibrational frequency for out-of-plane motion of R in MR, is 40-50 cm-I (table 4), being amenable to observation.The weak secondary features in tetracene-Xe, (n = 1 and 2) and in fluorene-R, (R = Ar, Kr and Xe) correspond to excitations of 30-50 cm-l and are tentatively attributed to intermolecular vibrational excitations in S,, which are in accord with the model calculations. MICROSCOPIC SPECTRAL SHIFTS The energetic shift, 6 v , of the electronic origin of the So -+ S, excitation of a well- characterized MR, VDW complex relative to the 0-0 transition of the bare M molecule corresponds to the spectral shift induced by a well-defined microscopic solvent structure. The quantitative data for these spectral shifts (table 1) of aromatic hydrocarbons exhibit the following features : (1) The spectral shifts induced by Ne, Ar, Kr and Xe atoms are all towards lower energies.Such red spectral shifts are assigned conventionally to the change in dis- persive interaction^.^^ It is a quite simple matter to extend the Longuet-Higgins- Pople formalism 24 to large MR, complexes bypassing the dipole-dipole expansion by the use of a multicentre monopole expansion.25 The red dispersive spectral shift for MR, can be expressed in the form dvdis = a ~((R,b>(x,),(y,),{C~a),{Ejk),~), where a is the polarizability of the R atom, R a b is the distance between the ath C atom located at (xcr,y,,z, = 0) and the R atom, (Cja} are the molecular-orbital coefficients, (Ejk} are 7t -+ 7t * excitation energies, I? stands for an average excitation energy of R, while the explicit form of the function F( ) is derived by second-order perturbation theory.(2) The microscopic spectral shifts exerted by a single R atom reveal a linear dependence on the polarizability of the perturbing atom dvdis cc tc (fig. 13) as is expected for dispersive interactions. (3) The spectral shifts of MR, complexes are only approximately additive per added R atom (fig. 14). Deviations of the order of 10-20% from additivity of Svdis do not seem to originate from three-body dispersive interactions, but rather from the occup- ation of geometrically inequivalent sites by the successive R atoms, which complex with M. The dependence of 6v on the nature of the electronic excitation of the aromatic hydrocarbon has not yet been elucidated. In general, one expects that the dispersive contribution to 6 v will increase with increasing oscillator strength, f, of the elec- tronic t r a n ~ i t i o n .~ ~ * ~ ~ This expectation is borne out by the observation of small spectral shifts for the So -+ S , transition of ovalene-R, (R = Ar and Kr) complexesEVEN e t al. 1 69 0 1.0 2 .o 3.0 4 .O ~ r / l O - ~ ~ cm3 FIG. 13.-Dependence of the red spectral shifts 6v of the vibrationless So+Sl excitations of tetracene- R1 and fluorene-R1 (R = Ne, Ar, Kr and Xe) molecules on the polarizability 0: of the rare-gas atom. The solid straight line, which passes through the origin, corresponds to the relation Sv = Act. The proportionality constants A are very close for the two families of complexes; however, other MR1 families are expected to be characterized by different A values.(withf% which are considerably lower than the 6v values for tetracene-R, complexes (where,f= lo-'). In complexes consisting of alternating non-substituted hydrocarbons bound to heavy rare-gas atoms, the dispersive contribution to 6v is expected to dominate. An additional electrostatic contribution to 6v, originating from dipole-induced-dipole interactions (DIDI) is expected to be exhibited for complexes containing polar molecules, which undergo an appreciable change in their dipole moment upon elec- tronic excitation. This is the case for aniline-Ar, (n = 1 and 2) complexes, where the dipole moment of the aniline molecule is I .5 D * in So and 2.4 D in S,.26 The aniline- Ar, complexes exhibit a large red spectral shift (table 1). We have adopted a simple electrostatic model for the evaluation of the contribution 6vDIDI from DIDI to 6v using the electron density in the So and S, states 26 and locating the Ar atom at 3.45 A above the centre of the ring, resulting in SvDIDi % -4 cm-'.This contribution from DIDI is in the right direction; however, the major contribution to 6v originates from dispersive terms. An additional contribution to 6v for So -+ S1 transitions is expected to originate from short-range repulsive interactions in S1, which result in spectral shifts to higher energies. For He and Ne complexes dispersive interactions are weak and these repulsive interactions are important. Blue spectral shifts exhibited for He complexes essentially originate from this effect. Up to this point spectral shifts were considered within the framework of a frozen nuclear picture.The possible effects of nuclear motion on 6v are intriguing in this * 1 D FS 3.33356 x 10-30Cm.170 LARGE VAN DER WAALS MOLECULES n FIG. 14.-Dependence of the red spectral shifts 6v of the vibrationless So+S1 excitations of the tetracene-R,, [R = Ar (O), Kr (0) and Xe (A)] and of the fluorene-Ar,, (0) molecules on the co- ordination number n. The straight lines are provided for the sake of a visual demonstration of the additivity of the spectral shifts per added R atom. context. The simplest effect in this category involves the modification of the vibra- tional frequencies in MR,, relative to those of the bare molecule. We are not aware of any information concerning this effect. INTRAMOLECULAR DYNAMICS Interstate and intrastate intramolecular dynamics of MR, complexes are of intrinsic interest for the understanding of the effects of external perturbations on electronic relaxation phenomena and for the elucidation of the nature of internal energy flow in weakly bound systems. The exploration of interstate dynamics of VDW molecules provides information on microscopic solvent perturbations on electronic relaxation (ER) from photoselected states of well-characterized MR, complexes.A prediction 28 of the theory of intra- molecular ER in large molecules, corresponding to the statistical limit, is that the ER rate of the electronic origin is practically unaffected by an " inert " solvent, which does not modify the energy levels and the intramolecular coupling.As is apparent from the lifetime data assembled in table 6, the lifetimes z of the vibrationless S, excitations of tetracene-Ne,, tetracene-Ar, (n = 1-4) and pentacene-Ar, (n = 1 and 2) complexes are close to and somewhat longer than the lifetime zo of the S1 origin of the correspond- ing bare molecules, which is in accord with theory.28 On the other hand, the lifetimes of tetracene-Kr,, pentacene-Kr, and tetracene-Xe, reveal a dramatic shortening of z relative to zo(table 6). This is attributed to the external heavy-atom effect (EHAE) onEVEN e t a l . 171 TABLE 6.-DECAY LIFETIMES FROM THE VIBRATIONLESS S, STATE OF MR, MOLECULES molecule z/ns T TNe, TArl TAr, TAr3 TAr, TAr, TAr6 TAr, TKr, TG2 TKr3 TKr, TXe, TXe2 P PAr, PAr2 PKr, 19 33 29 35 17 21 27 21 26 7 8 9 9 <3 t 3 19 22 19 6 T = tetracene, P = pentacene; accuracy of lifetimes f 10%.the S, + T, intersystem crossing 394 manifesting the modification of the non-adiabatic coupling in M. Two scrambling mechanisms inducing the EHAE can be operative,29 i.e. (I) mixing with neutral excitations of the heavy R atom, and (11) mixing with charge transfer states. Mechanism (I) is cumulative with respect to n, while mechanism (11) is determined by M-R pair interactions. The observation (table 6) that the lifetimes of tetracene-Kr,, (n = 1-4) molecules are very close, showing only a slight variation with increasing n, implies that for these complexes mechanism (11) is dominating. The understanding of intrastate reactive vibrational predissociation (VP) 28 and non-reactive vibrational energy flow 28 in large MR, complexes is still in the embryonic stage.In a vibrationally excited MR, VDW complex, characterized by a small value of n, the following mechanisms may be operative : (i) Non-reactive processes: when the intramolecular vibrational energy of M is lower than the binding energy of R, vibrational energy flow from M to the VDW bonds can occur. The low-frequency M-R modes may accept energy in small portions, as the vibrationally excited M molecule may cascade down via the intramolecular vibrational ladder. In view of the relatively high D values for M-R, complexes (table 2), this non-dissociative mechanism is operative for the 314 cm-' vibrational excitation of tetracene-R, (R = Ar and Kr). (ii) Reactive processes : when the intramolecular vibrational excitation of M exceeds D, reactive VP can occur, as demonstrated for tetracene-Kr complexes excited above 1200 ~ m - ' .~ For MR, complexes with a large number of R atoms clustering around M, the mechanism of vibrational energy relaxation may undergo a qualitative change. Even when the intramolecular vibrational energy of M exceeds the M-R binding energy, non-dissociative vibrational relaxation can occur via energy transfer to the intermolecular " phonon modes " of the large cluster. This problem bears a close analogy to vibrational relaxation of a guest molecule in a host matrix, except that the matrix is of a finite size. It will be interesting to monitor the " transition " from a172 LARGE VAN DER WAALS MOLECULES reactive molecular-type VP process to a solid-state-type relaxation process with increasing coordination number of the complex.The notion of low-frequency “phonon modes” of large VDW complexes brings up some intriguing questions regarding the establishment of relations and correlations between the energetic and dynamic features of “ free ” large molecules and of these molecules in condensed phases. Studies of the energetics and dynamics of VDW complexes and of large molecules embedded in large clusters ’ O will bridge the gap between these two cultures. This research was supported in part by the United States Army through its Euro- pean Research Office, and by the United States-Israel Binational Science Foundation (grant no. 2641), Jerusalem, Israel. D. H. Levy, Annu.Rev. Phys. Chem., 1980,31, 197. D. H. Levy, in Advances in Chemical Physics, ed. J. Jortner, R, A. Levine and S. A. Rice (Wiley Interscience, New York, 1981), vol. 47, part I, p. 323, A. Amirav, U. Even and J. Jortner, Chem. Phys. Lett., 1979, 67, 9. A. Amirav, U. Even and J. Jortner, J. Chem. Phys., 1981, 75, 2489. A. Amirav, U. Even and J. Jortner, J. Phys. Chem., 1981, 85, 309. A. Amirav, U. Even and J. Jortner, J. Chem. Phys., 1981, 74, 3745. A. Amirav, U. Even and J. Jortner, Chem. Phys., in press. S . Leutwyler, U. Even and J. Jortner, Chem. Phys. Lett., in press. M. J. Ondrechen, Z . Berkovitch-Yellin and J. Jortner, J. Am. Chem. Soc., 1981, 103, 6586. lo T. R. Heyes, W. Henke, H. L. Selzle and E. W. Schlag, Chem. Phys. Lett., 1980,77, 19. l1 W. Klemperer, Ber. Bunsenges. Phys. Chem., 1974, 78, 128. l2 K. V. Chance, K. H. Bowen, J. S. Winn and W. Klemperer, J. Chem. Phys., 1979, 70, 5157. l3 K. C. Jackson, P. R. R. Langridge-Smith and B. J. Howard, Mol. Phys., 1980,39, 717. l4 A. Amirav, U. Even and J. Jortner, Chem. Phys., 1980, 51, 31. Is A. Amirav, U. Even and J. Jortner, J. Chem. Phys., 1981, 75, 3370. l6 A. Amirav, U. Even and J. Jortner, Chem. Phys. Lett., 1981, 83, 1. A. Amirav, U. Even and J. Jortner, Chem. Phys. Lett., 1980, 71, 21. l8 R.E. Smalley, L. Wharton and D. H. Levy; J. Chem. Phys., 1977,66, 2750. l9 A. Herman, S. Leutwyler, E. Schumacher and L. Wosto, Helv. Chim. Acta, 1978, 61, 453. 2o U. Boessel, H. J. Neusser and E. W. Schlag, 2. Naturforsch., Teil A , 1978, 33, 1546. 21 M. A. Duncan, T. G. Dietz and R. E. Smalley, J. Chem. Phys., 1981,75, 2118. 22 A. D. Crowell and R. B. Steele, J. Chem. Phys., 1961, 34, 1347. 23 J. 0. Hirschfelder, C. F. Curtiss and R. B. Bird, in Molecular Theory of Gases and Liquids 24 H. C . Longuet-Higgins and J. A. Pople, J. Chem. Phys., 1957, 27, 192. 25 C. A. Coulson and P. L. Davis, Trans. Faraday Soc., 1952, 48, 777. 26 J. R. Lombardi, J. Chem. Phys., 1969,50, 3780. 27 R. E. Smalley and J. B. Hopkins, personal communication. 28 J. Jortner and R. D. Levine, in Aduances in Chemical Physics, ed. J. Jortner, R. D. Levine and 29 S. P. McGlynn, T. Azumi and M. Kimoshita, in Molecular Spectroscopy of the Triplet State 30 A. Amirav, U. Even and J. Jortner, Chem. Phys. Lett., 1980, 72, 17. (Wiley, New York, 1954), p. 168. S. A. Rice (Wiley Interscience, New York, 1981), vol. 47, p. 1. (Prentice Hall, Engelwood Cliffs, N.J., 1969).

ISSN:0301-7249    

DOI:10.1039/DC9827300153

出版商:RSC    

年代:1982

数据来源: RSC

摘要:

GENERAL DISCUSSION Prof. J. Jortner (Tel-Aviv University) said : The studies of the electronic-vibrational excitations of dimers of aromatic molecules provide important information pertaining to the intermolecular interactions, which determine the exciton band structure in molecular crystals. For the electronic origin of the (AB) dimer, two excitonic com- ponents are expected whose energies are E1,2 = Acf + DA + (AD/2) If [(AD/2)2 + V2]1’2 where Aef is the free-molecular excitation energy, AD = DB - DA represents the difference between the changes of the electrostatic intermolecular energies in the excited state and the ground state, with DA and D, corresponding to the excitation of A and B, respectively. V is the intermolecular excitation transfer integral multiplied by the nuclear Franck-Condon factor.As pointed out by Levy, for the T-shaped dimer of tetrazine IADl 9 I VI and the level splitting essentially measures the difference between the two environmental shift terms. For the planar tetrazine dimer AD = 0 and the excitonic splitting is 2V but, however, the transition to only one excitonic component is allowed. Levy et al. have reported that “ symmetry breaking ” of the planar dimer by complexing with He results in two transitions separated by AE = 0.2 cm-l. It will be extremely interesting to determine whether this splitting originates from the diagonal contribution AD induced by the presence of the He atom, or rather from the off-diagonal contribution V. Proceeding to higher energies one expects to encounter the interesting phenomenon of cooperative electronic-vibrational excitations (CEVE) in dimers. When the electronic excitation is accompanied by a single vibrational excitation, two distinct types of zero-order excitations can be considered : (1) The electronic and vibrational excitation are both localized on the same molecule. (2) The electronic excitation is localized on one molecule, while the vibrational excitation is localized on the second molecule.The cooperative double excitation (2) is energetically split from the con- ventional excitation (1) by the difference of the vibrational molecular frequencies in the ground and excited states. Configurational interaction between the cooperative and the conventional excitations induced by the off-diagonal excitation transfer spreads the intensity for the transition to the two zero states (1) among four components. This phenomenon of CEVE is well documented in molecular crystals providing information concerning the exciton band structure.Studies of CEVE in dimers and in larger clusters will be interesting. E. I. Rashba, J. Expt. Theor. Phys., 1966 23, 708; J. Klafter and J. Jortner, Chem. Phys. 1980, 47, 25. Prof. D. H. Levy (University of Chicago) said: After hearing Prof. Jortner’s interesting suggestion of possible cooperative electronic-vibrational excitations, we examined some old data for evidence of such a process. The frequencies of bands produced by this effect are predictable from the known ground- and excited-state vibrational frequencies of tetrazine. Our data are stored so that a search after the fact retains the full sensitivity of the apparatus.To the limit of our sensitivity, we were unable to find any clear evidence for bands produced by cooperative excitations.1 74 GENERAL DISCUSSION Certainly such excitations must take place, but the cross-sections appear to be very small. Prof. J. M. Lisy (University of Illinois) said: One unresolved question in the molecular-beam studies of Van der Waals molecules has been the failure to observe structural isomers. Thus the observation of isomers of tetrazine dimer is quite intriguing. In particular, a natural question is which isomer is the most stable and by how much? A clue to the orientation of the tetrazine in the T-shape structure may be given in the structure of the benzene-HCl complex determined by Flygare et aZ.2 The HCl centre of mass is 2.627 A above the benzene C, axis with H atom pointing towards the benzene molecule.In the T-shaped tetrazine dimer, a similar situation may occur with the H atom of one tetrazine molecule pointing towards the centre of the other. D. H. Levy, C . A. Hayman and D. V. Brumbaugh, Furuduy Discuss. Chem. Soc., 1982,73,137. ' W. G. Read, E, J. Campbell, G. Henderson and W. H. Flygare, J. Am. Chern. Soc., 1981,103, 7670. Dr. A. Tramer (University of Paris, Orsay) said : An important result of this work is a direct evidence for the broadening of the non-predissociated levels of the complex due to the vibrational redistribution : energy flow from internal (molecular) to external (complex) modes. I would like to consider a more general problem: whether in some systems the vibrational redistribution between discrete levels does not play the role of the primary step in vibrational predissociation ? The simplest example is that of an atom-diatom system (like I2 - Ar) where more than onevibrational quantum of 1-1 vibration are necessary for predissociation, Since all theoretical models suggest a strong propensity for Av = 1 coupling, predissociation may be considered as a sequential process: transition from the " cold " v-level to the discrete manifold corresponding to v - 1 quanta in the 1-1 mode with the energy excess in external modes and then to the (u - 2) dissociative continuum (fig.1). If it is so : (i) the homogeneous linewidth in absorption (fluorescence excitation) spectrum does not characterize the predissociation rate but that of (more rapid) vibrational re- distribution, (ii) energy disposal in products would depend rather on the intermediate than on the initially excited state.There exist some experimental data suggesting such a sequential behaviour. (i) In the infrared dissociation of the (HF), clusters of the type the homogeneous width of absorption bands is of the order of 20 cm-'. It seems highly improbable to assume that the cyclic complex dissociates with a simultaneous break of two strong hydrogen bonds on the timescale of 0.3 ps. A more plausible mechanism would be initial energy flow from the H-F stretching mode to the low- frequency ring modes followed (on a longer timescale) by the ring reclosing with elimination of the HF monomer.(ii) The fluorescence spectrum observed under the excitation of the 66l level of the tetrazine-argon complex is composed of: (a) the resonance emission from initially excited state of the complex, (b) the emission from lower lying T&i2 state of the complex with large amount of vibrational energy in external modes and (c) the emission from the (still lower) 16al level of the free tetrazine molecule. The 6&+T6ii2 redistribution and 6d1-+16a1 predissociation processes may be either parallel or sequential but highlyGENERAL DISCUSSION 175 V v - 1 v - 2 FIG. 1 .-Vibrational levels on an atom-diatom complex. selective population of the 16a' state of the disso ciation product suggests a stepwise process : 6d1+i6a2+16a1.Since all processes take place on the nanosecond time-scale, the direct evidence for the sequential (or parallel) process might be deduced from time-resolved experiments. Note added in proof: Recent 'data of Ramaekers et aL3 suggest the sequential behaviour. l M. F. Vernon, J. M. Lisy, D. J. Krajnovich, A. Tramer, H. S. Kwok, Y. R. Shen and Y. T. Lee, Faraday Discuss. Chem. SOC., 1982, 73, 000. (a) J. E. Kenny, D. V. Brumbaugh and D. H. Levy, J. Chem. Phys., 1979,71,4757; (6) D. H. Levy, personal communication. J. J. F. Ramaekers, J. Langebar and R. P. H. Rettschnick, Chem. Phys. Lett., in press. Dr. U. Even and Prof. J. Jortner (Tel-Auiv Uniuersity) said: We have recently obtained new information on electronic-vibrational excitations of the following classes of large Van der Waals molecules : (a) M-R Van der Waals molecules consisting of a rare-gas atom (R) bound to a large aromatic molecule (M), (6) M.X Van der Waals molecules consisting of medium- sized molecules (X = water, ammonia, methane, carbontetrachloride, methanol, acetonitrile and benzene) bound to a large aromatic molecule (M -- fluorene and tetracene), (c) The H2P.Ar molecule consisting of argon bound to free-base porphine (H,P), ( d ) P.X molecules consisting of porphyrins (P = free-base porphine or zinc- octaethylporphine) bound to medium-sized molecules (X = water, acetonitrile and pyridine). The large Van der Waals molecules were synthesized in pulsed supersonic jets of He (stagnation pressure 1500-2000 Torr expanded through a 600 ,urn nozzle) seeded176 GENERAL DISCUSSION with the two molecules which form the complex. This preparation method allows for effective internal cooling of the Van der Waals complex.We have explored intravalence electronic excitations of these complexes. The microscopic spectral shifts, dv, for the electronic origin of the first spin-allowed S,+S, transition of M or of P in these Van der Waals complexes originates from the follow- ing additive effects: (1) Short-range repulsive interactions in S, resulting in a spectral shift towards higher energies. These small blue spectral shifts are expected to dominate Sv when the dispersive contributions are small, being exhibited in He-M complexes. (2) Dispersive interactions. The red, dispersive, spectral shifts induced in a non- polar aromatic molecule bound to rare-gas atoms Ar, Kr and Xe, or to non-polar molecules, e.g.CH, and CCI,, can be quite well rationalized by the relation Sv = Aa, where a is the polarizability of the ligand (table 1). Although this correlation is expected to hold only for small non-polar ligands, it appears to be roughly obeyed even for a large ligand such as benzene (table 1). (-3) Polar interactions. These can result in the enhancement of either attractive or repulsive electrostatic interactions in the S, state of M-X, where X is a polar molecule. TABLE RED SPECTRAL SHIFTS FOR THE ELECTRONIC ORIGIN OF TETRACENE - X VAN DER WAALS COMPLEXES X Gv/cm-' - (6v/a)/l 024 cm - Ar - 38 24 (3-34 - 68 26 NH3 - 105 48 HzO -113 75 ccl4 - 180 18 C6H6 - 350 (35) For the case of complexes containing the non-polar tetracene (T) molecule, we have observed large red spectral shifts for T-H20 and T-NH3, which are considerably higher than Sv for T-CH, (table 1).The enhancement of the red spectral shift is attributed to the stabilization of SL by dipolar interactions. In the flu0rene.X complexes (table 2), where the M molecule is polar and undergoes a change in its dipole moment upon electronic excitation, a repulsive contribution is exhibited by the polar ligand on the S, state. This repulsive contribution to Sv practically cancels out the attractive, dispersive contribution for X = H20, NH3 and CH,OH, and results in pronounced blue shift for X = CH3CN. This repulsive dipolar contribution roughly increases with the size of the dipole moment, ,u, of the polar ligand.(4) Intramolecular configurational modifications induced by Van der Waals bind- TABLE 2.-sPECTRAL SHIFTS FOR THE ELECTRONIC ORIGIN OF FLUORENE ' x VAN DER WAALS COMPLEXES CH4 - 78 0 NH3 - 17 1.47 H20 0 1.82 CH30H 15 1.7 CHjCN 157 3.6GENERAL DISCUSSION 177 ing. We have found that the spectral shift of the So+SIx transition (the Qx band) of the free-base porphine-Ar (H,P.Ar) complex is 6v = +9 cm-l to the blue. This observation is rather surprising as the spectral shift on a nn* transition of a large aromatic molecule induced by the binding of an Ar atom is expected to be dominated by red dispersive interactions. As the SO+Slx transition of H2P is characterized by a low oscillator strengthf= 0.01 ,l one expects a modest dispersive contribution to dv, so that other contributions to 6v will not be masked by the dispersive contribution and could be amenable to observation.The experimental blue shift presumably originates from medium-induced changes in the geometry of the two internal N-H bonds induced by the binding of Ar, which modifies the energy levels of H2P. It should be emphasized that the physical origins of the contributions to the spectral shifts (1)-(4) considered herein pertain to intravalence excitations, involving a minor charge expansion of M in its electronically excited state. For extravalence Rydberg- type transitions of Van der Waals complexes, the dominant contribution to the spectral shifts will originate from short-range repulsive interactions in the excited state. Regarding the vibrational excitations in the S1 state of large Van der Waals complexes, we found it interesting to study the excitations involving the intermolecular nuclear motion of R with respect to M in M-R, or of X with respect to M in M-X, while the internal vibrations of M are not excited.Three types of such intermolecular vib- rational excitations were observed by us: (A) In-plane motion of a rare-gas atom along the surface of a large aromatic molecule. The So+Sl excitation of the tetracenedAr complex at 38 cm-I above the electronic origin of the bare tetracene molecule was reinvestigated in He jets under superior conditions of internal cooling, revealing the splitting of the spectral features into two components split by 2 cm-l. The second high-energy component of this doublet is tentatively attributed to the excitation of a low-frequency in-plane motion of Ar parallel to the long axis of the tetracene molecule.(B) Out-of-plane motion of a rare-gas atom with respect to the surface of the aromatic molecule. In fluorene-R complexes we have observed vibrational excitation at 40 & 2 cm-l for fluorene-Ar, at 30 5 4 cm-l for fluorene-Kr, and at 40 & 4 cm-l for fluorene.Xe, while for tetracenesxe a vibrational excitation at 32 & 4 cm-' is exhibited. These vibrational excitations are assigned to the out-of-plane motion of R. The experimental frequencies are in accord with the results of numerical simu- lations of potential surfaces of large MR1 complexes, which result in the frequencies of 40-60 cm-I for the out-of-plane motion for a heavy R atom (Ar, Kr or Xe) in a direc- tion perpendicular to the plane of the tetracene molecule. In the fluorene-benzene complex we have observed a well characterized vibrational pro- gression which corresponds to the intermolecular nuclear motion of the two large molecules constituting the complex (fig.2). The vibrational frequency is 28 c111-l. The electronic-vibrational coupling strength is S N" 0.4, indicating a finite configur- ational change in the intermolecular fluorene-benzene equilibrium separation upon SO+Sl excitation of the fluorene molecule. L. Edwards, D. H. Dolphin, M. Gouterman and A. D. Adle, J. MoZ. Spectrosc., 1971, 38, 16. (C) Intermolecular nuclear motion of large ligands in M-X complexes. Prof. F. A. Gianturco (University ofRome) said: In relation to the results reported by Even and co-workers, it seems of some interest to further comment on their observed anomalous behaviour of large Van der Waals molecules with He and Ne atoms.These authors report, in fact, that the energetic shifts, 6v, of the electronic origin of the So+Sl excitation of a well characterized MR, Van der Waals complex relative to178 GENERAL DISCUSSION F 0-0 FB 0-0 FB (28) FB 10 2950 2960 2970 2980 2! wavelength18i 30 FIG. 2.-Fluorescence excitation spectrum of the fluorene molecule and of the fluorene-benzene com- plex in pulsed supersonic expansions of the He. Stagnation pressure p = 2000 Torr, nozzle temp- erature T = 150 "C and nozzle diameter D = 600 pm. The electronic origin of the So+S1 transition of the bare fluorene molecule is labelled F 0-0, while the electronic origin of the So+S1 transition of the complex is labelled F B 0-0.The numbers in parentheses represent the vibrational excitations of the complex (in cm-') above its electronic. the 0-0 transition of the bare molecule M are all towards lower energies (red shifts) when the molecule contains Ne, Ar, Kr and Xe atoms. On the other hand, blue spec- tral shifts are exhibited by complexes with He atoms, a fact which they say could be attributed to the increased importance of repulsive interactions as opposed to the dis- persive terms which are in turn becoming weaker for complexes containing helium atoms. We investigated the structure of simpler complexes containing one 0, molecule and an R atom (R = He, Ar) and also found that their equilibrium geometries strongly suggest the increased importance of repulsive terms when He is concerned. The latter, smaller atom in fact gets closer to the centre-of-mass of the molecule and samples more strongly the highly directional repulsive forces between the oxygen atoms.Thus the Cmh geometry (collinear configuration) shows a minimum position of ca. 7.6 au for ArO, and ca. 7.2 au for HeO,, with well depths of ca. 3.4 x au in the former case and ca. 0.79 x au in the latter. The C,, geometry, however (" perpendicular configuration ") exhibits equilibrium distances from the centre of mass of ca. 6.8 au for ArO, and of ca. 6.2 au for HeO,. The corresponding well depths are ca. 5.2 x in the former system and ca. 1.3 x Our present study therefore seems also to provide further evidence that the genera: trend of the present spectral shifts strongly relates to the relative and specific interplay of short-range and dispersion interactions, hence to the " size " of the charge cloud in the rare-gas partners.in the latter.GENERAL DISCUSSION 179 Dr. B. Soep (University of Paris, Orsay) said : Prof. Jortner and collaborators have reported the interesting observation of sequence vibrational transitions in the Van der Waals modes of rare-gas-tetracene molecules. The presence of vibrational transitions in the optical spectra of rare-gas molecular complexes is usually manifested by only very faint vibrational This supposedly arises from the similarity of both Van der Waals potentials in the ground and excited states.We (N. Halberstadt, C. Jouvet and myself) have found unusually long and intense vibrational progressions in glyoxal (CHO-CH0)-H,, D, complexes when excited in the lAu+lAg transition. Van der Waals complexes appear as satellite bands in the molecular spectra pro- duced in supersonic expansion. Their intensity rises as the pressure of the complexing gas is increased. Thus the band assignment is usually done from the pressure dependence of their intensities. However, bands exhibiting the same pressure effect can either be assigned to vibrational transitions or to chemical isotopes. In fig. 3, we present the fluorescence excitation spectrum of glyoxal-H, and glyoxal- 100 50 0 v1cm-l FIG. 3.-Fluorescence excitation spectra of complexes of glyoxal with Hz (A) and Dz (B) Hz and D2 complexes in the 8; spectral domain.D, molecules. Each vibrational transition of the uncomplexed molecule is accom- panied by satellite peaks (a, b, c, d and e). We attribute the (h), (c), ( d ) and (e) peaks to a vibrational progression in the Van der Waals mode (O-tO, 1,2,3), and the (a) peak to a hot band (130). For glyoxal-H, we have determined Lo = 17 cm-' and cox = 1.4180 GENERAL DISCUSSION b-a c-b d-c e-d f-e 1 2 3 4 5 n 1 2 3 4 v‘+ 1 Avlcrn-’ [= v(n + 1) - v(n), v(u’ + 1) - v(u’)] FIG. 4.-Interpretation of H2-glyoxal Van der Waals progressions. cm-l for the frequency and anharmonicity of the van der Waals mode in the elect- ronic excited state, from fig. 4. If this interpretation is correct we expect for glyoxal-D, a reduction of the frequency by a factor ql’,, where q is the ratio between the reduced masses ‘I = p (glyoxal-D,/p(glyoxal-H,) = 1.98, while the anhar- monicity should be reduced by the factor q.We expect then w = 12.2 cm-l and wx = 0.7 cm-l for glyoxal-D,. From our results presented in fig. 3 we get w = 12 cm-l and wx = 1 cm“, which agrees with the above prediction within experimental error. These results confirm the attribution of the satellite peaks of fig. 3 to a vibration pro- gression in the Van der Waals mode. The existence of such progressions must arise from a profound change in the com- plex geometry caused by an important modification of the electronic distribution in the excited molecular state. A Franck-Condon factor simulation predicts a dis- placement of the van der Waals equilibrium distance of 1.7 A.As the hydrogen is most likely attached above the molecular plane, the corresponding stretching vibration could not be so perturbed. Rotational contours of the vibrational progres- sions are under investigation. Preliminary results imply the other possibility of an hydrogen displacement parallel to the molecular plane, in the excited state, in accord- ance with the electronic distribution in excited ~ t a t e s . ~ D. H. Levy, Adu. Chem. Phys., 1981,47, 323. A. Amirav, U. Even and J. Jortner, J. Chem. Phys., 1981, 75, 2489. N. Halberstadt and B. Soep, Chem. Phys. Lett., 1982, 87, 109. C. E. Dykstra and H. F. Schaefer 111, J. Am. Chem. SOC., 1976,98,401. Prof. J. Jortner (Tel-Aviv University) said: Dr.Soep is correct in stating that the appearance of vibrational progressions of intermolecular vibrational modes in the electronic spectra of large complexes is quite rare, indicating that the intermolecular potential surfaces in the So and in the S, electronic states are similar. We have observed a well characterized vibrational progression of 28 cm-l, clearly exhibiting the 0-1 and 0-2 members for the fluorene-benzene complex. The relative intensity of the 0-1 transition relative to the 0-0 is 0.4, indicating a substantial change of the intermolecular equilibrium configuration.GENERAL DISCUSSION 181 Dr. P. G. Burton (University of Wollongong) said: Prof. Jortner has illustrated energetic shifts of the electronic origin of S,+S, excitations of a molecule M complexed with one or more rare-gas atoms (MR,), compared to the isolated molecule value, and raised the question about modification of the vibrational frequencies in MR, relative to those of the bare molecule.I should like to draw attention to an initial survey undertaken by Meyer et al.' of the modification of the H, potential curve by the proximity of a helium atom. Fig. 8 of their paper gives results which indicate that for collinear geometries, the internal H2 potential curvature increases near re, while it decreases for perpendicular geometries, when a series of fixed intermolecular distances, R, are sampled. In the absence of a full solution to the internal vibrational dynamics of such a complex, we might compute the internal H2 frequencies for a range of angles at each separation of the He and H2, and perform an angular average on the computed vibrational frequencies to estimate the effect of zero-point and higher frequency changes of the H2 at each value R on the isotropic HeH, potential.These internal frequencies thus depending parametrically on R could be averaged over any dimer vibrational level wavefunctions. Relying on the much greater frequency of internal vibration compared to the dimer vibration, this should provide a reasonable estimate to the influence of Van der Waals complex- ation on isolated molecule frequencies, provided of course the underlying electronic potential is sufficiently reliable, and lead to an effective vibronic dimer potential. Any difference between the rigid-rotor electronic potential and such an effective vibronic potential should be most marked for the H2 case, due to its large vibrational amplitude. We are in the process of performingjust such calculations at Wollongong, based on new potentials we have computed for He-H, and for the more complicated H2-H, case.In the latter case, we believe that modifications to the H2 zero-point energies in Van der Waals dimers is at least in part responsible for the isotropic component of our rigid-rotor H1-H2 electronic potentials lying slightly deeper in the Van der Waals minimum than semiempirical potentials. Although this discrepancy (ca. 1 cm-' for the variational PNOCI, ca. 7 cm-' for CEPA2-PNO) is a very small fraction of the computed total energy in our (superposition corrected) supermolecule calculations (1 : 100 000), it represents a significant fraction of the H2-H2 well depth.However to complete such an analysis of the influence of internal zero-point energy changes on the effective isotropic potential is a time-consuming process with the large basis sets we are using, because of the large range of internal coordinate and angular variations that must be completed at each R. W. Meyer, P. G. Hariharan and W. Kutzelnigg, J. Chem. Phys., 1980, 73, 1880. P. G. Burton and U. E. Senff, J. Chem. Phys., 1982,76, 6073. Prof. J. Jortner and Dr. U. Even (Tel-Auiu University) said: The problem of the modification of the intramolecular vibrational frequencies of M in MR, complexes relative to those of the bare molecule is of considerable interest not only in triatomic Van der Waals complexes, as alluded to by Dr.Burton, but also in large complexes. Preliminary evidence pertaining to this problem was obtained recently from the spec- troscopy of the He complexes of 1,4-dihydroxyanthraquinone [H,H] and of its deuter- ated analogue [D,D], studied in our laboratory by G. Smulevich, M. Marzocchi, U. Even and J. Jortner. The electronic origin of the S,+S, transition of the He[H,H] complex is blue-shifted by Av = 5 & 1 cm-' from the electronic origin (0-0) of the S,+S, transition of the bare [H,H]. This observation can tentatively be attributed to the effect of short-range repulsive interactions. An intriguing deuterium isotope effect on the microscopic solvent shift of He[H,H] was observed. We have found that182 GENERAL DISCUSSION Av increases from 5 1 cm-' for He[H,H] to 11 5 1 cm-' for He[D,D].This effect may manifest to modification of the zero-point energies of [H,H] and [D,D] upon binding of an He atom and can be expressed in terms of third-order differences between the zero-point energies of the He[H,H] and He[D,D] complexes relative to the zero- point energies of the bare molecule in s, and in s,. Prof. T. E. Gough, Dr. M. Keil, Mr. D. G. Knight and Prof. G. Scoles (University of Waterloo) said: We would like to give an example of the infrared spectral shift upon attaching rare-gas atoms to a " guest " molecule. These shifts are much smaller than those observed in electronic spectroscopy, but likewise clearly show an evolution from the isolated molecule to one attached to many rare-gas at0ms.l The experiment utilizes the optothermal spectroscopy technique developed by Gough et al., Upon laser-induced photodissociation of Van der Waals molecules, we observe a reduction of molecular-beam inten~ity,~ since the photofragments are scattered beyond the detector viewing angle of 1.0".Modulation of the C.W. line- tuned CO, laser provides a chopped bolometer signal corresponding to the Van der Waals absorption spectrum. The molecular beam is formed by expanding a 1 % SF6 in Ar gas mixture through a room-temperature 17 ,urn diameter nozzle. After being skimmed, the molecular beam is crossed with a 7.5 W laser beam, and after a further flight path of 149 mm, impinges on the 2 K bolometer. Rotational cooling of the SF, v3 mode-is sufficient that no absorption signal of the bare SF, molecule is seen: only absorptions due to Van der Waals molecules are observed.In fig. 5 we show the evolution of absorption features from low ( p , = 1050 Torr) to high ( p , = 6500 Torr) source pressures. For comparison, we show at the very top of fig. 5 low- and high- resolution spectra of SF6 in an Ar matrix. At low pressure, the single spectral feature is red-shifted by ca. 2 cm-' relative to the bare SF6 molecule. As the source pressure is increased, a second feature appears. This second feature is red-shifted by ca. 4 cm-l and displaces the first feature at p s = 3100 Torr. At yet higher source pressures, the peak absorption again shifts to the red, resulting in a fairly narrow peak, red-shifted by ca.10 cm-I relative to the bare SF,. This highest- pressure peak ( p , = 6500 Torr) lies at the absorption maximum of the matrix-isolated SF6 m01ecule.~*~ Other spectra taken between p s = 3900 Torr and p s = 6500 Torr (not shown) demonstrates a rather sudden appearance for the high-pressure peak at 939 cm-l. The results presented here show that the stepwise evolution from Van der Waals molecules, through large clusters, to condensed phase species, may conveniently be followed for vibrational transitions. Further work will be needed to identify the number of Ar atoms bound to the SF6 " guest " molecule. This would enable one to estimate the coordination number necessary to prepare a localized " matrix-isolated '' environment. We thank Paul Rowntree and Karen Fox for their help in gathering the experi- mental data.U. Even, A. Amirav, S. Leutwyler, M. J. Ondrechen, Z. Berkovitch-Yellin and J. Jortner, Faraday Discuss. Chem. Suc., 1982, 73, 153. * T. E. Gough, R. E. Miller and G. Scoles, AppE. Phys. Lett., 1977,30, 338. T. E. Gough, R. E. Miller and G. Scoles, J. Phys. Chem., 1981, 85, 4041. R. V. Ambartzumian, Yu. A. Gorokhov, G. N. Makarov, A. A. Puretzky and N. P. Furzikov, Proc. VKOLS Laser Spectroscopy Conf., ed. B. P. Stoicheff, A. R. W. McKellar and T. Oka, (1981), p. 439. B. I. Swanson and L. H. Jones, J. Chem. Phys., 1981, 74, 3205.GENERAL DISCUSSION 183 I I I 2600 f l 1050 I 5 940 945 9 laser frequencylcm- 0 FIG. 5.-Photodissociation spectra of SF,.Ar,, clusters observed at the indicated source pressures (in Ton).The spectra were recorded as a laser-induced attenuation of the molecular beam intensity for various COz laser lines (data points). Matrix spectra are from ref. (4) and (5), at low- and high- resolution, respectively. The dashed vertical line corresponds to the bare SF6 absorption at 948.5 cm-'. A 17 pm room-temperature nozzle was used for expanding a gas mixture of 1 % SF6 in Ar. Dr. A. Amirav, Dr. U. Even and Prof. J. Jortner (Tel-Auiu Uniuersity) said: The interesting data of Keil and Scholes on the infrared spectra of SF6 in supersonic expansions of At at high stagnation pressure provide information on clustering of Ar atoms on SF6. For some time we have been interested in the effects of selective and gradual solvation on excited-state energetics and dynamics of large molecules in clusters in order to bridge the gap between the photophysics of isolated molecules and of the same molecules in condensed phases.We have conducted studies of electronic spectroscopy of large clusters of Ar, each containing a single aromatic molecule184 GENERAL DISCUSSION (anthracene, tetracene or pentacene), which acts as a fluorescent pr0be.l These clusters were synthesized in supersonic jets of the seeded Ar, expanded through a 150 pm nozzle and stagnation pressures p = 1500- 13 000 Torr. Fig. 6 portrays the laser- induced fluorescence (LIF) excitation spectra of the electronic origin of the " isolated " tetracene molecule (at p = 180 Torr), of the small tetracene-Ar, (n = 1-8) clusters (at p = 710 Torr), while a further increase of the stagnation pressure t o p > 3300 Torr exhibits the LIF spectra of tetracene in large clusters of Ar.The red spectral shifts of the electronic origin (fig. 7) are dominated by dispersive interactions, reaching an I I A 1 p=710 Torr 0 *- 4460 4550 4f 50 wavelength/A FIG. 6.-Fluorescence excitation spectra of tetracene (T) in a seeded supersonic beam of Ar expanded in the pressure rangep = 180-8300 Torr, expanded through a 150 pm nozzle. In all cases the lowest- energy spectral features are shown. The spectrum at 180 Torr is dominated by the electronic origin of the bare T molecule, while the weak satellite band corresponds to T-Ar. The spectra at 710 and 1200 Torr reveal well defined spectral features which could be assigned to TAr,, complexes with n = 1-8. The spectra in the high-flow supersonic expansions at 3350 and 8300 Torr correspond to the electronic origin and to the first 314 cm-' vibrational excitation of large TAr,, clusters. asymptotic value of -705 5 cm-' at p > 6000 Torr, which is comparable to spec- tral shifts in matrix-isolated large molecules. The total spectral widths (fig. 7) of the absorption bands (f.w.h.m.) exhibit a maximum in the vicinity o f p N" 2000 Torr, and are independent of the stagnation pressure at p > 3000 Torr. It appears that the low-pressure range p < 3000 Torr corresponds to the buildup of the first coordination layer, where the effects of inhomogeneous broadening, due to the distribution ofGENERAL DISCUSSION 185 - I ] ! I ; " I I 1 I 5000 10000 1500C p/Torr FIG. 7.-The spectral shifts, relative to the0-0 transition of the bare molecule, and the widths (f.w.h.m.) of the 0-0 transition of tetracene in Ar clusters synthesized at the stagnation pressures p 1=1 2000- 14 000 Torr. cluster sizes, are severe. These effects of inhomogeneous broadening are drastically manifested by the appearance of the maximum in the spectral widths. The satur- ation of the spectral shift and linewidth at p 3 6000 Torr indicates the completion of the buildup of the second coordination layer. In this high-pressure rangep >, 6000 Torr, we have applied the technique of laser fluorescence line-narrowing to provide evidence that the line-broadening of large tetracene-Ar clusters is essentially homo- geneous, presumably originating from phonon coupling effects. A. Amirav, U. Even and J. Jortner, Chem. Phys. Lett., 1980,72, 16.

