摘要:

Dare 1964 1964 I955 1965 I966 1966 1967 1967 1968 1968 1969 1969 1970 1970 1971 1971 1972 I972 1973 1973 1974 1974 1975 1975 1975 1977 1977 1977 I978 1978 1979 1979 1980 1980 1981 1982 i9ai FARADAY DISCUSSIONS OF THE CHEMICAL SOCIETY Subject Chemical Reactions in the Atmosphere Dislocations in Solids The Kinetics of Proton Transfer Processes Intermolecular Forces The Role of the Adsorbed State in Heterogeneous Catalysis Colloid Stability in Aqueous and Non-aqueous Media The Structure and Properties of Liquids Molecular Dynamics of the Chemical Reactions of Gases Electrode Reactions of Organic Compounds Homogeneous Catalysis with Special Reference to Hydrogenation and Bonding in Metallo-organic Compounds Motions in Molecular Crystals Polymer Solutions The Vitreous State Electrical Conduction in Organic Solids Surface Chemistry of Oxides Reactions of Small MolecuIes in Excited States The Photoelectron Spectroscopy of MoIecules Molecular Beam Scattering Intermediates in Electrochemical Reactions Gels and Gelling Processes Photo-effects in Adsorbed Species Physical Adsorption in Condensed Phases Electron Spectroscopy of Solids and Surfaces Potential Energy Surfaces Radiation Effects in Liquids and Solids Ion-Ion and Zon-Solven t Interactions Colloid Stability Structure and Motion in Molecular Liquids Kinetics of State Selected Species Organization of Macromolecules jn the Condensed Phase Phase Transitions in MoIecular Solids Photoelect rochernis try High ResoIution Spectroscopy Selectivity in Heterogeneous Catalysis Van der Waals Molecules Oxidation Precipitation 43 1 Volume 37 38 39 40 41 42 43 44 4.5 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 * 66 67 68 69 70 71 72 73 * Not available; for current informrrfion on prices, etc., of auaiiuble uofolumes, please confact fhe Morkering Oficer, Royal Society of Chemistry, Bidingtoan House, London WI V OBN stating whether or nof yuu are a member of the Society.Dare 1964 1964 I955 1965 I966 1966 1967 1967 1968 1968 1969 1969 1970 1970 1971 1971 1972 I972 1973 1973 1974 1974 1975 1975 1975 1977 1977 1977 I978 1978 1979 1979 1980 1980 1981 1982 i9ai FARADAY DISCUSSIONS OF THE CHEMICAL SOCIETY Subject Chemical Reactions in the Atmosphere Dislocations in Solids The Kinetics of Proton Transfer Processes Intermolecular Forces The Role of the Adsorbed State in Heterogeneous Catalysis Colloid Stability in Aqueous and Non-aqueous Media The Structure and Properties of Liquids Molecular Dynamics of the Chemical Reactions of Gases Electrode Reactions of Organic Compounds Homogeneous Catalysis with Special Reference to Hydrogenation and Bonding in Metallo-organic Compounds Motions in Molecular Crystals Polymer Solutions The Vitreous State Electrical Conduction in Organic Solids Surface Chemistry of Oxides Reactions of Small MolecuIes in Excited States The Photoelectron Spectroscopy of MoIecules Molecular Beam Scattering Intermediates in Electrochemical Reactions Gels and Gelling Processes Photo-effects in Adsorbed Species Physical Adsorption in Condensed Phases Electron Spectroscopy of Solids and Surfaces Potential Energy Surfaces Radiation Effects in Liquids and Solids Ion-Ion and Zon-Solven t Interactions Colloid Stability Structure and Motion in Molecular Liquids Kinetics of State Selected Species Organization of Macromolecules jn the Condensed Phase Phase Transitions in MoIecular Solids Photoelect rochernis try High ResoIution Spectroscopy Selectivity in Heterogeneous Catalysis Van der Waals Molecules Oxidation Precipitation 43 1 Volume 37 38 39 40 41 42 43 44 4.5 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 * 66 67 68 69 70 71 72 73 * Not available; for current informrrfion on prices, etc., of auaiiuble uofolumes, please confact fhe Morkering Oficer, Royal Society of Chemistry, Bidingtoan House, London WI V OBN stating whether or nof yuu are a member of the Society.

ISSN:0301-7249    

DOI:10.1039/DC98273FX001

出版商:RSC    

年代:1982

数据来源: RSC

摘要:

Faraday Discuss. Chem. SOC., 1982, 73, 7-17 THE LENNARD-JONES LECTURE Intermolecular Binding BY JOHN A. POPLE Department of Chemistry, Carnegie-Mellon University, Pittsburgh, Pennsylvania 15213, USA. Received 26th April, 1982 1. INTRODUCTION It is most appropriate that this first Lennard-Jones lecture should be given at a Faraday Discussion connected with intermolecular forces. Lennard-Jones fathered the subject through his early work on realistic empirical potentials and their relation to macroscopic properties of gases. The famous " 6-12 potential " u(r) = Ar-12 -Br-6 (1) for the interaction energy of rare-gas atoms has long since been a standard starting point for statistical-mechanical work on all phases of matter.' The potential was based on a combination of long-range interaction energies (initially believed to be proportional to inverse sixth powers of the distance on empirical grounds) and another short-range part introduced to take account of repulsive effects which ultimately prevent close approach of colliding molecules.In addition to his work on empirical force fields, Lennard-Jones played an import- ant role in the development of molecular-orbital theory,2 now widely used for the non- empirical investigation of force fields. In this lecture, I shall emphasize current tech- niques for investigating the interaction energy of closed-shell diamagnetic molecules, using the orbital theory. In the second section, the role of partitioned theories, which attempt separate computation of long- and short-range parts of the potential, will be compared with the general approach treating the interacting molecules as one large system.This will be followed in the third and fourth sections by some illustrative applications to interactions of small molecules, both moderately weak ones such as hydrogen bonds and also strong interactions which lead to complexation into a unified molecule. 2. THEORETICAL METHODS FOR INTERMOLECULAR FORCE FIELDS The history of the application of quantum mechanics to the computation of inter- molecular force fields has proceeded in two broad directions. The first, in part following the philosophy introduced by Lennard-Jones potentials, attempts to par- tition