ISSN:0301-7249    

DOI:10.1039/DC9827300173

出版商:RSC    

年代:1982

数据来源: RSC

摘要:

Faraday Discuss. Chem. Soc., 1982, 73, 187-203 Rotationally Inelastic Scattering of Hydrogen Molecules and the Nan-spherical Interaction BY UDO BUCK Max-Planck-Institut fur Stromungsforschung, 3400 Gottingen, West Germany Received 30th November, 198 1 Molecular-beam scattering experiments on state-to-state differential cross-sections for rotational transitions of hydrogen moIecules are a sensitive probe for both the isotropic and the anisotropic interaction potential with other atoms and molecules. New experimental results on the weakly coupled system D2 + Ar allow reliable determination of the repulsive part of the complete potential. It is compared with the well known D2-Ne potential and recently proposed model potentials which predict the general behaviour but not the details.For the important H2-H2 interaction differential cross-sections for 0+2 transitions in HD + D2 and Dz + H2 collisions are presented. The evaluation of these data leads to repulsive anisotropic potential terms which show the same shift of 0.1 8, to smaller R-values with respect to the best ab inirio calculation found previously for the isotropic interaction. Scattering experiments provide a direct tool for the determination of the interaction forces of Van der Waals molecules. Elastic scattering experiments essentially probe the isotropic potential whereas rotationally inelastic processes are due to the angular dependent non-central forces. For the determination of isotropic interactions as well as the attractive part of the complete potential surface many other methods are available.' However, the large-angle differential cross-sections for single rotational transitions are one of the few sources for information on the repulsive anisotropy of the system under study.For the interactions of neutral species cross-sections of this type are only available for few systems, namely for H, and its isotopes scattered from He,, Ne3-5 and Ar,6*7 for Na, scattered from He,8 Ne '3" and Ar,','' for N,, 0, and CO scattered from He l2 and for CH, scattered from He,', Ne l 3 and Ar.I3 The only molecular system studied in this way is H, + H, in its different isotope configurations HD + HD,14 HD + D2 and D, + H2.18 Among these, the hydrogen-molecule- rare-gas interactions and the hydrogen-molecule dimers are the best studied systems, involving as they do the analysis of many experimental properties.In particular, high- precision diffusion coefficients l9 and other transport phenomena,,' the spectros- copy of the Van der Waals c ~ r n p l e x e s , ~ ~ - ~ ~ bulk relaxation data,,, and molecular beam experiments on total total differential 26 and integral cross-sections with oriented molecules 27 have been used. In addition, these small systems are well suited to perform highly reliable calculations of the ab initio type with configuration inter- action included 28-30 or of the model type where the dispersion attraction is added to the SCF repulsion with suitable corrections for the overlap r e g i ~ n . ~ ' - ~ ~ The theoretical analysis of rotationally inelastic transitions of HD and D, molecules scattered from rare-gas atoms in the energy range from 30 to 90 meV gave results as follows. (i) The inelastic cross-sections are peaked in the backward direction, that is, the coupling essentially occurs in the repulsive part of the anisotropy. (ii) If we expand the interaction potential in Legendre polynomials V(R9.J) = VO(R) + ~,(R)~z(c0s.J)188 INELASTIC SCATTERING OF HYDROGEN MOLECULES where R is the distance between the atom and the centre of mass of the molecule and y is the angle between the molecular axis and R, two terms are sufficient to describe the scattering process for the homonuclear species.(iii) The 0 + 2 transitions of D2 are directly related to the repulsive V,(R) term. Since the coupling is weak we were able to construct an inversion procedure based on the exponential distorted wave approxim- ation which directly gives V2(R) once CJ (0 -+ 2; 3) is k n ~ w n .~ ~ ~ ~ ~ The magnitude of the coupling is determined by V, where 0 depends quadratically on V,. However, the range is given by the elastic wavefunctions which can be approximated by the classical turning point Ro of Yo. Therefore, Vo also has to be known for the analysis. If V,, and thus R,, decreases, generally a larger value of V, is probed and the inelastic cross- sections are approximately determined by the ratio V2/V0. (iv) The 0 + 1 transitions of HD are produced by the U,(R) term of the potential which is mainly sensitive to the Yo(&) term of the homonuclear interaction. Only a small dependence (ca. 10%) on the V2 term is left.Combining the rotationally inelastic transitions with the total differential cross- sections, we were able to derive a very reliable complete interaction potential for H,-Ne. Similar considerations hold for the scattering of hydrogen dimers. The analysis of the total differential cross-section and the 0 --+ 1 transitions of HD in HD + D, collisions leads to a very reliable isotropic potential for this system. The comparison with several ab initio potentials indicated that the surface calculated by Mayer and Schaefer 30 was the best one, although the agreement is not perfect. In the present contribution we summarize the state of the H,-Ne surface and present a similar analysis for D,-Ar. Based on the angular dependence of the 0 --f 2 transitions of D, + Ar collisions and a reliable Vo from the total differential cross- section, a complete surface is constructed.The results are compared with predictions from model potentials and also with the most recent version of the D,Ne surface. For the hydrogen dimer we summarize the present state of the attempt to derive a unique interaction potential for this important molecule-molecule system which correctly predicts the numerous data in the literature. We will, in particular, discuss the influence of the recently measured differential cross-sections for 0 -+ 2 transitions in HD + D, [ref. (17)] and D2 + H2 [ref. (IS)] collisions on the determination of the anisotropic part of the surface. It turns out that a particular combination of initially rotating molecules not only allows one to determine the usual anisotropy which corres- ponds to V , in the atom-diatom case but also the term which asymptotically corres- ponds to the quadrupole-quadrupole interaction.This term plays an important role in phase transitions of the molecular hydrogen solid at low temperatures and high pressures.38$ 39 EXPERIMENTAL The experiments have been carried out in a crossed-molecular-beam apparatus which has been described el~ewhere.~,~ The two colliding beams, which cross at 90°, are produced as nozzle beams from two differentially pumped chambers. The angular dependence is measured by rotating the source unit with respect to the fixed detector assembly. The scattered particles are detected by a doubly differentially pumped mass spectrometer operat- ing at pressures <lO-'O mbar.* Elastic and inelastic events are separated by time-of-flight analysis of the scattered particles using the pseudorandom chopping technique.The flight path has been extended to 589 mm: that is to say it is larger than in the previous experimental arrangement. In this way the velocity distributions of the scattered particles were measured with a resolution of cu. 2%, where effects from the finite ionization volume, the shutter * 1 bar = lo5 Pa.U . BUCK 189 function and the channel width have been included. Collection and processing of the data were executed by a minicomputer. The pumping facilities for the two beam sources permit the use of pressures up to 150 bar with 18 pm diameter nozzles. Therefore, the two intersecting beams not only have velocity distributions with a full width at half maximum of better than 5% but also expand in such a manner that nearly all molecules are in their lowest available rotational states.In addition, the accumulating time for measuring time-of-flight spectra could be reduced by a factor of 25 compared with expansions at 2 bar stagnation pressure. The actual beam conditions for three scattering experiments are given in table 1 . For the TABLE BEAM DATA parameter 0-Dz Ar HD n-D, o-D, n-H2 ~~ nozzle diameter/pm 10 source pressure/bar 192 source temperature/K 304 peak velocity/ms- I 2 042 speed ratio, S 29 rotational temperature/K 70 fraction in j = 0 0.89 j = 1 . . . . j = 2 0.1 1 100 3 304 560 21 ... ... . . . ... 50 20 10 80 164 306 1930 2040 31 30 < 10 68 1.00 0.60 .. . . 0.33 .... 0.07 18 150 303 2 131 37 56 0.95 0.05 .... 20 70 305 2 874 39 115 0.24 0.75 0.01 Ar secondary beam the pressure has been kept at low values in order to avoid condensation. Similar considerations hold for the cooled HD beam. The homonuclear hydrogen molecule beams at room temperature are not limited by condensation. Since there are two nuclear spin modifications of normal (n)-hydrogen molecules these beams consist of 5 o-D2(i p-H2) and + p-D2($ o-H,) for the even and odd rotational states, respectively. In order to increase the contribution in j = 0, the gas lines could be fed to converter cryostats to produce pure o-D, or p-H, beams. The degree of rotational population of the different beams has been obtained by a careful energy balance, which also accounts for real gas e f f e ~ t s , ~ ~ ' ~ and a suitable extrapolation of Raman studies for these nozzle beams at lower stagnation pressures.4o The time-of-flight spectra (see for instance fig.6 and 8) obtained at different laboratory angles are normalized to their maximum intensity after subtracting a large background contribution (signal-to-background ratio ca. 1 %). The accumulating time varies between 2 and 4 h. The peaks which appear at flight-times corresponding to inelastic transitions are clearly resolved. The procedure to derive cross-section ratios from these data has been discussed in detail in ref. (5). Briefly, as a first step the distribution functions in the final velocity space are calculated by a Monte-Carlo procedure for the elastic and possible inelastic transitions using the angular and velocity spread of the two intersecting beams and the trans- mission function of the time-of-flight analyser.As for the second step the measured spectra are fitted to the calculated distribution functions. If the half-width of the distribution func- tion is smaller than the corresponding energy loss due to the inelastic transition, the problem can be solved in a simple manner. The fitted amplitude Aif is directly related to the cross- section oif(g,g) by where K is a constant with respect to velocity variables and only depends on the laboratory deflection angle 6, p i is the relative population of the initial state, Jif the Jacobian for the transformation from the centre of mass (CM) to the laboratory system, g the relative velocity and 8 the CM scattering angle.The bars indicate average values within the small distribution function. It may happen that several individual cross-sections contribute to one energy-loss peak. If we normalize these values to the elastic contribution we can derive cross-section ratios. To obtain absolute inelastic cross-sections the total differential signal is used to- gether with a suitable calibration based on a calculation. Depending on the information190 INELASTIC SCATTERING OF HYDROGEN MOLECULES available for a system we will present the data as cross-section ratios or absolute inelastic cross-sections in the further course of this work. RESULTS HYDROGEN-MOLECULE-RARE-GAS SCATTERING RESULTS FOR H,-Ne For this system the complete potential surface has been accurately determined from molecular-beam scattering data.5 The high-resolution total differential cross- section for D2 + Ne and the rotationally inelastic cross-section for the 0 + 1 tran- sition of HD + Ne, together with the second virial coefficients as constraints for the long-range attractive part, were used to determine the isotropic part of the potential.For the anisotropy, the angular dependence of the 0 -+ 2 transitions of D, + Ne has been used together with the integral cross-section for different orientations 27 as constraints for the attractive part. The main input data are essentially sensitive to the position, slope and magnitude of the repulsive part of the potential as shown in a direct inversion pr0cedu1-e.~~ The accuracy of the isotropic potential was nicely confirmed by a comparison of the high-precision diffusion data l9 with predictions of several potential models. Among all H,-rare-gas systems this potential was the only one which predicted the measured data correctly.For the anisotropy small dis- crepancies existed in the attractive part. Thus a new measurement of the hyperfine dimer spectrum for H,-Ne by the magnetic-beam resonance technique was used to slightly modify the V, term in the attractive part.23 Fig. 1 shows a comparison of a calculation of the differential cross-section for the 0 + 0 and 0 + 2 transitions of D2 + Ne based on this modified V, value 23 and the V, value of ref. ( 5 ) with the measured 10’ 1 oo * I LI 10- ’ 0 50 100 150 9.1 a FIG.1.-Measured differential cross-sections for O+O and 0+2 rotational transitions of Dz + Ne and calculations based on the potential surface of ref. (5) with slightly modified attractive anisotropy [ref. (23)] (see fig. 4). E = 84.9 meV.U. BUCK 191 data. The agreement is excellent, so we can consider this potential (see also fig. 4) as being well-understood in both the attractive and repulsive regions. It can be used as a good test case for the accuracy of ab initio calculations. RESULTS FOR H2-Ar In contrast to the case of H,-Ne, the H2-Ar potential is only well known in the attrac- tive part. A careful determination of this potential by Le Roy and Carley from spec- troscopic data of the Van der Waals complexes ,' was confirmed by total differential cross-sections 26d and low-energy integral c r o s ~ - ~ e c t i o n ~ .~ ~ ~ Therefore, it is very interesting to perform a similar analysis as for D2-Ne where essentially data are used 10 n C c) ." c1 Y Y 1.0 0 . l l I I I 1 I I I I I 1 10 20 30 40 50 laboratory deflection angle,O/" FIG. 2.-Measured total differential cross-sections for Dz + Ar in the laboratory system. The calculations (solid line) are based on the potential of this work (see table 3 and fig. 3). E = 83.2 meV. which are sensitive to the repulsive part of the interaction. Fig. 2 shows a measure- ment of the total differential cross-sections at the energy of E = 83.2 meV.7 The diffraction oscillations are clearly resolved and serve as a precise measure for the zero point of the isotropic potential and the form of the repulsive wall. The large-angle part, which has been obtained up to 8 = 70°, gives the slope of the repulsion.In- elastic 0 + 2 transitions have been measured at E = 85.0 meV and are given in table 2. Because of large background problems there are only three points available. First, we compare our data with the best potential of Le Roy and Carley,,I called BG, where the collapsed diatom limit has been incorporated in the interaction potential. It is of the general Buckingham-Corner form Vi (R)= Aiexp(-BiR) - (C6JF + C8iR-8)f(R)192 INELASTIC SCATTERING OF HYDROGEN MOLECULES with There are four free parameters, Ai, pi, C,i and Csi, for each potential i = 0, 2. For convenience two of these parameters, A i and Csi, are replaced by the parameters of the potential minimum, Rmi and ci, the distance and the well depth, respectively.The calculated total differential cross-sections based on this potential show very small deviations in the oscillatory regime and are slightly too large in the monotonic large- angle part. Such behaviour is to be expected, since this potential is not able to TABLE 2.--a(0-+2)/0(0-+0) FOR D2 + Ar AT E = 85 meV IN PERCENT O(lab)/" I ~ ~ ~ ( C M ) / " 8,,,,(CM)/" experiment this work LC a TT RS 60 72.7 72.2 6.0 f 1.2 6.1 5.6 3.4 5.2 70 85.5 85.5 8.9 & 1.0 8.5 7.8 4.6 7.1 80 98.3 98.8 10.3 f 1.2 10.8 10.0 5.9 8.8 a Ref. (21), BC3 experiment; ref. (31), model; ref. (33), model. predict the diffusion coefficients of Dunlop and Tengro~e,'~ which also probe the repulsive part.The calculated inelastic cross-sections are lower than the measured values (see table 2). Since, on the other hand, the deviations are rather small we decided to use the same potential form and to determine mainly the parameters p1 by the large-angle measurements, keeping the other three parameters as close as possible to the values determined by spectroscopy. The result of such an analysis is a potential which differs in RmO = 3.60 0.03 A by 0.8% and in Po = 3.4 & 0.2 A-' by 5.8%. Then, following the procedure described in detail in ref. ( 9 , the complete potential surface is obtained from the inelastic 0 --f 2 differential cross-sections. To solve the coupled equations, the coupled-states approximation has been used. With a basis of j = 0, 2, 4 convergence was reached.The final result for the two potentials is dis- played in fig. 3 and table 3. The calculations based on this result are shown in fig. 2 and table 2 in comparison with the experiment. As expected, the new isotropic potential Vo is a little softer than Le Roy's potential, whereas V, is nearly the same. Since, as discussed in the introduction, the inelastic transitions depend on both V, and Yo, the lower value predicted by the BC, potential of Le Roy is easily understandable.* A calculation of the diffusion coefficient gives a value of 0.813 cm2 s-l in reasonable agreement with the experimental value of 0.824 cm2 s-l. Therefore, we think that this potential is also very reliable concerning both the isotropic and the anisotropic part. What still has to be done is the incorporation of the spectroscopic data of the infrared absorption and the hyperfine structure of the Van der Waals molecules in the evaluation of the data.This will be done in a forthcoming paper and ought not to change the results very much, since the attractive part has been kept nearly fixed in the present procedure. DISCUSSION We now have two reliable potential surfaces of the H2-rare-gas interaction, which makes it interesting to compare the results. This is done in fig. 4, where we plot the * This conclusion also holds for the cross-section ratio, since the elastic cross-section introduces only a much weaker additional dependence on Vo.U. BUCK 193 40 30 20 2 .- 4 4 3 g 10 u 0 -10 I I I 2.0 3.0 3.5 distance/W 0 FIG. 3.-Potential curves for D,(H,) + Ar: (-) this work; (- .-) ref. (21); ( x ) ref. (31); (0) ref. (33). result for H2-Ne on the same scale as the H,-Ar result of fig. 3 and the ratio V,/Vo for the two potentials. The two potential surfaces are very similar. The potential well for the isotropic part is roughly a factor of 10 deeper than for the anisotropic part. Both potentials cross just above the zero point and Y2 is generally much smaller than V,. A comparison of the absolute potentials shows the expected behaviour. The TABLE 3.-POTENTIAL PARAMETERS FOR D2 f Ar Vo 6.307 3.595 3.175 1686.18 3.40 16.676 191.71 Vz 0.711 3.725 3.335 719.55 3.69 1.6738 3 1.746 value of the position of the potential minimum is shifted by ca. 0.3 A and the minimum itself is larger by a factor of two for the H,-Ar system, obviously due to the larger polarizabilities.The same shift is also observed in the repulsive region. An import- ant quantity for the interpretation of rotationally inelastic data is the value of V, at the classical turning point of Yo, which can be expressed by the ratio V,/VO and which is roughly responsible for the strength of the transition. The comparison shows the surprising result that this ratio is larger for H,-Ne than for H2-Ar. Therefore, 0 -+ 2 inelastic transitions are larger for D2-Ne than for D2-Ar at the same collision energy, which can be seen directly by looking at the measured time-of-flight spectra. Since no ab initio potentials are available in the literature. we compare our results194 INELASTIC SCATTERING OF HYDROGEN MOLECULES with predictions of the so-called Hartree-Fock dispersion m ~ d e l , ~ ’ .~ ~ where the cal- culated repulsion is added to the calculated dispersion with some suitable corrections for the vanishing contribution of the dispersion in the overlap region. In the Tang- Toennies (TT) model 31 this is done by the Drude model, whereas Ahlrichs et aL3, use an empirical damping function for all coefficients. In a refined version Rodwell and Scoles (RS) 33 introduce individual damping functions for each coefficient. First we LO 30 20 % E 8 . - .d C w 10 0 -10 I I I 0.21 D z + A r / X I 1 1 2.5 3.0 3.5 i distance/%i FIG. 4.-Potential curves for D2(H2) + Ne: (-) ref. (5) and (23); ( X ) ref. (31); (0) ref. (33). Insert: Ratio of VJV0 for the best experimental potentials of fig. 3 and 4 as a function of the classical turning point Ro( Vo) expressed by the values Vo of the isotropic potential.compare the predictions of these models for the measured ratio of the 0 -+ 2 transition to the 0 -+ 0 transition of D, + Ar. Both predictions are too low, ca. 40% for the TT model and 15% for the RS model, indicating that the ratio V2/ V, is too low. An inspection of the individual potentials in comparison with our best-fit results in fig. 3 shows that the deviations for the TT model are essentially due to V, being too large, while the RS model gives too small V, values. The comparison of the D,-Ne potentials shows deviations for the TT model from the experimental best-fit potential which partly compensate each other for the prediction of 0 -+ 2 transitions and a good agreement for the RS model.Thus the predictive character of the two models is satisfactory if the requirement for accuracy is not too strong. The high-precision measurements presented here can deviate appreciably, as shown for D, + Ar and the TT model.U. BUCK 195 HYDROGEN-MOLECULE DIMER SCATTERING THE INTERMOLECULAR POTENTlAL The interaction potential for a diatom-diatom system can, as in ref. (30), be expanded in complete orthonormal sets of sperical harmonics Wl,YZ,R) = 2 A I , I , L ( W 2 , R ) 2 (4m1~2m2l~J2W YAlrnI(4) rnz,z(+2) Y*Irn(R) . u 2 L mlrnlni where Y, and Y, are the internuclear vectors of the two molecules and R is the separation of the centres of mass. (a b c dl * - - .) is the usual Clebsch-Gordon vector coupling coefficient.Fig. 5 gives the leading potential terms for H2-H, as obtained from the 10 a 6 4 2 0 - 2 f ’ I I I I 2.2 2.6 3 .O 3.4 3.0 RIA FIG. 5.-Potential terms for H2 + H2 in spaced fixed coordinates of the ab initiu calculations of ref. (41) and (42). The dashed line is the isotropic potential determined from ref. (16). The arrow marks the position of the classical turning point of Aooo at E = 89.1 meV. most elaborate ab initio calculation by Meyer and S ~ h a e f e r . ~ ~ It is an improved version of the results published in ref. (30) which is based on a six-term rigid-rotor expansion at vibrationally averaged distances ri = 1.449 a.u. of the ground state u = 0, j = 0. These results have been corrected first for the different vibrationally averaged distances for differentj-states (vib-rotor) and second for a 19-term expansion at small distances.42 The first term, Aooo/(4n)3’2, is exactly the isotropic term.The leading anisotropic terms are The first corresponds to the Vz term of the = A022 and A224.196 INELASTIC SCATTERING OF HYDROGEN MOLECULES diatom-atom potential and is responsible for the 0 +- 2 transitions of one molecule. The second is asymptotically due to the long-range quadrupole-quadrupole interac- tion of the two molecules and couples only two rotating molecules. There are numerous experimental investigations of this system, ranging from equilibrium properties of gases and solids 43 and transport and relaxation phenom- ena 20*24 to scattering experiments 25-27 and the spectroscopy of the dimer.22v44 The isotropic interaction potentials, which are in most cases derived, differ appreciably from each other and suffer from the fact that they are mostly determined by one measured quantity. However, there are some general trends which converge to the same solution. In a comparative study Rulis and Scoles 26c found that the potentials derived by Silvera and Goldman 43 from solid-state equilibrium data also fitted their total differential cross-sections.Independently we ended up with nearly the same potential from the measurement of the total differential cross-section for D2 + H2 and the differential cross-sections for the 0 -+ 0 and 0 -+ 1 transitions of HD scattered from D2.* This potential is also plotted in fig. 5. All these measurements are mainly sensitive to the near-repulsive part of the potential from the minimum upwards.As is clearly seen there is a small but significant difference from the ab initio potential of Meyer and Schaefer, being essentially a shift of the zero point by 0.08 A to lower values. A very recent multiproperty analysis of high- and low-temperature virial coefficients, bound- state energies and low-energy integral and differential cross-sections confirms this conclusion, giving a potential zero of Ro = 3.018 A [ref. (45)], in close agreement with our value of A, = 3.00 0.03 A. All experiments which essentially probe the attractive part of the potential, such as low-energy integral cross-sections 25a-c or transport data,20 are predicted within 510% by the ab initio potential.No other calculated potential is precise enough to reproduce all these data, which are sensitive to the isotropic potential. A very similar conclusion holds for the experimental test of the anisotropic terms. A comparative study of rotational relaxation showed that none of the published potentials were able to fit these data.24 Up to now only the potential surface of ref, (41) and (42) gave good agreement for the rotational relaxation 42 and tensor cross- sections in the field dependence of transport p h e n ~ m e n a , ~ ~ as well as for low-energy transport data 2o and the integral cross-sections with oriented molecules.44 It should be noted that for those quantities 42,46 which are essentially sensitive to the repulsive part of the interaction the correction due to the nineteen-term expansion proved to be significant in reaching agreement with the measured quantities.We summarize: the corrected potential surface of Meyer and Schaefer is the best description of all the data available, with the exception of the repulsive part of the isotropic potential near the zero point, which should be a little softer. MEASUREMENTS AND RESULTS To test the anisotropy of the H2-H2 potential by microscopic properties we have performed measurements of the differential cross-section for 0 -+ 2 transitions for the HD + D2 system at 70.3 meV l7 and the D, + H2 system at 89.1 meV.18 We will discuss both systems separately since the analysis showed that they are sensitive to different parts of the potential surface. * A reevaluation of the total differential cross-section for D2+H2 presented in ref.(16) gave the parameters for the potential model of ref. (16) with the damping function exp { - [2.29(C8/C6)1'2/R - 112} : A = 101.4 eV; p == 2.779 A-1, y = 0.08 A-2; C, = 7.264 eV A6; c8 = 36.008 A8 eV; C,, = 225.56 eV; e = 2.92 meV and R, = 3.436 A.U. BUCK 197 HD + D2 The measured time-of-flight spectra are shown in fig. 6. The possible transitions are listed in table 4 and the arrows in fig. 6 mark these transitions in the time-of-flight spectra. The spectra are characterized as follows: (i) At small angles the elastic transition (0) dominates. (ii) The contribution of the 0 -+ 1 transition of HD(1) increases with increasing scattering angle and exceeds that of the elastic transitions at 1 .o 0.8 -5 0.6 h !3 -5 0.4 8 75! 0.2 G o u a .- G I 30" 5 0" Loo I 012 3 I 01 2 3 60" 0 1 2 3 - loops flight time FIG.6.-Measured time-of-flight spectra for HD + D2 at different laboratory scattering angles The numbered arrows mark the transitions listed in table 4. large angles. (iii) The 0 -+ 2 transitions of D2 give almost no detectable contribution over the measured angular range. (iv) Transition (3), which consists of the 0 -+ 2 transition of HD and the simultaneous transition 0 -+ 1 of HD and 0 --f 2 of Dz, increases with increasing scattering angle. Now we have to explain why transition (3) is much larger than transition (2) although it needs 50% more energy to be excited. In addition, we would like to know to which part of the potential surface transition (3) is sensitive. To answer these questions, we first have to transform the H2-H2 potential to the HD + D2 coordinate system.The details of this procedure have been described in ref. (16). TABLE 4.---TRANSITIONS FOR HD + D2 energy loss/ no. meV transition (HD, D2) 0 0.00 oo+oo, 01-+01,02+02 1 11.06 00-+10,01+11,02-+12 10.91 02-20 2 22.20 004 20 3 33.1 1 00+20,01-+21,02-+22 33.26 OO+ 12198 INELASTIC SCATTERING OF HYDROGEN MOLECULES Then we have to perform close-coupling calculations based on this transformed potential. The calculations were performed as described in ref. (16) and (30). Only open channels have been retained in the close-coupling basis expansion, which means that the relatively large cross-sections, which have been measured in the experiment, are considered converged within a few percent.The calculations were performed for the latest version of the ab initio potential of Schaefer and Meyer 41942 and a potential surface where only the Aooo term was replaced by the experimentally determined rigid- rotor potential derived from ref. (16) * keeping all the other terms of the ab initio potential and performing the necessary transformations. The results are given in table 5 for two characteristic angles. In addition we artificially changed the AZo2 TABLE CALCULATED ROTATIONAL TRANSITIONS FOR HD + D2 AT E = 67.1 meV a potential $(CM)/" 00 10 20 12 02 remark ref. (42) and (43) 60 1.875 0.545 0.035 0.013 0.011 - 180 0.321 0.524 0.117 0.033 0.015 ref. (42) and (43) 60 1.710 0.505 0.042 0.020 0.022 reduced Aooo: ref.(16) * 180 0.398 0.444 0.122 0.044 0.022 isotropy ref. (30) 60 1.854 0.547 0.036 0.017 0.015 - 180 0.303 0.560 0.102 0.042 0.023 ref. (30) 60 1.846 0.556 0.026 0.010 0.004 reduced 0.67 4202 180 0.358 0.588 0.094 0.027 0.007 anisotropy a The initial state is always 00; the values are in A2 sr-l. term of the H2-H, interaction by 33% in order to study the influence of a pure anisotropic change of the potential. The potential surface used is a previous version of the ab initio potential of Meyer and Schaefer 30 where the anisotropy is somewhat larger. The calculations confirm the experimental result. The fact that the 00 -+ 02 transition of D2 is excited with much less probability than the transitions 00 --+ 20 and 00 --f 12, where essentially HD is excited, is obviously due to the different coupling elements.The former transition is only caused by the coupling element Ah22, whereas AiO2 (which is larger than Ah,,) and A;21 are reponsible for the latter ones.7 In addition, successive transitions of Aj = 1 steps might also be possible for these transitions, which are caused by the much larger Aiol coupling term. The results of the calculations for the 0 -+ 2 transitions based on different potential variations are summarized as follows: (i) Decreasing the isotropic part Aooo by ca. 30% leads to larger cross-sections for 0 -+ 2 transitions by approximately the same amount. Only the 00 --f 20 transition (HD) in the backward direction is less affected. As discussed in detail for atom-molecule scattering, this general behaviour is caused by probing the anisotropy at smaller R values where it is larger.(ii) Decreasing the anisotropic part A202 of the H2-H, interaction by 33% leads to a decrease of all 0 -+ 2 transitions, again with the exception that the backward scattering of the 00 --+ 20 transition is less affected. We conclude that 0 -+ 2 transitions in HD + D2 collision are determined by equal amounts of the isotropic and the anisotropic potentials. However, in the extreme backward direction the 00 --+ 20 transitions of HD are more influenced by the A'lOl t Note that the prime (') is used for the transformed potential in HD + D2 coordinates. For * See previous footnote. details and a figure, see ref. (16).U . BUCK 199 term, which causes double transitions and which mainly depends on the isotropic potential.As described in detail in ref. (16), this behaviour leads only to small changes in the transition probability when varying the isotropic potential, since its influence on AAoo and Aiol almost compensate each other. Fig. 7 shows the comparison of the measured data with calculations based on the B(lab)/" I I I I I I I 0 50 90 130 170 deflection angle, SF,(CM)/O FIG. 7.-Comparison of measured differential cross-section ratios for HD + D2 with calculations based on the ab initio potential of ref. (41) and (42). For the transitions involved see table 4 and text. E = 70.3 meV. ab initio potential of ref. (41) and (42). We have plotted the expression a3/(ao + a,) with a3 = po [O(OO -+ 20) + a(O0 -+ 12)] + p1a(Ol -+ 21) + p2402 --+ 22) 0 0 = po a(O0 -+ 00) + p,a(Ol -+ 01) + p,a(02 + 02) o1 = po ~ ( 0 0 -+ 10) + plo(Ol + 11) + p2 [a(02 -+ 12) + a(02 --+ 20)].This quantity is easily derived from the time-of-flight distribution. For the calculations we have to sum all the different components. The comparison clearly shows deviation at large angles but agreement at intermediate angles. The agreement at intermediate angles indicates that the ratio of the anisotropic potential to the isotropic one is correct. However, the disagreement at large angles (probably being determined by successive 0 -+ 1 transitions) indicates that the isotropic potential is not correct. Using the experimental fit potential for Aooo would give larger values for the theory. To get agreement with experiment the anisotropy has also to be lowered.200 INELASTIC SCATTERING OF HYDROGEN MOLECULES Dz 3- H2 To check the conclusion of the last section we have tried to measure the very weak 0 -+ 2 transitions of the homonuclear system.We started with the system 0-D, + n-H, at E = 89.1 meV. The beam data are given in table 1. The result of a time-of- flight spectrum at the laboratory angle of 6' = 60" which corresponds to 9. = 160" in the centre-of-mass system is shown in fig. 8(b). An inelastic peak, which is due to the I 00-0020 02 1 01-01 21 I t I ~ ~ I ~ I ~ I L ~ ~ ~ ~ I ~ I ~ ~ 8 , a l l i l l l , l . , i l i , , , , , . . o r 20 0 300 200 3 00 flight time/,us flight timeips FIG. &-Measured time-of-flight spectra for D2 + Hz at E = 89.1 meV and &lab) = 60". The inelastic transitions are also shown in 5-fold enlargement.The solid lines are the results of a Monte- Czrlo simulation of the spectra with the cross-section ratios given by the ab initio potential of ref. (41) and (42). (a) o-Dz +p-H2. (b) o-Dz + n-H2. 0 -+ 2 transition of D,, is clearly resolved. To demonstrate this fact this portion of the spectrum is enlzrged by a factor of 5 in the figure. Since the scattering partner is n-H,, 75% of these molecules are i n j = 1 before and after the collision. A careful inspection of the coupling matrix elements 30 shows that such a transition 01 -+ 21 is due to both the AZo2 and the A,,, tzrm. The quadrupole-quadrupole term enters because H2 rotates before and after the collision. To separate this term from the usual anisotropy we performed a similar experiment for 0-D, + p-H, where the H2 molecules do not rotate before and after the collision.The result is presented in fig. 8(a). Now two very weak inelastic peaks appear which are due to the excitation of D,(OO -+ 20) and H,(OO -+ 02). The striking feature is that the 00 -+ 20 transition is much less in intensity than the same transition 01 -+ 21 with a rotating collision part- ner. This is obviously caused by the different matrix elements which are responsible for this transition. Thus it is possible to extract both anisotropic terms from these two sets of data. The calculation based on the ab initio potential of Schaefer and Meyer 4 1 3 4 2 is shown in fig. 9. The measured behaviour is confirmed. All inelastic transitions are smaller by more than one order of magnitude compared with the elastic transitions.The 01 --+ 21 transitions are approximately a factor of two larger than the 00 --f 20 tran-U. BUCK 20 1 sitions. The peaking in the forward direction clearly stresses that this difference is essentially due to the A,,,-term which is the largest anisotropic contribution at long- range distances and which is responsible for the small-angle scattering. To compare these calculations with the measured points we have carefully evalu- ated the measured time-of-flight spectra as described above. Since the amplitudes are 0 30 60 90 120 150 180 angle/" FIG. 9.-Calculated differential cross-section for rotational transitions of il(Dz)j,(H,)-ti:(Dz)i;(H,) at E = 89.1 meV based on the ab initio potential of Meyer and Schaefer [ref.(41) and (42)]. The experimental points are normalized to the elastic cross-sections and have been corrected for the measured level population: 0,01+21; A, 00-+20; 0,00-+02. rather small it proved very important to simulate the experimental distribution func- tions. The solid lines in fig. 8 are based on such a simulation with the cross-section ratio from the ab initio potential of ref. (41) and (42). The final results normalized to the calculated elastic cross-sections are displayed in fig. 9. In contrast to the large- angle result for HD + D2, the points are reproduced by the ab initio potential within their experimental error. This result is in agreement with the investigations on the rotational relaxation 42 and tensor cross-sections in transport properties 46 for H2 + H2, which were all predicted correctly by the ab initio potential.According to the discussion of atom-molecule scattering we know that 0 -+ 2 transitions of homo-202 INELASTIC SCATTERING OF HYDROGEN MOLECULES nuclear molecules are determined by the ratio of the anisotropy to the isotropic potential accounting for the fact that the coupling range is changed by the classical turning point. Since we know from several studies that the Aooo term has to be lowered 16*43945 compared with the ab initio calculation we conclude also that the anisotropic terms and A224 have to be shifted by the same amount. Indeed first calculation with a complete potential surface shifted by 0.1 8, gave the same results for the inelastic transitions." As for the accuracy of the final result it should be noted that the cross-sections depend quadratically on the potential, which means that even with experimental errors of 20% the potential is known with an accuracy of 10%.This result also has interesting consequences for the phase transitions in solid H2,38*39 which were not predicted correctly by the theoretical quadrupole interaction. The experiment required a smaller interaction in agreement with our conclusion. Almost the same result was obtained by the analysis of the hyperfine spectrum of the hydrogen- molecule dimer, which mainly probes the attractive interaction. The measured quantities could only be explained by shifting the potentials, both the isotropic and the anisotropic ones, to smaller values.22 SUMMARY The measured angular dependence of 0 --f 2 rotational transitions of D2 in D2 + H2 collisions proved to be a sensitive probe for the repulsive anisotropic part of the H2-H, interaction potential.Depending on the rotation of the scattering partner H2 both the normal anisotropy A202(R) and the quadrupole-quadrupole term A,,,@) are determined. As for the isotropic part AOo0, a shift of 0.1 8, to smaller R values is found compared with the most elaborate ab initio potential computed by Meyer and Schaefer. Note that the inelastic cross-sections are correctly predicted by this potential. However, since the cross-sections are sensitive to the ratio of anisotropic to isotropic forces, the necessary correction for the isotropic part found in earlier investigations has also to be applied to the anisotropic forces.I thank Drs F. Huisken, H. Meyer, A. Kohlhase and D. Otten for their contri- butions to this paper. I also thank Dr J. Schaefer for performing the close-coupling calculations for HD + D, and D2 + H2. I am grateful to Prof. R. B. Le Roy for helpful correspondence on his H,-Ar potential and to Prof. R. T. Pack for providing me with a copy of his program for calculating diffusion coefficients. Faraday Discuss. Cheni. SOC., 1982,73. W. R. Gentry and C. F. Giese, J. Chem. Phys., 1977, 67,5389. U. Buck, F. Huisken, J. Schleusener and H. Pauly, Phys. Reu. Lett., 1977,38, 680. U. Buck, F. Huisken and J. Schleusener, J. Chem. Phys., 1980,72, 1512. ' J. Andres, U. Buck, F. Huisken, J. Schleusener and F. Torello, J.Chem. Phys., 1980, 73, 5620. J. Andres, U. Buck, F. Huisken, J. Schleusener and F. Torello, in Electronic and Atomic Collisions, ed. N. Oda and K. Takayanagi (North-Holland, Amsterdam, 1980), p. 531. ' U. Buck, H. Meyer and R. B. Le Roy, to be published. * K. Bergmann, U. Hefter and J. Witt, J. Chem. Phys., 1980,72, 4777. lo U. Hefter, P. L. Jones, A. Mattheus, J. Witt, K. Bergmann and R. Schinke, Phys. Rev. Lett., l1 J. A. Semi, A. Morales, W. Moskowitz, D. E. Pritchard, C. H. Becker and J. L. Kinsey, J. l2 M. Faubel, K. H. Kohl and J. P. Toennies, J. Chem. Phys., 1980, 73, 2506. l3 U. Buck, A. Kohlhase and H. Meyer, to be published. l4 W. R. Gentry and C. F. Giese, Phys. Rev. Lett., 1977, 39, 1259. K. Bergman, U. Hefter, A. Mattheus and J. Witt, Chem.Phys. Lett., 1981, 78, 61. 1981,46,915. Chem. Phys., 1980, 72, 6304.U . BUCK 203 l5 W. R. Gentry, in Electronic and Atomic Collisions, ed. N. Oda and K. Takayanagi (North- l6 U. Buck, F. Huisken, J. Schleusener and J. Schaefer, J. Chem. Phys., 1981, 74, 535. l7 U. Buck, F. Huisken, G. Maneke and J. Schaefer, J. Chem. Phys., 1982, to be published. l8 U. Buck, F. Huisken, A. Kohlhaw, D. Otten and J. Schaefer, J. Chem. Phys., 1982, to be l9 R. D. Tengrove and P. J. Dunlop, 8th Int. Symp. on Thermophysical Properties 1981, preprint. ’O see L. Monchick and J. Schaefer, J. Chem. Phys., 1980,73, 6153. R. J. Le Roy and J. S. Carley, Adv. Chem. Phys., 1980, 42, 353. ” M. Waaijer, M. Jakob and J. Reuss, Chem. Phys., 1982,63, 257. 23 M. Waaijer and J. Reuss, Chem. Phys., 1982, 63, 263. 24 R. M. Jonkman, G. J. Pragsma, I. Ertas, H. F. P. Knaap and J. J. M. Beenakker, Physica, 1968, 38, 441. ’’ (a) W. Bauer, B. Lantzsch, J. P. Toennies and K. Walaschewski, Chem. Phys., 1976, 17, 19; (b) R. Gengenbach, Ch. Hahn, W. Schrader and J. P. Toennies, Theor. Chim. Acta, 1974, 34, 199; (c) D. L. Johnson, R. S. Grace and J. G. Skofronik, J. Chem. Phys., 1979, 71,4554; ( d ) J. P. Toennies, W. Welzand G. Wolf, J. Chem. Phys., 1979,71, 614; (e) R. Gengenbach and Ch. Hahn, Chem. Phys. Lett., 1972, 15, 604. 26 (a) M. G. Dondi, U. Valbusa and G. Scoles, Chem. Phys. Lett., 1972,17, 137; (b) J. M. Farrar and Y. T. Lee, J. Chem. Phys., 1972, 57, 5492; (c) A. M. Rulis and G. Scoles, Chem. Phys., 1977, 25, 183; ( d ) A. M. Rulis, K. M. Smith and G. Scoles, Can. J. Phys., 1973,56, 753. Holland, Amsterdam, 1980), p. 807. published. 27 L. Zandee and J. Reuss, Chem. Phys., 1977,26, 345. 28 W. Meyer, P. C. Hariharan, and W. Kutzelnigg, J. Chem. Phys., 1980, 73, 1880. 29 G. A. Gallup, Mol. Phys., 1977, 33, 943. 30 J. Schaefer and W. Meyer, J. Chem. Phys., 1979, 70, 344. 31 K. T. Tang and J. P. Toennies, J. Chem. Phys., 1978,68, 5501; 1981,74, 1148. 32 R. Ahlrichs, R. Penco and G. Scoles, Chem. Phys., 1977, 19, 119. 33 W. R. Rodwell and G. Scoles, 1981, preprint. 34 K. C. Ng, W. J. Meath and A. R. Allnatt, Mol. Phys., 1979, 37, 237. 35 J. van Kranendonk, 1981, preprint. 36 R. B. Gerber, V. Buch and U. Buck, J. Chem. Phys., 1980, 72, 3596. 37 R. B. Gerber, V. Buch, U. Buck, G. Maneke and J. Schleusener, Phys. Reu. Lett., 1980,44,1397. 38 R. J. Wijngaarden and I. F. Silvera, Phys. Rev. Lett., 1980, 44, 456; I. F. Silvera and R. J. 39 A. Lagendijk and I. F. Silvera, to be published. 40 H. P. Godfried, I. F. Silvera and J. van Straaten, 12th Int. Symp, on Rarefied Gas Dynamics, 41 W. Meyer and J. Schaefer, to be published. 42 J. Schaefer and B. Liu, to be published. 43 I. F. Silvera and V. V. Goldman, J. Chem. Phys., 1978, 69, 4209. 44 J. F. C. Verberne, Ph.D. Thesis (University of Nijmegen, The Netherlands, 1979). 45 G. T. Conville, J. Chem. Phys., 1981, 74, 2201. 46 W. E. Kohler and J. Schaefer, J. Chern. Phys., 1981, in press. Wijngaarden, Phys. Rev. Lett., 1981, 47, 39. Charlottesville, 1980, paper 102.

ISSN:0301-7249    

DOI:10.1039/DC9827300187

出版商:RSC    

年代:1982

数据来源: RSC

摘要:

Furuduy Discuss. Chem. SOC., 1982, 73, 205-220 The He-N2 Anisotropic Van der Waals Potential Test of a Simple Model Using State-to-state Differential Scattering Cross-sections BY MANFRED FAUBEL, KARL-HEINZ KOHL AND J. PETER TOENNIES Max-Planck-Institut fur Stromungsforschung, 3400 Gottingen, West Germany AND KWONG TIN TANG AND YAT YAN YUNG Department of Physics, Pacific Lutheran University, Tacoma, Washington 98447, U.S.A. Received 3rd December, 198 1 For He-N, the total and thej = 0 4 2, 1 4 3 rotationally inelastic differential cross-sections have been measured at a collision energy of Ecm == 27.3 meV and at centre-of-mass angles between 5 and 50". These new data have been used to test the validity of an anisotropic potential recently calculated using the model of Tang and Toennies.Theoretical close-coupling cross-sections based on the potential are found to be in good agreement with the data. By studying the effect of small modific- ations in the potential it has been possible to estimate that the model potential predicts the anisotropy with only a negligible error but understimates the well depth of the spherical potential by ca. 20 %. INTRODUCTION The anisotropy of molecular potentials is of great interest for an understanding of crystal structure and also the solid, liquid and gaseous equations of state.2 The anisotropy of molecules also has a direct effect on gas dynamic transport coefficients and rotational relaxation times.2 Recently developed molecular-beam experiments of essentially two types, in which (1) the cross-section anisotropy of polarized molecule beams and (2) the rotationally inelastic differential cross-sections are mea~ured,~ have made possible a precise measurement of the anisotropy of the Van der Waals potential.The latter experiments, first successfully carried out for TIF in small angle collision^,^ have since been extended to larger scattering angles, first for Li+-H2,6 H +-H, ' and then for He-H,, Ne-H, collisions using time-of-flight techniques. These scattering experiments have made it possible critically to test ab initio CJ calculations of anisotropic Van der Waals potentials. They have also stimulated the development of semi-classical and semi-empirical lo models, which rely only on easily calculated asymptotic interaction parameters. For the rare-gas-H, systems the model of Tang and Toennies has been able to predict not only the anisotropy for He-H, and Ne-H,," but also the full potential hypersurface in good agreement with all of the available ab initio and experimental data.Unfortunately, until recently l3 beam experiments were not available to test the model for more typical molecules. In this paper we report on extensive measurements of the He-N, rotationally inelastic differential cross-sections at E,, = 27.3 meV in the range of centre-of-mass scattering angles 5-5Oo.l4 Parallel to this development it has also recently been206 He-N, ANISOTROPIC VAN DER WAALS POTENTIAL possible to apply the Tang-Toennies model to calculate the anisotropic Van der Waals potential of He-N2.15 In the present paper we report on a comparison of inelastic differential cross-sections calculated for the model potential with the new experimental data.The overall agreement is good and well within the errors of the input para- meters. The small remaining differences between the model potential and the true potential are estimated. time-of-f tig ht pr I ma ry beam -v- chopper FIG. 1 .-(a) Thz principle of the crossed supersonic molecular beams time-of-flight apparatus shown schematically. (b) The actual design of the machine, incorporating six differential pumping stages for the production of two well defined supersonic beams. A helium partial pressure of Torr in the mass-spectrometer detector is sustained by four additional differential vacuum stages. Horizon- tal and vertical scattering angles are selected by motion of the ball bearing supported flanges " v " and " h." The flight-path length of 165 cm allows 1 % velocity resolution.FAUBEL, KOHL, TOENNIES, TANG A N D YUNG 207 EXPERIMENTAL APPARATUS The apparatus is shown schematically in fig.l(a) and to scale in the side view machine assembly drawing in fig. l(b). Two nearly monochromatic molecular beams, a He primary and the N2 target beam, are formed in nozzle expansions. After collimation by skimmers and additional differentially pumped collimators the two beams intersect at the centre of the apparatus. Scattered He atoms are observed by the mass-spectrometer detector at the right- hand side of fig. l(a) or (b), respectively. The scattering angle is changed by rotating the detector part of the machine with respect to the vertical axis indicated in fig.l(b). For observing individual rotational transitions at a given scattering angle the flight time, and thus the energy change of scattered molecules, is measured. To do this a 12 cm diameter chopper wheel with four 1 mm wide slots is rotated through the He beam at 24 000 r.p.m. The count rate of scattered particles arriving at the detector is then recorded as a function of the time delay from the passage of a chopper slit through the helium beam. A typical time- of-flight spectrum is shown in fig. 2. The time resolution and thus the energy-level resolution is limited by apparatus effects such as the finite chopper opening time of 7 ps and the ratio of the ionizer length AL !z 1.5 cm to the length of the total flight path, which is L = 165 cm and thus allows a velocity resolution of 1 % corresponding to an energy resolution of 0.6 meV at a He energy of 30 meV.A second, and more serious, kinematical limitation on the energy resolution4 results from the finite velocity spreads and angular divergences of the colliding beams. Hence, the scattering experiment was optimized in the sense of chap. 1V.B of ref. (4). The actual beam distributions of the present experiment, which are either affecting the resolution or the evaluation of the scattering experiment, are, for brevity, all summarized in table 1. At 8LAB = 20" the resulting overall energy resolution is 0.8 meV for a collision energy of 27 meV. TABLE 1 .-EXPERIMENTAL PARAMETERS component nozzle conditions spreads (f.w. h.m.) ~~ helium primary beam diameter: DN = 20 pm angular: Aa = 0.75" in-plane pressure: temp.: To = 88 K po = 100 bar velocity: A u/v, = 0.7% at u, = 950 m s-l AP = 2.0" out-of-plane ~ N2 pressure: po = 8 bar velocity: Au/u, = 5% at u, = 750 m s-' target beam diameter: DN = 100 pm angular: Aa = 2" temp.: To = 300 K AD = 2" rotational population [ref. (1 6)] ji L- 0 : 52% ji = 1 : 33% ji = 2 : 15% detector I. velocity: Avlv = AL/L = 1% at L = 165 cm angular: Aa = 0.4" AP = 0.2" The narrow beam collimation and the long flight path lead to a drastic loss in scattering intensity. Therefore, the He partial pressure in the detector had to be lowered to less than the signal density of 10- l6 Torr. As shown in fig. l(b) the detector vacuum is sustained against the main chamber pressure of Torr by four differential pumping stages along the flight path.the average detector count rate was ca. 10 counts s-' and measuring times of typically 10 to 30 h were required to obtain the TOF spectra shown in fig. 2 and 3. With an estimated He detection efficiency of208 He-N, ANISOTROPIC VAN DER WAALS POTENTIAL I I I I I I I I I 170 0 1800 1900 2000 flight time/ls FIG. 2.-Example of a time-of-flight spectrum of He scattered from N2. The spectrum was accumul- ated in a measuring time of 14 h. For the collision energy of E,, = 27.3 meV and the scattering angle OLAB = 20" (equivalent to aC, = 17.5") the flight time of elastically scattered He is 1750 p s . Well separated from the elastic peak He which has excited thejl := 0 toj' = 2 rotational transition of Nz and lost an energy of 1.5 meV (12 cm-l) arrives at a flight time of 1800 ps, Additional inelastic peaks from the 1 4 3 , 2+0 and 2+4 rotational transitions are also observed as indicated by the arrows.CROSS-SECTION MEASUREMENTS With the He and N2 beam velocities listed in table 1 the centre-of-mass collision energy is E,, = 27.3 meV An example of a TOF spectrum measured at a laboratory scattering angle of 20" is shown in fig. 2. It shows a fast peak of elastically scattered helium atoms at a flight time of 1750 ps. At 1800 f i s an even larger second peak is found with an energy loss of 1.5 meV equal to the level spacing between t h e j = 0 and t h e j = 2 rotational levels of 0rth0-N~. The shoulder on the right-hand side of the 0+2 transition peak results from the 1-23 tran- sition of para-NZ.Also observed, and indicated by the arrows in fig. 2, are the much weaker 2+0 and 2+4 rotational transitions of thej, = 2 state of ortho-N,, which, as listed in table 1, accounts for 15% of all N2 beam molecules. Higher rotational transitions could not be observed and are estimated to have transition probabilities of < 3% in all measured TOF spectra. The variation of TOF spectra with scattering angle is shown in fig. 3 for a number of out- of-pIane laboratory scattering angles in the range from 16 to 47". At most angles the 0+2 and 1+3 rotational transition probabilities are quite small, of the order to 10 to 20%. However, at certain specific angles OLAB = 20, 28 and 47" and at larger angles the inelastic transition probabilities are comparable to the elastic probabilities.Using the N2 initial rotational state populations of table 1 the TOF spectra were corrected to yield state-to-state differential laboratory cross-sections. These were then transformed into the centre-of-mass system and are shown in fig. 4 as the total and the 0+2 and 1+3 rotationally inelastic differential cross-sections. The systematic average error in the absolute calibration in AZ sr-' is estimated to be 5%. The accuracy of the scattering angles is 0.2".FAUBEL, KOHL, TOENNIES, TANG AND YUNG 209 t ime-of-flight FIG. 3.-He-N2, Ecm = 27.3 meV time-of-Aight spectra at various laboratory scattering angles. Here the amplitude of the largest peak is normalized to unity.Locations of the 0+2 and 1+3 rotational transitions are labelled by vertical lines. cross-210 He-N, ANISOTROPIC VAN D E R WAALS POTENTIAL Individual points have statistical errors of ca. 10% fcr cross-sections t 5 A2 sr-' and 3% for Iarger cross-sections. All cross-sections show very well resolved Fraunhofer diffraction oscillations, but as an indication of the very shallow well of the He-N2 potential no characteristic rainbow struc- ture is present. The absolute amplitude of the oscillations in the rotationally inelastic 0+2 and 1+3 transitions is slightly quenched as compared with the total differential cross-section. The 1+3 transition differential cross-section is smaller than the 0+2 cross-section, and its oscillatory structure is almost proportional to the 0+2 cross-section with the proportionality ratio of ca.0.7. This value agrees to within 10% with the ratio predicted by the infinite- order sudden factorization rule.14 Remarkably, however, the total cross-section oscillations are shifted almost exactly 180" out-of-phase with respect to the inelastic cross-sections. THEORETICAL THE POTENTIAL MODEL The He-N, Van der Waals interaction potential used to fit the data is reported in ref. (1 5) and was calculated from the potential model described in ref. (9) and (1 1). This model potential is in some respects similar to the Buckingham potential and related models proposed in the past. In keeping with these, the short-range potential is assumed to be of the Born-Mayer form, with potential parameters, however, determined from an accurate SCF calculation.The long-range potential is given by the usual perturbation expansion of which all the terms are included. However, the asymptotic divergence of the full expansion is accounted for in a simple approximate fashion. In the intermediate region a new term is introduced to account for the per- turbation of the dispersion forces by the repulsion. The dispersion coefficients can be accurately expressed in terms of integrals over the frequencies of the dynamic multipole polarizabilities of the isolated partners. The dynamic polarizability for He was derived from accurate CI calculation^.'^ For N2 they have been estimated by an SCF procedure." For the explicit form of the rather lengthy integrals over the appropriate combinations of dynamic polarizabilities which give the dispersion coefficients we refer to ref.(15) and (18). The long-range dispersion potential is expanded in R, the distance between centres-of-mass, where y is the angle between R and the molecular axis. the C,,(y) are expanded in Legendre functions up to P4 To account for the anisotropy CJy) = C&[1 + ~::)P,(cosY) + ri2P4(cosy)l. ( 1 4 The numerical values of the C,", and r\\) are listed in table 2. To remove the asymp- TABLE 2.-sOME OF THE DISPERSION COEFFICIENTS c;,, AND I-\:) VALUES USED FOR CALCULATING THE LONG-RANGE He-N, POTENTIAL 2n = 6 10.2 0.101 0 2n = 8 185 0.486 - 0.045 2n = 10 4 360 0.486 -0.045 2n = 12 13 3519 0.486 - 0.045 ~~ a 1 a.u. = 0.529 A (length); 1 a.u. = 27.21 eV (energy).F A U B E L , KOHL, TOENNIES, TANG AND YUNG 21 1 totic divergence of the dispersion series each of the terms C,,(y)/R2" has been truncated by a function : The repuIsive part of the He-N, potential was obtained by calculating the SCF potential energies in the range 4 < R/a.u. < 8 for the orientation angles y = 0, 45 and 9U0.15 The SCF energy values were also fitted to a 3-term Legendre expansion in y, each coefficient of which could be well approximated by Born-Mayer radial func- tions: with the values A,n and P2n being given in table 3.TABLE 3.-SCF BORN-MAYER RADIAL POTENTIAL VALUES A,n AND FOR He-N2 l5 term A,A/a.u. Dz A /a - u . 2R = 0 134.7 2.094 211 = 2 217.9 2.122 21, = 4 91.86 2.178 These two asymptotic regions of the He-N, potential are joined together by the following prescription which takes account of the effect of electron overlap on the dis- persion coefficients.This correction has been estimated semi-classically from the energy splitting induced in Drude model molecules when coupled to each other by the SCF interaction : Here the derivatives with respect to R are taken for the SCF potentiaI eqn (3). The factors M (He) and M (NJ stand essentialIy for the quotient w,/k of the frequency wo and the force constant k of the Drude electron gas model employed for the isolated He and N2 molecules. Their values can be determined from the known C, constants of the He-He and the N2-NI potential, the static pdarizabilities of He and of N, and the (Drude model) effective number of electrons. The value of M for He is M(He) = 0.044, and for N2 it was found to be almost independent of the orientation angle y with an average value of M (N,) = 0.130.15 With this correction the complete model potential reads :212 He-N, ANISOTROPIC VAN DER WAALS POTENTIAL was used, and the first three radial potential functions u21 (R) for the He-N, interaction are shown in fig.5 as smooth lines. I 1 2.5 3.0 3.5 4.0 4.5 5 .O - 4 RIA FIG. 5.-Radial terms vo, u2 and u4 of the He-N, Van der Waals potential as predicted by the Tang- Toennies potential model (smooth lines). Also shown (dashed lines) are the uo and u2 terms of an experimental He-N2 potential obtained by Keil, Slankas and Kupperniann from fitting total differential cross-sections only. The spherical part uo of this potential has a well depth of E' = 1.62 meV and an R& = 3.83 A.Remarkably, in the region of the repulsive barrier the u, ( R ) potential term is considerably larger than the spherical term while the u4 term has only a sig- nificant effect in the potential well region and at the beginning of the repulsive barrier. For comparison, in fig. 5 we also show (dashed lines) the uo and the u2 term of a recently published He-N, potential derived by Keil, Slankas and Kuppermann l9 entirely from total differential cross-section measurements at E,, = 64 meV. This potential is an " effective " potential for the total differential cross-section in that it accounts indirectly for the quenching due to inleastic processes. There is a sur- prisingly large discrepancy between the spherical part uo of this previous potential and the uo part of the present potential model, which we will discuss later.COMPUTATION OF CROSS-SECTIONS Differential elastic and rotationally inelastic cross-sections were computed from this predicted He-N, interaction potential for comparison with the experimentalFAUBEL, K O H L , TOENNXES, TANG AND YUNG 21 3 data. The experimental collision energy of E,, = 27.3 meV was low enough for a close-coupling calculation including 5 rotational states with 25 coupled channels to give fully converged cross-sections. For the details of the essentially exact quantum close-coupling procedure we refer to ref. (20) or to the review article in ref. (21). Since the N2 molecular beam used in the scattering experiments contained both 1 O2 1 0' 1 oo 30 60 90 120 150 180 8 cml' FIG.6.-5 states, 25 channels close-coupling elastic, rotationally inelastic and total differential cross- sections calculated from the Habitz, Tang and Toennies potential for He-N2at E,, = 27.3 meV. The initial rotational stage of Nz is (a) ji = 0 and (6) ji = 1 . ortho-(j, = 0,2) and para-(ji = 1)N2, with the populations given in table 1 , cross- section calculations had to be performed for the initial states ji = 0,l and 2. Fig. 6(a) and (6) show these theoretical cross-sections for the dominant N, fractions j i = 0 and j i = 1, respectively, and for the full angular range from a,, = 0 to 180". Con- vergence of the calculations was checked at every 10th partial wave by a 64 channel calculation. Additionally, 4 channel (2 states) calculations showed only minor changes214 He-N, ANISOTROPIC V A N DER WAALS POTENTIAL in the cross-sections of 10% in the experimentally important region from 0 to 60".In further test calculations it was also found that the zi4 term of the potential influences the cross-sections by < 10% for angles 9 (60". The cross-sections shown in fig. 6 are therefore believed to converge also with respect to the potential expansion. At small scattering angles, up to ca. 50", the elastic and Aj = 2 rotational tran- sitions are largest, and show the characteristic diffraction oscillations of the experi- ment. For larger angles the higher Aj = 4 and Aj = 6 cross-sections become pre- valent, thus giving rise to the rotational rainbow phenomenon observed in Na, + rare-gas 22*23 collisions.Also shown in fig. 6 are the total differential cross-sections for j i = 0 and j i = 1 . Compared with the rather large differences seen between the individual Aj, and in particular in the Aj = 0 elastic transitions of the different initial states, the total cross-sections are almost identical for j i = O,.ji = 1 and also for the j i = 2 state which is not shown here explicitly. DISCUSSION COMPARISON OF THEORY AND EXPERIMENT For the comparison with the measurements, shown in fig. 7, it was more conven- Examination ient to compare the inelastic 0+2 and 1+3 and the total cross-section. 0 10 20 30 40 50 60 9 cml' FIG. 7.-Comparison of the predicted theoretical (smooth curves) with the measured total (O), 0+2 (0) and 1+3 (U) differential cross-sections for He-N2 at Ecm = 27.3 meV.of fig. 7 reveals that all characteristic features of the experimental cross-sections, e.g. oscillatory decreasing total cross-sections, amplitude of inelastic cross-sections and relative phase shift between the diffraction oscillations of the total and the rotationally inelastic cross-sections, are well predicted by the theory. However, slight deviations exist in the absolute magnitude and in the position of the diffraction maxima. Part of this discrepancy might arise from the small experimental averaging of the cross-sections over a finite range of centre-of-mass energies and scattering angles. The effect of this convolution of the apparatus resolution with the ideal theoretical cross-section is considered separately in fig. 8. Here, for numerical simplicity, theFAUBEL, KOHL, TOENNIES, TANG AND YUNG 215 " total " differzntial cross-section generated from an effective spherical potential for He-N, l9 was used and was convoluted both with the experimental centre-of-mass scattering angle and the energy resolution function.This centre-of- mass resolution function was approximated by a two-dimensional Gaussian distribution with the vacancies in E,, and a,,, and their covariance calculated from the haIf widths of the molecular-beam spreads and the detector aperture angles given in table I. The comparison of the convoluted (smooth line) with the original (dashed line) total cross- section at E,, = 27.3 meV in fig. 8 shows a considerable quenching onIy in the region I I I I I I0 20 30 4Q 50 60 $C,/" FIG.8.-Effect of the angular and energy averaging by the present experiment. The (dashed line) theoretical total cross-section calculated for Ecm = 27.3 meV from the spherical KSK potentiaI was convoluted with the apparatus Ecm - aCm weighting function, resulting in the (smooth line) convoluted cross-section. Comparison with the experimental total cross-section only (e) shows the KSK spherical potential is in similarly good agreement as was the full HTT potential in fig. 7. of the minima, which is especially apparent fur the first two very deep and sharp diffraction minima at 8,, = 11.6 and 18.3". Only minor changes are, however, observed in the position and in the amplitude of the diffraction maxima which are decreased by the averaging of the present experiment by <lo%.Thus, the sig- nificantly larger deviations between the experiment and the model potential close- coupling cross-sections found in fig. ? cannot be ascribed to the finite resolution of the experiment. SENSITIVITY TESTS FOR VARIATIONS OF THE POTENTIAL In order to estimate the magnitude of the errors in the model potential that could account for the remaining discrepancies between experiment and theory, a number of sensitivity test cross-section calculations were made with slightly changed potentials. A first test has already been given in fig. 8 where the spherical He-N, potential of Keil, Slankas and Kupperman (KSK) and not the Habitz, Tang and Toennies (HTT) v0 was used. As was seen in fig. 5 the KSK spherical potential is quite different from the zio term of the present potential model by a 7% smaller R,.Because of the small u2 term of the KSK potential, the (not shown) KSK rotational inelastic cross-sections are more than a factor of 5 too small. Nevertheless, as seen in fig. 8 the KSK effective216 He-N, ANISOTROPIC VAN DER WAALS POTENTIAL vo potential gives approximately the same fit to the experimental total cross-section points as did the close-coupling total cross-sections of the HTT potential in fig. 7. This seems to indicate that with strongly asymmetric potentials the total scattering cross-section structure is preferentially weighting the perpendicular orientation Y(R,y = 90°> 2i u,(R) - 3 u2(R) of the potential. As can be easily estimated from the HTT potential in fig. 5, this potential is quite close to the effective KSK uo potential.Next, the influence of the non-spherical potential deformation on the inelastic cross-sections was checked. As aiready stated, the effect of the v4 part of the potential on the experimentally observed Aj = 2 cross-sections is negligible. To see the in- fluence of a change in the 27, term the HTT potential model was shifted in R by an amount a : u; ( R ) = u2 (R - a). The potential plots, fig. 9(a), show for a = +0.08, -2 L --I \-/I- \ -4 4 2.5 3.0 3.5 4.0 4.5 5.0 R i A FIG. 9.-Variations of the HTT potential used for the sensitivity test calculations shown in the subsequent fig. 10 and 11. (a) uXR) = vZ(R - a); (b) ui(R) = bv,(R), vz, uq omitted. -0.08 and -0.24 A how this produces slight right- or left-hand shifts of o2 (A) and is thus increasing or decreasing the potential asymmetry.Fig. 1 O(g)-(d) compare the close-coupling total and 0+2 inelastic differential cross-sections obtained by these changes in u2 with the respective experimental cross-sections of fig. 7. Fig. lO(a> shows that a shift of v2 (R) by only +0.08 A, i.e. ca. 2% of R,, to larger R leads to a drastic (more than a factor of two) increase in the 0+2 cross-section and also to a noticeable shift in the locations of the diffraction pattern of the 0+2 as well as of the total cross-section. Shifts of u2 (R) to smaller R by similar amounts of a = -0.08, -0.16 and -0.24 A, shown in fig. TO(b)-(d), lead to a steady decrease of the 0+2 cross- section. For these decreasingly smaller aspherical deformations, however, the total cross-section remains nearly unaffected.In the 0+2 cross-section only the absolute magnitude is changed, but neither the overall structure nor the position of the maxima of the oscillations are effected. Thus, the experiment has been carried out under conditions where not only the ratio of the total to the ineIastic cross-sections is chang-FAUBEL, KOHL, TOENNIES, TANG AND YUNG s 0 217 0 * 0 0 [-JS y/(cclp/qq FIG. lO.-(u) to ( d ) show sensitivity test calculations of cross-sections for the v2 potential term vari- ations illustrated in fig. 9(a). (u) a = f0.08, (b) a = -0.08, (c) n =- -0.16, ( d ) a 1 0 . 2 4 A. Smooth lines are the theoretical total and 0-+2 rotations1 differential cross-sections, and 0 the respective experimental points.Compared with the changes induced in the theoretical 0+7 cross- section, the total cross-section is rather insensitive to these potential variations., ing rapidly, but also the phase relation between ths 0 4 2 and total differential cross- section undergoes rapid changes with small changes in the potential asymmetry. The unexpected shift of the phase of the inelastic cross-sections at a critical poten- tial deformation can be rationalized lb from a consideration of the Fraunhofer diffrac- tion from a ring-shaped aperture in a diaphragm.24 Here, as in inelastic forward218 He-N, ANISOTROPIC V A N DER WAALS POTENTIAL scattering, only partial waves within an up?er and lower bound can contribute to the diffraction. For this case a rather sudden transition occurs from the diffraction pattern of the full circular aperture (or equivalently a disc) to a similar but 180" phase- shifted pattern when the annular opening is made narrower than 20% of the outer radius.Rather crude, but analytic, inelastic scattering models like the energy sudden in- elastic Fraunhofer scattering theory for a slightly deformed hard sphere predict 25 that the rotationally inelastic cross-sections should be proportional to the square of the deformation amplitude 6. When assigning the deformation amplitude of this model to the difference of the repulsive potential V (R,y = 0) and the V (R,y = 90") orientations (at 20 meV) the present value of the deformation would be 6 = 0.6 8, for the HTT-N2 potential. A shift of u2(R) by a = -0.24 8, as used for the cross- section variation of fig.10(d) reduces the deformation amplitude to 8% 0.23 & corresponding to a change of the original deformation 6 = 0.6 A by a factor of 2.6. Thus, within the inelastic Fraunhofer model a reduction of the 0+2 cross-section by a factor of (S/S')* = 6.8 would be predicted, in good agreement with the close coupling result shown in fig. 10(d). From this a2 relationship, the amplitude deviation of ca. 15% between experimental 0+2 cross-section and the HTT potential prediction, fig. 7, would be removed by a maximal decrease of the deformation of only 7% to 6 z 0.56 A, or, in terms of u2 (R), by a shift of a z -0.02 to -0.03. The total cross-section, however, as is seen in fig. lO(b)-(d), would only be negligibly changed, and thus the dis- crepancy between the experimental and predicted total cross-section would remain.Therefore, in a further sensitivity test calculation the vo term of the potential was changed by replacing uo (R) = bu, (R). This variation of the vo potential, shown in fig. 9(b) for b = 1.2 and 2.0, linearly decreases the potential well without affecting seriously the location of the repulsive barrier. The effect on the total cross-section is shown in fig. 11, and is again compared with the experimental points. In relation to the dashed line cross-section from the HTT original u,, b == I .2 increases the total cross-section by 10 10 0 I ?,A 3 10 8 F. 'd, 10' I I I I I I I I I I 0 10 20 30 40 50 60 9. cmlc FIG. 11.-Sensitivity of the experimental (0) total differential cross-section to the uo part of the HTT potential.The oscillation amplitude of the theoretical cross-sections (dashed for the original and smooth lines for the varied potential) increases when the parameter 6 , controlling essentially the well depth of the potential, is increased. uo'(R) = buo(R), u2 and u4 omitted. The uo potential term was varied as shown in fig. 9(b).FAUBEL, KOHL, TOENNIES, T A N G A N D Y U N G 219 approximately the right amount, and b = 2.0 gives too large an oscillation maxima. From this we estimate that the well depth of the present potential model is ca. 20% and at the very most 30% too shallow. CONCLUSIONS The comparisons between experimental cross-sections and calculations based on the potential of Habitz, Tang and Toennies reveal that the model provides an overall good description of the true potential.This agreement has been further substantiated by calculations of diffusion and viscosity coefficients,26 which after account is taken of the relatively large anisotropy agree remarkably well (< 1 %) with the published mea~urements.~~~~' The test provided by the differential cross-sections is an extremely critical one since the data is found to be sensitive to errors of only 0.02 8, in R and ca. 0.1 meV in energy in the potential-well region. The errors in the model potential are very small in the repulsive range but the well depth appears to be cu. 20% ( ~ 0 . 3 meV) but definitely not more than 30% too shallow. In this connection it is gratifying to note that the model is based on estimated dispersion coefficients which were reduced by ca.20% to bring them into agreement with the best semi-empirical previous esti- mates, and without this correction the model might well be in better agreement with experiment. Thus the agreement is roughly within the estimated errors of the input parameters of the model potential. The present work indicates for highly anisotropic systems similar to He-N, that spherical potentials obtained from total cross-sections are quite different from the u, spherical term in the standard Legendre expansion. This is a striking departure from the experience for H, where the anisotropy is only a small perturbation and the experi- mental 29 and theoretical 30 evidence indicated that total cross-sections were only sensitive to no.However, the present study does suggest that it may be possible to generate good effectively spherical potentials which can account for the anisotropy and that such a potential possibly comes close to that for the most probable broadside approach. It remains to be seen if this rule holds for molecules where the potential is lowest for a collinear approach 31 for which the collision probability is less than for the broadside approach. Another interesting generalization coming from the model potential is the observ- ation that, just as found for H,-rare-gas systems," the reduced anisotropic potential U,(X,>/E,, where x2 = R/Rm2, has the same form as the reduced isotropic potential vo(xo)/eo. This is quite remarkable when one examines the apparent big differences in the potentials, as shown in fig.5. This suggests that the previously proposed law of corresponding isotropic and anisotropic potentials may have a general validity. This law has so far not been tested experimentally. Now that the Tang-Toennies model has been found also to work well for He-N,, the same frequency-dependent polarizabilities can be used to predict an anisotropic potential for Ne-N, and the important prototype system Ar-N, and eventually N,-N, once the SCF calculations have been done. Work along these lines is in progress. The experiments, on the other hand, have been extended to He-0,, CO, CH,. With some apparatus improvements it should be also possible to study N,-Ar and N,-N,. We thank R. T. Pack (Los Alamos) and F. A. Gianturco (Rome) for many stimul- ating discussions, G.Drolshagen and A. Obst for help in carrying out some of the potential calculations and M. S. Bowers for calculations of transport properties. K. T. Tang thanks the National Science Foundation for grants CHE-7809808 and PRM-7921430 and the Research Corporation for additional funding.220 He-N, ANISOTROPIC VAN DER WAALS POTENTIAL T. Kihara, Intermolecular Forces (Wiley, New York, 1978). J. 0. Hirschfelder, C. F. Curtiss and R. 13. Bird, Molecrrlar Theory of Gases and Liquids (Wiley New York, 1954). J. Reuss, Ado. Chem. Phys., 1976, 30, 369. M. Faubel and J. P. Toennies, Adc. ,4r. Mol. Hiys., 1977, 13, 262. J, P. Toennies. Discuss. Faraday Suc., 1962, 33, 96. €3. E. van den Bergh, M. Faubel and J . P. Toennies, Faraday Discuss.Chem. Sac., 1973, 52, 203. ’ K. Rudolph and J. P. Toennies, J . Chenr. Pfiys., 1976, 65, 4483; H . Schmidt, V. Herrnann and F. Linder. Chem. Phys. Lett., 1976, 41, 365. U. Buck, F. Huisken and J. Schleusener, J. Chem. PAys., 1978, 68, 5654; W. R. Gentry and C. F, Giese, 1. Chem. Phys., 1977, 67, 5389. K. T. Tang and J. P. Toennies, J. Chetn. Phys., 1977, 66, 1496. For errata see J. Chem. Phys., 1977,67, 375 and 1978,68,786. lo R. Ahlrichs, R. Penco and G. Scoles, Chem. Phys., 1977, 19, 119, l1 K. T. Tang and J. P. Toennies, J . Chem. Phys., 1978, 68, 5501 and 1981, 74, 1148. l 2 K. T. Tang and J. P. Toennies, 1. Chenz. Phys., to be published. l3 M. Faubel, K. H. Kohl and J. P. Toennies, in Rarefied Gas Dynamics, ed. S. S. Fisher, Prog. l4 M. Faubel, K. H. Kohl and J. P. Toennies, Book of Abstracts, 12fh Znt. Conf. Elect. At. Col- lS P. Habitz, K. T. Tang and J. P. Toennies, Chem. Phys. Leu., 1982, 85, 461. l6 M. Faubel and E. R. Weiner, J. Chenz. Phys., 1981, 75, 641. Astronaut. Aeronaut., 1981, 74, 862. lisions, ed. S. Datz (ORNL, Oakridge, Tennessee, 1981), p. 395. W. Meyer, Chern Phys , 1976, 27, 27. P. Coulon, R, Luyckx and H. N. W. Lekkerkerker, J. Chem, Phys., 1979,71, 3452. B. H. Choi and K. T. Tang, J . Chem. Phys., 1975, 63, 1775. 1979) chap. 8, p. 265-299. i9 M. KeiI, J. T. Slankas and A. Kuppermann, J. Clrem. Phys., 1979, 70, 541. 21 D. Secrest, in Atum-Molecule Collisiuii Theory, ed. R. B. Bernstein (Plenum Press, New York, 22 K. Bergmann, U. Hefter and J. Witt, J . Chem. Plzy~., 1980, 72, 4777. 23 J, A. Serri, C . H. Becker, M. B. EIbeI, J. L. Kinsey, W. P. Moskowitz and D. E. Pritchard, 24 M. Born and E. Wolf, PrincQ1e.s of Opfics (Pergamon Press, Oxford, 6th edn, 1980), p. 417. 25 J. S. Blair, in Lectures ia Theovef icd Physic7 (The University of Colorado Press, Boulder, 1966), 26 M. S. Bowers and K. T. Tang, to be published. 27 J, Kestin, S . T. Ro and W. A. Wakeham, J . Clzem. Phys., 1972, 56, 4036. 29 A. Kuppermann, R. J. Gordon and M. T. Coggiola, Fwuday Discuss. Chem. Soc., 1973,55, 145. 30 P. McGuire, Chem. Phys., 1974, 4, 483. 31 G. Ewing, Annrr. Rec. Phys. Clem., 1976, 27, 553. J. Chenr. Phys., 1981, 74, 5116. vol. VIIIC, p. 343ff. P. S . Arora and P. J. Dunlop, J . Chem. Phys., 1979, 71, 2430.

ISSN:0301-7249    

DOI:10.1039/DC9827300205

出版商:RSC    

年代:1982

数据来源: RSC

摘要:

Faruday Discuss. Chem. SOC., 1982, 73, 221-233 Gas-phase Properties and Forces in Van der Waals Molecules BY E. BRIAN SMITH AND ANDREW R. TINDELL Physical Chemistry Laboratory, South Parks Road, Oxford OX1 3QZ Received 22nd December, 198 1 The application of inversion methods developed to elucidate the intermolecular forces of spherically symmetric molecules to anisotropic interactions is investigated. It is found that the inversion procedures can be used to extract an effective, temperature independent, potential-energy function which can accurately reproduce both second virial coefficient data and gas-phase transport properties (calculated by the Monchick-Mason method). The nature of the averaging processes that may lead to such an effective potential is examined. In the last decade a notable advance has taken place in our quantitative under- standing of the forces which bind the simpler Van der Waals molecules.In particular our knowledge of the intermolecular forces for monatomic species, especially those that bind the inert gas dimers, is now rather satisfactory. There are a number of factors which have aided this progress.' First new experimental determinations of bulk properties have been made which, in important cases, show that earlier results are seriously in error. to be in error by up to 8%-a fact that had misdirected the search for consistent potential-energy functions for many years. Accurate measurements of the second virial coefficients at low temperatures played an important part in aiding the rejection of over-simplified potentials.* Secondly, improved semi-empirical calculations have enabled the intermolecular energy at large separations to be determined ac~urately.~ Thirdly, it has become possible to measure properties previously thought to be inaccessible.In particular the fine structure of the U.V. spectra of the inert gas dimers has given valuable information about the potential ell.^*^ Important information has also been attained from differential cross-sections measured using molecular Thus early gas viscosity data at high temperatures were found beams. ' energy can now be made. potential energy is the Axilrod-Teller triple-dipole term Fourthly, corrections for non-pairwise additive contributions to the molecular It appears that the major contribution to the many-body - v12,( 1 + ~COSO~COSO~COSO~) UcDDDt, - ____ r3 3 3 12r23r13 where rI2, r23 and ~ 1 3 are the sides of the triangle formed by three molecules, and el, 8, and 0, are the corresponding angles between the sides.The coefficient ~ 1 2 3 may be estimated by the relation 3uC6 v=-- 4(47%) where OL is the polarizability and C6 is the leading coefficient in the dispersion energy. * We will use the term potential for intermolecular pair potential-energy functions.222 GAS-PHASE PROPERTIES Fifthly, with the availability of computers, methods of analysing experimental data have been improved. It is one such development in this area, the use of inversion methods, that will be the main subject of this paper. In recent years inversion methods have been developed for both the transport and equilibrium gas phase properties of spherically symmetric molecules.'o These enable such properties, taken over a reasonably wide range of temperatures, to give direct information about potential functions.This information is quite specific and appears in general to be essentially unique. This is surprising as in the expressions for therrnophysical properties the potential is usually deeply buried under layers of integration, and in some cases it can be shown formally that truly unique potentials cannot result from inversion. The power of these methods has done much to enhance the value of gas-phase data in the determination of intermolecular forces.11 This success for isotropic interactions suggests that these or similar inversion methods might be of value in the study of anisotropic intermolecular forces.Few investigations of this type have yet been undertaken. Maitland et al.lZ investigated the inversion of gas viscosity for monatomic-diatomic gas mixtures, calculated using the infinite-order sudden approximation, They found the potential which they obtained on inversion was not similar to the unweighted spherical average, nor was it related to the potential function corresponding to any particular angular orientation. In this paper we examine a number of important questions which are raised by the application of the inversion methods to molecules which lack spherical symmetry. (i) Does the inversion of thermophysical data arising from an anisotropic inter- action using, in unmodified form, the methods developed for isotropic functions give rise to a single " effective " (temperature-independent) spherically symmetric function which will represent such data over the temperature range for which they are acces- sible? (ii) Are such " effective " potentials likely to be significantly different when obtained by the inversion of different thermophysical properties? (iii) Can the averaging processes that would lead to such effective potentials be understood? This would enable the prediction of an effective potential, given the form of the anisotropic function.It is to these and other related questions that this investigation is directed. In performing test calculations we have used a model anisotropic potential, the two- centre Lennard-Jones function (usually referred to as the di-LennardJones potential).In order to work in real rather than reduced units we have based our calculations on a specific potential with parameters that have been proposed for nitr~gen.'~ THE INVERSION METHODS Thermophysical properties P(T), determined as a function of temperature, T, are usually related to potential functions U(r) by integration over one or more subsidiary variables. The problem is to unfold the resulting integral to obtain the intermolecular separation Y as a function of U from P as a function of T . In some cases a formal inversion is possible. Even when this is so the sensitivity of the inversion procedure to errors in the experimental thermophysical data may cause the method to be in- effective.14 On the whole the more formal transform methods have not proved valuable. However, a number of rather complex procedures have been developed which work very effectively in practice.The methods can usually be justified for limiting types of intermolecular potentials, for instance for steep repulsive forces as might occur at short range. However, it is usually not possible to justify them form- ally or even to appreciate why they work so well over the whole range of molecularE . B . SMITH AND A . R. TINDELL 223 separations. in terms of the property under investigation.15 Thus In general they define a temperature-dependent characteristic length u" y"(r> =ffmx To take a specific example for gas viscosities, we may define i = [Q('~')/n]* where the collision integral CF2J' is related to the viscosity coefficient q by the equation 5 (mnkT)* q = - - 16 Q(2*2)(T) where m is the molecular mass.For potential functions of the form U = a/rm we can show that U(17) = kTG(rn), where G(m) is a number determined only by m and the sign of a. For long-range attractive forces when a is negative and m = 6, G = --0.58. Realistic potentials contain both attractive and repulsive terms and can be imagined to be comprised of a number of simple inverse power terms so that the effective exponent m is a continually changing function of separation. For realistic potentials, therefore, C is a temperature-dependent function G(T) ranging from 0.85 at high temperatures when repulsive forces dominate to a limiting value of -0.58 at very low temperatures where long-range forces determine the viscosity and the r - 6 term is most important.Despite this temperature variation, G(T) is remarkably similar for all realistic poten- tials and it is the stability of this function that leads to the effectiveness of the inver- sion method9 The procedure is to calculate an approximate G,(T) from a simple model potential U,,(r) such as the Lennard-Jones (12-6) function. Then we write For repulsive forces with m = 8-18, G lies in the range 0.86-0.82. U,(F) = Go(T) kT obtaining points on a new potential function Ul(v") which proves to be closer to the true potential than Uo(r). A new inversion function G,(T) may be calculated from Ul(v> and the process repeated for a number of iterations. In order to perform the calculation an estimate of the we11 depth q'k is required.In practice a number of trial values of ~ / k are employed and the one that leads to the inversion potential that best accommodates the data is selected. For highly precise data this procedure usually gives U ( r ) with a mean deviation of better than 1 % [excluding the regions where V(r) is close to zero]. If the data are not of the highest quality more than one iteration is not usually advantageous. Similar inversion methods have been used to treat thermal conductivity and diffusion data each of which can, like viscosity, be related to first approximation to a single collision integral. Thermal diffusion is determined largely by a ratio of collision integrals and an inversion method devised to relate this property to the steep- ness of the potential function, though related to those for other transport data, is different in detail.16 When an inversion technique is applied to second virial coeffi- cient data, B(T), the definition of r7 becomes lo a definition that can be justified in the repulsive region by use of perturbation theory.The methods have been tested by appIying them to data caIcuIated from a known potential and examining the accuracy with which this function can be recovered.'*224 GAS-PHASE PROPERTIES T H E CALCULATIONS The di-Lennard-Jones potential used in this work consists of two Lennard-Jones centres each interacting with parameters Elk = 33.46 K and Q = 0.3266 nm. The anisotropy of the system is defined by I* = Z/a where I is the separation cf the two centres (see fig.I). In this work we took a range of I* centred on the value proposed FIG. 1 .-Definitions of separations for site-site potential model. for N2, I* = 0.3292.13 The two-centre di-Lennard-Jones potential may be expressed as The classical second virial coefficients were calculated from the anisotropic potent- ial by means of the expression (see fig. 2 for definition of the ang!es) B(T) = - N " 41, r1,2dr12 /(+wJl /-0nsine2d4 ~~12(r12,e1,e2,s,)dv~2 which is valid for linear molecules. vI2 = v1 - v2 and f i 2 = exp(- U,,/RT) - 1 is the Mayer f fun~fi0n.l~ This integration was performed using a non-product al- gorithm method which required a total of 16 128 function evaluations to give the second virial coefficient accurate to ca. 0.02 cm3 rno1-l. FIG. 2.-Definition of angles of orientation for site-site potentials.The transport properties were calculated using the Monchick-Mason approxim- ati0n.l' Mason and Monchick proposed a simple scheme whereby the collision integ- rals may be calculated. The principal assumptions made in their approach are that first, the energy transfer between translation and internal modes does not affect the transport properties, and second, the relative angular orientation of the molecules remains unchanged throughout the collisional process. They suggested therefore that the calculation should be performed by computing the fixed-zngle collision integrals B~'*'~'(T,co) for the collision of two molecules at the fixed relative orientation w. This is then the calculation of the collisional integrals for the potential at fixed orientation U(v,co).The collision integral for the non-spherical potential is then ob- tained quite simply by averaging the fixed orientation collision integrals with equal weight over all space. ThusE. B. SMITH AND A . R. TINDELL 225 This integration was carried out using a program based on that of Price 2o which uses a five-point Gaussian-Legendre integration in d(cos8,) and d(cos0,) and a five-point Gaussian-Chebyshev integration in cp. This calculation took ca. 120 s on the CDC 7600 computer of London University to calculate the collision integrals for 40 temperatures. Since in general, inversion of data over a limited temperature range only defines the potential function over a limited range of Y, some form of extrapolation is required if the full iterative inversion procedure is to be employed, and in this work we use two types of extrapolation which are very different in form.These were the Lennard- Jones 18-6 function and the BBMS function 21 proposed for argon on the basis of the analysis of the U.V. spectra of the argon dimer. Two different transport properties were calculated from the collisional integrals. These were viscosity, which depends on Q(2q2)(T), and self-diffusion, which depends on a different collisional integral SZ('Pi)(T). These were calculated for temperatures over the range 50-2000 K. The second virial coefficients were calculated in the tempera- ture range 70-700 K. EFFECTIVE SPHERICAL POTENTIALS The inversion procedures described above which have been developed for spheri- cally symmetric potentials may be applied to the properties calculated for an aniso- tropic model potential.We have investigated their application to the viscosity, self- diffusion and second virial coefficient data obtained using the di-Lennard-Jones potential with the parameters given above and anisotropies in the range I* = 0.8 16 46- 0.6584. The significance of these values can be more readily appreciated if we note that a value of I* z 0.5 has been proposed for fluorine and I* z 0.7 for bromine and carbon dioxide. For values of I* < 0.50 each inversion gave a spherical potential function which was capable of reproducing the original data to high accuracy. For the degree of anisotropy estimated for nitrogen molecules, I* = 0.3292, the viscosity and self-diffusion coefficients were reproduced to &0.2% and the second virial coefficient data to k0.5 cm3.The quality of fit deteriorates for higher levels of anisotropy but remains adequate for values of I* up to the highest we have investig- ated. The most striking feature of the results is that the potentials produced on inversion of the three properties investigated are all extremely similar to each other as shown in fig. 3-6. It should also be noted that these potentials are very different (except at long range where the anisotropy appears to be no longer significant) from the unweighted spherical average also shown in fig. 3-6. For example in the case of I* = 0.3292 the potentials from inversion are deeper by 15% than the spherical average, which has characteristic parameters of CT = 0.370 nm and Elk = 87 K.It was found in this case that within the uncertainties involved in the inversion procedure, all three properties were consistent with a single potential function which had a well depth of Elk = 100 & 3 K and a collision diameter CT = 0.360 & 0.02 nm.22 This function, which is shown in fig. 4 as the continuous line, reproduces the virial coefficient data to a standard deviation of 1.6 cm3, the viscosity coefficients to 1.6% and self-diffusion coefficient data to 1.1 %. These errors, though greater than when the properties are calculated from the potential produced by inverting that property alone, would be found acceptable if the method were being applied to real experimental data.226 130 110-0 90 70- 50 30- GAS-PHASE PROPERTIES o + - 0 x 4.- ' -k - 90 c :+ X + 0 + x x+ I I I I I I 430' o!,, ' 0.38 0.42 0.46 rlnm FIG. 3.-Results from the inversion of properties calculated from the di-LennardJones potential using I* = 0.1646: x , viscosity results; 0, virial results; +, unweighted spherical average of the anisotropic potential. The parameters for the potentials obtained by inversion at the other values of anisotropy investigated are given in table 1, together with the parameters for the TABLE 1 .-POTENTIAL PARAMETERS OBTAINED BY INVERSION OF VISCOSITIES AND SECOND VIRIAL COEFFICIENTS anisotropy I* = l/a potential parameters obtained on inversion virial viscosity (Elk) /K ajnm (&/k)IK o/nm potential parameters for unweighted spherical potential ~~ ~~ 0.1646 120 f 2 0.337 f 0.001 120 & 3 0.337 f 0,001 0.3292 100 & 3 0.360 f 0.002 100 i 3 0.360 4 0.001 0.4938 90 5 0.377 + 0.002 90 & 5 0.375 %.0.003 0.6584 80 f 6 0.398 & 0.003 75 & 7 0.398 & 0.004 118 0.339 87 0.370 59 0.412 40 0.459E . B . SMITH AND A . R. TINDELL 227 54 FIG. 4.-Results from the inversion of properties calculated from the di-LennardJones potential using I* = 0.3292: x , viscosity results; 0, virial results; U, Felf-diffusion results: f, unweighted spherical average of the anisotropic potential. corresponding potentials obtained by taking an unweighted average over all mutual orientations. The potentials obtained by inversion are, except at the lowest value of anisotropy , very different from the spherical average, the differences becoming more marked at higher anisotropies.For all but the highest value of anisotropy investig- ated the inversion potentials obtained from different properties are remarkably similar. We can conclude that: (i) For diatomic molecules with a degree of anisotropy comparable to that which has been proposed for fluorine an effective spherically symmetric potential can be defined which can be used to calculate to high accuracy both second virial coefficients and transport properties (obtained using the Monchick- Mason approximation), and (ii) these potential functions can be obtained directly by the application of established inversion techniques. The conclusions, though as far as they concern transport properties they are subject to the validity of the Monchick-Mason approximation, are quite unexpected. To the extent that they offer the hope of correlating the thermophysical properties of a sub- stance using only a simple potential function they may be of practical value.They could provide an economical method of estimating accurately a wide range of thermo- physical data when only limited measurements are available. On the other hand they228 GAS-PHASE PROPERTIES 1 I I I I I I I 1 0.36 0-40 0.44 0.48 0.52 r/nm FIG. 5.-Results from the inversion of properties calculated from the di-Lennard-Jones potential using I* = 0.4938: 0, viscosity results; x , virial results; +, unweighted spherical average of the anisotropic potential. indicate that attempts to extract information about molecular anisotropy from the thermophysical properties we have investigated may not be straightforward.THE RELATIONSHIP BETWEEN THE EFFECTIVE POTENTIALS AND We have noted that the isotropic effective potential obtained by inversion Uinv(r) is THE ANGULAR-DEPENDENT POTENTIAL FUNCTION very different from the spherical average given by This result is not unexpected and was indeed noted by Parker and who calculated viscosities from the spherical average of an anisotropic function and showed them to be considerably different from those calculated directly from the anisotropic potential. The question which we need to answer is what kind of averaging pro- cedure will lead from the anisotropic potential to the ‘‘ effective ” isotropic potential uinv(r)-E. B. SMITH AND A. R.TINDELL 229 I I I I I I I I 0.38 0.42 0.46 0.50 - 90' rlnm FIG. 6.-Results from the inversion of properties calculated from the di-Lennard-Jones potential using I* -- 0.6584: x , viscosity results; 0, virial results; +, unweighted spherical average of the anisotropic potential. AVERAGES PERFORMED A T FIXED SEPARATION There are several kinds of average which may be performed on the di-lennard- Jones system. The first type, the averaging of energies which is performed at fixed separation of the centre of masses, r, of the diatomics, may be divided into two classes; one a Boltzmann type of averaging and the other a so-called free-energy averaging method. The relevant equations for these two methods are (i) Boltzmann Averaging (ii) Free-energy Averaging where q is the angle-dependent potential and To is a weighting temperature.The so-called free-energy averaging corresponds to the way in which the potential function of an isotropic molecule is weighted when second virial coefficients are cal- culated. However, the function employed in the inversion of second virial coefficients,230 GAS-PHASE PROPERTIES [B + T(dB/dT)], is related to an average (exp(- U/kT)(l + U/kT)). Although this relationship offers an alternative approach, which we have investigated, the results do not justify further discussion here. Unfortunately these procedures lead to a different effective potential at each temperature, which is not the simple result obtained in the previous section. The effective potential obtained by inversion can be discussed in terms of an apparent weighting temperature To, which is itself a function of separation. In other words the value of U, given by the effective potentials, at any separation Y can be ob- tained from the anisotropic potential by performing (for instance) a free-energy average with the appropriate value of TO.We have obtained T@(r) for our effective potentials by an iterative procedure. The function is illustrated in fig. 7 for the case 400 \ v - 200 - - l o o t \ \ V \ 2 R/10 nm FIG. 7.-Effective temperature (To) from free-energy averaging of the di-LennardJones potential with 1* = 0.3292: V, values extracted from potential obtained by inversion; broken line, values obtained from the inversion relationship. I* = 0.3292. It falls steeply with increasing r tending to low values at long range where the anisotropy of the potential function is no longer important [and where the potential.obtained by inversion tends to the unweighted spherical average of eqn (l)]. Examination of the inversion methods enables us to see how a relation between To and r could arise which could lead to an effective, temperature independent potential function. The inversion procedures seek to relate a characteristic separation, F, derived from the value of a property measured at temperature T with the energy kT by means of the inversion functions. Thus we can associate each value of the separation, 7, with a particular temperature T, to define an effective temperature independent potential. The explicit relationships linking the properties and r" used in the inversion procedures enable us to evaluate T as a function of F.The result isE . B . SMITH AND A . R. TINDELL 23 1 included in fig. 7. It can be seen that the temperature calculated from this model behaves in the same way as the derived effective temperature Te falling with increasing r. However, the function obtained from the inversion equation is much less steep and is not in good quantitative agreement with To. Thus although the inversion methods lead us to expect a single function linking the effective temperature with separation, the particular function they indicate does not appear to be consistent with the values obtained directly. We have also evaluated the weighting temperatures TO for potentials with differing anisotxopy.The values are very different but all fall off with approximately the same functional forms, To again tending to low values at separations greater than that at which the potential energy is at a minimum. We also investigated the form of the effective temperature function for two potentials of different atom-atom functional forms (12-6 and BBMS) 21 but with the same atom-atom parameters. The behaviour of To is again of the same general form but is quantitatively different for the two cases. The ef€ective temperature averaging method is of value in providing an insight into where anisotropy is important and how it varies with separation, but unfortunately it proves unsuccessful in a predictive capacity when used for calculating the effective spherical potential from the anisotropic potential.The results of the application of this type of analysis to the Boltzmann averaging technique will not be included here, since it was found that To defined by this average behaved in a similar manner to that described above for the values obtained from the free-energy averaging method. AVERAGES PERFORMED AT FIXED TEMPERATURE In the Monchick-Mason l9 technique for evaluating the collision integrals for anisotropic systems these integrals are evaluated for fixed orientations, and the final overall collision integral is a simple average of the fixed orientation integral. The evaluation of the second viriaI coefficients for an anisotropic potential can be carried out by a similar procedure. These considerations may enable us to understand why the effective potentials obtained from the different properties are essentially the same.They also indicate an alternative way in which the averaging of anisotropic potential functions may be discussed. We regard the anisotropic function in terms of the component fixed orientation potentials Uw, which are functions only of separation. For each of these potential functions we may calculate a ‘‘ property ” P,(T) (second virial coefficient or transport property) at a selected temperature T. The Monchick- Mason transport properties and the second virial coefficients are then a simple average of these fixed-angle properties over all orientations. For each property PJT) evaluated at a fixed orientation (u) we can obtain a characteristic separation rw using the inversion relation fox transport properties and the corresponding expression fur second virial coefficient data.A simple average of these values of Fw and their corresponding energies U, lead to a new averaged potential function for both sets of properties. This type of averaging was carried out for two levels of anisotropy I* = 0.1646232 GAS-PHASE PROPERTIES and 0.3292. In the former case the potential obtained by the above procedure for second virial coefficient data led to a potential very similar to the effective potential obtained by direct inversion. It reproduced both second virial coefficient and trans- port property data to high accuracy. However, at the higher level of anisotropy the potential obtained was sufficiently different to suggest that this averagbg procedure is less valuable at these higher degrees of anisotropy. Some insight into this method of averaging may be gained by recognising that the collision integral derived by the Monchick-Mason method may be expressed Rg$(T) = (Q$'9"'(T)) = (&>.Thus the inversion method, if exact, would lead to a value of Y = (Y:)*. A similar analysis produces the conclusion that the inversion of second virial coefficient data leads to a value of r" given by These two averages are sufficiently similar to explain the agreement between the potentials obtained by the inversion of both transport properties and second virial coefficient data. In addition they may suggest a reason why in general the potential obtained by inversion of virial coefficient data lies at slightly higher values of Y than does that from the inversion of transport properties. CONCLUSIONS The most striking finding of this work is that there can exist, for diatomic molecules of moderate anisotropy, an effective potential which will reproduce both second virial coefficients and gas-phase transport properties, calculated using the Monchick-Mason scheme, over a large temperature range.It is hoped that these effective potentials will be of value in calculating thermophysical data. Investigations that would lead to an understanding of how the effective potential arises from the averaging of the anisotropic interactions have not, however, led to any definite conclusions. Two methods of averaging the anisotropic potential to give the " effective " isotropic potential have been tried.The first, at fixed separation, proved to be unsatisfactory since the effective temperature used in the averaging could not be predicted independently. The second method of averaging based on examination of " properties " calculated from fixed orientation potentials did seem to produce an effective spherical potential which, at modest degrees of anisotropy, could both reproduce the second virial co- efficients and also could be related to the potential produced on inversion. However, this method also proved to be unsatisfactory as the degree of anisotropy is increased, although there does appear to be some evidence that it does reproduce the repulsive portion of the " effective '' potential. However, the preliminary results suggest that there may be underlying 'simplifications which, al- though not fully understood at the present time, could be of considerable value in elucidating the behaviour of non-spherical molecules.The quantitative investigation of anisotropy is in its infancy. The authors acknowledge the S.E.R.C. for an award to A. R. T. and thank Prof. J. S. Rowlinson, Dr J. A. Barker and Mr B. H. Wells for advice.E . B . SMITH AND A . R. TINDELL 233 G. C. Maitland, M. Rigby, E. B. Smith and W. A. Wakeham, Intermolecular Forces: Their Origin and Determination (Clarendon Press, Oxford, 1981). ’ R. A. Dawe and E. B. Smith, J. Chern. Phys., 1970, 52, 693. ’ M. A. Byrne, M. R. Jones and L. A. K. Staveley, Trans. Faraday Soc., 1968, 64, 1747; R. D. Weir, I. Wynn-Jones, J. S. Rowlinson and G. Saville, Trans. Faraday Soc., 1967, 63, 1320. G. Starkschall and R, G. Gordon, J. Chern. Phys., 1971, 51, 663. Y . Tanaka and K. Yoshimo, J. Chem. Phys., 1970, 53, 2012. E. A. Colbourn and A. E. Douglas, J. G e m . Phys., 1975,65,1741. B. M. Axilrod and E. Teller, J. Chem. Phys., 1943, 11, 299. D. W. Gough, G. C. Maitland and E. B. Smith, ilzfol. Phys., 1972, 24, 151. Smith, A. R. TindeII, B. H. Wells and F. W. Crawford, Mu!. Phys., 1981, 42, 937. G. C. Maitland and E. B. Smith, Chern. Suc. Rev., 1973, 2, 181 ; ref. (l), chap. 9. l2 G. C. Maitland, V. Vesovic and W. A. Wakeham, to be published. l3 P. S. Y . Cheung and J, G. Powles, MuZ. Phys., 1976,32, 1383. l4 G. C. Maitland and E. B. Smith, Mol. Phys., 1972, 24, 1185. l5 G. C. MaitIand and E. B. Smith, Pmc. 7th Symp. on Therrnophysicd Properties (Am. SOC. l6 E. 3. Smith, A. R. Tindell, B. H. WelIs and J. L. Brun, Physica, 1981, 106c, 117. ’ J. M. Parson, P. E. Siska and Y. T. Lee, J. Chem. Phys., 1972,56, 151 1. lo H. E. Cox, F. W. Crawford, E. 3. Smith and A. R. Tindell, Mul. Phys., 1980, 40, 705; E. B. Mech. Eng., 1977), p. 412. J. 0. Hirschfelder, R. F. Curtiss and R. B. Bird, MoZeculur Theory of Gases and Liquids (Wiley, New York, 1954), p. 148. l8 A. H. Stroud, Approximate Calculation of Multiple Integrals (Prentice-HaI1, New Jersey 1971). l9 L. Monchick and E. A, Mason, J. Chem. Phys., 1961,35, 1676. 2o S . L. Price, Chem. Phys. Lett., 1981, 79, 553. 21 G. C. Maitland and E. 3. Smith, MaZ. Phys., 1971,22, 861. 22 Preliminary results for one degree of anisotropy have been reported: E. 3. Smith, D. J. 23 G, A. Parker and R. T. Pack, J. Chern. Phys., 1978, 68, 3585. Tildesley, A. R. Tindell and S. L. Price, Chem. Phys. Lett., 1980, 74, 193.

ISSN:0301-7249    

DOI:10.1039/DC9827300221

出版商:RSC    

年代:1982

数据来源: RSC

摘要:

Faraduy Biscuss. Chem. Suc., 1982, 73, 235-240 Van der Waals Molecules and Condensed Phases BY J. A. BARKER IBM Research Laboratory, 5600 Cottle Road, San Jose, California 95 193, U.S.A. Receitled 1 1 th January, 1982 Spectroscopic studies of Van der WaaIs mofecules have recentIy provided detailed and precise information on the forces between molecules and in particuIar rare-gas atoms. This information can be used to calculate properties of condensed phases, thus providing a stringent test of theoreticd and computational methods. Condensed phases may themselves be regarded as very large Van der Wads molecules, and smaller Van der WaaIs molecules (clusters) play an important role in the nucleation and growth of condensed phases. This paper reviews these questions with particular reference to rare gases.The rare gases play an important role as test cases in many areas of chemical physics, and the use of spectroscopic methods to determine intermolecular potential- energy functions is no exception. Historically the potential-energy function for interaction of two ground-state argon atoms was determined with high accuracy (better than 1 % in the depth of the potential) almost simultaneously by two indepen- dent methods. Barker, Fisher and Watts (BFW) used experimental data on solid argon at 0 K, gaseous argon and liquid argon, together with the assumption that the Axilrod-Muto-Teller triple-dipole interaction was the only significant many-body interaction, to determine a multi-parameter potential function. At almost the same time Maitland and Smith (MS) used a Rydberg-Klein-Rees analysis of the spectro- scopic data of Tanaka and Yoshino on the vibrational levels of the argon dirner together with data on the second virial coefficient to determine another argon-argon potential, which proved to be almost identical with that of BFW.The close agree- ment of these two potentials demonstrated convincingly the value of the spectroscopic data, and incidentally provided strong evidence for the validity of the assumption concerning many-body interactions. Subsequently these potential functions were shown to be consistent with the argon- argon differential scattering cross-section data of Lee and c o ~ o r k e r s , ~ - ~ who also derived a quite similar potential function from their data. More recently Aziz and Chen derived another potential using the rotationally resolved spectroscopic data of Colbourn and Douglas and a fresh analysis of gas-phase data.For consideration of condensed-phase properties at pressures which are not too high the differences between these potentials may be regarded as small, and attention will be confined here to the BFW potential. It should be emphasised that the modern argon-argon potentials differ in depth by ca. 18% from the earlier 6-12 potential. The 6-12 potential may be regarded as qualitatively representing the properties of rare gases but it is quite unsatisfactory quantitatively. Accurate potential functions are also known for other like pairs of rare-gas atoms; details are given in the excellent book by Maitland et aZ.* It is interesting to note that in the determination of an accurate xenon-xenon potential the spectroscopic23 6 VAN DER WAALS MOLECULES AND CONDENSED PHASES data of Freeman et a1.I' played a much more important role than in the argon case, partly because of inconsistencies in the bulk-phase data for xenon.CONDENSED-PHASE PROPERTIES Knowledge of an accurate potential-energy function provides a powerful predictive tool. The properties of the substance in question in gaseous, solid and liquid states can be calculated by appropriate methods. In the case of argon this programme has been carried through. Low-density gas properties (second virial coefficients, gas transport properties) depend only on the interactions of pairs of molecules. They are the subject of another paper in this Discussion, by E.B. Smith, and will not be discus- sed here. At somewhat higher gas densities interactions of three molecules at a time become significant, and this is reflected in the values of the third virial coefficient. A comparison of calculated and experimental third virial coefficients for argon is shown in fig. 1. If only two-body interactions are included the calculated values are smaller 3000 2000 N 0 100 200 300 400 500 temperature/K FIG. 1.-Third virial coefficients of argon. The circles are experimental data of Michels et aZ.I3 The dashed curve is calculated with the BFW potential alone, the dotted curve includes the triple- dipole interaction, the dash-dotted curve adds third-order dipole-quadrupole interactions, the solid curve adds fourth-order dipole interactions. than the experimental values by almost a factor of two at low temperatures. Inclusion of the triple-dipole three-body interaction corrects much of this discrepancy, and the third-order quadrupole and fourth-order dipole terms much of the remainder.Note that there is substantial cancellation between the latter two terms. In the condensed phases this cancellation is even closer, so that their net contribution is very small and will be neglected. In fig. 2-4 a comparison with experimental data of some calculated properties of solid argon at low temperatures is shown; they are the Debye parameter, which is a measure of the specific heat, the pressure-volume relationship at pressures up to 20 kbar,* and the thermal expansion.Fig. 5 compares calculated and experimental * 1 bar = lo5 Pa.J . A . BARKER 237 100 95 90 k4 3 E -E ---. Y cd 85 ' 80 75 70 o Q 2.5 5.0 7.5 10.0 12.5 15.0 TIK FIG. 2.-Debye parameter for argon as function of temperature. potential and triple-dipole interaction. of Finegold and Phillip~.'~ Solid curve, calculated with BFW Circles, experimental values from specific heat measurements 23 22 21 20 19 18 17 16 V/cm3 mol-' FIG. 3.-Pressure-volume data for solid argon near 0 K. and triple-dipole interaction. Solid curve, calculated with BFW potential Squares and circles, experimental measurements (on two different samples) of Anderson and Swen~on.'~238 VAN DER WAALS MOLECULES AND CONDENSED PHASES 0.001 1 5 O.OOO! -u" 4s O.OOO[ I 1 1 I I I TIK FIG. 4.-Low-temperature integrated thermal expansion of solid argon.The solid curve is calculated for the BFW potential and the triple-dipole interaction, the error bars and triangles respectively are experimental data of Peterson et ~ 1 . ' ~ and Tilford and Swen~on.~' 14 - 12- E ' 0 - 2 + l.7 . 0 - % 6 - 4 - 2 - 20.0 22.0 24.0 V/cm3 mol-I FIG, 5.-Pressure-volume relation for solid and fluid argon on the melting line. culated with BFW potential and triple-dipole interaction. et UZ.,'~ Crawford and Daniels l9 and Stishov and Fedositov.20 Solid curves, cal- Circles, experimental data of FlubacherJ . A . BARKER 239 pressures for solid and fluid argon at high temperatures along the melting line, and fig. 6 shows a comparison of the calculated and experimental radial distribution function of liquid argon at 85 K.Agreement between calculated and experimental values is uniformly excellent.; this is a stringent test of both theoretical and experimental 3.0 4.0 5.0 6.0 7 .O RIA FIG. 6.-Radial distribution of liquid argon at 85 K. Solid curve, experimental neutron-diffraction result of Yarnell et al. ;21 circles, calculated with BFW potential and triple-dipole interaction. techniques. Phonon frequencies have also been calculated and compared with values derived from inelastic neutron scattering, again with excellent agreement. These are the vibrational frequencies of the giant Van der Waals molecule which the crystal comprises. These comparisons are intended to show the range of applications of the kinds of potential-energy functions that can be derived from the study of Van der Waals molecules. MOLECULES, CRYSTALLITIES AND DROPLETS The simplest Van der Waals molecules are dimers, but there exist also trimers, tetramers and so on up to macroscopic crystals and liquid droplets.No doubt it is a matter of taste at what point one stops speaking of molecules and uses a more neutral term like cluster. In any event clusters ranging in size from a few to a few hundred atoms are important for the nucleation of condensed phases, and can be observed for example in supersonic nozzle experiments. Garcia and Torroja l2 have made Monte Carlo calculations of the free energy of argon clusters using the 6-12 potential and made comparisons with the results of nucleation experiments. The agreement was not quantitatively satisfactory.In view of what has been said here about the limit- ations of the 6-12 potential this is perhaps not surprising. Etters et aZ.13 have cal- culated the ground-state energies of clusters using the BFW potential with three-body interactions. However, for comparison with a nucleation experiment one needs more than this: one needs the free energy of the clusters as a function of temperature. Calculations of this kind are currently being made in our laboratory by D. Romeu using the Monte Carlo method. As part of this work vibrational frequencies of clusters are being calculated and these results will be published at a later date.240 V A N DER WAALS MOLECULES A N D CONDENSED PHASES J. A. Barker, R. A. Fisher and R. 0. Watts, Mol. Phys., 1971, 21, 657.Y. Tanaka and K. Yoshino, J. Chem. Phys., 1970,53,2012; Y . Tanaka, K. Yoshino and D. E. Freeman, J. Chem. Phys., 1973, 59, 5160. P. E. Siska, J. M. Parson, T. P. Schafer and Y. T. Lee, J. Chem. Phys., 1971,55, 5762. R. A. Aziz and H. H. Chen, J. Chem. Phys., 1977,67, 5719. E. A. Colbourn and A. E. Douglas, J. Chem. Phys., 1976, 65, 1741. G. C. Maitland, M. Rigby, E. B. Smith and W. A. Wakeham, Intermolecular Forces (Clarendon Press, Oxford, 1981). J. A. Barker, R. 0. Watts, J. K. Lee, T. P. Schafer and Y . T. Lee, J. Chem. Phys., 1974, 61, 3081 ; J. A. Barker, M. L. Klein and M. V. Bobetic, ZBM J. Res. Dev., 1976,20, 222. lo D. E. Freeman, K. Yoshino and Y. Tanaka, J. Chem. Phys., 1974,61,4880. N. G. Garcia and J. M. S. Torroja, Phys. Rev. Lett., 1981, 47, 186. l’ R. D. Etters and R. Danilowicz, J. Chem. Phys., 1979,71,4767. l3 A. Michels, J. M. H. Levelt and W. de Graaff, Physica, 1958,24, 659. l4 L. Finegold and N. E. Phillips, Phys. Rev., 1969, 177, 1383. l5 M. S. Anderson and C. A. Swenson, unpublished results. l6 0. G. Peterson, D. N. Batchelder and R. 0. Simmons, Phys. Rev., 1966, 150, 703. l7 C. R. Tilford and C. A. Swenson, Phys. Rev. Lett., 1969,22, 1296. la P. Flubacher, A. L. Leadbetter and J. A. Morrison, Proc. Phys. SOC. (London), 1961,78, 1449. l9 R. K. Crawford and W. B. Daniels, J. Chem. Phys., 1969, 50, 3171. 2o S. M. Stishov and V. I. Fedositov, JETP Lett., 1971, 14, 217. ’’ J. L. Yarnell, M. J. Katz, R. G. Wenzel and S. H. Koenig, Phys. Rev. A , 1973, 7, 2130. ’ G. C. Maitland and E. B. Smith, Mol. Phys., 1971, 22, 861. ’ J. M. Parson, P. E. Siska and Y. T. Lee, J. Chem. Phys., 1972,23, 1511.

ISSN:0301-7249    

DOI:10.1039/DC9827300235

出版商:RSC    

年代:1982

数据来源: RSC

摘要:

Faraday Discuss. Chern. SOC., 1982, 73, 241-256 Anisotropic Intermolecular Potentials and Transport Properties in Polyatomic Gases * BY FREDERICK R. W. MCCOURT Departments of Chemistry and Applied Mathematics, University of Waterloo, Waterloo, Ontario, Canada N2L 3G 1 AND WING-KI LIU Waterloo, Ontario, Canada N2L 3Gl Department of Physics, University of Waterloo, Received 7th December, 198 1 It is widely accepted that the ordinary bulk transport properties of polyatomic gases, such as the shear viscosity or the thermal conductivity, are predominantly determined by the isotropic part of the intermolecular interaction, so that it is difficult to utilize them directly as sources of or tests for intermolecular anisotropies. It is also widely known that the effects of external electric and magnetic fields on these same transport properties depend critically upon the anisotropies, e.g.they are non- zero only for anisotropic potentials. Thus, external field effects on transport properties, particularly their temperature dependence, provide an important source of data for testing and/or determining intermolecular anisotropies. Before such data can be utilized, however, it is necessary to have available exact relationships between the effective cross-sections determining these phenomena and the S-matrix elements obtained from scattering calculations, together with reliable numerical pro- cedures for calculating the S-matrix elements. An illustration of Liouville space methods for carry- ing out the first step of this procedure is given for one of the cross-sections occurring in the thermal conductivity.The infinite-order sudden approximation is utilized to show the importance of taking into consideration the full potential surface when analysing ordinary bulk properties of systems. Results are given for binary mixtures of N2 and Ar. The results of a full close-coupled calculation of the thermal conductivity Senftleben-Beenakker effect (SBE) for the H,-He system are presented and discussed briefly. The past decade has seen a concerted effort at elucidating the nature and quanti- fying the parameters of the interaction potentials between noble-gas at0ms.l A vast array of experimental results, ranging from molecular-beam integral and differential scattering data3 and Van der Waals ~pectra,~ on the one hand, to temperature- dependent transport-property rnea~urements,~ on the other hand, have been utilized with considerable success in refining our knowledge of these interactions.It is prob- ably safe to say that the stage has now been reached where extensive discussion can be focused upon the remaining problem of finding a “ best ” representation of atom- atom potential functions. In contrast to the situation just described for atom-atom interactions, however, rather little is yet known of the details even of atom-diatom interactions, especially of the anisotropic portions of such intermolecular potential-energy surfaces. For a few special systems, such as the extensively studied hydrogen-isotope-rare-gas and hydro- gen-chloride-rare-gas systems, for which high-quality spectra have been obtained and analysed 7 9 8 and/or state-selected molecular-beam integral cross-section scattering data have become available,’ the full potential-energy surface can be considered to be * Research supported in part by the National Science and Engineering Research Council (NSERC) of Canada and by NATO (research grant no.145.81).242 POTENTIAL ANISOTROPIES A N D TRANSPORT PROPERTIES well characterized. Note also that for the H2-He system for which there are no bound states, and hence no collision-induced spectrum, there is available, in addition to state- selected molecular-beam scattering data,' an ab initio potential surface obtained at the full CI level." For almost all other atom-diatom systems, however, the (anisotropic) potential-energy surfaces remain largely undetermined.Ideally, it would be desirable to have a complete set of state-to-state differential scattering cross-sections measured at a series of beam energies but, although a good start in this direction has been made recently l1 in Gottingen, it will probably be some time before such measurements become routine. For a few relatively weakly aniso- tropic systems, such as N,-He, Ne and CO-He, Ne, for example, it is still possible to obtain reasonable estimates of the isotropic part of the potential-energy surface by combining beam total differential scattering cross-section data l2 with transport- property measurements. In general, however, it will be difficult to extract in this manner reliable isotropic components because the anisotropic terms, even though they enter only weakly, do affect both the total differential scattering cross-section and the transport phenomena in a non-simple way.It has been recognized for a number of years that the temperature dependence of relaxation phenomena, such as rotational and vibrational relaxation,13 nuclear mag- netic relaxation 14.*5 and depolarized Rayleigh and rotational Raman collision broadening,16*17 provides a useful source of information on potential anisotropies. However, accurate data over sufficiently extensive temperature ranges have not yet become available, especially for binary mixtures of diatomic species with noble gases. Such data are quite sensitive to intermolecular interaction anisotropies, and so it is important for these studies to be made.In much the same class as relaxation phenomena are the Senftleben-Beenakker effects '* (SBE) (the influence of externally applied electric and magnetic fields on the transport properties of polyatomic gases) which provide a direct measure of collisional changes of tensorial molecular polarizations produced under non-equilibrium con- ditions in gaseous systems. Again, it has long been recognized that in principle such field-effect measurements would provide information useful for obtaining anisotropic potential terms. The major problem in doing this has been, until recently, the lack of a detailed connection between the SBE effective cross-sections and the S-matrix elements of molecular scattering thcory. One such detailed connection I9v2O presents the shear viscosity SBE-effective cross-sections essentially in terms of Arthurs- Dalgarno 21 S-matrix elements in the total-,/ representation.Using these expressions, tests have been made of both thc infinite-order sudden 22 approximation (IOSA) procedure for the N,-He system 23 and of the centrifugal sudden 22 approximation (CSA) procedure for the H2-He system.24 It was found that both the IOSA and the CSA procedures (for their respective cases) gave reliable results for the effective decay (or " diagonal ") cross-sections but that, unfortunately, both procedures failed to give reliable results for effective production (or '' non-diagonal ") cross-sections. Hence, if production cross-sections are desired, it is necessary to employ a close-coupled (CC) calculational routine in order to obtain reliable results for them.The present paper presents an example employing the Liouville space formalism to relate an effective cross-section to !+'-matrix elements, then discusses the use of IOSA caIculations to clarify the role of inelastic contributions to the field-free shear viscosity coefficient qmix(0) for a binary mixture of N, with Ar and, finally, presents results for the thermal conductivity SBE cross-sections for the H,-He system via CC calculations for two specific potential surfaces.F. R. W . MCCUURT AND WING-KI L I U 243 EXAMPLE OF THE USE OF THE LIOUVILLE SPACE FORMALISM Althoughfuvmal connections between macroscopic observables in nun-equilibrium gas-phase systems and (generalized) Boltzmann collision integrals have been estab- lished for some time,25 no realistic dynamical calculations of these quantities have been attempted until very recently due to the absence of exact expressions relating the collision integrals to quantities which could be conveniently extracted from scattering calculations. By contrast, great advances in the theory of pressure-broadening of spectral lines had been made, firstly by fan^,^^ who employed the Liouville space form- alism to achieve a density expansion of the relaxation matrix, expressing the lowest- order result in terms of the t-matrix of scattering theory, and secondly by Ben- R e ~ v e n , ~ ' who took advantage of the rotational invariance of the system to give a systematic derivation of the relaxation matrix eIements in terms of t-matrix elements in the total-l representation.and approxi- mate 28 quantum-mechanical scattering computations of the relaxation parameters. Even though the equivalence of the relaxation matrix of Fano in the impact approxi- mation at low density and the generalized Boltzrnann collision operator was estab- lished z9 in 1971, the procedure introduced by Ben-Reuven has onIy recently been generalized to take into account symmetry considerations 30 and the systematic deriv- ation of the SBE collision integrals l9 in terms of SJ-rnatrix eIements so that appropriate quantum-mechanical scattering calculations can now be performed. As the Liouville space derivation of the SBE cross-sections has been well-docu- mented in the literature," the details will not be reproduced here.It is important, however, to point out here that the present approach differs from earlier treatments 31 in the following respects. (i) No dynamical approximation (such as the distorted- wave Born approximation which has been shown to be inaccurate) 32 has been made at the outset * so that the resulting expressions are exact ones; only hvhen convergent CC calculations are inconvenient or infeasible would dynarnical approximations be introduced. (ii) The more familiar spherical tensors were utilized throughout the derivation rather than the less familiar Cartesian tensors, both since the algebra of the former is well documented and widely employed, 2nd since it is yery easy to use rotational invariance arguments with them. (iii) Appropriate tensorially-coupled Liouville basis vectors have been introduced in terms of which any operator (of both h e a r and angular momenta) can be expanded.This final feature allows the deriv- ation of all SBE cross-sections to be carried out in a unified manner. As an illustration of the use of the Liouville space technique, the rotational heat flux effective cross-section will be discussed in detail. Thiscross-section isdefined by 33 t This paved the way for subsequent exact where the subscripts A, B refer to the linear molecule and its collision partner, respectively, @~'\Ool (2k,/C,",,>-iAc W, (2) is a first rank tensor, with k , Boltzmann's constant and C,",, the rotational heat capacity for species A. Moreover, AE represents the deviation of the rotational * Although the work mentioned as ref.( 3 I d ) is an exception to this statement, the basic approach is quite different from the present one. t The superscripts pqst on is an outer product of a pth rank irreducible tensor in W with a yth rank irreducible tensor in J multiplied by the sth and trh members of sets of orthogonal polynomials in W 2 and J2, respectively. [ J] 'L !2( W') Ro' "( J '), where L:''( W') is an associated Laguerre polynomial. indicate that Thus, for example, a'''' = [ W ]244 POTENTIAL ANISOTKOPIES AND TRANSPORT PROPERTIES energy of the linear molecule from its thermally averaged value (in units of k B T O with To the absolute equilibrium temperature of the system) and is a reduced velocity. atom-molecule collisions is represented by the colliding pair and (* * colliding system via Further, the generalized Boltzmann collision operator for fiAH is the average relative speed of By transforming to relative and centre-of-mass reduced velocities y and 3 for the denotes an (absolute) equilibrium average, u/, II? Y A g r- Y B y where yA and yB are mass factors given by yi2 mi/'(m, + m,) eqn (1) can bc cast into the form An operator CD in ordinary Hilbert space is considered to be a vector I@}) in the Liouville space in which an inner product is defined by ( ( @ ~ v / ) ) = tr@fv/, rn = ( 2 7 ~ ) - ~ (1 01 - s @ S * ) is a tetradic operator on the Liouville space (the bars on S and 1 imply delta functions for energy conservation); k!' is the ath spherical tensor component of the relative wavevector k and w is the (isotropic) operator defined by exp{ -p&,/2pkBT0}p, where p is the equilibrium rotational density matrix.The factor 3/2 associated with ( 5 4 comes from an integration over 3* It is worth noting that ( 5 ~ is also the cross-section that contributes to sound absorption through the volume viscosity. The remaining task is to express the Liouville vectors in eqn (4) and (5) in terms of coupled Liouville basis vectors in the total4 representation scheme, from which the matrix elements of m can be read out directly.27 Thus, the isotropic operator AE becomes 34 /A& >> = SdEk z: (2J + 1)I(jWKjWl+; 00 >> AEj (6) jlJ and the first-rank operator A&] has an expansion where and [x, * * x,] denotes the product (2x1 -t- 1)(2x2 + I ) - * (2x, + 11, the rest of theF . R .W. MCCOURT AND WING-KI LIU 245 notation being the same as that of ref. (19). basis vector corresponding to the coupling scheme 1(jJf)Jf[(jiZi)Ji]+ ; KQ )) is the Liouville jf + If = Jf, ji $- fi 7 Ji, Jf - Ji == K whcrej, f and J are the rotational, orbital and total angular momenta, respectively, and Q is the projection of K onto a space-fixed axis. Substitution of eqn (6) and (7) into eqn (4) and (5) immediately yields the final results with energy-dependent cross-sections cr&',jI&) given by with the energy-dependent cross-sections o:)(j',jlE,J being the same ones dctermining the W s l ) contribution to the shear viscosity of a diatom-atom mixture. It is given explicity as where the SJ-matrjx elements are as defined by Arthurs and Dalgarno.thermal averages (. *) are given by Finally, the Primes denotc post-collisional values. derived in a similar manner and will be discussed el~ewhere.~~ All other relevant SBE cross-sections can be IOSA CALCULATIONS AND ORDINARY TRANSPORT PHENOMENA The IOSA procedure was first utilized for the calculation of transport coefficients by Parker and Pack 35 for the C0,-He system. They pointed out the importance of retaining the anisotropic terms in thc potential function when analysing high-precision measurements of transport coefficients, especially at high temperatures. This point has been remade in a recent article by Maitland et Q Z . ~ ~ in a study of inelastic contri- butions to general transport collision integrals (essentially generalizations of the omega integrals of Hirschfelder et ~ 7 1 .) ~ ~ using the IOSA. However, neither group attempted to extract information on potential anisotropies (although Parker and Pack did recommend that this be done). An attempt at extracting potential anisotropies from the inherently more sensitive SBE measurements was reported for the N,-He system in ref. (23). A preliminary anisotropic N,-He potential obtained from total differential scattering data by Keil et aZ.12 turned out to be too weakly anisotropic, giving at 77 I(, for example, a decay246 POTENTIAL ANISOTROPIES A N D TRANSPORT PROPERTIES cross-section G(0200A)~~ for the angular momentum second-rank irreducible tensor polarization Oo2O0 = J J which is more than a factor of four smaller than the experi- mental value.38 The two anisotropy parameters in the potential form are basically associated with the depth and position of the potential minimum.The decay cross- section G(0200A)~~ was found to be highly sensitive to the anisotropy in the position and much less sensitive to the anisotropy in the depth of the minimum. A modified version of the potential [labelled KKM3 in ref. (23)] which gives a value of 6 ( 0 2 0 0 A ) ~ ~ within the experimental error bars, gives at 85 K, for example, a value Go(1200A)~~ = 16.8 A2 for the decay cross-section associated with the position on the magnetic field by pressure ( H / p ) axis at which the longitudinal thermal conductivity SBE for the N,-He system attains half its saturation value: This value should be compared with the experimental value of 13.0 A' reported by Heemskerk et al.39 The KKM3 potential will require further testing over an extended temperature range and for other phenomena and may well need further fine tuning of the anisotropy parameters.It is still necessary to make a detailed check of the quantitative accuracy of the IOSA for the calculation of the inelastic contributions to the ordinary transport collision cross-sections and for the calculation of tensorial decay cross-sections for several typical systems where the IOSA may be expected to be reasonable. A preliminary comparison 23 of CC and IOSA results for decay and production SBE cross-sections at a total energy E = 58 cm-' for the N,-He system shows the IOSA at its poorest since such calculations are known to be more reliable at high relative kinetic energies (in the entrance channel) than at low relative kinetic energies.22 An anisotropic potential-energy surface that has been more extensively studied is that proposed for N,-Ar by Pattengill et al.40 For example, Pack 41 has shown that the IOSA procedure gives inelastic contributions to the ordinary transport cross-sections with < 57; error (for low j-values) when compared with CC results.Since it can be expected that the IOSA should be better for N2-He than for N,-Ar, it can be argued indirectly (based upon Pack's results 41 for the N2-Ar system) that the IOSA results of ref. (23) for N,-He should also be quite reliable. One of the major problems encountered in the analysis of transport data in order to extract information on anisotropic potential-energy surfaces is the degree of entangle- ment of the inelastic contributions in the measured effect.Consider, for example, the shear viscosity of a binary mixture of a polyatomic gas with a monatomic gas. The field-free viscosity for such a mixture, ignoring tensorial polarization contributions, is given by kB TO vmix(0) = 7 {x~W%)AB + ~A~B[~+YAG(~OB),, - 2G(ioOi)AB vABAAB - XAXB G2(%il)AB (14) with xA, xB the mole fractions of polyatomic and monatomic species, respectively, fiAB (8k~T,/Ep)' the mean thermal speed and subscripts AA, AB or BB on the effective cross-sections indicating (binary) collisional changes of the second-rank irreducible velocity polariz- ations I--- - 1 (D:o G wi wi.F . R. W . MCCOURT AND WING-KI LIU 247 In addition to the pure gas cross-sections G(~OA)AA and G(~OB)BB appearing in eqn (13) and (14), there are only two independent quantities, designated by Gf) and Gh2) in ref.(19). Specifically, the cross-sections G(igt)AB, G($gg)AB and G($8i)AB are related to (5:) and Gh2) by G($gi)AB = G:) + 6;) Tllese cross-sections are a 10 related to the “ interaction ” viscosity vAB(0) and the At2 parameter of Kestin et aL4, by and Table 1 gives values of Gy), Gh2), A;, and vAB(0) for the N,-He system at 298 K both for the original KSK potential l2 and for our KKM3 modified potential 23 as well as for the values extracted from the experimental results of Kestin et aZ.42 As can be seen from table 1, the KKM3 cross-sections are closer to the experimental values than the KSK cross-sections.Since the isotropic potential term for both the KSK TABLE 1 .-SHEAR VISCOSITY CROSS-SECTIONS AND THE INTERACTION VISCOSITY FOR THE N2-He SYSTEM AT 298 K KSK 5.862 0.573 1.140 167.3 KKM3 6.282 0.616 1.144 155.6 experiment 6.757 0.658 1.136 145.7 a Kestin ef al. 42 and KKM3 potentials is the same, this difference is clearly due to the presence of additional anisotropy in the KKM3 potential surface. Table 2 Iists IOSA values of Gt), Gt), vAB(0) and A12 both for the isotropic part of the LJAR potential (columns labelled “ elastic ”) and for the full LJAR potential of Pattengill et aZ.40 Clearly, both 6;) and Gr) contain sizeable inelastic contri- TABLE 2.-sHEAR VISCOSITY EFFECTIVE CROSS-SECTIONS AND THE INTERACTION VISCOSITY FOR THE N2-Ar SYSTEM FROM IOSA RESULTS T/K q / A 2 q?/A* A ?2 VAB(O)/PP elas.full elas. full elas. full elas. full ~~ 77.3 47.11 47.83 44.68 44.68 1.107 1.089 52.8 52.8 250 26.99 24.83 24.88 22.93 1.076 1.078 170.4 184.9 298 25.45 22.56 23.45 20.90 1.075 1.081 197.6 221.7 3 50 24.27 20.58 22.37 19.12 1.075 1.087 224.2 262.4248 POTENTIAL ANISOTROPIES AXD TRAFSPORT PROPERTIES butions for temperatures above 250 K. However, the effect of the inelastic contri- butions from 5;) is masked by the fact that in qmix(0), (3:) appears effectively only through the dimensiodess parameter A&. The difference between the solid and dashed curves in fig. 1 illustrates for a temperature of 298 K the effect on qmix(0), as a function of the mole fraction of the poIyatomic species, of neglecting the anisotropic contributions to the potential surface for N2-Ar.Experimental points obtained by HeIlernans el have been placed on the figure for comparison. The differences illustrated here, while not large (being typically less than 5 "/o) are nonetheless signi- ficant, especially when it is realized that the error bars on the experimental points are smaller than the size of the triangles in the figure. It is known4' that the diffusion coefficient for the binary gas mixture is more sensitive to inelastic contributions than is the shear viscosity coefficient. In fact, the cross-section for the diffusion coefficient is given simply by 44 220 - - 210- a Y -_ h p 2 0 0 - - 190 - - 180- \ \ \ t 0 0.2 0.4 0.6 0.8 XNZ FIG. 1.-Dependence of the shear viscosity of binary mixtures of Nz and Ar on the mole fraction of N2 for the full LJAR potential of Pattengill et Tri- (-) and for its isotropic part (- - - -).angular points are from Hellemans et al.43 T = 298 K.F . R . W . MCCOURT AND WING-KI LIU 249 so that at 298 K, e.g. the LJAR potential gives G(i;i)AB == 27.4 A2, while its isotropic limit gives = 30.9 A2, differing from the full value by 12.8%. These cross- sections correspond to diffusion coefficients of 0.212 cm2 sW1 and 0.188 cm2 s-l, respectively, which should be compared with the accepted value 45 of 0.197 cm2 s-l for the N,-Ar system at 298 K. Another illustration of the importance of using the anisotropic potential surfaces can be obtained from the results displayed in table 1. At 300 K, the diffusion cross-section can be obtained from eqn (17) with y i = 0.874 97 for the N2-He system as G(lOA)AB = 3.350 A2 for the KSK potential and G(lOA)AB = 3.590 A2 for the KKM3 potential, giving rise to diffusion coefficients of 0.859 cmZ s-l and 0.801 cm2 s-l, respectively, to be compared with the experimental value 46 of 0.713 cm2 s-l.Just as for the shear viscosity, the inclusion of additional anisotropy in the N,-He surface (required for agreement with the 77.5 K GT SBE decay cross-section for N2-He) has led to better agreement with the experimental value of the diffusion coefficient, although the agreement on the whole is still not impres- sive. Although diffusion measurements are less accurate than shear viscosity rneasurcments, they are certainly sufficiently accurate to detect such differences.CLOSE-COUPLING CALCULATIONS AND SENFTLEBEN-BEENAKKER EFFECTS The effect of an externally applied magnetic field on transport phenomena (SBE) is much more directly sensitive than ordinary transport phenomena to the anisotropic parts of intermolecular interactions since it vanishes for isotropic interactions. It had been hoped that the CS and 10s dynamical approximation schemes would prove to be useful for the calculation of the decay and production cross-sections entering into the field effects. Unfortunately, it is now known 2 3 ~ 2 4 that both the IOSA and the CSA schemes fail to give accurate values for the production cross-sections. This means that it is necessary for the time being to use CC procedures in order to obtain reliable production cross-sections, and this severely limits the number of systems that can be studied.Calculation of the shear viscosity effective cross-sections for the H2-He system have becn reported recently 24 for two different potential surfaces. Similar calculations for the thermal conductivity SBE effective cross-sections for the same potential surfaces 4 7 9 4 8 will be reported here for the o-H,-He system. A discussion of the potential surfaces and of the scattering calculations themselves can be found in ref. The quantity of experimental interest in a calculation of the present type is the slope of the saturation value of a longitudinal thermal conductivity SB effect as a function of mole fraction of the polyatomic species A in the limit that xA goes to zero (the infinite dilution limit).This quantity can be written as (24). where - dWlzl is given by 4* dXA (xA=O)250 POTENTIAL ANISOTROPIES AND TRANSPORT PROPERTIES where r == (2C$,/5kB)". Lists of values of the unlike-molecule effective cross-sections appearing in eqn (18) and obtained from CC calculations with the MVAW and RS H,He potentials are given in tables 3 and 4 for temperatures ranging between 77.5 and 450 K. Further, the temperature dependence is displayed for the MVAW potential in fig. 2 and 3. In contrast to the somewhat simpler pure-gas expression for TABLE 3 .-EFFECTIVE DECAY CROSS-SECTIONS FOR THE THERMAL CONDUCTIVITY AND THERMAL DIFFUSION SBE: MVAW AND RS o-H2-He POTENTIALS a 77.5 100 150 200 250 298 350 400 450 19.65 19.15 18.80 18.32 17.43 17.00 16.43 16.01 15.47 15.07 14.42 14.00 13.05 12.67 11.64 11.27 10.22 9.87 19.67 19.22 18.75 18.31 17.38 16.95 16.35 15.94 15.29 14.90 14.06 13.69 12.49 12.15 10.87 10.57 9.30 9.04 25.16 24.04 23.50 22.94 21.80 21.26 20.80 20.07 19.58 19.11 18.69 18.17 17.66 17.18 16.69 16.25 15.70 15.27 - 7.263 - 7.094 - 6.908 - 6.740 - 3.360 -6.197 - 5.965 -5.805 - 5.640 -5.481 - 5.330 -5.197 -5.038 - 4.906 -4.787 -4.632 - 4.496 -4.356 19.44 19.16 18.48 18.17 17.17 16.83 16.29 15.93 15.56 15.23 15.08 14.63 14.50 14.00 13.96 13.39 13.41 12.75 The upper entry for each temperature is the result for the MVAW potential and the lower entry is the result for the RS potential.All cross-sections have units A'. AA/A, three rather than two production cross-sections, G(i:iEi)AB, G ( i ~ ~ ~ ~ ) A B and G(i!:Ai)AB, and four rather than three decay cross-sections, G( 1001A)AB, G( 1010A)AB, G(ig$i)AB and Go( 1200A)AB, appear in the expression for the thermal conductivity effect.Moreover, due to incomplete cancellation, the structure of the lOlOi produc- tion cross-sections is more complicated than for the pure-gas case. All three thermal conductivity production cross-sections involve the production of the relative velocity vector polarization from what is commonly called in the literature on field effects the Kagan vector polarization. In the pure-gas case, two terms of three in the coupling of the translational heat flux to the Kagan vector, both of which involve the production of the relative velocity vector polarization, exactly cancel, leaving only the third term, which is a multiple of the shear viscosity production cross-section G@).However, in the binary-mixture case, because of the presence of different mass factors in each of the terms, an analogous exact cancellation does not occur. In fact, the resultant difference existing in this case dominates over the viscosity production term over the entire temperature range considered in the present study. At 298 K, for example, the values in tables 3 and 4 give -2-1 dXA (x,=O) z -1.60 xF . R. W . MCCOURT AND WING-KI LIU 25 1 which, if compared with the corresponding quantity - d-1 ==: -5 x 1 0 - 5 dx, (x, = 0) for the shear viscosity $BE at 298 K, indicates that it will likely be quite difficult to measure the thermal conductivity SBE for H,-He mixtures. Unfortunately, in this case, as opposed to the shear viscosity case,24 no particular advantage would accrue were p-H,-He mixtures to be studied rather than n-H,-He mixtures.TABLE 4.-EFFECTIVE PRODUCTION CROSS-SECTIONS FOR THE THERMAL CONDUCT IV17 Y AND THERMAL DIFFUSION SBE: MVAW AND RS o-H,-He POTENTIALS 77.5 100 150 200 250 298 350 400 450 -0.0018 --0.0017 - 0.0064 -0.0063 -0.0244 - 0.0235 -0.0374 -0.0356 - 0.0429 -0.0405 - 0.0445 - 0.041 6 --0.0437 -0.0405 --0.0415 - 0.038 1 -0.0384 - 0.0349 0.0568 0.0543 0.0523 0.0496 0.0373 0.0354 0.01 96 0.0192 0.0070 0.0082 0.0003 -0.0013 - 0.0074 -0.005 1 --0.0112 - 0.0087 - 0.0 1 40 -0.01 12 - 0.0207 0.0199 -0.0190 -0.0183 - -0.0142 - 0.01 38 - 0.0089 -0.0087 - 0.0054 - 0.0052 - 0.0029 - 0.003 1 0.0013 0.0003 - 0.0007 0.0005 I 0.0000 -0.0016 -0.0578 -- 0.0569 -0.0601 - 0.0588 ' - 0,0575 -0.0554 - 0.0493 -0.0469 - 0.0447 - 0.0422 -0.0427 -0.0399 0.0409 -0.0397 -- 0.0387 -0.0359 - 0.0359 0.0327 The upper entry for each temperature is the result for the MVAW potential and the lower entry is the result for the RS potential.All cross-sections have units of A2* The rotational relaxation cross-section and similar purely inelastic decay cross- sections for scalar polarizations are, for the H2-He interaction, about one order of magnitude smaller than the production cross-sections G(i!&gi)AB, G(i!gAi)AB, G ( ~ ~ & ~ ~ ) a B [and approximately equal to G(i;i)AB], roughly two orders of magnitude smaller than pure reorientation decay cross-sections like 6(0200A),,, and about four orders of magnitude smaller than typical elastic cross-sections up to 300 K.Thus, apart from non-classical sound absorption, such cross-sections can be completely neglected at and below 300 K. This means that the approximate relation given by Kohler and 't Hooft 44 is well satisfied for the H,-He system so that the G ( ~ ~ ~ ~ ~ ) A B cross-section can be used to calculate the production cross-section governing the field effect on diffusion and entering into the field effect for thermal diffusion. These results have also been given in table 4 and illustrated in fig. 3 for the MVAW potential, At 300 K, the MVAW and RS potentials give for o-H,-He thc values G(i!::i)AB = -0.0445 and -0.0416 A2, respectively, which should be compared with the experimentally determined 49 absolufe calm (for n-H2-He) of 0.0542 A2.252 POTENTIAL ANISOTRUPIES AND TRANSPORT PROPERTIES t 0 50 100 150 200 250 300 350 LOO T/K FIG.2.-Temperature dependence of the thermal conductivity and diffusion SBE effective decay cross-sections for the MVAW 47 o-H2-He potential. 1000A)AB, L-.-- (a) GC!i!2)AB, (b) 6 0 (12*OA) 1200A A 8 , (c) G;(loooA (4 w::OO:i)AB, (e) - G(Eww3 Neither potential appears to be sufficiently anisotropic. Moreover, ,Go( 1200A)A, from the same experiments 49 has the value 18.5 A2, and both potentials again give results that are considerably lower. The one remaining question that can be answered from the present calculations concerns checking that the signs of the production cross- sections presently calculated are indeed correct.This can be done by making use of the exact relation 44 The sign of G ( ~ ~ ~ > , B is known from flow-birefringence measurements to be positive in all known cases and, indeed, this has been confirmed in the calculations presented in ref. (24) for the €€,-He case. For the MVAW potential, a comparison of the results obtained with the current sign and with the sign of the ~ ~ ~ ( j ’ , j / E ~ ) cross-sections reversed, at 100, 298 and 450 K is given in table 5. Clearly, the only internally con- sistent results are those having the present sign. SUMMARY AND CONCLUSIONS The relative simplicity of the LiouviIle space approach for the derivation of the exact relationship between bulk effective cross-sections and the S matrix of molecularF . R. W .MCCOURT AND WING-KI LIU FIG. 3.-Temperai cross-sections for 2 3t 253 ! I 1 - 50 100 150 200 250 300 350 401 TIK Lure dependence of the thermal conductivity and diffusion SBE effective production the MVAW 47 o-Hr-He potential. (a) 1 0 2 G ( : % ) ~ ~ , (h) ~O%(~!%)AB, (c) l O % @ X ) * R , ( d ) 3 oza;::;&. TABLE TEST OF THE SIGN OF THE THERMAL CONDUCTIVlTY PRODUCTION CROSS-SECTION USING EQN (20) ~ _ _ _ _ _ _ _ _ _ _____________ ~ present sign for O4K(j’[,jlEk) opposite sign for oqK(j’j/E)k T / K 1.h.s. eqn (20)/A2 r.h.s. eqn (20)/Az 1.h.s. eqn (20)/Az r.h.s. eqn (20)/A2 100 -0.00 14 -- 0.001 4 ’ - 0 .oooo -0.0014 298 ’ - 0.0095 - U.OU95 - O.0093 - 0.0095 450 -0.0126 --0.0129 ’- -0.0128 - 0.01 29 scattering theory has been illustrated for the decay cross-section for the rotational heat flux.Similar expressions have been obtained and are currently available for all effective cross-sections appearing both in ordinary bulk transport phenomena and in the SBE and gas-phase relaxation phenomena. The use of these expressions together with either CC calculations or dynamicaI approximation schemes such as the CSA or IOSA provides the means whereby bulk phenomena can be utilized as sources of detailed information on anisotropic intermolecular potential surfaces. IOSA calculations have been utilized for the N,-Ar system to illustrate the influence of the anisotropic parts o f the potential-energy surface on the ordinary bulk shear viscosity qmix(0) for a simple binary mixture. It has been clearly demonstrated254 POTENTIAL ANISOTROPIES A N D TRANSPORT PROPERTIES here and even earlier in ref.(35) that the inelastic contributions to the shear viscosity cross-sections contribute sufficiently to the shear viscosity coefficient that any purely isotropic potential extracted from an analysis of such bulk transport data will not represent the isotropic part of the actual potential-energy surface well [see also ref. (36)]. Conclusions arrived at by the application of IOSA calculations for " heavy '' systems such as N,-Ar, C0,-He etc., will also apply to the use of CSA calculations made for " light " systems such as HF-He, HF-Ar or HCl-Ne, for example. Of course, CC calculations could be employed for such studies as well, but it is likely that such calculations will only be performed for systems where the anisotropy is reason- ably weak, typified by the hydrogen isotopes and their binary mixtures with the noble gases.For the hydrogen-isotope-noble-gas mixtures, it is doubtful that ordinary bulk transport measurements would yield any useful information on the anisotropic parts of the potential-energy surfaces. For these systems, however, SBE measure- ments and relaxation phenomena should prove sufficient for testing critically the detailed surfaces obtained by Van der Waals spectroscopic studies and the inversion of molecular-beam scattering data. Further, the application of recently developed sensitivity analysis techniques for use in quantum scattering theory by Eno and Rabitz 51 should eventually allow information on the various parts of the full potential surface to be extracted in an efficient and sensible way, utilizing all available experimental information simultaneously.At present, of course, we are only at the threshold of development of practical procedures for carrying out such detailed analyses. None- theless, this gives us sufficient reason for considerable optimism regarding the role of high-precision bulk transport data obtained over extensive temperature ranges. Even though, as has been pointed out above, both the CSA and IOSA procedures fail to give accurate results for the SBE effective production cross-sections, they do give reliable results for the SBE effective decay cross-sections, including those for purely angular-momentum-dependent polarizations. Thus, the highly sensitive (H/p)+ measurements and measurements of relaxation phenomena, such as n.m.r., collisional broadening of spectral lines, including the depolarized rotational Rayleigh and the rotational Raman lines, can be analysed using IOSA and CSA procedures.For very light systems, for which CC calculations are feasible, so that the production cross- sections can also be accurately calculated, field effect saturation data can be included in the analysis. Preliminary calculations for two available H,-He potential surfaces indicate that they both are insufficiently anisotropic. It would be useful to determine from such SBE data more information about the anisotropic parts of the potential surface, but this awaits the availability of more SBE and relaxation measurements over an extended temperature range before it will become worthwhile.F. R. W. M. thanks Prof. J. J. M. Beenakker and his group at the Huygens Laboratorium in Leiden, The Netherlands, for their hospitality in the period during which much of this paper was written. We also thank Dr W. E. Kohler of the Institut fur Theoretische Physik in Erlangen, Germany for pointing out an error in our original fig. 3. G. C. Maitland, M. Rigby, E. B. Smith and W. A. Wakeham, Intermolecular Forces: Their Origin and Determination (Oxford University Press, Oxford, 1981). U. Buck, Rev. Mod. Phys., 1974,46, 369; U. Buck, F. Huisken, J. Schleusener and J. Schaefer, J. Chem. Phys., 1980,72, 1512; J . Andres, U. Buck, F. Huisken, J. Schleusener and F. Torello, J. Chem. Phys., 1980,73, 5620; U. Buck, F.Huisken, J. Schleusener and J. Schaefer, J. Chern. Phys., 1981, 74, 535.F . R. W . MCCOURT AND WING-KI LIU 255 L. Frommhold, Adv. Chern. Phys., 1981, 46, 1; Y. Tanaka and K. Yoshino, J. Chem. Phys., 1970,53,2012; 1972,57,2964; 1973, 59, 564. G. C. Maitland and E. B. Smith, Chern. Soc. Reu., 1973, 2, 181; R. A. 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ISSN:0301-7249    

DOI:10.1039/DC9827300241

出版商:RSC    

年代:1982

数据来源: RSC

摘要:

Faraday Discuss. Chem. Soc., 1982, 73, 257-273 Atom-Molecule Interactions from Muhiproperty Analysis An Integrated Study of the Dynamics for Oxygen-Rare-gas Systems BY FRANCO BATTAGLIA AND FRANCU A. GIANTURCO Gruppo di Chimica Teorica, Nuovo Edificio Chirnico, Universitii di Roma, 00100 Rorna, Italy AND PIERGIORGIO CASAVECCHIA, FERNANDO PIRANI AND FRANCO VECCHIOCATTIVI Dipartimento di Chimica, Universitg di Perugia, 06100 Perugia, Italy Receked 30th Nouember, 198 1 The interaction potential between the ground electronic state of the oxygen molecule and the helium atom is obtained through a numerical fit of data from collision experiments and is given via an effective, spherically symmetric analytic form. The latter expression, together with similar ones obtained previously for the 0,-Ar and 02-02 systems, is then used within a quantum-mechanical model to treat vibrationally inelastic processes and to generate various partial integral cross-sections at several energies.A comparison with experimental relaxation data is then carried out for the three cases examined; the results show generally good agreement for the He-02 and Ar-02 systems, suggest a greater influence of rotational inelasticity for the Or-02 case and allow us to assess the importance of Van der Waals complex formation on the relaxation mechanism. I . INTRODUCTION Collisional effects have constantly plagued the experimental studies of intramole- cular dynamics in “ isolated ” molecules. It is important to understand in these processes the interplay between intramolecular dynamics and intermolecular perturb- ations induced by collisions.In this context one should attempt to extract micro- scopic information regarding the selective effects of collisions, ultimately exploring monoenergetic collisions which affect intramolecular dynamics for either large molecules or weakly bound Van der Waals complexes. Significant progress in this direction has recently been accomplished, for example by exploiting some features of supersonic beam expansions which make it possible to study energy-resolved collisional effects in the proximity of the nozzle sources.1*2 Another useful classification of the basic molecular processes separates those processes occurring on a single electronic potential energy surface (PES) from those which involve at least two distinct electronic configurations.In the former case one focuses attention on intrastate dynamics, which deal with some forms of rotational and/or vibrational predissociation as well as the intramolecular vibrational energy re- distribution that takes place between bound vibrational levels of a large molecule or of a weakly bound adduct, as in the examples discussed below. The basic molecular processes provide a general framework for the elucidation of the nature of a variety of interesting chemical phenomena. It is well known that complex chemical events in large molecuIes can be described in terms of several258 02-R A R E-G A S COL L l SI ON A L 1 NT ER A CT I ON S parallel and/or sequential basic molecular relaxation processes. The effects of medium perturbations, for instance, on intermolecular and intramolecular dynamics are of importance in understanding the role of collision-induced processes in excited- state reactivity in the condensed phase.Moreover, collisional processes can be explored on the microscopic level by considering the role of Van der Waals bonding on vibrational prediss~ciation,~ which results in vibrational relaxation for one or for all of the weakly bound partner^.^ The induction of collision-controlled electronic relaxation can also be investigated by a similar approach. In particular, of con- siderable current interest are the intrastate predissociation processes that involve rotational and vibrational de-excitation of weakly bound complexes. Here the presence of an attractive well plays a key role, as shown by the simplest example of an atom and a diatomic molecule at an energy not sufficient to excite the vibration of the molecule.As the partners approach each other the attraction increases the available kinetic energy, with the result that the diatomic molecule can be excited while the atom is now bound by the attractive force. ’ The resulting species is only quasibound since it can predissociate by transferring the “ local ” vibrational excitation of the diatomic molecule back to the translation, thus heavily affecting V-T processes.6 This mechanism is quite general, since even in the absence of chemical attraction the long-range Van der Waals forces support bound states, and hence give rise to such “ local ” quasibound states as those described.In a predissociation via V-T or R-T processes, the energy required to promote the translation to a dissociative state is provided by the de-excitation of either the vibration or the rotation.’ Another possibility is provided by the inverse rotational predissociation (IRP), a process involving collisions between constituents with given angular momentum hJ and specific energy ca. En, which give rise to the emission of an infrared photon to a lower bound state of the complex. It can be envisioned as the decay from a trans- lational continuum via an intermediate state to a radiative continuum, of consider- able importance in a~trophysics.~ For IRP involving atom-diatom collisions, with the diatom being in a vibrationally excited state, the decay channel can involve non- radiative vibrational predissociation.An example is provided by the recent sug- gestion that the low-temperature vibrational relaxation of 1, by He occurs via an orbiting resonance of the complex HeI,(B 311, v) decaying into the dissociation channel He + I,@ 311, v - 1). In the present work we investigate all the above possibilities for collisional pro- cesses involving O2 molecules and rare gases (He, Ar) wherein the full interaction is obtained via a quantitative analysis of several experimental findings; moreover, the structure of the corresponding Van der Waals complexes is examined in order to provide a basic understanding of the low-temperature behaviour of the vibrational relaxation in the gaseous mixture. The integrated use of experiments and theoretical models will thus give us, in the examples discussed below, a novel method for the interpretation and prediction of the collisional behaviour of such Van der Waals complexes.Since some of the systems have already been discussed in the available literat~re,~.~ in section 2 only new results regarding the He-0, system are presented in detail and in section 3 the theoretical model that allows us to obtain vibrationally inelastic cross- sections under various collisional conditions is discussed. In section 4 we offer a comparison between the specific case examined here and also draw conclusions as to the present model’s capabilities in treating vibrational relaxation of Van der Waals complexes.F . BATTAGLIA et al. 259 2. POTENTIALS FROM MULTIPROPERTY ANALYSIS The potential functions used here for the Ar-02, 0,-O2 and He-0, systems are obtained by a combined analysis of different properties: for the Ar-0, and O,-O, systems the integral scattering cross-sections, the second virial coefficients and some spectroscopic features were analysed; for the He-0, system, the integral and total differential cross-sections together with the second virial coefficients were analysed.Note that all the experimental data are sensitive mainly to the spherical average of the potential: therefore they are able to provide the first term of the usual Legendre expansion of the PES for an atom-diatom system: co W , r , y ) = c VI(R,r)Pi.(cosy) (1) 1 = 0 where y is the orientation angle, that is the angle between the diatom axis, Y, and the position vector of the atom with respect to the centre-of-mass of the molecule, R.For the analysis of these three systems a multiparameter potential model, with enough flexibility to reproduce efficiently the complete set of analysed data, was chosen. The Exponential-Spline-Morse-Spline-Van der Waals (ESMSV) model was used: Vo(R) = &f(x); x = R/R, (2) f(x) = A exp[--(x - l)] x< x1 x2 < x < x3 x3 < x < x4 x4 < x = exp[al + (x - xl)(a2 + (x - x2)[a3 + (x - xl)a41>l x1 < < x2 = exp[-2/3(x - l)] - 2 exp[-p(x - l)] = b1 - (X - X3)(b2 + (X - X4)[b3 4- (X - X3)b4]} (3) -cbx-6 - c8x-8 - clOx'o - c6 = C6/&R:; c8 = Cs/&RA; Cl0 = ClO/&RI,O. While details of the analysis for the Ar-0, and 0,-0, systems have been reported and discussed The best-fit potential parameters for the spherical component of the interaction in the three systems are listed in table 1.Relative integral cross-sections for He-0, collisions were measured by Butz et a1.l' in the range of collision velocities between 0.2 and 2.6 km s-'. They are reported in fig. 1 as a function of LAB velocity. These results show a glory maximum located at ca. 0.6 km s-', and they provide experimental evidence for the existence of at least one bound state in the He-0, Van der Waals molecule, as will be discussed below. Luz- zatti ut al." also performed measurements of the absolute value of the integral cross- sections in the 0.8-2.3 km s-' velocity range. These cross-sections are also shown in fig. 1 as a function of the centre-of-mass (c.m.) velocity, and are the results of two independent experiments: one for the scattering of a helium beam by an oxygen target and the other for the scattering of an oxygen beam by a helium target.The very good agreement between these data and those of the previous investigation'O (see fig. 1) has allowed us to place the integral cross-section for He-0, over a wide range of collision velocities on an absolute scale. Recently Keil et a1.12 measured the total differential cross-section for the same system at a relative collision velocity of 0.56 km s-l and managed to resolve some of the diffraction oscillations. It is important to remember that all these scattering data are mainly sensitive to the low- energy repulsive branch of the potential curve, with the exception of the integral cross-section in the glory region which is sensitive to some characteristics of the potential surface near the equilibrium distance.13 The second virial coefficient for the He-0, gaseous mixture, BHe-02(T), is available the analysis for the He-0, system is described here.260 O2-RA R E - G A S CO L L I SI ONA L INTERACT I ON S TABLE 1 .-SPHERICALLY AVERAGED, EFFECTIVE POTENTIAL PARAMETERS (IN ATOMIC UNITS) FOR THE SYSTEMS EXAMINED.FOR THE MEANING OF THE PARAMETERS, SEE TEXT AND EQN (2) AND (3). parameter 0 2 - 0 2 Ar-02 He-02 4.19( - 4) 7.45 6.30 0.8490 12.53 7.81( +1) 1.56(+3) 4.06( $4) 0.63 0.75 1.05 1.48 Spline coefficients: a1 4.4724 a2 - 15.4739 a3 -24.5323 a4 94.9043 bl - 0.9270 b3 - 1.4387 64 - 3.8364 b2 1.8660 4.23(-4) 7.03 6.45 0.6246 13.69 7.50( + 1) 1.61(+3) 4.48( $4) 0.72 0.86 1.10 1.50 3.3626 -23.0071 - 66.5509 -418.3314 -0.7741 1.5351 5.2294 - 4.205 1 9.26( - 5) 6.50 6.00 1.3208 12.46 1.10( + 1) 3.55( +3) 1.73( +2) 0.61 0.85 1.10 1.45 5.1388 - 20.901 0 - 35.1579 - 152.2673 -0.7964 1.6878 4.1422 - 3.6674 only in the 90-80 K temperature range.14v15 However, it is possible to obtain BHe-02 ( T ) over a more extended range of temperature by using the following rela- tionship : B H e - 0 2 ( T ) = 3 [ B H e - H e ( T ) + B 0 2 - 0 2 ( T ) 1 - E H e - O z ( T ) and assuming for the values measured by Brewer and Vaughn l6 for He-Ar gaseous mixtures.The coefficients for pure He and 0, are taken from Levelt- Sengers et al.” This approximation, which also provides good agreement with the values that were directly measured for the He-0, ~ y s t e m , ~ ~ ” ~ is based on the fact that 0, and Ar show very similar gaseous properties, e.g.the second virial coefficients for Ar and 0, are practically equal over a wide temperature range.” It therefore becomes possible to have the required second virial coefficients available up to a temperature of 500 K. All these data were simultaneously analysed in order to obtain the potential para- meters for the spherical component of the interaction reported in table 1. Since all these data are minimally sensitive to the short- and long-range interaction, the A , a , C,, C, and C,, were kept at fixed values during the best-fit procedure. For the repulsive walls the values assumed were obtained following Smith’s work and taking the 02-02 short-range repulsion as proposed by Cubley and Mason.19 The c6 parameter was chosen at the value calculated by Starkshall and Gordon,” while c, and Clo were obtained assuming for the c,/c6 and c,oC6/c,’ ratios the values of 4.43 A2 and I .3, respectively, which are the theoretical values for the He-Ar system.The differential [I(O)sinO] and integral [ Q(u)] cross-sections were calculated within the JWKB approximation. For the fitting to the Butz et al. and Keil et al. data, theF . BATTAGLIA et al. 26 1 500 LOO 300 5 ; 200 8 0 .- Y I m c c! on + ._ 100 dplL- 1 2 3 collision velocity/km s-l FIG. 1 .---Integral cross-sections as function of collision velocity for the He-02 system. The circles of curve (a) report the Gottingen data lo as a function of LAB velocity and on a relative scale, fitted by the present potential (continuous line).The data of curve (b) refer to absolute measurements as a function of velocity and are also fitted via the present potential. centre-of-mass cross-sections were convoluted to the laboratory system taking into account the experimental conditions reported in the relevant papers. The best-fit Q(u) are reported as a solid line in fig. 1. In fig. 2, Z(0)sinO in the c.m. system cal- culated with the present potential (solid line) is compared with that calculated with the best-fit spherical MSV potential by Keil et al.', (dashed line). The slight dis- crepancy as to the maximum of the oscillations is actually an improvement of the fit quality. The B(T) coefficients were calculated for a spherical potential following Hirschfelder et al.,, with two quantum corrections. Finally, in fig.3 the derived spherical potentials for He-0,, Ar-0, and 0,-0, are plotted together around their well regions. The long-range tails for Ar-0, and 0,-0, are very similar, while in the short-range region the repulsions lead to a smaller R, for Ar-0,. The well depths for Ar-02 and 01-02 are almost the same. As expected, the He-0, system shows weaker interaction. The energy levels of the bound states for these systems were calculated using the WKB approximation. The Ar-0, system shows 7 vibrational levels with the rota- tional quantum number K = 0, while 0,-0, shows 6 vibrational levels for K = 0. The total number of the rotational levels is limited because of rotational predissoci- ation: 23 in the Ar-0, system, the maximum value of K allowed for the ground vibra- tional level, u = 0, is 44, which decreases when going to higher vibrational levels.One eventually gets a maximum K value of 9 for u = 6. Among all these roto- vibrational states, 41 levels are above the dissociation limit. For the 0,-02 system, if the electronic, nuclear and spin symmetries are neglected, the situation is very similar to that of Ar-0,, as is expected from the similarity of the potentials, while for262 O,--RARE-CAS COLLISIONAL INTERACTIONS I I 1 I .. .. the He-U, system only one vibrational state is possible. The existence of this state has been shown experimentally by the detection of a glory maximum in the velocity dependence of the integral cross-section measured by Butz et The maximum value of K allowed for He-0, is 6 and only K = 5 and K = 6 are above the dissocia- tion limit.While for the He-02 and 0,-02 systems all the available experimental data sample only the spherical component of the interaction, for Ar-0, there exist some scattering 24 and spectroscopic 25 measurements which are also rather sensitive to the anisotropic I_,-..L-I 3.5 10 4 5 RIA FIG. 3.-Effective potentials for all the systems studied here: Ar-02 (-1, He-02 (- - -), 02-02 (- * - -). Only the radial region arouna the Van der Waals wells is shown.F . BATTAGLIA id. 263 component of the interaction. This allowed us to obtain an approximate V2(R) component of the potential energy surface, as discussed el~ewhere.~’~ For the He-0, system, Keil et d .1 2 found a small discrepancy in the amplitude of the dif- fraction oscillations in the differential cross-section, between the experimental data and one of their best-fit curves. They attribute this discrepancy to the influence of the potential anisotropy. In our opinion this discrepancy is only an artefact of the potential model they assumed, since it is seen to have been reduced by our present potential. A better chance of obtaining an accurate angular dependence of the interaction in the He-02 system can be found by using the recent experimental re- sults of Faubel et aLZ6 on the rotationally inelastic cross-sections at thermal energy. A further study, which fits these highly accurate partial cross-sections and generates a fully anisotropic PES which includes the presentIy discussed effective spherical potential for He-02., has recently been completed 27 and confirms the good accuracy attained by the present fit of several sets of experimental data.3 . VTBKATIONAL INELASTlCITY AND COLLISIONAL RELAXATION Once such knowledge of the interaction potential for the systems under study here is obtained, it becomes interesting to assess their reliability by theoretical and com- putational means. In the case under discussion, one possibility is offered by the experimental data that are available on the vibrational relaxation, over extended ranges of temperature, of 0, moIecules 28 and of their mixtures with rare gasesat9 We therefore analyse the use of the above effective potential forms within a quantum- mechanical treatment of the collision problem, as well as the succeeding calculations of their corresponding relaxation times.Energy transfer processes for the case of a structureless particle colliding with a target with internal structure may readily be classified. This leads to translational- internal energy transfer (T-I), where I stands for rotational (R), vibration (V) and electronic (E) motions, or for a combination of these. Insofar as one restricts the study to thermal (E,,,, 5 0.1 eV) and hyperthermal (E,,,, 5 10 eV) collision energies, the Born-Oppenheimer separation of electronic and nuclear motions is an acceptable initial approximation. Adiabatic interaction potentials can therefore be used to provide the required dynamical coupling between the states which are employed to expand the total wavefunction : where, in an uncoupled representation, the xI.are diatomic vibrational functions, the mj are rotational functions, and the uJ are unknown translational functions describing the relative motion of the partners. If the index x ~ { j , u, J ) , then any initial condition xin is defined by a set of the above quantum numbers, and the relevant potential function causes transitions between target internal states and consequent energy transfer within the usual conservation energy requirements for the isolated system. I n the case of Van der Waals molecules, the uJ also describe locdly a set of bound states and pseudobound states of the weakly bound complex; hence J contains the required quantum numbers for their classification.Unfortunately, it is still not possible to carry out a rigorous calculation within the above quantum-mechanical scheme for those excitation processes which involve rotational and vibrationaI states of the target. The reason for this is that the com- puting time increases as ca. N6, N being the number of rotational states included in eqii (4) where all the energetically accessible states should be added. Even if we264 0,-RARE-GAS COLLISIONAL INTERACTIONS neglect the vibrational states and include only the rotationalIy open channels in a coupled representation, in order to go from low collision energies (E,,,, x 1.0 eV) to hyperthermal values (Ecoll w 2-3 eV) one must increase the computing time by a factor > 102.30 Moreover, to know the correct dependence of the computed PES on the internal coordinate Y as a function of the collision coordinate R, is by no means a simple problem even in the adiabatic approximation, and therefore only for a very few systems has such a systematic study been carried out in detaiL3’ Computing through the known PES the relevant matrix elements for vibrational coupling adds, in any event, more equations to the purely rotational problem and therefore makes matters even worse as far as computing time goes.Therefore, for all the above reasons, one must still rely on simpler theoretical models fur the prediction of vibrational excitation cross-sections, and extract those modeIs from the large body of experimental information already available from beam e~perirnents.~~ The simplest and one of the most successful models is based on a surprisingly obvious prediction for the quantum-mechanical transition probabilities of a forced harmonic o ~ c i l l a t o r , ~ ~ * ~ ~ where one starts by averaging the potential over all angles in order to simplify the scattering problem: where x = Y - reqJ and Y,, is the equilibrium distance of the oscillator.One could then numerically calculate the classical trajectory for elastic scattering (i.e. neglecting energy losses during encounters) using V(R,x = 0), hence yielding R(t). Substituting R ( t ) into eqn ( 5 ) generates Veff(t,x), which can then be expanded in a Taylor series: By neglecting all higher-order terms in the above series and including only the first- order terms, the dynamicaI equation of motion reduces to : ,iiX + kx = F,(t) (7) where One would have then obtained a prescription for the coupling between relative motion and molecular oscilIations.As pointed out by Carruthers and N i e t ~ , ~ ~ the most probable energy transferred to the oscillator also coincides with the total energy being transferred, which is given by : where Po + is the excitation probability from state lo} to state Ik} and Aw is the oscil- lator frequency. Thus a close correspondence exists between classical and quantum- mechanical results provided that second-order terms can be neglected, as appears to be the case in many situation^.^^ In a time-independent quantum-mechanical picture the driving force plays its role through the usual coupling matrix elements between target states: V”,*Oo = (4 I/,,f(RJ)I~? (9)F .BATTAGLIA et al. 265 where the integration is over the internal coordinate, x. Since one is now dealing with an averaged potential, the rotational coupling is disregarded for the time being and the expansion (4) contains only vibrational functions to represent target states. The total angular momentum is now a good quantum number for the whole system, hence the unknown radial coefficients can be written as: %, Z(RPl(R4 which gives rise to the familiar set of coupled radial equations: Each coefficient is obviously subject to the usual boundary conditions that ensure flux conservation and provide the correct free-wave behaviour beyond a radial value R = Rfin where the potential Veff has essentially vanished. In the present case the latter matching was performed beyond R z 15.0 a.u.and the relevant numerical procedure employed was one which has been used by us p r e v i o u ~ l y . ~ ~ * ~ ~ It is clear from the above model that the dynamical influence on vibrational energy redistribution during collisions is controlled by the size and radial dependence of the matrix elements on the r.h.s. of eqn (10). In order to extract these from the effective potential forms generated in section 2, let us rewrite the truncated series of eqn (6) as follows: Veff(R,O)ex~ [aeff~l where the averaging of eqn ( 5 ) has been performed without following it with a classical choice for R(t), i.e. the linearized forced oscillator is here represented via an exponential coupling term in the full effective potential.The corresponding stretching parameter, cceff(R), is the logarithmic derivative taken around req; now the relevant matrix ele- ments of eqn (9), once the target functions are chosen to be either harmonic oscillators or Morse oscillators, are rapidly evaluated over many eigen~tates.~~ Previous comparisons with other atom-diatom systems 40 indeed showed that the present model employed for each II component in eqn (1) constitutes a good approxi- mation to the true form of that part of the PES which presides over the coupling between vibrational states, at least for transitions involving the lower-lying eigenstates. Moreover, the construction of an effective potential for oxygen molecules interacting with rare gases seems to indicate that a major role is played by the spherical part of the PES in low-energy collision^,^^^^^^ hence suggesting that the possible knowledge of aeff(R) could already provide useful indications on the vibrational low-energy behaviour for Ar and 0, as projectiles, together with the generally “ softer ” nature of the He-0, interaction (dashed line).Because of the smaller size of the latter rare gas, one also sees that the oxygen target, when interacting with it, behaves as a smaller- sized target. Each S-matrix element obtained by propagating the radial coefficients u,,,(R) of eqn (10) up to the outer matching can now generate the corresponding inelastic opacities at each collision energy examined : of -, ,,(kZ) = ( ~ ~ ~ i / k t ) ( 2 1 + l ) ~ ~ u , u t - sf -+ ut(kZ>12 k t = 2PU(Evo - E, + kZ0/W = 2P(Ec0,1 - 0 (12) (1 3) where266 0 2 - ~ A R E - G A s c o L L I s I o N A L INTER A c T I o N s The initial state of the target has been assigned here the arbitrary subscript v,.The unitarity of the full S-matrix was checked via the usual detailed balance accuracy, while numerical convergence was tested with the v'-summation on the r.h.s. of eqn (10) by including all open channels available up to a maximum of eight eigenfunctions. The corresponding 0 can be considered accurate within 1-2% of their value. Finally, each partial integral cross-section was obtained in the usual way by summing over contributing partial waves: In order to compare computed quantities with some experimental measurements, one needs now to perform the corresponding kinetic analysis.30 One of the standard approaches focuses on a two-state model, whereby the rate constants pertaining to direct and inverse processes between the initial and final vibrational levels can be related to each other when dynamical equilibrium is reached.For a gas in a trans- lational equilibrium, the rate constants are then obtained as the Laplace transforms of a function proportional to the relevant inelastic cross-sections : 37,38 where F(p) = EQ i +f(E). At a pressure of 1 atm, the vibrational relaxation time z is thus given, in units of s, by the following simple expression: p~ == ( y i 4 f + y J - - + i ) - l (16) where Ji} and If) are the two vibrational levels considered. In the classical study by Landau and Teller,41 a simple expression relates the logarithm of p z to the temperature T of the gas undergoing sound dispersion experiments. A standard LT plot would then give linear behaviour for p z as a func- tion of T-113.Any observed departure from such a model in the real-life collisional transfer should then show up as a non-linear dependence in specific ranges of tem- perature. Moreover, the specific features of the PES employed, like its dependence on A, on r and on their relative orientation 7, will also affect results in the linear region by yielding relaxation of these particular systems. It is also well known that the repulsive region of the potential surface is the one mostly responsible for the largest values of the required matrix elements of eqn (9), at least as far as neutral species arc.concerned, where only weak dispersion forces act in the long-range regions. Vibrational energy transfers require substantial distorsion of the molecular charge distribution by the incoming atom, and therefore inelastic events are most likely to happen within the region of charge overlap between bound electrons. An approximate stretching parameter could therefore be obtained by considering theshape of the PES at the classical turning point, Rcla, for the collision energy under study. If one now defines the logarithmic derivative of the effective potential plus the centrifugal barrier that pertains to the chosen impact parameter at a specific RC1,, one can write: where the Veff is now V,,, + Z(1 4- 1)/2yR2 for each partial wave appearing in eqn (10).This new approximate coupling parameter obviously depends on the collisionF. BATTAGLIA et al. 267 1 .o 0.8 2 0.6 . e s W 0 . 4 0.2 0 I' I I \ ;i \ He-02 \ - \ \ \ L . J '. 3.0 3.5 ._ '. 1 - . - I RIA r- - -1, 0 2 - 0 2 (- ' - *). FIG. 4,-Repulsive walls for the effective potentials discussed in this work : Ar-0, (-), He-0, energy as well as on the shape of the PES within the repdsive region. A possible refinement could be introduced by modifying Ecoll according to the value of the well depth, ahhough this modification is Iikely to have little effect for Van der Waals complexes with rather shallow wells and for which the repulsive wall often appears in a logarithmic scale as linearIy dependent on R. The existing differences in the shape of the repulsive regions for the systems studied here are evident from fig. 4, where a linear scaIe clearly shows the similar computed relaxation times which are too fast or too slow with respect to the values found through measurements, or which exhibit very different slopes when compared with experimental data.In conclusion, both the detailed mechanism by which energy is transferred during collisions and the specific features of the PES at various geometries will greatly affect the magnitude and slope of computed LT plots, which in turn will provide an additional test for the reliability of the full potential surface considered. 4. RESULTS AND DISCUSSION A comparison between some of the computed excitation cross-sections is shown The energy E in fig.5 for oxygen molecules interacting with He and Ar projectiles.268 02-R A R E - G A S C 0 L L I S 1 0 N A L I N T ERA CTION S is here defined as E = Ec0,, - E t h , where E t h is the threshold energy value for the final level; in the case of If) = 11) it corresponds to 0.1923 eV. As seen in fig. 5, helium is a more effective projectile in transferring energy to the vibrational modes of the target, in spite of the fact that its coupling between molecular levels induced by the potential surface is rather small, in fact smaller than in the case 1 . c n c, ._ g 10-2 '2 0 w W W Q 7 lo"* 6 10-6 I I 0 2 I I 4 6 EIeV FIG. 5.-Partial integral cross-sections for vibrational excitation of O2 molecules by He and Ar projectiles. The energy E corresponds to the energy available above the relevant thresholds.Ar-02 (- - -), He-02 (-). of Ar projectiles. Some selected values for the stretching parameter at various collision energies and for different partial waves are shown in table 2 for the two rare gases and indicate the direct influence, within the present model, of the shape of the potential on vibrational coupling. Table 3 also reports full matrix elements, at a given value of R, between various levels and for the three different collision projectiles. In spite of the smaller range of its potential and the weaker vibrational coupling, one sees from the tables and figures that the He projectile manages to produce larger cross-sections, mainly because of its lighter mass and therefore shorter collision times that enforce more effectively the sudden condition on the relevant c ~ o s s - s ~ c ~ ~ ~ ~ s .~ Fig. 6 presents similar calculations for the 02-0, system and compares them with the results for argon projectiles. In spite of their similar values of D, and of matrix elements, as seen in tables 2 and 3, the above systems show marked differences in their cross-section behaviour at intermediate energies (ca. 1.0 < E/eV < 4.0). The 0, projectile appears to be less efficient in causing vibrational excitations of the 0, target, especially within the ranges of energies where the higher repulsive part of the potential is sampled.F . BATTAGLIA el a/. 269 TABLE 2.--STRETCHING PARAMETERS, D1(Ecoll), AS A FUNCTION OF PARTIAL WAVE AND COL- LISION ENERGY, FUK He AND Ar PROJECTILES ON 0 2 TARGETS (IN ATOMIC UNlTS) ~ ~ ~~ ~~ E 0.5 eV 2.0 eV 3.0 eV 1 Ar We Ar He Ar He 0 I0 20 40 60 80 120 130 260 0.952 0,951 0,947 0.930 0.876 0.805 _- -- - 0.954 0.940 0.880 0.690 0.479 0,240 - - 0.968 0.967 0.966 0.959 0.941 0.91 7 0.849 0.708 - 0.957 0.952 0.93 I 0.853 0.737 0.598 0.343 0.173 I 0.970 0.969 0.968 0.963 0.950 0.932 0.882 0.776 0.595 0.958 0.953 0.93 8 0.880 0.79 I 0.680 0.445 0.235 0.146 '"t 1 I I I I I 0 2 I, 6 EleV FIG.6.-Same as in fig. 5 for 0, and Ar as projectiles.270 Oz-RA RE- GAS COL LIS IONA L INTER A CTTONS T K 8000 I03 : - - 1-0 2-1 E c 2 10’ r - 6 - . 3 - 4 I -L-L 0.05 0.10 0 15 TJIIC3 FIG. 7.-LT plots of computed and measured relaxation times for Ar-Q and (- - -) Ar-02 (computed), (-) 0 2 - 0 2 (computed), (- 0 --) Ar--02 (measured), (measured ).O,-O, mixtures: (- --) 02-02 Since numerical accuracy was achieved within similar levels in both cases, a pos- sible explanation could be provided by the small differences in the repulsive walls of their respective PES; at higher collision energies the repulsive potential used here for the OZ--O, interaction becomes “ softer ” than in the Ar-0, case. The corresponding differences in stretching parameters cause the matrix elements to become smaller in the case of the former than in the latter, hence the slower increase of O,-O, inelastic cross-sections when the collision energy increases. The marked variations seen in fig. 6 can then be attributed to their changes in the form of the effective potential and to their strong influence in the present model, where no other physical effects are introduced in the more complex molecule-molecule relaxation process.The corresponding relaxation times are compared with experiments 28*29.43 in fig. 7 , for both Ar and O2 projectiles, over the temperature range 1000-8000 K. One TABLE 3.-cOUPLYNG MATRIX ELEMENTS, vU,”t(R) (IN ATOMIC UNITS), FOR THE THREE “ STRUCTURELESS ” PROJECTILES STUDIED IN THE COLLISlONAL EXClTATlON OF 0 1 TARGETS vu,LA R ) R He Ar 0 1 v 0 , m 4.7 0.043 64 0.061 62 0.061 58 v o , ,(R) 4.7 0.136 1(-2) 0.920 S(-3) 0.922 6(-3) vo ,d R) 4.7 0.104 1 ( - - 3 ) 0.60021-4) 0.600 8(-4) v1,m 4.7 0.062 62 0.092 48 0.092 40 V2,d R ) 4.7 0.077 82 0.1 I5 22 0.115 21F. BATTAGLIA et al. 27 1 sees that the computed p z values for Ar-0, are very close to measured values, both in magnitude and slope, and follow the simple linear behaviour of the LT plot.The 02-02 results, on the other hand, produce relaxation times that are slower than ex- pected and therefore indicate that a simple spherical interaction is not sufficient to describe vibrational relaxation in this system. In fact, the faster relaxation of pure 0, molecules, as opposed to their behaviour in the presence of argon, indicates the greater availability in the first case of rotational energy exchanges that speed-up the slower vibrational decay which would occur uia only V-T transfer processes.44 Moreover, the highly anisotropic interaction of O2 molecules forming weakly bound adducts is disregarded in our model, which thus underestimates relaxation times even more as the temperature decreases.The possibility of adduct formation at low T is, however, taken into account in our model for those cases where the effective potential used is a more realistic description of the whole interaction. This is shown in fig. 8 and 9, where the computed results for He-0, and Ar-0, are compared with the available measurements both in the high-T region and at low temperatures. Fig. 8 reports relaxation times for He and Ar colliding with oxygen and shows very good agreement between computed values and experimental findings. This is lo3 102 E c, v) 3 10' Q 100 TIK 8000 1000 300 I I I / ' I ,/P T+/K+ FIG. 8.-Same as in fig. 7 for the Ar-02 and He-02 systems. true for the He-0, mixture down to ca. 500 K, where the presence of pseudobound states (resonances due to potential shape) increases the interaction and favours V-T transfer as the temperature decreases.Experiments seem to show departure from linear behaviour and our calculations confirm this effect. Such an effect is shown even more clearly by the results plotted in fig. 9, where the low-Tdata for Ar-0, are well reproduced by our calculation. On the other hand, the272 02-R A R E- GAS C 0 L L I S I 0 N A L INTER ACT I 0 N S T+/K+ FIG. 9.-Low-temperature behaviour of computed relaxation times for 02-02 and Ar-0, compared with available measurements: (-) O,-O, (computed), I- - -) Ar-02 (computed), (- -) OrOz (measured), (- 0 -1 Ar-O.! (measured). 0,--0, data remain linear in the range observed and are only qualitatively given by our computed p z for (1 +O> relaxation.Here again, the spherical effective potential is missing the orientational effects between approaching O2 molecules and therefore yields computed cross-sections that remain too small down to very low values of T. In conclusion, we feel that the effective potentials obtained by fitting a large body of data regarding simple systems could be used for computing vibrational inelasticity during collisions and can provide interesting information on the mechanism which presides over these processes. In particular, diatomic moIecules mixing with rare gases are chemically inert systems where the interaction is dominated by repulsive terms and only weakly affected by long-range forces. Therefore, the vibrational relaxation behaviour at low T and at high T can be effectively described by the simple, linearized coupling given in our model, once a fairly accurate knowledge of the full interaction is available.Elastic scattering events are the major outcome of collisional encounters between these partners, with other energy exchanges mainly appearing as small perturbations and therefore likely to be well represented within our model. P. C . thanks Dr A. Lagan5 for computational help. The financial support of the Italian National Research Council (CNR) is gratefully acknowledged. G. M. McCIelland, K. L. Saenger, 5. J. Valentini and D. R. Herschbach, J. Phys. Chem., 1979, 83,947. J. Tusa, M. Sulkes and S . A. Rice, J. Chem. Phys., 1979, 70, 3136. J. A. Beswick and J. A. Jortner, J.Chem. Phys., 1978, 69, 512.F . BATTAGLIA etal. 273 G. Ewing, Chem. Phys., 1978, 39, 253. R. A. Marcus, J. Cheni. Phys., 1956, 24, 966. G. Drolshagen, F. A, Gianturco and J. P. Toennies, f. Chem. Phys., 1980,73, 501 3. R. D. Levine, J . T. Muckermann, B. R. Johnson and R. B. Bernstein, J. Chem. Phys., 1968,49, 56. 3. Brunetti, G. Liuti, E. Luzzatti, F. Pirani and F. Vecchiocattivi,J. Chem. Phys., 1981,74, 6734. F. Pirani and F. Vecchiocattivi, Chem. Phys.. 1981. 59. 387. H. P. Butz, R. Feltgen, H. FauIy and H. Vehrneyer, Z. Phys., 1971, 247, 70. l1 E. Luzzatti, F. Pirani and F. Vecchiocattivi, Mol. Phys., 1977, 45, 1279. l2 M. Keil, J. T. Slankas and A. Kuppermann, J. Chen7. Phys., 1979,70,541. l3 F. Pirani and F. Vecchiocattivi, Mol. Phys., in press. j4 A, van Itterbeek and W. van Doninck, Pruc. Phys. Sue. London, Sect. B, 1949, 62, 62. l5 C . M. Knobler, J. J, M. Beenakker and H. F. P, Knaap, Physica, 1959, 25, 909. l6 J. Brewer and G. W. Vaughn, J. Chem. Phys., 1969, 50, 2960. l7 J. M. H. Levelt-Sengers, M. Klein and J. S. Gallagher, in AIP Handbook (McGraw-Hill, New York, 1972). F. T. Smith, Phys. Rev. A , 1972, 5, 1708. l9 S. J. Cubley and E. A. Mason, Phys. Fluids, 1975, 18, 1109. ?O J. Starkshall and R. G. Gordon, J . Chem. Phys., 1471, 54, 663. 21 K. T. Tang, J. M. Norbeck and P. R. Certain, J. Chern. Phys., 1977, 64, 2063. 22 J, 0. Hirschfefder, C. F. Curtis and R. B. Bird, The Molecular Theory of Gases and Liquids 23 G. Herzberg, Molecular Spectra and MoIecular Structiire (Van Nostrand, Princeton, I950), vol. 24 J. J. H. van de Biesen and C . J. N. van de Meijdenberg, to be published. 25 G. Henderson and G. Ewing, J . Chenr. Phys, 1973, 59, 2280. 26 M. Faubel, K. H. Kohl and J. P. Toennies, XII ZCPEAC, Book ofAbsstracts (Gatljnburg, 1981), 27 F. A. Gianturco, personal communication. 2s D. R. White and R. C. Millikan, J. Chem. Phys., 1963, 39, 807. 29 D. R. White and R. C. Millikan, J. Chern. Phys., 1963, 39, 1803. 30 E.g., see: F. A. Gianturco, The Transfer of Molecular Energies by Collision (Springer-Verlag, (Wiley, New York, 1954), chap. 6. I, p. 425. p, 935. Berlin, 1979). P.Hebitz, K. T. Tang and J. P. Toennies, Chern. Phys. Lett., 1982, in press. 31 M. Faubel and J. P. Toennies, Ado. A t . Mol. Phy~., 1977, 13, 229. 33 M. S. Bartlett and J. E. Moyal, Proc. Cambridge Philos. Soc., 1949, 45, 545. 34 E. H. Kerner, Can. .I. Phys., 1958, 36, 371. 35 E. P. Carruthers and M. M. Nieto, Am. J. Phys., 1965, 33, 537. 36 C. E. Treanor, J. Chem. Phys., 1966, 44, 2220. 3i e . g . , see: F. A. Gianturco and U. T. Lamanna, Chen7. Phys., 1979, 38, 97. F. Battaglia and F. A. Gianturco, Chem. Phys., 1981, 59, 397. 39 F, H. Mies, J. Chem. Phys., 1964, 40, 523. 40 F. Battaglia and F. A. Gianturco, Chenr. Pliys., 1981, 55, 283. 41 L. Landau and E. TelIer, Phys. Z. Sowjetunion, 1936, 10, 34. 42 A. S. Dickinson, to be pubIished. 43 R. C. Millikan and D. R. White, J. Chenz. Phys., 1963, 39, 3209. 44 F. E. Nikitin, Dokl. Akad, Nauk SSSR, 1960, 132, 395.

ISSN:0301-7249    

DOI:10.1039/DC9827300257

出版商:RSC    

年代:1982

数据来源: RSC

摘要:

GENERAL DISCUSSION Dr. P. G. Burton (Uniuersity of Wollongong) said: My comment relates to the H2 dimer intermolecular potential ; specificially at this stage the isotropic component of this potential. A comment is made concerning the leading anisotropy of this potential following Reuss's paper. We have been intrigued from the theoretical point of view by the odd characteristics of the small but important discrepancy between the recent ab initio potential of Schae- fer and Meyer and several semi-empirical H2-H2 isotropic potentials. Of the latter, the apparently good agreement of the Silvera and Goldman " solid H, " pair potential and the independently derived " total differential scattering " pair potential of Buck et ~ 1 . ~ combined with the multiproperty-fit potential of Mc- C ~ n v i l l e , ~ suggests that the low part of the repulsive wall of the Schaefer-Meyer potentials is too " hard " or repulsive, and that for example their ab initio R, value (zero crossing of the isotropic component of the potential) of 3.09 A is too great by 0.07-0.10 A.Reference to Buck's fig. 6 illustrates how the Schaefer-Meyer potential would have to be adjusted inwards to agree with these semi-empirical potentials, and we were struck by the apparently rapid onset at ca. 3.5 A of such an adjustment to the ab initiu potential. It was difficult to see from the theoretical point of view what might have been missing from the ab initiu calculations that could account theoretically for such a difference. One possibility of course was a deficiency in the theoretical models upon which experimental data fits were based; that the choice o f " damping " terms in the HFD models to interpolate between the long-range Van der Waals coefficient expansion of the potential and the short-range Hartree-Fock data typically underestimated the distance of separation beyond which it is appropriate to replace the interaction potential by a purely asymptotic multipole form.The use of unzform-fields to compute atomic or molecular polarizability is ultimately inappropriate to the real case of a localised, nearby perturbation (characterised by radiating lines of force), and accord- ing to Koide s significant modification of the conventional symptotic expansion should extend well beyond an intermolecular separation of 8 ~ 0 , where in any case overlap and other effects (which must be determined through supermolecule computations) are already significant.However, since the Schaefer-Meyer potentials are more repulsive in the low part of the repulsive region of the isotropic component than all the recent semi-empirical potentials, a second possibility which we have addressed in our own H2-H, calcu- lations was that the basis sets used in the Schaefer-Meyer electronic structure com- putations of the H,-H, are unable to account for more than ca. 92% of the electron correlation energy of H,, and the additional electron correlation beyond that accessible in their calculations might have softened the molecular interaction. I n contrast to the approach of Schaefer and Meyer, where different basis sets are used in the computation of different features of the overall interaction, cf.ref. (7), we have consistently used the same basis for each H, throughout our supermolecule calculations, and this basis set accounts for 96.3% of the electron correlation energy of an isolated H2 molecule.* With such a basis of 39 functions per H, we compute the R, value for the isotropic component of the interaction to lie between 2.96 and 3.02 A,276 GENERAL DISCUSSION depending on whether CEPA2-PNO or the PNOCI method was used to allow for correlation effects in the supermolecule caIculations (see comment on the paper by Stone and Hayes concerning these methods). We can see then in our results for H,-H, that there is a justification from the theoretical point of view for adjusting the Schaefer-Meyer potential to be more attractive in the low part of the repulsive wall. What is certainly not apparent from our calculations is why the significantly increased correlation within each H, and between the H2 molecules that we find in our calculations (compared with the Schacfer-Meyer results) shouId not extend right out to the range of R = 11.0 a,, where the asymptotic mdtipole form can take over.In our calculations, the softening of the low repulsive part of Vooo is intimately associated with a deepening of the computed well depth. This effect extends out beyond 8ao, making our rigid-rotor H,-H, potentials significantly deeper than all the recent semi- empirical determinations of the isotropic component of the potential [E,(PNOCI) 2 27.5 cm-I, cm(CEPA2-PNO) = 33.6 cm-’1.Whether more than six different angular conformations need to be explicitly treated to define the spherical average potential * precisely enough in the region of the Van der Waals minimum, whether the rigid-rotor approximation itself is too severe due to the large vibrational amplitudes of H2 in the H,-H, complex and “ intramolecular ” vibrational energies (including the zero points) are modified by the interaction, or whether the experiments are especially sensitive to a preferred geometry or geometries of interaction, for which the effective potential is very different from our isotropic potential (which is determined by equal weighting of all angles in the spherical averaging), remains to be seen, J. Schaefer and W, Mcyer, J.Chent. Phys., 1979, 70, 344; J. A. Schaefer, personal communic- ation of 1980 results which onIy slightly modify the isotropic potential. 1. F. Silvera and V.V. Goldman, J. Chem, Plzys., 1978, 69, 4209. (0) M. G. Dondi, U. VaIbusa and G. Scoles, Chrtir. Phys. Letf., 1972, 17, 137; ( / I ) 3. M. Farrar and Y.T. Lee, J . Chem. Phys., 1972, 57, 5492; (c) A. M. Rulis and G. Scoles, Chem. Phys., 1977, 25, 183 ; ( d ) A. M. Rulis, K. M. Smith and G. Scoles, Cmi. J . Phys., 1973,56, 753. G . T. McConville, J . Chem. I’hys., 1981, 74, 2201. A. Koide, J. Phys. B, 1976, 9, 3173. P. G . Burton and U. E. Senff, J. Chent. Phys., 1982,76, 6073. P. G. Burton, F. D. Gray and U. E. Senff, Mol. Pliys., submitted for publication. ’ W. Meyer, P. C . Hariharan and W. Kutzelnigg, J .Chem. Pliys,, 1980, 73, 1880. Dr. J. M. Wutson (Universify of Waferloo) said : I fully agree with Prof. Klemperer that anisotropic iiitermolecular forces at small distances are not well represented by 1-1 low-order Legendre expansion; we have obtained indirect evidence for this when fitting potential surfaces for the rare-gas-HC1 systems simultaneously to molecular-beam spectra and rotational line-broadening cross-sec1ions.I * z However, I do not think that the isotopic substitution experiment proposed by Prof. Klemperer would provide a probe of the high-order Legendre terms. Odd A,j transitions in systems such as He + I4Nl5N and Ar + HD arise from odd-order Legendre terms in the intermoIecular potentials, and these odd-order terms themselves arise because the geometrical centre of the diatomic molecule does not coincide with the centre of mass of the isotopically substituted species. The dominant contribution to odd Aj transitions in these systems arises from theP,(cos 0) term in the potential due to the centre-of-mass transf~rmation,~ and the most important contribution to this comes from the isotropic term in the potential for the unsubstituted species.* We followed Schaefer and Meyer, using thc same geometries and the same weighting co- efficient s ,GENERAL DISCUSSION 277 Nevertheless, there is a need for information on the high-order anisotropic terms in intermolecular potentials, and there are a number of experiments which would probe these directly. (1) Direct observation of high A j transitions in inelastic scatter- ing experiments.The j = 0-4 transition in an atom-diatom system, for example, is mainly sensitive to the P,(cos 0) anisotropy. (2) High-resolution spectra of Van der Waals complexes. McKellar’s recent spectra of rare-gas-H, complexes may be at a sufficiently high resolution to determine a P,(cos S) contribution to the potential anisotropy. (3) Observation of product state distributions for vibrational pre- dissociation of Van der Waals molecules. The fraction of rotationally hut fragments produced will depend strongly on the higher-order terms in the potential. ’ J. M. Hutson and 3. J. Howard, Moi. Phys., 1981,43, 433. J. M. Hutson and B. J. Howard, Mol. Phyi;., 1982, 46, 769. H. Kreek and R. J. Le Roy, J. Chem. Phys., 1975, 63, 338. Prof. U.Buck (Max- PlaPrck-Institut fur Stromungsforschung, Gotringen) said : I do not think that an isotopic substitution will heIp in probing higher-order terms in the Legendre expansion and I agree completely with the comment of Dr. Hutson on this subject. We extensively studied the systems HD-Ne and D,-Ne, both experimentalIy and theoretically, and found that 80% of the anisotropy for HD, the P,(cos 0) term, results from the slope of the isotropic potentiaI term of the homonuclear system 1 ~ 2 due to the shift of the centre of mass from the centre of symmetry. U. Buck, F. Huisken, J. SchIeusener and J. Schaefer, J. Chem. PJzys., 1980, 72, 1512. J. Andres, U. Buck, F. Huisken, J. Schleusener and F. Torello, J . Cliern. f h y s . , 3980,73, 5620. Prof. J. J. M. Beenakker (University ofleiden) said : T want to draw attention to the fact that measurements of transitions out of low j states might have another window on the potential surface than the transitions that are of importance in studying bulk properties.Prof. U. Buck (Max-Plunck-lnstitut fiir Stromuizgsforschung, Gottingen) said : In reply to Prof. Beenakker’s remarks concerning the use of only low rotational states and low energies in beam experiments I would like to make two comments: First, the use of the lowest rotational state j = 0 as initial state is the best possible choice. The well known factorization formula shows that any other transition starting from different j can be directly related to the one starting from j = 0 with some angular coupling coefficients. Although this relation is based on the infinite-order sudden approximation it seems to be of much rnure general validity as shown in many calcu- lations.In my contribution collision energies between 30 and 90 meV have been used, which correspond roughly to the range from liquid-nitrogen to room temperature, a range which is quite normal for macroscopic properties. ‘ See D. Kouri, in Atom-Molecule Collision Theory, ed. R. B. Bernstein (Plenum Press, New York, Secondly, the final j-state depends obviously on the collision energy. 1979), p, 324. Dr. M. Faubel (Max-Planck-Institut fur Srromungsforschung, Giittingen) said : I share Prof. Buck’s view in his repIy to Prof. Beenakker’s comment. The factorization formula derived for energy sudden rotational collisions of an atom with a diatomic molecule is definitely telling you that you should start from the rotational ground state in order to obtain the maximum possible information on the potential.Tt would therefore be very interesting to see if a corresponding factorization can be derived for say the collision of two diatomic molecules,’ which might be considered as mure typical in studying bulk properties. However, near-resonant transitions between rotational2 78 GENERAL DISCUSSION o r rotational and vibrational states (for example), will, presumably, require measure- ments (and accompanying theoretical investigations), with excited initial states. See L. H. Beard et al., J . Chem. Phys., 1982, 76, 3623. Dr. G. C. Maitland, Dr. V. Vesovic and Dr. W. A. Wakeham (Imperial College, London) said: Prof.Buck refers to confirming the accuracy of the isotropic part of potentials by evaluation of diffusion coefficients and comparison with experimental data. For many years it has been the widely held view that gas transport coefficients, particularly viscosity and diffusion, are insensitive to the anisotropic part of the inter- molecular potential and so only give information about the spherically symmetric part of the interaction. However, recent work in our laboratory has shown that this is not necessarily the case. The paper by Smith and Tindell, and the subsequent dis- cussion remarks, will show clearly that transport coefficients for anisotropic systems are not determined solely by the isotropic potential. The point we wish to emphasise here is that even when the full anisotropy of the potential is taken into account, extreme care must be taken in the method of calculating transport cofficients.The calculation of dilute gas transport properties, which is relatively straight- forward for monatomic systems,' is complicated in the case of polyatomic molecules by (i) the non-spherical nature of the intermolecular pair potential; (ii) the occurrence of inelastic collisions. Although a formal semi-classical kinetic theory for dilute polyatomic gases was developed in the 1960s by Wang Chang, Uhlenbeck and De Boer (W.C.U.B.),2 it has not been possible to carry the full calculation scheme through for any realistic potential model. This is because the evaluation of the state-to-state differential scattering cross- sections, I(j'+j), for binary collisions which occur in the expressions for the transport coefficients must in general be performed by a solution of the full Schrodinger wave equation for the process.The computation time and costs involved are prohibitive, certainly for routine calculations. This led to the development of approximate methods, the most famous of which is the Mason-Monchick approximation (M.M.),3 which makes use of physically reasonable assumptions to simplify both the W.C. U.B. expressions and the dynamics of the collision process. Recently various sudden approximations have been developed which involve simplifications of the full close- coupling equations so as to reduce the computational effort required to obtain the scattering cross-sections I(j'+-j).The W.C.U.B. expressions may then be evaluated exactly, subject only to the sudden approximation for I . We have evaluated the transport properties of a series of model atom-diatomic molecule systems in this way, treating the collision dynamics within the infinite-order sudden approximation (I.0.S.A.).4 These calculations show the influence of the anisotropy of the inter- molecular pair potential on different transport coefficients and enable the accuracy of the widely used M.M. approximation to be assessed. They have been carried out over the temperature range 200-1000 K for a series of hypothetical atom (A)-rigid- rotor (BC) systems of varying rotor rotational temperature Or (= h2/2I,,k) interacting through a series of model potential-energy surfaces differing markedly in their degree of anisotropy.A comparison to first-order in the W.C.U.B.-I.O.S. and M.M. approximations is given in table 1 for two hypothetical molecular systems at 500 K. This shows that use of the M.M. approximation can lead to considerable errors, especially where large rotational energy-level spacings and/or highly anisotropic potentials are involved. The major conclusions of the complete study may be sum- marked as follows: (i) The M.M. approximation is good only for evaluating viscosity coefficients for small potential anisotropies and low Or. (ii) The deviations of M.M. from W.C.U.B.4.O.S. are in all cases greater for diffusion than viscosity. (iii) TheGENERAL DISCUSSION 279 deviations of M.M. from W.C.U.B.-I.O.S.increase with the degree of potential anisotropy and with rotational energy-level spacing but are relatively insensitive to the value of the thermodynamic temperature. (iv) The diffusion coefficient for rotational energy (Drat) can be significantly different from that for mass (D) in a dilute gas. The ratio D,,,/D can be both greater or less than unity, unlike in the M.M. TABLE 1 .-COMPARISON OF W.C.U.B./I.O.S.A. AND M.M CALCULATIONS OF TRANSPORT COEFFICIENTS FOR HYPOTHETICAL ATOM-RIGID ROTOR GASES AT 500 K potential surface A anisotropy parameter a 0.06 B 0.28 C 0.54 molecular system ' He-N2 ' ' He-C02 ' ' He-N2 ' ' He-C02 ' ' He-N2 ' ' He-C02' WK 2.87 0.56 0.995 1 .oo 0.74 0.97 0.41 0.8 1 ,,pi.M./21 W.C.U.B. 12 d 1.01 1 .oo 0.88 1.01 0.53 0.94 A a8M.M. 1A"W.C.U.B.e 1-01 1 .oo 1.18 1.05 1.29 1.15 By2. M :/Dy2. C. U . B. C rot l D Y * u. B. 1.01 1.01 1.05 1.08 0.74 1.11 DW .C.U .B. a Based on rate of change of classical turning point with relative orientation. Hypothetical; Self-diffusion coefficients. Collision integral ratio A* Z 5npDI2/3ql2 where n = number density and D,,, is the diffusion coefficient for rotational energy; ratio is unity in M.M. names used merely indicate molecular masses of species involved. p = reduced mass. approximation. Interaction viscosities. approximation where it is equal to 1.0. (v) Evaluation of thermal conductivity from viscosity using the M.M. approximation is in general unsatisfactory, in agreement with recent studies for real polyatomic gases.5 In the light of these conclusions, we would like to ask Prof.Buck and Dr. Faubel which method of calculation was used in the evaluations of transport coefficients reported in their papers and to comment on the extent to which these calculations provide a true test of the proposed potentials. G. C. Maitland, M. Rigby, E. B. Smith and W. A. Wakeham, Intermolecular Forces: Their Origin and Determination (Clarendon Press, Oxford, 1981). C. S. Wang Chang, G. E. Uhlenbeck and J. De Boer, Studies in Statistical Mechanics (North Holland, Amsterdam, 1964), vol. 11, part C. E. A. Mason and L. Monchick, J. Chem. Phys., 1961,35, 1676; 1962, 36, 1622. G. A. Parker and R. T. Pack, J. Chem. Phys., 1978, 68, 1585. G. C. Maitland, M. Mustafa and W. A. Wakeham, J. Chem. SOC., Favaday Trans. I , 1983,79, in press (2/752).Prof. U. Buck (Max-Planck-Institut f u r StrBrnungsforschug, Gottingen) said : In response to the question of Dr. Maitland I would like to say that the calculations of the diffusion coefficient for H,-Ne and H,-Ar mentioned in the paper have been performed only with the spherical component of the interaction potential. We assume that inelastic collisions and the anisotropic component make a negligible contribution (< 1%) to the collision integrals. This behaviour is confirmed by the calculation of viscosity cross-sections for the H, system by Kohler and Schaefer using close-coupling methods on the complete ab initio potential surface.' W. E. Kohler and J. Schaefer, J. Chem. Phys., 1982, in press.280 GENERAL DISCUSSION Prof F. R. W. McCourt (University qf Waterloo, Canada) said : In partial response to the query in the comment by Drs.Maitland, Vesovic and Wakeham on Prof. Buck's paper and in support of the H2-Ne potential obtained by Prof. Buck, I would like to report on one result of calculations that I have been making using the potential surface of Rodwell and Scoles.' Prof. Buck has shown us the excellent agreement between his experimental H,-Ne surface and the semi-empiricai surface of Rodwell and ScoIes, and he has indicated the rather good agreement between the diffusion coefficient DI2 measured by Trengove and Dunlop at 300 K and that calculated using the centrifugal sudden approximation and the isotropic part of his H,-Ne surface. For the hydrogen- isotope-rare-gas interactions, the anisotropies are very weak and so the value of D,, will indeed be largely determined by Vo.of D1, = I , 171 cm2 s-' at 300 K is only 0.4% lower than the value of Trengove and Dunlop of Ol2 = 1.176 cm2 s-l. I have performed a full close-coupled calculation of Ol2 and the interaction viscosity pAB(0) for the H,-Ne system using the RodwelI-ScoIes potential surface and obtained values of D,, = 1.176 cm2 s-I at 300 K and pAB(0) = 13.28 pPa s at 298 K, to be compared with the Trengove and Dunlop value for D,, and the value pAB(0) = 13.05 pPa s obtained by Clifford et aZ.4 from their experimental measurements. The calculated value of pAB(0) Iies within the estimated uncertainty of &2% estimated by Clifford et al.' Thus it appears that the Rodwell-Scoles-Buck surface for H,-Ne is quite reliable. Buck's value W.R. Rodwell and G. Scoles, J. Phys. Chem., 1982,86, 1053. 1981, preprint. A. A. Clifford, J. Kestin and W. A. Wakeham, Ber. Bunsenges. Phys. Chem., 1981,85,385. ' R. D. Trengove and P. J. Dunlop, 8th International Symposium on Thermophysical Properties, ' U. Buck, personal communication. Prof. R. J. Le Roy (Unicersity qf Waterloo) said: One of the more intriguing results in Prof. Buck's paper ' is the observation that at a given collision energy, D2(j = 0+2) inelastic transitions occur much more readily in collisions with Ne than with Ar. Since internal-rotational predissociation of a Van der Waals molecule may be thought of as an inelastic transition occurring below threshold, his result is in excellent accord with our prediction that internal-rotational predissociation level widths of H,-inert-gas complexes increase from H,-Xe to H,-Kr to H,-Ar.These results appear to present a paradox, since the splitting of energy levels of the H,-inert-gas complexes caused by the potential anisotropy, and hence the effective anisotropy strength determined from the discrete spectra of these molecules, increases with the size of the inert-gas partner. This apparent discrepancy is explained by the facts that these two types of pheno- mena are sensitive to different parts of the anisotropy strength function V,(R), and that the functions associated with the heavier inert-gas partners are displaced to smaller distances relative to the position of the zero of the isotropic potential. In particular, the level energies observed in the discrete spectra depend mainly on the anisotropy in the classicalIy allowed region at distances greater than the zero of the isotropic potential.In this region V,(R) is mainly negative, preferring a collinear relative orientation, and (as does the potential as a whole) its strength increases with the size of the inert-gas partner. On the other hand, the matrix elements which govern the rotational inelasticity and predissociation processes depend mainly on V,(R) at shorter distances, where it is positive (preferring a T-shaped configuration). and grow rapidly with decreasing R. The trends observed for these properties there- fore simply reflect the fact that the change in sign of V2(R) occurs at relatively smaller distances for the heavier inert-gas p a r f n e r ~ .~ , ~ Note that application of this argument to the diatom bond-length-dependent partGENERAL DISCUSSION 28 1 of the potential leads to the prediction that the ease of vibrational excitation of hydro- gen by inert-gas collision partners increases from Xe to Kr to Ar. U. Buck, Furuday Discuss. Chem. SOC., 1982, 73, 187. R. J. LeRoy, G. C. Corey and J. M. Hutson, Furuday Discuss. Chem. SOC., 1982, 73, 339. R. J. LeRoy and J. Van Kranendonk, J. Chem. Phys., 1974,61,4750. (a) R. J. LeRoy, J. S. Carley and J. E. Grabenstetter, Furuday Discuss. Chem. SOC., 1977, 62, 169; (6) J. S. Carley, Furuday Discuss. Chem. SOC., 1977, 62, 303. Prof. U. Buck and Mr. H. Meyer (Max- Planck-Institut fur Strornungsforschung, Gottingen) (communicated) : The question of the conformality of angular-dependent potential surfaces can now be answered on the basis of very precise experimental re- sults for the hydrogen-molecule-rare-gas systems.The reduced potentials for H,-Ne and H,-Ar, which are essentially derived from state-selective differential cross-sections (O+O and 0+2 rotational transitions) and the spectroscopy of the bound dimer, are 8 6 w 2 4 a= x . 4 3 2 c-( ._ c.’ a 0 -1 I I I 0.8 1.0 1.2 distance, R/R, FIG. 1.-Reduced potential surfaces for the isotropic (Vo) and the anisotropic ( V2) potential term: ( a - .) H2-Ne; (-) H2-Ar. displayed in fig. 1. In spite of the large differences in the potential well depths co = 2.84 meV and E , = 0.27 meV for H,-Ne and c0 = 6.31 meV and E, = 0.71 meV for H,-Ar, the curves are rather similar.However, they are only conformal with respect to the rare-gas atom and not with respect to the isotropic potential Vo or the aniso- tropic potential V,. This result is in contrast to the result of the Tang-Toennies model, where a complete conformality is found. Note that the potential models used in evaluating our data are different for the two systems.2 82 GENERAL DISCUSSION Prof. F. A. Gianturco (Uniuersity of Rome) said: In relation to the paper by Faubel et al. on the determination of the He-N, interaction from experimental data and from theoretical models, I think it of interest also to report a similar study that we have carried out for another system, the 0, molecule interacting with He atoms, where we have carefully fitted experimental data via theoretical calculations of infinite-order sudden (10s) partial and total differential cross-sections.These very accurate measurements (carried out, in fact, at the same laboratory in Gottingen) observed the total differential cross-section over a wide angular range at a collision energy of ca. 26.8 meV, together with the partial, rotationally inelastic differential cross-section under the same conditions and for the (1+3) excitation process. Since we had already obtained an effective, averaged potential for this very system,, an attempt was made at generating its anisotropic component by analogy with the be- haviour of the 0,-Ar system for which detailed studies had also been made by our g r o ~ p . ~ , ~ Thus, various anisotropic coefficients were introduced in an Exp-Spline- Morse-Spline Van der Waals potential from the interaction, by writing each of its required parameters P in the following form : P = B [I + a,~,(cos R - r)] (1) and by choosing the values of B required from the averaged 0,-He interaction and the ap from their values for the 0,-Ar full interaction.Calculations were then carried out within the 10s approximation of the dynamics, and the individual parameters were thus adjusted to fit the energy-averaged experi- mental 'data transformed in the centre-of-mass system used to perform the calcu- lations. Fig. 2 reports the actual fit obtained, where one sees that a very high accuracy was achieved down to rather small scattering angles. It is worth noting, in fact, that the relative values of the total cross-section (TDCS, curve a) with respect to the partial cross-section (IDCS, curve b) are not adjusted at all but are those directly produced by experiments as well as theory, once the latter was fixed at the arbitrary units of the former when determining the TDCS.One can also observe the following: (i) The P values in eqn (1) finally used are all very close to the initial choice from the average 0,-He interaction. (ii) The 0,-He anisotropy for the well depth and position turns out to be markedly larger than the one derived from the 0,-Ar interaction, thus producing rather different values of ap in eqn (1). (iii) The anisotropic characteristics of the wall region and of the dispersion tail appear instead to be similar in both cases, at least within the sensitivity of the present procedure.(iv) Ab initio calculations for the C,, and C,, geometries of the 0,-He were carried out by Jaquet and Staemmler at various internuclear distances and exhibited shallower minima than those suggested by our present calculations. M. Faubel, K. H. Kohl and J. P. Toennies, unpublished results. F. Battaglia, F. A. Gianturco, P. Casavecchia, F. Pirani and F. Vecchiocattivi, Faraday Discuss. Chem. SOC., 1982,73,257. F. Pirani and F. Vecchiocattivi, Chem. Phys., 1981, 59, 387. F. Battaglia and F. A. Gianturco, Chem. Phys., 1981, 59, 397. M. Faubel, K. H. Kohl, F. A. Gianturco and J. P. Toennies, unpublished results. R. Jaquet and V. Staemmler, personal communication. Dr. M. Faubel (Max-Planck-Institiit fur Stromungsforschung, Gottingen) said : The infinite-order sudden approximation, because of its numerical speed, is ideally suited to the purpose of fitting a potential to experimental cross-sections.We therefore checked the accuracy of this approximation in the present range of experimentalGENERAL DISCUSSION 283 0 0 D U 0 lo-$- 1 1 1 1 1 , ~ , ~ 1 , , , , 1 , , , , 1 , 1 1 1 I 1 1 , , 0.0 10.0 20.0 30.0 4 0.0 50 0 @cm FIG. 2.-Total differential cross-sections (a) and partial, inelastic differential cross-sections (b) for the O,-He system. The continuous lines refer to calculations within the 10s approximation, while the open circles are from experiments. The collision energy was 268 meV and the inelastic process refers to the ( j , =: l+jf = 3) rotational excitation of 02.energies and scattering angles by comparison of 10s cross-sections * with the close- coupling He-N, results given in this paper. With the “ HTT ” potential and at E,, = 27.3 meV the general agreement between the “ respective ” 10s and CC cross-sections was in the order of some 10% in the scattering angle range from 10 to 50”. At scattering angles below lo”, where the influence of the potential attractive well is important, the TOSA can be expected to frtil, and indeed does. For example the first diffraction minimum in the j = O-+j’ = 2 cross-section is too small by an order of magnitude. Keeping this limitation on the angular range of validity in mind the IOSA is suitable to obtain realistic potential fits from the experimental He-N, as well as from the quite similar He-02 cross-section data.Prof. B. Schramm (Uniuersity of Heidelberg) said : We have calculated He-N, second interaction virial coefficients with the Habitz-Tang-Toennies potential using the classical formula B(T) = dVA/mrn ( I - exp[- V(R,y)/kT]}RZsiny dy dR. 0 0 * I thank Russel Pack for permission to use his 10s program.284 GENERAL DISCUSSION A comparison with experimental data taken from the literature 1 , 2 is shown in fig. 3. Most of the data from different sources agree very well. The second virial coefficient at 90 K was calculated from the measured excess quantity E = B(He-N,) - +[B(He) + B(N2)] using B(N,) = - 180 cm3 mol - '. Recent measurements 3*4 indicate that this value should be -204 cm3 mol-'. The virial coefficient at 90 K therefore has to be lowered from 12.4 to 0.4 cm3 mo1-'.The corrected value is shown in fig. 3. If quantum corrections As can be seen, the calculations do not fit the data. i 4 I 4 2 1 . 4 -1 X X X TIK FIG. 3.-Second interaction virial coefficients of He-N,. Experimental data from different sources are shown with different symbols. Dashed line : calculated with Habitz-Tang-Toennies (HTT) potential. Full line: calculated with Faubel's modification of the HTT potential. are taken into account the disagreement becomes even worse and is therefore not shown here. The modification of the Habitz-Tang-Toennies potential proposed by Dr. Faubel does not give better results, as can also be seen from fig. 3. In order to lower the calculated second virial coefficients the attractive bowl of the potential has to be enlarged, probably by making it wider.* J. H. Dymond and E. B. Smith, The Virial Coefficients of Pure Gases and Mixtures (Oxford ' T. N. Bell and P. J. Dunlop, Chem. Phys. Lett., 1981, 84, 99. University Press, 1980). G. Pocock and C . J. Wormald, J. Chem. SOC., Faraday Truns. I , 1975, 71, 705. B. Schramm and R. Gehrmann, J. Chem. SOC., Faraduy Trans. I , 1979, 75, 479. Dr. M. Faubel (Max-Planck-Institut fur Str6mungsforschung, Gottingen) said : Both Prof. Schramm and Prof. McCourt noted a serious disagreement between experimental second virial coefficients and the values calculated from the HTT model potential for He-N,. In applying our estimated correction for the potential well depth Prof. Schramm found that the discrepancy became worse.As is obvious from fig. 9(6) of our paper the oversimplistic procedure for deepening the potential well by multiplyingGENERAL DISCUSSION 285 the vo part of the potential by a constant factor of ca. 1.2 is unintentionally increasing the repulsive barrier part of the potential at the same time. This latter effect is counteracting the decrease of the virial coefficients obtained by only deepening the well but leaving the repulsive barrier unchanged. We expect that a potential fit to the scattering data will finally give a potential well 20% deeper than the model and with slightly inward-shifted uo and u2 repulsive terms with an essentially unchanged repulsive barrier " deformation " of potential. Mr. R. R. Fuchs (University of Waterloo) said: In this paper the measurements of total and state-to-state differential collision cross-sections are reported for He-N,, and these are used to test the validity of a corresponding semi-classical semi-empirical anisotropic intermolecular potential derived from the Tang and Toennies mode1.l This potential is hereafter referred to as the HTT potential.Our group has determined two semi-empirical anisotropic potentials (soon to be published) for this system of the Hartree-Fock-plus-dispersion type. These will be referred to as HFDl and HFD2, respectively. Other anisotropic intermolecular potentials for He-N, referred to are those of Keil et aL4 (KSK), and a modified version of KSK labelled KKM3 in ref. (5). For the above potentials the following physical quantities were evaluated: (i) the second virial coefficient B,,(T), (ii) the viscosity qmix(0), (iii) the binary diffusion co- efficient o,,, (iv) the total differential collision cross-sections (TDCCS), and (v) the state-to-state differential collision cross-sections (SSDCCS) ;* these were then com- pared with experiment.All the scattering calculations were made in the infinite-order sudden approximation (IOSA). Second virial coefficients were calculated [including the first quantum-mechanical correction to B12(T)] in the temperature range 90-748 K. These calculations agree very well with those of Schramm lo for the HTT potential. Some representative values of B12(T) are listed in table 2. The HFDl potential is found to be in very good TABLE 2.-sECOND VIRIAL COEFFICIENT BIZ( T ) IN Cm3 n l O l - T K 11 3.26 148.27 321 .78 523.2 598.2 exptl 10.43 15.32 21.7 22.41 5 0.19 21.73 3 0.42 HFDl 10.8 15.4 21.6 22.3 22.3 HFD2 14.8 18.6 23.6 23.8 23.7 HTT 20.4 23.8 27.8 27.8 27.5 KSK - 8.2 17.3 18.8 18.8 agreement with experiment throughout the temperature range studied, whereas the HTT and HFD2 potentials are ca.40% and 20% too large, respectively. KKM3 and KSK can be up to ca. 30% and 45% too low, respectively, in the temperature range studied. It is found that the HTT potential is in excellent agreement with experiment, whereas HFDl and HFD2 are not in such good agreement.? * We thank Dr. M. Faubel for his calculations of these latter two quantities using our potentials. t At 373 K preliminary results show HFDl to agree much better than HTT with experiment.Although morej-states must be included in this calculation than in the 298 K one, we doubt that the relative agreement will change. Thus it seems probable that although one potential is good at a given temperature, it may not be as good as others at another temperature. The mixture viscosity qmix(0) results at 298.15 K are listed in table 3.286 GENERAL DISCUSSION The binary diffusion coefficients Ol2 calculated are listed in table 4. These show the HTT and KKM3 potentials to be in better agreement with experiment than the HFDl and HFD2 potentials. Calculations of the TDCCSs show that both the HTT and HFD2 potentials agree well with experiment while HFDI is shifted towards higher angles. On the other TABLE 3.-VISCOSITY q.lmix(O) IN PUP mole fraction X,, T/K exptl l 1 HFDl HFD2 HTT 0.2 298 183.2 186.2 185.5 183.2 0.4 298 188.6 194.7 193.2 188.6 0.6 298 194.4 202.8 200.8 194.3 0.8 298 199.2 207.7 205.7 199.2 hand, calculations of SSDCCSs have the HTT and HFDl potentials agreeing well in both magnitude and phase, while HFD2 is not in as good agreement on the phase.In the paper in question Faubel et al. mention that the Yo part of the He-N, potential should be ca. 20% deeper than the Vo part of the HTT potential. In fact, the Vo parts of HFDl and HFD2 are 34% and 16% deeper than V, (HTT), respectively. The TABLE 4.-BINARY DIFFUSION COEFFICIENT D12 IN Cm2 S-’ AT T = 300 K exptl l2 HFDl HFD2 HTT KSK KKM3 0 1 2 0.713 0.824 0.807 0.745 0.859 0.801 reason however that HFD2 does not agree better than HTT with the scattering is that the V2 and V, parts of the potential are different for both these potentials.From the above results it is quite clear that a definitive anisotropic intermolecular potential for He-N, is not as yet available. For errata see J. Chem. Phys., K. T. Tang and J. P. Toennies, J. Chem. Phys., 1977, 66, 1496. 1977, 67, 375 and 1978,68, 786. P. Habitz, K. T. Tang and J. P. Toennies, Chem. Phys. Lett., 1982, 85, 461. C. Douketis, G. Scoles, S. Marchetti, M. Zen and A. J. Thakkar, J. Chem. Phys., 1982,76,3057. M. Keil, J. T. Slankas and A. Kuppermann, J . Chem. Phys., 1979, 70, 541. W. K. Liu, F. R. McCourt, D. E. Fitz and D. J. Kouri, J. Chem. Phys., 1981,75,1496. K. R. Hall and F. B. Canfield, Physica, 1969, 47, 219. J. Brewer and G. W. Vaughn, J.Chem. Phys., 1969,550,2960. T. N. Bell and P. J. Dunlop, Chem. Phys. L e f f . , 1981, 84, 99. R. J. Witonsky and J. G. Miller, J . Am. Chem. SOC., 1963,12,282. J. Kestin, S. T. Ro and W. A. Wakeham, J. Chem. Phys., 1972,56, 4036. lo B. Schramm, this Discussion. l2 P. S. Arora and P. J. Dunlop, J. Chem. Phys., 1979, 71, 2430. Mr. C. Douketis, Dr. M. Keil and Prof. G. ScoIes (University of Waterloo) said: The experiments of Faubel et al. forcefully demonstrate the usefulness of state-to-state scattering data when investigating anisotropic intermolecular potentia1s.l Since such detailed studies are very difficult to perform, however, it would be useful to know the information content of less detailed measurements. In particular, we would like to know whether differential cross-sections (DCS), without resolving elastic from rotation-GENERAL DISCUSSION 287 ally inelastic transitions, contain information on the anisotropy of the Van der Waals potential.As discussed earlier,2 little anisotropic information is obtained from these experi- ments unless the " total " DCS show pronounced damping of the oscillatory structure normally present for atom-atom scattering. The paper by Faubel et al. makes this point quantitatively. On the other hand, more strongly anisotropic potentials do result in significant damping of the DCS oscillations, and consequently have been investigated by total DCS measurement^.^ We have used a model potential, based on the HFD procedure and extended to anisotropic systems, to compare to experimentally obtained potentials for He + C02.3 This is done in the same spirit as the comparison of Faubel et al.with the Tang- Toennies model.' In fig. 4 we compare the experimentally determined anisotropic I I I I 3 4 5 6 rlA FIG. 4.-Intermolecular potentials for He + COz for orientation angles of y = 0 and 90". The experi- mental potential (--) is the EGMSV potential obtained from the total DCS measurements of ref. (3); the HFD potential (- - - -) is obtained by extending the model of ref. (4). potential with the HFD potential. These potentials are seen to be in qualitative agreement, even though the model HFD potential has not been optimized. These results show that total DCS data exhibiting damping of the diffraction oscil- lations reasonably may be expected to contain information on the potential anisotropy.Secondly, both the results of Faubel et al. and those just discussed indicate that the respective anisotropic model potentials would serve as good starting points for analyses of experiments sensitive to the anisotropic potential.288 GENERAL DISCUSSION 0.30 0.20- 0.10- M. Faubel, K. H. Kohl, J. P. Toennies, K. T. Tang and Y. Y. Yung, Faraday Discuss. Chem. SOC., 1982, 73, 205. M. Keil, J. T. Slankas and A. Kuppermann, J. Chem. Phys., 1979, 70, 541. M. Keil, G. A. Parker and A. Kuppermann, Chem. Phys. Lett., 1978,59,443; G. A. Parker, M. Keil and A. Kuppermann, J. Chem. Phys., in press. C. Douketis, G. Scoles, S. Marchetti, M. Zen and A. J. Thakkar, J. Chem. Phys., 1982,76,3057. - Prof. Ph. BrCchignac (Uniuersity of Paris, Orsay) said: I understand from Dr.Faubel's paper that it is a dangerous procedure to derive an " effective " potential for anisotropic systems from total differential cross-sections only, while inelastic differen- tial cross-sections seem to be more reliable. However, such detailed data still need a lot of hard experimental work. 1 wish to point out that a significant amount of information is already contained in integral inelastic cross-sections, which, in spite of some averaging, have the big advantage that they can be obtained from bulk experi- ments (LIF or various kinds of double resonance). Fig. 5 shows the state-to-state I 0 I P 4 I I I I I 0 T I 4 6 8 10 FIG. 5.-Relative values of state-to-state integral inelastic crosa-sections for CO ( j = 7) - H2 as a function of finalj.The open circles correspond to theory, the full circles correspond to experiment. (a) T = 77 K; (b) T = 293 K. integral inelastic cross-sections obtained from an IRDR experiment for the Co-H, system.' The Aj = 1 and Aj = 2 cross-sections are very similar, and the higher Aj smaller. This fact appears to be the direct reflection of the shape of the intermolecular potential. Indeed the ab initio surface computed by Kochanski et al., is characterized by extremely similar R-dependence of the polynomial expansion coefficients u,,,(R) and v,,,(R) which are, respectively, responsible for Aj = 1 and Aj = 2 quantum jumps. Fig. 6 shows another set of integral inelastic cross-sections obtained from a steady- state IRDR experiment for the NO-NO ~ystern.~ The quantum jumps A j = 1, 2, 3GENERAL DISCUSSION 289 were all found to contribute significantly.This indicates that the anisotropy is not dominated by multipolar interaction. The dynamical calculations of the cross-sec- tions necessary to test specific surfaces have not yet been done. However, due to the lL-doubling of the 2rc electronic ground state of NO a more complete study of parity 1 -3 T -1 0 * 1 I T + 3 i CHANGE b i YES NO YES YES NO YES PARITY CHANGE FIG. 6.--Relative values of rate constants for NO('R~,~, j % 18.5) + NO as a function of angular momentum and parity changes. T = 300 K. changes accompanying the changes of angular momentum would be necessary, which would give a better insight into the corresponding intermolecdar potential.I Ph. Brkhignac, A. Picard-BerselIini, R. Charneau and J . M. Launay, Chem. Phys., 1980,53,165. D. R. Flower, J. M. Launay, E. Kochanski and J. Prissette, Chem. Phys., 1979, 37, 355. Ph. Brechignac, in preparation. Dr. R. Candori, Dr. F. Pirani and Dr. F. Vecchiocattivi (Uniuersity qf Perugia) and Prof. F. A. Gianturco (Univevbity of Rome) said : In the paper by Faubel et al.' experi- mental scattering cross-sections have been used to test the validity of a theoretical potential-energy surface (p.e.s.) for the He-N, system. We made a similar test using other scattering data from our Iaboratory and we found an uncertainty larger than that estimated by Faubel et al. We measured the absolute integral cross-sec- tions, for the He-N, system, as a function of the colIision ve10city.~ These results 5ave been obtained with a molecular-beam apparatus which has been described previously.4 The cross-sections are reported in fig.7 and are compared with the calcdation performed using the V,(R) component of the p.e.s. by Habitz et aL2 These data are expected to be almost completely sensitive only to the spherical average of the ~ . e . s . ~ As it is evident from the figure, both the value and the dope of the cal- culated cross-sections are in disagreement with the experimental results. Moreover, a disagreement is found also with the relative cross-sections by Butz et aL6 and the second virial coefficients calculated with this potentid show large deviations from the experimental These inconsistencies of the p.e.s. in the Vo(R) component could also be reflected in the angular dependence of the interaction. A much better p.e.s.can be obtained by a combined analysis of several experimental data, including inelastic differential and totaI cross-sections by Faubel et ul.,' as has been done for He-U,.8 M. Faubel, K. H. Kohl, J. P. Toennies, K. T. Tang and Y . Y . Yung, Farada~ Discuss. Chem. Soc., 1982, 73, 205. P, Habitz, K. T. Tang and J. P. Toennies, Chem. Phys. Lett., 1982, 85, 461. R. Candori, F. Pirani and F. Vecchiocattivi, unpublished results. V. Aquilanti, G. Liuti, F. Pirani, F, Vecchiocattivi and G. G. Volpi, J. Chem. Phys., 1976, 65, 4751. Such work is in progress.290 GENERAL DISCUSSION l Z O t ’ ti I I \ ‘\ \ ++++ \ \ \ - 1 I I 1 1.0 1.5 2.0 velocity/km s-l FIG. 7.-Effective absolute integral cross-sections for He-N, collision as a function of the Iaboratory velocity.The dashed line is the calculation performed with a theoretical potential (see text). The calculated cross-sections are convoluted to the experimental conditions. F. Pirani and F. Vecchiocattivi, Chem. Phys., 1981, 59, 387. H. P. Butz, R. Feltgen, H. Pauly and H. Vehmeyer, Z . Phys., 1971, 247, 70. ’ J. Brewer and G. W. Vaughn, J. Chem. Phys., 1969,50,2960. F. Battaglia, F. A. Gianturco, P. Casavecchia, F. Pirani and F. Vecchiocattivi, Furaduy Discuss. Chem. SOC., 1982,73,257. Dr. G. C. Maitland, Dr. V. Vesovic and Dr. W. A. Wakeham (Imperial College, London), said: We have been examining a similar question to that considered by Smith and Tindell : how an effective spherical potential obtained by inversion of transport coefficients for atom-linear-molecule systems is related to the full anisotropic potential- energy surface.The system chosen for study was “Ar-C02” interacting with a hypothetical potential-energy function based on electron-gas calculations.’ Simulated diffusion and interaction viscosity coefficients were evaluated for this system over the temperature range 100-2400 K, which represents the maximum range over which experimental data might reasonably be expected to exist for any real system. The Mason-Monchick approximation,2 which other calculations 394 suggest should be reasonably accurate for this particular system, was used to evaluate the transport co- efficients. These data were then inverted by the standard procedures developed for monatomic systems to generate a spherical inverted potential V,(r) ; this is illustrated in fig.8. This confirms the conclusion of Smith and Tindell that within the M.M. approximation diffusion and viscosity coefficients lead to the same inverted potential. This effective spherical potential V,(r) is significantly different from the unweighted spherical average of the full potential from which the transport coefficient data were calculated, in line with the findings of Smith and Tindell for diatom-diatom systems.GENERAL DISCUSSION 29 1 120( 80( 4 O( & z n ( . . K -lo( - 20( - 30( -40( 0.30 0.34 0.38 0.42 0.46 0.52 r/nm FIG. 8.-Comparison of potential V,(r) obtained by inversion of model ' Ar-C02 ' transport co- efficients (0, viscosity; o, diffusion) with the unweighted spherical average ( S ) and two fixed orienta- tion components of the full anisotropic potential used to generate the data.It is also very different from the potentials corresponding to head-on ( y = 0') or side- ways-on ( y = 90') collisions (see fig. s), nor does it correspond to any simple fixed orientation potential. Various other weighted averages of the full potential, such as Boltzmann and free-energy averages, were investigated but none of these coincided with the inverted potential over a wide range of separations. However, when the data are plotted in reduced form (kV/e against r/a where E is the well depth and a the separation where V = 0, see fig. 9), the inverted potential is remarkably close to the spherical average of the full potential over the entire range.Hence it appears that the shape of V,(r) for atom-linear molecular interactions may be inferred from the full anisotropic potential V(r,y) but that there is no simple relation between the values of its scaling parameters, E and a, and those of V(r,y). We would like to ask Smith and Tindell whether they have observed a similar con- formality between V,(r) and the spherical average of the true potential in their work on diatom-diatom systems. Should such a conformality prove to be universal, it may have some connection with the observations reported in the earlier papers by Buck and by Faubel et al. of the near conformality of V,(r) and V,(r) for some sys- tems. Although this may prove to be simply a coincidence for the particular cases studied, it suggests that an investigation of how the E and 0 parameters for V,,(r) and292 GENERAL DISCUSSION w n L .z II n L W 2 I I 1 I 1 .o 1.25 1.5 1.75 r* = ria FIG. 9.-The data of fig. 8 plotted in reduced units [O, VF(r*)]. V,(r) are linked, if at all, may give further insight into the nature of the effective poten- tials obtained by inversion of bulk properties. G. A. Parker, R. L. Snow and R. T. Pack, J . Chem. Phys., 1976,64, 1668. G. C. Maitland, M. Rigby, E. B. Smith and W. A. Wakeham, Intermolecular Forces: Their Origin and Determination (Clarendon Press, Oxford, 1981). G. C. Maitland, V. Vesovic and W. A. Wakeham, to be published. V. Vesovic, PhD Thesis (University of London, 1982). Mr. C. D. Eley, Dr. E. B. Smith and Dr.A. R. Tindell (Uniuersity of Oxford) (communicated) : Following the findings of Maitland, Vesovic and Wakeham that the reduced unweighted spherical average of the anisotropic potential function ( Vo) was conformal with the potential obtained on inversion of transport properties calculated for an atom-diatom potential, we were interested to see if this was also applicable to our system of two interacting diatomic molecules. We calculated the second virial coefficients and transport properties from the di- Lennard-Jones potential for a wide range of anisotropy using the Mason-Monchick approximation.l These properties were then inverted to give an effective spherical potential. In fig. 10 we show, for a relatively high degree of anisotropy (comparable to that proposed for bromine), that the reduced effective spherical potential is confor- mal with the reduced unweighted spherical average of the anisotropic function.This phenomenon of conformality was investigated by a more quantitative pro-GENERAL DISCUSSION 293 ?x (8: QX -1.0 Ox xsdp 0 9 10 11 12 13 1.4 15 16 FIG. 10.-Comparison of the reduced unweighted spherical potential ( x j with the reduced effective spherical potential (0) for an anisotropy of 0.6584 (as defined in the paper by Smith and Tindcll). cedure in the following way. The unweighted spherical potential was rescaled by choosing suitable parameters for the well depth and collision diameter in order to obtain a best fit to the second virial coefficients calculated from the anisotropic potential function.These " best fit " parameters were then compared with those of the effective spherical potential obtained from inversion. The agreement, as shown in table 5, is extremely good and was found to hold for the wide range of anisotropy investigated. TABLE 5.-POTENTIAL PARAMETERS effective spherical rescaled V,, deviation of (e/k)/K lOa/nm /cm3 (&/k)/K 1 Ou/nm an isotropy potential (&/kj/K 1 Oo/nm calculation, B vo 0.3292 100 1 3 3.60 102.7 3.575 0.08 87.0 3.70 0.4938 90 4: 5 3.77 90.5 3.784 0.23 59.0 4.12 0.6584 80 1: 6 3.98 81 .o 3.963 0.29 40.0 4.59 10.02 10.02 10.03294 GENERAL DISCUSSION These results, then, support the observation of Maitland et al. that conformality exists between V, and the effective spherical potential produced on inversion.This, in view of the large differences in the two potentials (see fig. 6 of the paper by Smith and Tindell) is a rather surprising result. It cannot be explained for the atom-diatom system by the conformality of Yo and Yz which, as illustrated in fig. 11, are quite different in shape for r > Y,. L. Monchick and E. A. Mason, J. Chem. Phys., 1961 , 35, 1676. U* 0.5C 0.2: 0. c -0.25 -0.50 -0.75 -1.00 1 I I I 1.0 1.5 2.0 R* FIG. 1 1 .-Comparison of the reduced unweighted spherical potential V,, (full line) and the reduced second Legendre term, V2, (broken line) for an atom-diatom interaction for anisotropy of 0.3292 (as defined in the paper by Smith and Tindell). Prof. B. Schramm (University of Heidelberg) said: Some years ago we tried simul- taneously to describe second virial coefficients and viscosity data with a spherical potential function U(r).' We used modified Lennard-Jones potentials that were buiIt by connecting two (different) Lennard-Jones ( 4 6 ) potential curves at their minima.We chose one of the potentials and determined the potential parameters E and rmin with the help of the Boyle temperature [where B(TB) = 01 and of the second virial coefficient at the temperature 0.5TB. Then we calculated the viscosities and looked how far they were shifted from the data calculated in the same way with the Lennard- Jones (12,6) potential. Calculations of the same kind using Stockmayer potentials with different dipole parameters showed much larger shifts from the ca!culations with the Lennard-Jones (12,6) potential than all the spherical potentials.This indicates that second virialGENERAL DISCUSSION 29 5 coefficients and viscosities calculated with a Stockmayer potential cannot be re- produced with an effective spherical potential. It would be interesting to answer the following questions: Is the shape of the inverted effective potential significantly different from a modified Lennard-Jones potential? Can an effective potential be found from virial coefficients and viscosities that were originally calculated with a Stockmayer potential? Can an effective potential be found that describes second virial coefficients and viscosities of nitrogen within the experimental error bounds? ' B. Schramm, R. Wiesler and T. Merz, Ber. Bunsenges. Phys. Chem., 1975,79,1240. Dr. A. R. Tindell (University of Oxford) (communicated) : Schramm reports that it is not possible to fit second virial coefficients calculated from the Stockmayer potential to a modified form of the Lennard-Jones (n,6) potential, in which each branch of the potential has a different exponent n.He asks if it is possible to produce an effective spherical potential, using the inversion techniques advocated in our paper,3 which will reproduce the second virial coefficients calculated from a Stockmayer potential. We calculated virial coefficients from the Stockmayer potential using parameters ( p = 1.83 I), Elk = 400 K, o = 0.35 nm and t* = 0.5) which are in accord with those for an organic compound with high dipole moment (e.g. CH,Cl). We found it was possible to invert these virial coefficients and produce an effective potential capable of reproducing the second virial coefficients to within 1 cm3 over a wide temperature range 360-4000 K. The failure of the approach of Schramm probably results from the double (n,6) functional form of his potential.When the functional form is unconstrained as in the inversion method, there appears to be no difficulty in obtaining an effective spherical potential which can reproduce the virial coefficients, B. Schramm, R. Wiesler and T. Merz, Ber, Bunsenges, Phys. Cheiti., 1975, 79, 1240. W. H. Stockmayer, J. Chem. Phys., 1941, 9, 398, E. B. Smith and A. R. Tindel!, Faraday Discuss. Chem. SOC., 1982,13, 221. Pro€. J. J. M. Beenakker (University oJ'Leiden) said: I want to point to a dificulty that occurs in analysing gas-transport properties of polyatomic molecules.While for noble gases the expansion in '' Sonine polynomials " converges rapidly, the situation in this respect for polyatornics is more complicated. Direct expcrirnents on the non- equilibrium distribution function (Douma et a[.)' suggest that the scalar dependence on the molecular velocity will remain similar to the situation in noble gases. There are, however, strong indications that the situation for the dependence on the rotational state is more complicated. show for example that higher polynomials in the internal energy contribute loo/, and more to the volume viscosity. Experiments that are more directly measuring the influence of molecular orient- ation on transport propertics such as field effects and flow birefringence point in thc same direction.These difficulties will be absent in low-temperature properties of systems involving hydrogenic molecules as long as only one rotational state is excited, Recent realistic model calculations for N,-N, by Turfa For a survey of this last aspect see ref. (3). ' 13. S. Douma, H. F. P. Knaap and J . J . M. Bcenakker, C'hern. fhys. Left., 1980,74, 421. A. F. Turfa, H. F. P. Knaap, H. J . Thijsse and J. J. M. Beenakker,Physic.a, 1982, in press. E. Mazur, TI1esi.T (Leiden, 3981 ; Physicl-r, to be published.) Dr. E. B. Smith (Oxford University) said: A number of authors have raised the296 GENERAL DISCUSSION possibility that certain properties including second virial coefficients, gaseous viscosity coefficients etc.can be regarded as depending almost entirely on the " isotropic part " of the potential surface (the unweighted spherically averaged potential, Vo). 1 do not believe this to be the case and would endorse the comments of Dr. McCourt on bulk gas transport data. No thermophysical properties we have investigated appear to be determined, to any useful accuracy, by the isotropic part of the potential alone. The anisotropy plays an important role, although as the paper by Smith and Tindell shows this can be taken into account by employing effective potentials that are strikingly different from the isotropic contribution. Dr. S. L. Price (Cambridge Uniuersity) said: The problem which this paper has left unsolved, that is the relationship between the effective spherical potential and the anisotropic potential, is perhaps more than an interesting intellectual problem.At the moment, the accurate calculation of transport properties for even diatomic molecules is so computationally expensive that experimental transport coefficients can only be used to test proposed diatom-diatom intermolecular potentials, and cannot be used in an iterative fashion to help determine the potential parameters as part of a multiproperty analysis. I would like Dr. Smith's opinion as to whether a knowledge of the relationship between the effective spherical potential and the actual anisotropic potential would enable us to use some information from experimental transport in determining the potential. Also, might we hope that such work could lead to a cheap method of calculating transport properties, or do the authors already know that the errors would be too large for any such method to be useful? Dr.E. B. Smith (Oxford University) said: If we could understand the relationship between our effective spherical potentials and the anisotropic potential then this could indeed lead not only to a deeper understanding of the role of anisotropy but to an economical method of evaluating the transport properties of polyatomic molecules. (It is interesting to note in this context that the remote origins of the modern inversion methods for spherically symmetric potentials lay in a method devised to allow the transport properties to be calculated more economically.)' However, our calculations of transport properties were not exact and I believe that the use of transport properties in the elucidation of the forces between polyatomic molecules will be of limited value until we can assess the accuracy of the various approximate models employed.Extremely laborious calculations must be under- taken (both close coupled, and using the classical formulation of Taxman, as attempted by Evans).2 Unfortunately an accuracy of better than 1% for the bulk gas transport properties will be necessary if useful information about intermolecular forces is to be deduced. This accuracy presents a formidable challenge particularly in view of the fundamental difficulties that Prof. Beenakker has pointed out. J. 0. Hirschfelder and M. A. Ellison, Ann. N. Y. Acad. Sci., 1957, 67, 451. D. J. Evans, Mol.Phys., 1977, 34, 103. Dr. M. La1 (Unileuer Research, Port Sunlight) (communicated) : The procedure adopted by the authors for computing the second virial coefficient, B, is equivalent to the use of the equation B = -2nN {exp[-(U12(r))/kt] - l)r2dr I: where (U,,(r)) is the potential of mean force defined asGENERAL DISCUSSION 297 the angular brackets denoting the average taken over the configurational ensemble corresponding to a pair of molecules at fixed r, the energy of a state in the ensemble being equal to the sum of the internal energies of the two molecules (i.e. U' = U: + Ui for a state i). True inversion of B, therefore, should reproduce (UI2(r)). The in- version approach adopted by the authors is able to yield only the temperature- independent potential functions, whereas, as the authors rightly point out, for a high degree of anisotropy one would expect an appreciable dependence of the potential of mean force on temperature.This imposes a serious limitation on the applicability of the method to systems with large I*. Would the authors indicate the upper limit of 1" beyond which the inversion procedure will break down? Another class of systems possessing temperature-dependent potentials of mean force is that of chain molecules. In such systems the temperature dependence is due not only to the orientational effects but also to the variability of the internal energy of the molecules with their configurational states.'v2 M. La1 and D. Spencer, J. Chem. Soc., Faraday Trans. 2, 1973, 69, 1502. M. La1 and D. Spencer, J.Chem. SOC., Faraday Trans. 2, 1974, 70, 910. Dr. E. B. Smith and Dr. A. R. Tindell (University of Oxford) (communicated): As noted by Dr. La1 the expression for the second virial coefficient at a particular temper- ature T (exp[-(U(r))/kT] - l)r2dr leads to a definition of ( U ( r ) ) , Thus second-virial-coefficient data can be used to define a series of effective isotropic potential functions which depend on temperature. Our inversions are entirely consistent with this observation. However, the important and unexpected conclusion of our paper is that there exists one isotropic temperature-independent potential function which will reproduce thermo- physical properties over a wide temperature range. Furthermore as shown in our fig. 6 this conclusion is equally valid at high degrees of anisotropy.Just how significant the anisotropy can be in the systems we investigated can readily be seen in fig. 6 and is reflected in the enormous difference between the isotropic potential derived from inversion of the properties and the unweighted spherical average of the di-Lennard Jones function. Prof. A. W. Castleman Jr, Dr. B. D. Kay, Dr. F. J. Schelling and Dr. R. Sievert ( University of Colorado) (communicated) : At several points in the discussion following the presentation of Barker the question was raised as to how the results presented for small Van der Waals molecules relate to the nucleation processes by which individual molecules aggregate to form a condensed phase. Alternatively, one may ask at what size do molecular clusters begin to evidence properties of the bulk phase.Several aspects of our research on both neutral and charged clusters address questions of this nature and have yielded new insight. Research involving neutral clusters formed in a supersonic jet has provided inform- ation on the mechanisms of cluster growth and on their structure. A study of the co- clustered nitric-acid-water system is particularly intriguing since the results have shown that properties normally associated with the condensed phase are displayed for very small numbers of molecules. High-pressure mass spectrometry experiments on charged species have also contributed significantly to our understanding of nucle-2 98 GENERAL DISCUSSION ation phenomena, and allow a quantitative interpretation of molecular interactions in Van der Waals clusters of electrolyte-containing systems. An analysis, in terms of classical nucleation theory, of the thermodynamic data obtained as a function of cluster size highlights the structural changes required on forming the liquid phase.Investigation of successive gas clustering has been directed at determining the nature of ion solvation and of ion-ligand orientational effects; these are experimentally difficult problems to study in bulk solution. A study of clusters produced by co-expanding nitric acid and water vapour has provided data particularly relevant to the above questi0ns.l Detailed experiments were performed on mixed clusters of these molecules and their deuterated analogue. Neutral clusters produced in a supersonic expansion are detected by employing a phase-sensitive electron-impact quadrupole mass spectrometer.During the course of this investigation, numerous mass spectra were taken at a variety of stagnation and ionization conditions using argon, helium and carbon dioxide as carrier gases at stagnation pressures ranging from 200 to 800 Torr. Nitric acid concentrations employed in the study range from 2.4 to 11.2 mol dm-3 and ionization voltages range from 15 to 100 eV. The general shape of the envelope of the intensity distributions shown in fig. 12 is FIG. 12.-Plot of the equivalent mole fraction of modes per cluster as a function of the number of nitric acid molecules in the cluster.GENERAL DISCUSSION 299 found to depend on the conditions in a complex manner, but in each case the observed mass spectra exhibit certain reproducible characteristic features which are independent of all experimental variables.An extensive study suggests that the proton associated with the clusters following ionization is contributed from a water molecule. In the case of the clusters containing one nitric acid molecule, it i s noted that there is a dis- tinct minimum at the cluster with five waters (fig. 12). For clusters with more than one nitric acid molecule and between one and six waters, clusters having less than a particular number of waters are not observed. Furthermore, the pure nitric acid clustcrs (dimer, trimer, etc.) are not detected, even through the dimer may be observed in experiments involving anhydrous nitric acid.The fact that thc position of the m inirna and onset compositions are independent of the stagnation temperature and pressure, carrier gas, acid concentration and ionizing voltage, strongiy suggests that they arise due to the intrinsic chemical nature of the system. Isotopic effects were ruled out, as identical results were obtained for the deuterated species. An attractive explanation for the position of the minima and onsets observed, and the attendant fact that clusters with greater than a certain nitric acid concentration are unstable, is that they are indicative of some change in the properties of the system. Such a situation could occur if a complex became sufficiently hydrated that it began to display properties normally associated with concentrated nitric acid solutions.These are known to be both thermally and photochemically unstable, giving rise to the evolution of NO, and oxygen. Plotting the equivalent mole fraction of water per cluster (associated with the unstable cluster sizes found in the experiments) as a func- tion of the number of nitric acid molecules in the cluster (insert, fig. 12), shows an approach to a specific limiting value. This value represents the critical composition range for which liquid solutions of nitric acid become unstable. This result indicates that neutral clusters begin to display liquid phase properties at very small sizes. Complementing ow research on the properties of neutral species are a number of studies of the thermodynamics of ion-molecule clusters, performed using the technique of high pressure mass spe~txometry.~*~ Such clusters bear a close relationship to neutral Van der Wads complexes, as a recent review paper discusse~.~ This work is also closely allied to the study of nucleation, in that the early steps in the addition of Iigand molecules to an ion may be approximately followed.The results of these studies, for example, made it possible to evaluate the limitations of the ciassical liquid drop model of ion-induced nucleation. By sampling the equilibrium cluster distri- bution as a function of temperature, the enthaIpy and entropy changes for clustering reactions may be extracted. Experimental hydration and ammoniation enthalpies determined in this manner were found to be in excellent agreement with the values predicted by the classical Thornson equation for clusters containing as few as four to six rnole~ules.~ The comparison with theory for the clustering of ammonia about several ions is given in fig.13(a).5 Predictions of entropy changes determined from the appropriate derivative of the free energy, as expressed by the Thomson equation, differed significantly from the experimental values, as fig. 13(6) shows, again for the clustering of NHJF5 More negative values are found experimentally than are predicted by theories based on bulk liquid properties. This finding reflects the more highly-ordered structure of smdl systems relative to the disordered nature of liquids. Such structure is useful in qualita- tively explaining the observation that while the ‘Thornson equation agrees with experiment for some systems, large discrepancies are found for others.Of related interest is the observation that for a variety of atomic and molecular ions of both positive and negative charge, the buIk heat of solvation is rapidly approached as the number of ligands In addition, stability breaks, reflected in values3 00 GENERAL DISCUSSION of the enthalpy changes as a function of cluster size, are closely related to known solution properties.2 In agreement with condensed phase studies, stabilization in- duced by the filling of solvation shells and by the formation of coordination com- plexes are obser~ed.~ These results indicate that the properties of small clusters are intimately related to those of the condensed phase, and that the study of their proper- ties provides a unique bridge between those of Van der Waals molecules and that of bulk solution.B. D. Kay, V. Hermann and A. W, Castleman Jr, Chem. Phys. Lett., 1981,80,469. A. W. Castelman Jr, P. M. Holland, D. M. Lindsay and K. I. Peterson, J. Ant. Chem. Soc., 1978, 100, 6039. I I I 5 10 100 I( 01 1 n I 00 FIG. 13.-(u) Gas-phase ammoniation enthalpie~:~ a, theory; V, Na+ ; A, K+ ; 0, Rb" ; El, Bi' ; 0, NHZ, (b) Gas-phase arnmoniation entropies [symbols as in (a)].GENERAL DISCUSSION 301 R. G. Keesee, N. Lee and A. W. Castleman Jr, J . Am. Chem. SOC., 1979,101,2599. P. Hobza and R. Zahradnik, “Van der Waals Systems”, in Topics in Current Chemistry, 1980,93, A. W. Castleman Jr, P. M. Holland and R. G. Keesee, J. Chem. Phys., 1978,68, 1760. N. Lee, R.G . Keessee and A. W. Castleman Jr, J. Colloid Interface Sci., 1980,75, 555. 539-90. ’ P. M. Holland and A. W. Castleman Jr, J. Chem. Phys., 1982,76,4195. Dr. G. C. Maitland (Imperial College, London) (communicated): The Parker and Pack IOSA calculation of transport coefficients referred to in the paper of McCourt and Liu makes the further assumptions that the energy exchange on collision is small (E = E’) and that the summation over the differential scattering cross-sections I(j‘+ jlO), for the rotational transitionj’tj, in the expressions for the transport cross-sections can be carried out t o j ’ = co. It is hence entirely equivalent to the Mason-Monchick (MM) approximation. However, the maximum value ofj’ is strictly restricted by the energy conservation expression Ej.max = E j + E where E~ is the energy of the rotor in statej and E is the initial relative kinetic energy of the atom and the rotor.Our recent calculations using the IOS approximation taking this into account have shown that making the additional MM assumptions can lead to considerable errors in the transport cross-sections, especially for high potential anisotropies and large rotational energy level spacings. Are the N,-Ar calculations of McCourt and Liu carried out using the Parker-Pack procedure or have they evaluated the full unapproximated 10s expressions? Use of the latter procedure would be expected to lead to higher values for both mixture viscosity and binary diffiusion coefficients. The papers by McCourt and Liu and by Smith and Tindell adopt different views on the ability of bulk thermophysical properties to give detailed information on potential function anisotropy.Smith and Tindell are pessimistic about their usefulness in this respect; however, all their calculations are based on the MM approximation. McCourt and Liu do see a role for bulk properties in this area although they do not demonstrate the extent to which they may be used to define the angular dependence as opposed to the average magnitude of the anisotropy. I share their optimism but for a different reason. Our full 10s calculations show that the deviations from MM-type calculations are very different for diffusion and viscosity coefficients and extremely sensitive to the anisotropic part of the potential. This suggests that dzferences between bulk transport properties could provide useful information on non-spherical interactions in the future.However, exploitation of this sensitivity is dependent on progress being made in two areas: (i) All present calculations are based on the Wang C hang-Uhlenbeck first-order solutions of the Boltzmann e q ~ a t i o n . ~ There are indi- cations that the uncertainties in these expressions could be significantly greater than those in the experimental data which are now available for polyatomic systems. There is a need, therefore, to extend the rigorous kinetic theory expressions for polyatomic gases to at least second-order. (ii) Although there are sound reasons for believing that the 10s approximation is in most cases a good one for transport cross-sections, confidence in its use would be increased by direct tests of its accuracy against either close-coupling calculations in the quantum-mechanical limit or trajectory calculations in the classical limit.We are currently pursuing both these targets. G . A. Parker and R. T. Pack, J. Chem. Phys., 1978,68, 1585. V. Vesovic, Ph.D. Thesis (University of London, 1982). C. S. Wang Chang, G. E. Uhlenbeck and J. De Boer, Studies in Statistical Mechanics (North Holland, Amsterdam, 1964), vol. 11, part C .302 GENERAL DISCUSSION Prof. F. R. W. McCourt and Dr. W-K. Liu (Uniuersity of Waterloo, Canada) (communicated): Dr. Maitland raises a number of interesting points, two of which we wish to respond to here. (1) Both the N,-He and the N,-Ar calculations reported in our discussion paper were evaluated using exact expressions relating the transport-relaxation cross-sections to the S-matrix elements of scattering theory; the S-matrix elements were calculated using the infinite-order sudden approximation.Thus, no approximations of the type introduced by Monchick and Mason have been employed in our work. (2) Apart from the pioneering effort of Shafer and G ~ r d o n , ~ we are aware only of our own crude attempt to extract information about the anisotropic parts of inter- molecular potential surfaces from bulk transport and relaxation phenomena. In both cases, use was made of phenomena which depend crucially upon potential anisotropies : in the case of Shafer and Gordon use was made of nuclear magnetic relaxation data, sound absorption data and spectral line-shape data, while in our case use was made of the magnetoviscosity Senftleben-Beenakker effect.To illustrate the type of con- clusion reached in both these efforts, it is instructive to consider a Lennard-Jones potential representation suggested some years ago by Pack.' It can be demonstrated that the relaxation cross-section determining the value of the ratio of magnetic field strength to gas pressure, H/p, at which the longitudinal shear viscosity coefficients attain half their saturation values is rather sensitive to the angular variation of as reflected by the value of the anisotropy parameter uM, and much less sensitive to the angular variation of as reflected by the value of the parameter aD. Thus, if R,, and co have been accurately determined from some other experimental results, magnetoviscosity measurements can be utilized to obtain a meaningful value of aM, for example.In the case of H,-He studied by Shafer and G ~ r d o n , ~ R,, and E~ are accurately determined from molecular- beam scattering data (although it is to be remembered that they did not employ a Lennard-Jones potential form!). In the case of N,-He we4 had assumed that R,, and E~ were accurately determined from molecular-beam total differential scattering data,6 but following the work of Faubel et u I . ~ it seems clear that this is not the case, so that an experimental " best " potential will clearly only be obtained if data from a large number of phenomena are analysed simultaneously. Insufficient sensitivity studies have been carried out to this date to allow us to know just how useful data from various phenomena will be for this purpose.&(e) = &()[I $- aDP,(COS e)] Much has yet to be done. W-K. Liu, F. R. McCourt, D. E. Fitz and D. J. Kouri, J. Chem. Phys., 1979, 71, 415. ' E. A. Mason and L. Monchick, J. Chem. Phys., 1961,35, 1676; 1962, 36, 1622. R. Shafer and R. G. Gordon, J. Chem. Phys., 1973,58, 5422. W-K. Liu, F. R. McCourt, D. E. Fitz and D. J. Kouri, J. Chem. Phys., 1981,75, 1496. R. T. Pack, Chem. Phys. Lett., 1978, 55, 197. M. Keil, J. T. Slankas and A. Kuppermann, J. Chem. Phys., 1979, 70, 541. Soc., 1982, 73, 205. ' M. Faubel, K. H. Kohl, J. P. Toennies, K. T. Tang and Y . Y . Yung, Faraduy Discuss. Chem. Prof. F. R. W. McCourt (University of Waterloo, Canada) said: Dr.Maitland has mentioned in his comment problems related to the relative importance of uncertainties in experimental measurements and those inherent in the lowest-order Chapman- Enskog solutions of the Wang Chang-Uhlenbeck-de Boer (WCUB) equation. I am basically in agreement with what he says regarding the need for careful and detailed consideration of higher-order solutions. There are, however, other aspects of theGENERAL DISCUSSION 303 theoretical treatment at this lcvel which, although mentioned in the monograph by Maitland et al.' and familiar to the transport aficionado, will not be familiar to many of those attending this symposium. I speak of the type of treatment needed in order to deal with rotating molecules, with their consequent degenerate states.One aspect is that the WCUB equation applies strictly only to molecules possessing non-degenerate internal states (e.g. most vibrational states) and, in particular, does not allow for phenomena which may depend upon the orientation of the rotational angular momentum of the colliding molecules. Thus all tensorial phenomena are excluded. This does not, however, render the situation entirely grave. It has been shown many years ago that, if tensorial polarization corrections are neglected, it is appropriate to use the WCU B formulation provided that the collision cross-sections appearing in the WCU B results are interpreted as degeneracy-averaged collision cross- sections. An estimate of the error involved in doing this can be had by examining the saturation values of the appropriate Senftleben (for paramagnetic gases) or Senft- leben-Beenakker (for diamagnetic gases) effects: these are typically 0.5% or less contributions.Unfortunately, at the 0.1% level of reproducibility even polarization contributions cannot be neglected. Another aspect related to higher-order corrections and mentioned at this sym- posium by Prof. Beenakker is that the dependence of transport phenomena on internal energy or, for linear diamagnetic molecules, equivalently on J 2 may be much more sig- nificant for polyatomic species than our experience with the Sonine polynomial expansion in monatomic gases would indicate. indicates that for pure N, higher polynomials in J 2 contribute of the order of lox to rotational relaxation and virtually explain all discrepancies between the lowest-order calculations and experiment. I t is entirely possible that a similar behaviour will be found also in the thermal conductivity and a number of the field effects.The question of scalar expansion terms has to be thoroughly reconsidered. A recent paper from the Leiden group G. C. Maitland, M. Rigby, E. B. Smith and W, A. Wakeham, Itztermolecular Forces: Their Origin and Determination (Clarendon Press, Oxford, 1981). F. R. McCourt and R. F. Snider, J, Chem. Phys., 1964, 41, 3185. A. F. Turfa, H. F. P. Knaap and J . J . M. Beenakker, Physica, 1982, 112A, 18. Prof. Ph. Brkchignac (University o j Paris, Orsay) said : As mentioned in Prof. McCourt's paper one of the properties related to the anisotropic interactions is the pressure broadening (PB) of spectroscopic lines, I would like to comment about the sensitivity of PB measurements to determine the anisotropy, particularly in systems like N,-He where the IOSA works well.The first thing that a potential should be able to reproduce is the magnitude of the PB cross-sections. However, it is well known that it is possible to find several potential surfrtces fitting the experimental value with completely different shapes. So that this magnitude is not by itself an observable vury sensitive to the interaction. But more can be learned from the variation of the PB cross-sections with temperature or rotational quantum number, J. Concerning the latter the variation is explicitly given within the IOSA by the factorization relationships due to Goldflam, Green and Kouri.' The PB cross-section of an i.r.line for a diatomic molecule is expressed as:304 GENERAL DISCUSSION where the set of primary cross-sections are the individual inelastic cross-sections out of levelj =- 0. The algebraic coefficients decrease sharply with x and all the odd coefficients are zero. Then the PB cross-section is nearly constant with,j, except for the very first transitionsj 7 0+1 a n d i = 1 -+2, and the amount of variation is related to the magnitude of This behaviour can be illustrated by the CO- He system, interesting because both theoretical calculatioiis and experimental data are available. Two ah initio potentials exist: the electron-gas surface reported by Green and Thaddeus (GT) and the recent surface by Thomas, Kraemer and Dicrcksen (TKD) computed with large SCF and CJ calculations with extended basis sets.Fig. 14 shows how different these surfaces I I I I I I I r Rlau 8 10 I I I I I I i 6 8 10 Rlau FK. I4,-Plot of the first three Coefficients of the Legentlre polynomial expansion for thc potential energy surfaces of ref. (3) (solid line) and ref. (4) (dashed line), are, for both of the isotropic and anisotropic parts. Note that the magnitudes of the PB cross-sections obtained by close-coupling calculations from the two surfaces are almost the same. However, the difference is reflected in the individual inelastic cross- sections, as apparent in table 6. The larger value of (70-r2 obtained from the GT surface leads to a bigger variation withj of the PB cross-section, although it is still not very strong.Thus PB data able to discriminate hctween the two surfaces have to beGENERAL. DISCUSSION 305 TABLE 6.-INDIVIDUAL INELASTIC CROSS-SECTIONS potential TKD GT %+1/A2 13.7 10.0 c0--t21A2 7.2 18.0 d l ) ( O , l ;O, 1)/d1)(4,5;4,5) 1.07 1.17 very accurate. Owing to the high-resolution capabilities of diode-laser spectroscopy the measurements achieved by Picard-Bersellini et are quite reliable. The value d l ) ( O , l ; 0,1yd1y4,5; 43) = 1.08 + 0.03 obtained in this work seems to favour the TKD surface. In conclusion, the rather poor sensitivity of PB data to the anisotropic interaction in similar systems has to be compensated by high-accuracy measurements. ' R. Goldffam, S . Green and D. J. Kouri, J. Chem. Phys., 1977, 67, 4149.S. Green and P. Thaddeus, Astrophys. J . , 1976, 205, 760. L. D. Thomas, W. P. Kraemer and G. H. F. Djercksen, Chern. Phys., 1980,51, 131. A. Picard-Bersellini, R. Charneau and Ph. Brechignac, J. Mu/. Srruct., in press. Mr. A. J. B. Cruickshank (Ui7icersit.y of Bristol) (commiriicated) : Battaglia and Gianturco refer to specifications for the effective, spherically symmetric, potentials for the oxygen-argon and the oxygen-oxygen systems. Although their paper is con- cerned Iargely with vibrational relaxation processes, the potentials cited [eqn (2) and (31, with table I ] may be relevant to solving a quite different problem, namely that the liquid-phase mixture argon 2 oxygen exhibits an unexpectedly large, positive voIurne of mixing. Smith and Tindell, in their paper, argued that effective spherical potentials for diatomic molecules are obtained from experimental data in a notably consistent way, by the use of their inversion procedure.Barker, in the next paper, asserts that con- densed-phase properties of the rare gases may be accurately predicted from the inter- action potentiah deduced from the low density gas-phase properties, provided only that account is taken of the triple-dipole three-body' interaction. It is therefore tempting to wonder if effective spherical potentiah for diatomics might not be as useful in relation to the condensed phases as Smith and Tindell have shown them to be in relation to gas-phase properties. We have already the Barker, Fisher and Watts potential fur Ar-Ar, and now Battaglia and Gianturco present effective spherical potentials for Ar-0, and 02-02.I would therefore ask Dr. Barker for his view on whether it might be worthwhile to attempt a new approach to the liquid-phase system Ar + O2 using the potentials listed above. Dr. E. B. Smith and Dr. A. R. Tindell (Oxfurd Uniuersity) (communicated): Mr. Cruickshank raises the question of how far our effective spherical potentials might, after appropriate non-pairwise contributions have been incorporated, account for the properties of condensed phases. We would be surprised if they were useful for the solid-state but have made no calculations in this area. However, we have embarked on computer simulation studies to test how far these potentials can reproduce liquid- state properties.Our first studies have compared the properties of di-Lennard-Jones Auids with those calculated using our effective potential and have found that the internal energies agree to within 4%. Dr. J. A. Barker (ISM, San Jose) (conrmtmicated): The suggested study of the Ar -f- U2 liquid-phase system would undoubtedly be of considerable interest. The306 GENERAL DISCUSSION unexpectedly large positive volume of mixing could be due to differences in shape of the three isotropic pair potentials. On the other hand it could also depend on the anisotropy of the potentials involving 0, or on an exotic many-body interaction in 02. Certainly one will never known until one tries, and it would make sense to examine the simplest possibility first. The calculation could be done by perturbation theory, by the Monte Carlo method, or by both methods. This is a nice example of the kind of question concerning condensed phases to which potential studies can contribute.Conversely this kind of experimental fact can also contribute to our knowledge of potentials. Prof, G . E. Ewing (Indiana Uniwrsify) said: I find the calculated vibrational relax- ation times for O,-O, puzzling. Why should they be two orders of magnitude faster than for 0,-Ar? Infrared spectra of the Van der Waals molecules 0, - - * 0, and O2 * * Ar suggest that they have comparable barriers to internal r0tation.l That is to say, there is nothing dramatically different in the intermolecular potential surfaces of 02-02 and 0,-Ar. Might the answer have something to do with the reduced moment of inertia (for internal rotation) of 02-02 which will be half that of 0,-Ar? An efficient vibration-rotation relaxation channel may therefore be open for O,-O, but closed for 0,-Ar.It will be interesting to see how more detailed calculations can resolve this puzzle. G. Henderson and G. Ewing, J. Chem. Phys., 1973, 59, 2230. Mr. B. H. Wells (Univeusily of Oxfurd) said: Tt is desirable that intermolecular pair potentials be based on the widest possible range of avaiIable data as even multi- property fits can sometimes prove surprisingly inadequate. This can be illustrated by reference to the spherically averaged potential of Battaglia et al. for oxygen. We have caIcuIated viscosity coefficients for oxygen using the potential they employ. The r.m.s. deviation of the calculated from the experimentaI viscosities was 8% i.e.the calculated viscosities are less than the experimental by an amount considerably greater than experimental error (ca. 1 %>. If, however, we make use of both transport data and second virial coefficients, we may employ an alternative method of obtaining an effective spherically averaged potential for oxygen. Experimental viscosities and second virial coefficients were fitted to a di-Lennard-Jones model The resulting best-fit parameters are E,B = 44.5 K, a,p = 0,309 nm, d* = d/omb = 0.29 where d is the distance between the two interaction sites on the same molecule. Jt is known that the unweighted Spherical average of such a potential gives a poor representation of gas-phase pro perties. However, the paper of Smith and Tindell suggests an alternative route to an effective spherical potential: second virial co- efficients were calculated using the di-Lennard-Jones potential and inverted as ex- plained by Smith and Tindell to obtain a spherically symmetric potential with para- meters c = 145 K, 0 = 0.333 nm (those of Battaglia e t d .are e = 132.3 K, 0 = 0.350 nm). The effective spherical potentiaI we obtain fits both the experimental second virial coefficients and viscosities, whereas, as stated above, that of Battaglia ef a[. does not fit the latter property. Total integral cross-sections for the scattering of oxygen were calculated using theGENERAL DISCUSSION 30 7 JWKB approximation. Both the potential of Battaglia et al. and the one we propose adequately reproduce the data of Brunetti et The effective spherical potential we propose has the virtue of not being constrained to any preconceived functional form.Such constraints may be partly responsible for the inadequacy of some multiproperty potentials. The method we employ for ob- taining an effective spherical potential could easily be applied to other diatom-diatom and atom-diatom systems. G. P. Matthews, C . M. S . R. Thomas, A. N. Dufty and E. B. Smith, J . Chern. Soc., Faraday Trans. I , 1976, 72, 238. ’ J. H. Dymond and E. B. Smith, The Virial Coeficients of Pure Gases and Mixtures (Clarendon Press, Oxford, 1980). B. Brunetti, G. Liutti, E. Luzzatti, F. Pirani and F. Vecchiocattivi, J. Chern. Phys., 1981, 74, 6734. Dr. P. Casavecchia, Dr. F.Pirani and Dr. A. Vecchiocattivi (University of Perugia, Italy) and Dr. F. Battaglia and Dr. F. A. Gianturco (University qf Rome, Italy) said : Concerning the remarks made by Prof. Ewing, it is worth noting that the differences between the vibrational relaxation times of 0,-0, and 0,-Ar are indeed related to their reduced moments of inertia, as we also state in the text of our paper. Because of the real potential-energy surface for 0,-0, being more anisotropic than for 0,-Ar, we expect that there should be a more efficient presence of (V,R)-T relaxation channels in the former rather than in the latter. Thus, the use of effective potentials to compute (V,R) relaxation times is likely to hold more realistically either with systems exhibiting weak anisotropy in their inter- actions or for those temperature ranges where vibrational inelasticity dominates, hence making the angular coupling a less effective process for relaxation whenever close (small impact parameter) encounters dominate the collisions.This also seems to be the result of our present calculations for the above systems. Going back now to the comments on our data by Mr. Wells, one should make clear at the outstart that the correct orthornormal expansion of a general function uniquely defines its spherical part, V A = ~ ( ~ ) . This is the correct spherical potential yielded by an unweighted spherical average of the converged expansion (1). Any truncated version of eqn (1) necessarily produces coefficients which depend on the level of truncation: l m a x V(r,Q) = 1=0 Therefore the efficiency of the summation on the right-hand side of Lqn (2) in represent- ing V(r,e) may conceivably be different in different regions of space.This means that one can define an efective spherical potential which depends on the chosen I.,,, and that may fit only some of the chosen molecular properties, especially if the experi- ments involved sample different regions of the whole interaction. One may also decide to use the prescription suggested by Smith and Tindell in their paper; then one obtains another way of defining an effective, spherical potential which may not agree with the previous results when different properties are fitted in order to test their respective qualities. As an example, our 0,-0, interaction is an effective spherical potential that represents the correct spherical coefficient of eqn (1) rather closely in the regions around the well and at the onset of the attractive tail, while is probably less realistic for the repulsive-wall representation of it.308 GENERAL DISCUSSION The one proposed by Wells, on the other hand, is fine-tuned for virial coefficients and viscosities, while it does not fit the spectroscopic data, as we found from WKB calculations carried out with their effective potential.Their unweighted spherical average, on the other hand, now fits those spectra but no longer reproduces the scatter- ing data. In conclusion, we feel that here we have both generated eflectiue spherical potentials that were each fine-tuned to slightly non-overlapping sets of properties and therefore could not completely agree with each other.Thus, any form of truncated expansion (2) is likely to produce different effective coefficients, which become stable against changes only after the level of convergence for expansion (1) is reached. Dr. J. M. Hutson (Uniuersity of Waterloo) and Dr. B. J. Howard (Oxford Uni- uersity) (communicated) : Obtaining anisotropic potential-energy surfaces by multi- property analysis is clearly advisable, since most experiments probe only part of the surface. When performing analyses of this kind it is particularly important to know which parts of the potential are well determined by the experimental data and which are not. by simultaneous least-squares fitting of potential parameters to moleculear-beam rotational spectra of the Ar-HC1 complex, pressure-broadening of HC1 rotational lines by Ar and second virial coefficients of Ar + HCl mixtures. The potential surfaces obtained in this way were well determined near the absolute minimum at the linear Ar-H-C1 geometry and on the repulsive wall, but the behaviour of the potential near the linear Ar-Cl-H geometry (0 = 180") was uncertain. the parameterisation chosen did not allow a secondary minimum at 0 = 180", but such a minimum was not excluded by the experimental data. More recently, information has emerged which suggests that there is actually a secondary minimum at 6 = 180". Most importantly, the MBER spectra of Ne-HCl demonstrate that a secondary minimum is present there, and it seems unlikely that Ar-HCl and Ne-HCl will be dissimilar in this respect. We have therefore obtained a new potential for Ar-HC1 (M5 potential,2 fig. 15), in which there are local minima at both linear geometries; the absolute minimum is still at linear Ar-H-Cl. The M5 potential reproduces the spectroscopic and virial data as accurately as the M3 poten- tial, but is to be preferred because of its closer resemblance to the Ne-HCl potential around 8 = 180". We recently obtained potential-energy surfaces for Ar-HCl In our M3 potential J. M. Hutson and B. J. Howard, Mol. Phys., 1981, 43, 493. J. M. Hutson and B. J. Howard, Mol. Phys., 1982, 45, 769. Prof. U. Buck (Max-Planck-Institut fur Stromungsforschung, Gottingen) (communi- cated) : The rainbow scattering of the total differential scattering cross-sections is usually used to fix the well depth of the isotropic potential. In order to check this re- lation for the potential surface derived by Hutson and Howard for Ar-HCl we per- formed a 9-channel coupled-states calculation on their M3 potential and compared the result with our measured cross-sections.l The M3 potential accurately predicts the experimental position of the rainbow maximum and the large-angle scattering, but not the amplitude of the rainbow minimum (see fig. 16). The new potential M5 with a secondary minimum at 8 = 180" should improve the fit to the scattering data since additional damping is introduced for the potential well depth. We have performed the same type of calculation for the M5 potential and the result is shown in fig. 16. The M5 potential not only gives a slightly better fit to the rainbow minimum but also predicts the general form of the large-angle scattering much better than M3. This isGENERAL DISCUSSION 309 3.0 1 I i I I I I 0 30 60 90 120 150 180 6 ; ' FIG. 15.-Contour plot of the fully optimised M5 potential for Ar-HC1. The absolute minimum is at the linear Ar-H-Cl geometry, and the absolute well depth is 180.5 cm-'. Contours are at 10 cm-' intervals relative to the absolute minimum. t L laboratory deflection angle, 8'" FIG. 16.-Experimentai and calculated angular distributions in the laboratory system for the poten- tial surfaces of Hutgon and Howard: t - ) M5,(---) M3. E - 88.1 meV.310 GENERAL DISCUSSION obviously due to a more realistic description of the potential anisotropy which is re- sponsible for the cross-sections at this angular range. This is a clear experimental indication that the subsidiary minimum at 0 = 180" is also correct for Ar-HCI. The fit could still be improved, but considering the experimental and possible theoretical errors (use of the CS approximation) present, the fit is satisfactory. An improvement could only be obtained by directly measuring the inelastic contributions. Our analysis gave differences in the cross-sections of more than a factor of two for 0+2 rotational transitions calculated using the M3 and M5 potentials. U. Buck and J. Schleusener, J. Chern. Phys., 1981,75, 2470.

ISSN:0301-7249    

DOI:10.1039/DC9827300275

出版商:RSC    

年代:1982

数据来源: RSC

摘要:

Foraday Discuss. Chem. SOC., 1982, 73, 3 1 1-324 Model Studies of Resonances and Unimolecular Decay of Triatomic Van der Waals Molecules BY BRUCE K. HOLMER AND PHILLIP R. CERTAIN Theoretical Chemistry Institute, Department of Chemistry, University of Wisconsin-Madison, 1 101 University Avenue, Madison, Wisconsin 53706, U.S.A. Received 14th December, 198 1 The wave equation appropriate to describing the resonance states of atom-diatomic Van der Waals molecules in body-fixed coordinates is presented. The Van der Waals bond is described by a square-well attraction whose depth can depend on the internal state of the diatomic. With this simple model, the analytic structure of the scattering matrix is expressed by an exact partial fraction expansion in terms of simple poles due to the bound, antibound and resonance states of the complex.The expansion leads to partial fraction expansions for the collisional time delay and the spectroscopic transition probabilities, and the results are compared to the Breit-Wigner parametrization. The square well is idealized further to a delta-function attraction, and the scattering matrix for coupled channels is obtained. Poles corresponding to shape and Feshbach resonances are identified, and an expression for the decay probabilities of a resonance into the open channels is obtained for the model. 1. INTRODUCTION Experimental studies of unimolecular decay processes are undergoing rapid development at the present time because of the selective excitation capabilities of lasers.' The energized state of the molecule thus produced can be viewed as a resonance state; i.e.a state in which the molecule has sufficient energy to dissociate but the dissociation process is slow compared to some relevant time scale such as the time required for the excitation energy to become randomized over the available degrees of freedom of the molecule. It is generally assumed that this latter process is very rapid, and that statistical theories can be applied to predict the rate constants and branching ratios for decay into the various open channels3 Triatomic Van der Waals molecules of the type Ar-H,, He-1, and He-LiH are particularly attractive candidates for theoretical study because of the simplicity of their internal structure and the possibility of detailed information about their energy surfaces.If the diatomic molecule is excited, either vibrationally or rotationally, the excitation energy is usually sufficient to break the weak atom-diatomic Van der Waals bond, thus giving very simple examples of unimolecular decay through resonance states. The quantities of interest are the energy of the resonance state, its lifetime and branching ratios to various decay products (channels). Molecular-beam spectroscopy is the richest source of experimental information about the unimolecular decay of Van der Waals molecule^.^ Crudely speaking, the positions of spectral lines give the resonance state energies, linewidths give lifetimes, and fluorescence of the decay products (either spontaneous or laser-induced) gives information about branching ratios.Molecular-beam scattering experiments can also be used to study unimolecular decay.5 Here the existence of the resonance state is revealed by characteristic features in the scattering cross-section as the relative energy of the colliding molecules312 RESONANCES AND UNIMOLECULAR DECAY is varied i n the vicinity of the resonance energy. The observcd cross-sections often can be fitted by the Breit-Wigner formula,6 which contains the resonance energy and lifetime as parametcrs. Th~oretically,~ the resonance states of triatomic Van der Wads molecules can be associated with poles of the atom-diatom scattering matrix which occur for complex values of the total energy W = E -- ih/22. (1) Here E is the energy of the resonance state and z is its lifetime. In addition, the branching ratios for decay into the open channels corresponding to vibration- rotation states of the diatomic can be determined by examining the residues of the scattering matrix elements at the pole positions and by requiring unitarity.The advantage of studying the complex poles of the scattering matrix, rather than simply computing it for physical valucs of the energy, is that if its analytic properties are simple enough, or if the resonance energies occur close enough to the real axis, all of its properties for real energies can be deduced from a knowledge of the pole positions and residues.* There are several numerical techniques for determining the poles of the scattering r n a t r i ~ . ~ ’ ~ The scattering matrix, however, can exhibit singularities in addition to resonance poles, such as poles and branch cuts arising from the form of the potential- energy surface and branch cuts due to inultiple thresholds.If these features occur close enough to the real energy axis, or close to the resonance poles, they can have observable effects on the scattering matrix evaluated for real energies and can compli- cate the interpretation of experimental data. In addition, since the scattering matrix is not directly observed in experiments, but rathcr is only one of several contributions to the observables, the shape of the resonance in the observable may be different from the shape of the resonance in the scattering matrix itself. Before we can apply numerical methods to the unimolccular problem with con- fidence that the complex energies so obtained have a clear physical significance, it is necessary to have some knowledge of the behaviour of the scattering matrix in the complex energy plane and how its singularities are related to observable quantities, A great deal is known about this in but we seek concrete examples that can be studied in detail.The purpose is partly tutorial, for the authors and, hopc- fully, for the readers. In the present paper we assume an idealized form for the atom- diatomic potential function ; namely, a hard-sphere repulsion and either a square-well or delta-function attraction, where the strength of the attraction can depend on the internal state of the diatomic. Although these potentials are not expected to account for experimental data, many of the features of the scattering matrix are expected to carry over into problems with realistic potentials.I n our work, wc generate two ‘‘ experimental ” quantities, the scattering time delay function and the spectroscopic transition moment, and study how their behaviour for real energies is related to the analytic structure of the scattering matrix, Before specializing the discussion to simplified potcntials, we outline in the next section the general formulation of the triatomic Van der Waals molecule wave equa- tion and define the scattering matrix. In the following section, we introduce the spherically symmetric square-well potential and study its propertics for states of vanishing total angular momentum. This is the simplest possible case to consider, and the singularities of the scattering matrix for this casc will persist in modified form in problems with non-spherical potentials or non-zero angular momenturn.In the last section, we allow the square well to become a delta function and study some aspects of thc problem of determining branching ratios for decay prodiicts.B . K . HOLMER AND P. R . CERTAIN 313 2. THE WAVE EQUATION Previous treatments of Van der Waals molecules have developed the wave equation within a space-fixed axis system.".'2 In our work, however, which closely follows that of Pack,13 we shalI adopt a body-fixed system, since this allows us greater flexibility in dealing with a range of anisotropic potentials. The internal coordinates are r, the diatomic bond length; R, the Van der WaaIs bond length measured from the centre of mass of the diatomic; and 0, the angle between the diatomic bond and the Van der WaaIs bond.With these coordinates, stationary-state eigenfunctions of total angular momentum (quantum numbers J , M ) have the form lyJ2f0 = i x:(R,r,e)DhM (.PY) (2) 51= -J where DhaTf is a representation coefficient of the rotation from space-fixed to body- fixed coordinates. The Euler angles, c1, /?, y, are determined by specifying that the body-fixed z-axis coincides with the Van der Waals bond, and that the diatomic lies in the body-fixed xz-plane. The 0 is an index which is used Iater to complete the labelling of the wavefunctjon. The internal wavefunctions 22 satisfy a set of coupled wave equations where is the angular momentum squared pf the diatomic expressed in body-fixed coordinates and is the quantum number ofj, and J,.In addition R*(J,Q) = [J(J+ 1) - Q(ln 5 I)]' (6) Finally, V(R,r,B) is the intermolecular potentiaI between the atom and the diatomic. The diatomic Hamiltonian possesses bound-state eigensolutions { E ~ , yn(r,O)) where n = {zi,l,Q) is a collective index for the vibrational quantum number u and the rota- tional quantum numbers, I and st, which will be explicitly indicated as the need arises. For fixed total energy E, it is convenient to define channel momenta for the open channels ( E > E J , and decay constants K , = -ik, for the closed channels (E < &,J.314 RESONANCES A N D UNIMOLECULAR DECAY The scattering matrix is defined by considering a process in which an incoming wave in the nth channel is scattered into all other open ~hanne1s.l~ Thus, we seek solutions of the asymptotic (R + co) form where Gi?(R) = k,4((b,,.exp(-i[k,,R - ( J + l’)n/23} -Si,,(E)exp( + i[k,,R - ( J + l’)n/2]}). (1 1) Here SnJ.n is an element of the scattering matrix and fi = (u,I,-Q}. Two important properties of the scattering matrix are unitarity SJ’(E)S’(E) = 1 (12) and symmetry in the channel indices s;l;n = s;;.If the asymptotic form of xg is assumed to be valid for all R, the functions Gin must satisfy the set of coupled equations. ) G?iw(R) J(J + 1) + /’(I’ + 1) - 2R2 ($ + k:T - R2 n-(J,n’)n+(J,n’ - 1) GEYpgy - 1(R) A-f-(J,sz’>n-(J,a’ + ’) Gi?r,n, [ R2 R2 + subject to the boundary conditions (1 1). intermolecular potential evaluated in the basis of internal wavefunctions qn(r,O).not appropriate for treating resonance states. it is adequate and so we turn now to the square-well potential. Here (n’l Vln”) is the matrix element of the For strongly asymmetric potentials the asymptotic form of the wavefunction is For the models we consider, however, 3 . THE SQUARE-WELL POTENTIAL In this section we shall study a square-well potential of the form V(R,r,e) = - - h2 co2(r,6) 0 < R < a 2P = o R > a. (1 5) Representatives from this class of potentials have been studied for a variety of purpose^.'^-'^ We shall make the further restriction that co2(r,0) is diagonal in the basis vn(r,e). This leads, of course, to a dramatic simplification in the coupled eqn (14) since it removes the coupling between wavefunctions with differing values of v and 1.At this stage, resonance states are characterized by quantum numbers J, u and I and the decay channels by the orientation quantum number R. We take one step further, however, and consider states with vanishing total angular momentum, J = R = 0. Even at this vastly simplified level, where only potential scattering is possible, the scattering matrix has a richness which merits detailed study.B . K . HOLMER AND P. R. CERTAIN 315 With all of the simplifications just described, the wave equation becomes R2 U(R)) G,(k,R) = 0 + where U(R) = -wz for 0 < R < a and vanishes otherwise, and we have dropped all superfluous indices. From this equation, we can obtain explicit expressions for the scattering matrix, the time delay and the spectroscopic transition probability.In potential scattering, the scattering matrix, S,(k), is determined by the phase shift, q,(k), zliz.'* where q,(k) is determined from the asymptotic form of the wavefunction Sdk) = exp[2iM)l (17) The collisional time delay T~ is defined by lo although in our numerical work we find it more useful to use the virial-theorem expression zl(k) = $lorn G:(k,R) + R- '("1 d R . dR Resonance states are characterized by large amplitudes in the region of the potential well and hence relative maxima in T ! . Finally, the spectroscopic transition probability for a bound-free, P-branch transi- tion is proportional to (21) r Z k k > = li," G,+l(lc,R)~(R)G,(k,R)dR where M(R) is the electric-dipole transition-moment function and G , + ,(lc,R) is a bound-state solution of eqn (16) with E = - h 2 d / 2 p .The transition probability to resonance states is expected to be large because of the large amplitude of G,(k,R). Previous studies l8 have focused on this amplitude, but we have chosen to study 2, because of the ambiguity associated with defining a unique amplitude and because 2, is closer to experimentally determined quantities. We choose the transition-moment fun'ction to have the form M(R) = M,( 1 - R/u)exp( -&/a) (22) for R < a and to vanish * for R > a. considerations of each quantity. With these definitions, we now turn to detailed A . THE SCATTERING MATRIX The scattering solutions of eqn (16) are (23) Gl(k, R) = A(k'R/k'a)j,(k'R) O<Rc<a = A[XkRj,(kR) + YkRn,(kR)] R > a * For a dipole transition operator M(R) which does not vanish for R > a, additional singularities will occur in the integrand of eqn (21).316 RESONANCES AND UNIMOLECULAR DECAY where krZ = kZ + wz,j,(x) and q ( x ) are spherical Bessel functions,’’ and X = -kajl,(k’a)n, +,(ka) -+ k’aj, + ,(k’a)n,(ka) Y = -kraj, + ,(k’a)j,(ka) t kujr(k’a)j, + ,(ka).Matching the asymptotic form of eqn (23) with eqn (I 8) gives X - i Y X + iY‘ Sl(k) = - Thus it is clear that, for real k, S,*(k)S,(k) = 1 . For complex k the poles of S,(k) correspond to the vanishing of the denominator ; hence where hll)(x) = j,(x) + in,(x) is the spherical Hankel f~ncti0n.l~ We define the com- plex energy which satisfies eqn (26) as k;hj”(k,a)jl + ,(kia) - k, j l ( k ~ a ) h ~ ~ l ( k n a ) = 0 (26) (27) W, = - - ki = E, - irJ2.For potentiah which vanish for R > a and are not too singular at R = 0, the only types of singularities which are possible in S,(k) are poles, and these may be classified as bound-state, antibound-state or resonance The bound-state poles occur for positive pure imaginary momenta (k, = kn), and correspond to wavefunctions which are exponentially decreasing for R --+ co. The antibound states (or virtual states) occur for negative pure imaginary momenta (k, = - i ~ i ) and correspond to wavefunctions which are purely exponentially increasing for R -+ co. (For potentials which do not vanish for R > a, it becomes probIematica1 to define “ purely ex- ponentially increasing ”.) Finally, the resonance-state poles occur in the fourth quadrant of the complex k-plane (k, = an - ifin), and correspond to purely outgoing wavefunctions. For every resonance pole, there is also a pole at -k* = -a, - ip,.For potentials which vanish for R > a, the S,(k) can be expressed exactly by an infinite product of pole terms h2 2111 k + k, S,(k) = exp(-2ika) II ~ n k - k,’ In this expression, all poles, not just the resonance poles, are included. We shall see that this representation for S, gives rise to partial fraction representations for zl and Z l in terms of the poles of S,. Numerical results for the poles of S,(k) are shown in tables 1 and 2 for two dif- TABLE 1 .-POLES OF S,(k) [SEE EQN (17) AND (18)] FOR THE SQUARE-WELL POTENTIAL DESCRIBED IN SECTION 3 Here ma = 1312 and I = 1.Only the two lowest resonance poles are shown. ~~ ~ pole Re(ka) Im(ka) bound states 1 0.0 5.2282 2 0.0 1.2147 antibound states 3 0.0 -3.5281 4 0.0 -0.6542 resonance states 5 6.541 5 - 1.2440 6 10.5482 - 1.4862B . K . HOLMER AND P . R. CERTAIN 317 TABLE 2.-POLES OF Si(k) FOR THE SQUARE-WELL POTENTIAL (SEE TABLE 1) Here ma = 36/5 and I = 2. Only the three lowest resonances are shown; there are no antibound states in this case. pole Re(ka) Im(ka) bound state resonance states 1 0.0 5.1706 2 0.6062 - 2.1761 3 2.0821 - 0.38 14 4 7.8143 - I .2660 ferent choices of well depths and angular momenta. These results are used below 10 represent the energy dependence of the time delay and transition probability. The maximum in the centrifugal barrier occurs at R = a, corresponding to (ka)' = 1(2 + l), so only resonance 3 in table 2 is a sharp tunnelling resonance.The remaining reson- ances in tables 1 and 2 occur above the barrier, with the exception of resonance 2 in table 2 which actually has an energy below threshold, Re(ka)2 < 0. B . THE T I M E D E L A Y Differentiation of eqn (28) for S,(k) yields 2 i k,a ka -t 7 (ka)[(ka)2 - (k,aY]' h - 3- If a resonance pole is close to the real axis, its contribution to the sum in eqn (29) reduces to the Breit-Wigner formula and all the other poIes contribute to a slowly varying background. Eqn (29) is valid, however, regardless of the position of the poles, and is exact if all poles are included. The low-energy behaviour of z,(k) for a cut-off potential such as ours is 2o q ( k ) - k"-' , k+O.(31) Since this behaviour is not explicit in eqn (29), the sum is slowly convergent for small k. We can remove this difficulty, however, by recognizing that since kz,(k) has only simple poles, then k-2'+'zl(k) also does and may be expressed as a partial fraction series of pole terms. This leads to the alternate expansion where the sum still contains all poles of S,(k). It is interesting that eqn (29) and (32) are very similar to results that are obtained in studies of the completeness of the set of Siegert wavefunctions." Numerical results for 7 l are shown in fig. 1 and 2 for the two potentials whose scattering matrix poles are given in tables 1 and 2. It is seen that eqn (32) is an excelIent representation of the exact behaviour over the energy range of the graphs.For larger values of (ka) the truncated pole representation fails and, in general, omission of low-energy poles in eqn (32) leads to a poor fit, since each term in the sum31 8 RESONANCES AND UNIMOLECULAR DECAY varies as kZ1-j for large k. is included in fig. l(a). For comparison the Breit-Wigner time delay, eqn (30), Since it does not include the background terms due to the 0 n c +a ._ $ -1.c W s v -2.0 - - _ _ _ _ _ _ _ _ - - - - - _ -- .- A. _ _ - - - _ _ - - - I , 1 1 1 0 1 , I 0 50 100 I L t : I d I I 1 I I , I I I 0 50 100 (ka)‘ FIG. 1.-Scattering time delay t&) [see eqn (1911 for the square-well potential described in section 3. Here wa = 13/2 and I = 1. (a) (-) Exact; (- - -) pole representation, eqn (32); (- - - -) 3reit-Wigner representation.All poles in table 1 are included. (b) Same as (a) except hara-sphere term, - 2/ka, added to Breit-Wigner representation. bound and antibound states, its threshold behaviour is incorrect. This can be approximately corrected by inclusion of hard-sphere time delay, -22/ka, as shown in fig. l(b) and 2(b). Fig. 2(a) shows the individual contributions of the poles to zl. FIG. 2.-Scattering time delay 71 [see eqn (19)] for the square-well potential. I = 2. to table 2. Here o a = 36/5 and (a) IndividuaI contributions of pole terms in eqn (32); the numbering of curves corresponds (6) 7&): (---) Exact; (- - -) pole representation, eqn (32); (- - - -) Breit-Wigner represen tation with hard-sphere correction.B . K . HOLMER A N D P . R . CERTAIN 319 Each contribution has the correct behaviour for small k, but diverges for large k.In fig. 2 it is seen that the incorrect large k behaviour of the contribution from the sharp resonance 3 is cancelled by the contribution from the " unphysical " resonance 2. c. SPECTROSCOPIC TRANSITION PROBABILITY The normalized bound-state wavefunction GI + ,(tc,R) is 3 G,,,(lc,R) = [:] Ni'+3hiyl(i~a) [t]jl+l(kwR), 0 < R < a where Eqn (21), (23) and (33) combine to give the following expression for the transition moment fi2Zl(rc,k) 4 N 2 -- pa2M,2 nka X 2 + Y 2 X i i"3hSyl(i~a) - 0 x2( 1 - x)exp(-x)j,+l(k,ax)jl(k'ax)dx)2. (36) Partial fraction expansions similar to eqn (29) and (32) can also be obtained for Zl(rc,k). The poles in Z,(rc,k) are due to the normalization factor, l/(X2 + Y2), and therefore the residues of 2, can be easily obtained.* If we let A,,(K> be the residue of [h2Zz,(~,k)]/pa2kf~ at the pole k = k,, of the scattering matrix, then h2-Wc,k) - A,,(rc)(ka)2' + pa2M,2 - (k,a)2'[(ka)2 - ( k , , ~ ) ~ ] (37) where the bound, antibound and resonant state poles are included, and the low- energy behaviour Z,(K,k) - k 2 ' + l , k -+ O (3 8) is explicitly included in the sum.The corresponding Breit-Wigner formula is l 8 where Z:(K,k) is a slowly varying function of k and the sum includes only resonance poles. Often the parameters in the Breit-Wigner representation are used to fit data; however, we shall use the calculated poles and residues without adjustment. The terms in eqn (37) and (39) are complex, although 2, is real, so in calculations only the real parts of the terms are included.Numerical results for Z,(lc,k) are shown in fig. 3 and 4 for A1 = - 1 transitions to the states in tables 1 and 2 from the lowest bound states of each potential. The * See previous footnote.320 RESONANCES AND UNIMOLECULAR DECAY 0 50 1 FIG. 3 .--Spectroscopic transition probability Z&c,k) [see eqn (2111 for the square-well potential described in section 3, Here wa = 13/2 and I = 1 for a A1 = - 1 transition from the ground state. (a) Individual contributions to eqn (37); the numbering corresponds to table 1. (b) Z I ( ~ , k ) : (-) exact; (- - -) Breit-Wigner representation, eqn (39); the poIe representation, eqn (37), is indistinguishable from the exact result. 0 10 20 30 (kaS FIG.4.-Spectroscopic transition probability Z&,k) [see eqn (21)] for the square-well potential described in section 3. Here ma = 36,"s and 1 = 2 for a A1 = -1 transition from the ground state. (-) Exact; (- - -) Breit-Wigner representation using only pole 3 in table 2; the pole remesent- ation, eqn (37), is indistinguishable from the exact result. individual pole contributions to eqn (37) are shown in fig. 3(a), where no hint of structure in the spectrum is seen. These individual terms superimpose, however, to produce the spectrum in fig. 3(6), where the pole representation is indistinguishabIe from the exact result. The sharp feature is due to the antibound state 4 in table 1 , which is quickIy damped by states 2 and 3. The Breit-Wigner representation, whichB.K . HOLMER AND P. R. CERTAIN 321 does not include these contributions, shows only the broad resonance. In fig. 4 the only visible feature is from the sharp tunnelling resonance, which is well described by the Breit-Wigner representation. In summary, we see that the pole representation of the time delay and the transition probability is very accurate so long as all contributing poles are included. Further- more, interference between bound and antibound states can produce sharp features not only in the scattering behaviour but also in the spectrum. It would be desirable to develop pole expansions which incorporate both the low- and high-energy k- dependence, thereby reducing interference effects between terms. 4. THE DELTA-FUNCTION POTENTIAL In order to illustrate several aspects of the coupled-channel problem, namely, the persistence of singularities from the uncoupled probIern and the calculation of branch- ing ratios into the various decay channels, we consider the delta-function potential 22*23 h2 Y(R,r,O) = -- i,(r,Q)d(R - a) 2P where the strength of the attraction, i(r,O), may depend on the internal state of the diatomic.In order to preserve simplicity, we assume that 3.(r,O) is independent of 8 and couples only a small number of vibrational states. In addition, we consider the rotationless case J = I = R = 0. Then the coupled wave eqn (14) reduce to subject to the boundary conditions where At-,, is a matrix element of A(r,U) in the basis of internal states. I n addition to the boundary conditions, the singularity in the potential introduces a cusp condition a t R = a where the first two terms express the difference in the derivative of the wavefunction at R = a.Before doing this, however, it is useful to consider the uncoupled problem to identify the singularities in the simpler case. The scattering eqn (41) and (42) may be solved for S = (S,,,). A . POTENTIAL SCATTERING Here we consider the wave equation which has the solution G(R) = Asin(kR) R < a = exp(-ikR) - S(k)exp(ikR) R > a. (45)322 RESONANCES AND UNIMOLECULAR DECAY Requiring G(R) to be continuous at R = a yields an expression for the constant A , while the cusp condition gives the expression for S(k) [k - 3, sin (ka) exp(-ika)] S(k) = [k - 2 sin (ka) exp(ika)] ' It is evident that S(k) is unitary for real k, but for given by solutions of the equation or The solutions of this equation are easily determined.k = 3, sin (ka) exp(ika) 2ika = 3,a[exp(2ika) - 11. complex k it can have poles (47) For l a > 1, there is a single bound-state pole for k lying on the positive imaginary axis. When 3,a = 1, this pole moves to the negative imaginary axis and becomes an antibound state for 0 < Aa < 1 . Thus, in contrast to the square-well potential, the bound and antibound states do not co-exist. An infinite number of resonance poles occur in the lower half of the k- plane in the vicinity of Re(ka) = mn, rn = 1,2,3 . . . B. THE COUPLED-CHANNEL CASE In order to obtain an explicit expression for the scattering matrix S, it is convenient to introduce matrix notation for eqn (41); thus, we define G = (G,"*(R)}, A = {A,#,,,}, k = {B,,~k,~} (48) whence the wave equations become - d2 + k2 + AB(R - a) ( dR2 (49) Further it is convenient to partition these equations according to the open (kf.> 0) and closed (k:. = -& < 0) channels. Thus, we have two sets of coupled equations, in an obvious notation d2 ki + X,,B(R - a) G,(R) + A,,B(R - a)Go(R) = 0 where the boundary conditions are Go(0) = G,(O) = 0 Go(R) = k;3[Ui-)(R) - U$+)(R)S] R > a Gc(R) = Ub-'(R)A R > a. (51) Here Ui*)(R) and Uh*)(R) are diagonal matrices with elements exp(&ik,R) and exp( *Q), respectively. A is a matrix of constant coefficients. The cusp condition for the closed-channel equations can now be written -(Dc - Acc)Gc(a) + ACOGO(4 = 0 (52) where D, is a diagonal matrix with elements IC, exp(lc,a)/sinh (rc,a). used to simplify the cusp condition for the open channeis to give This result may be (Do - Ao)tJi+)(a)k;3S - (Do - Ao)tUi-)k;3 = 0 (53)B .K . HOLMER AND P. R. CERTAIN 323 where Do is a diagonal matrix with elements k,exp(-iik,a)/sin (k,a), and A, = 1, + U D C - L)-'lL. (54) Provided (Do - &) has an inverse, the expression for S is Despite its apparent complexity, if the coupling matrix A is diagonal, S is readily seen to reduce to a diagonal matrix with elements given by the potential scattering result, eqn (46). Thus for small off-diagonal coupling, all of the scattering resonances obtained in the uncoupled case will appear near their original positions. In addition, the matrix (D, - AcJ-l is singular near the bound-state eigenvahes of the uncoupled case, and this gives rises to the expected Feshbach or compound-state resonances in S .In addition, there are branch point singularities at each threshold (xL, I- 0). The poles of S are given by the generalized cigenvalue equation W = E - ir/2 (56) where C is the eigenvector. The physical significance of C is revealed if the derivation of S is reviewed and it is realized that when eqn (56) is satisfied, there is a solution of the coupled equations of the form (Go is now a column vector) G,(R) = k,aUL+'(R)C (57) that is, a solution in which there is a purely outgoing wave in each open channel. This is the Siegert definition of the resonance wavefunctjon,24 and thus an intuitive expression for the probability for decay into channel u i s This allows a partiaI width for each channel to be defined as 2 4 3 2 5 so that This interpretation is superficial, however, in the same sense as is our choice of the final wavefunction in the study of the transition probabiIity in section 3 .That is, in an experiment it is not the resonance wavefunction which describes the decay, but rather a wave packet constructed by superposition of stationary states defined in the real energy axis in the vicinity of the resonance pole. A deeper analysis is required for an unambiguous interpretation of the decay probabilities. The authors thank the National Science Foundation (U.S.) for support of this They are also grateful to Prof. K. W. McVoy for an enlightening discussion work.and for many references. ' B. D. Cannon and F. F. Crim, J. Chern. Phys., 1981, 75, 1752. ' R. D. Levine, Quanfum Mechanics of Mclleculur &re Processes (Oxford University Press, Oxford, 19691, pp. 268-292. D. W. Noid, M. L. Koszykowski and R. A. Marcus, Annu. Rec. Phys. Chem., 1981, 32, 267. D. H. Levy, Annu. Rev. Phys. Chern., 1980, 31, 197.324 RESONANCES AND UNIMOLECULAR DECAY In fact, we are not aware that such experiments involving Van der Waals molecules have been reported. For a theoretical discussion, see J. A. Beswick, G. Delgado-Barrier and J. Jortner, J. Chem. Phys., 1979, 70,3895, For a molecular-beam study involving unimolecular decay in a " normal " molecule, see J. M. Farrar and Y . T. Lee, J, Chern. Phys., 1975, 65, 1414. R. G. Newton, Scattering Theory of Waves and Particles (McGraw-Hill, New York, 1966), p. 313. R. W. Numrich and K. G. Kay, J. Chem. Phys., 1979, 70, 4343; 1979, 71, 5352. See the Symposium issue Complex Scaling in the Spectral Theory of the Hamihonian, Inr. J. Quantm. Chem., 1978, 14. * R. G. Newton, J. Math. Phys., 1960, 1, 319. lo J. R. Taylor, Scattering Theory (John Wiley, New York, 1972), chap. 13 and 20. l i R. J. LeRoy and J. Scott Carley, Ado. Chern. Phys., 1980, 42, 353. l2 S-I. Chu, J. Chern. Phys., 1980, 72, 4772. l3 R. T. Pack, J. Chem. Phys., 1974, 60, 633. I4 H. M. Nussenveig, Nucl. Phys., 1959, 11, 499. l6 J. 0. Hirschfelder and K. T. Tang, J. Chern. Phys., 1976, 65, 470. l7 R. T. Pack and J. 0. Hirschfelder, J. Chern. Phys., 1980, 73, 3823. l8 M. S. Child, in MuZecuIar Spectroscopy (The Chemical Society, London, 1974), voI. 2, chap. 7. l9 J. D. Jackson, CCassicaC Electrodynamics (John Wiley, New York, 2nd edn, 1975), pp. 740-742. 'O L. D. Landau and E. M. Lifshitz, Quantum Mechanics (Pergamon Press, New York, 3rd edn, 21 D. L. Huestis, J. Math.Phys., 1975, 16, 2148. 22 W. J. Romo, Can. J. Phys., 1974, 52, 1603. 23 B. V. Carlson, K. W. McVoy and M. C. Nemes, Nucl. Phys. A, 1979,331, 117. 24 R. Yaris and H. S. Taylor, Chem. P h p . Lett., 1979, 66, 505. 25 C. W. McCrrrdy and T. N. Rescigno, Phyz Rep. A, 1973,20, 2345. K. W. McVoy, L. Helkr and M. BoIsterIi, Reu. Mod. Phys., 1967, 39, 245. 19771, pp. 542-543.

ISSN:0301-7249    

DOI:10.1039/DC9827300311

出版商:RSC    

年代:1982

数据来源: RSC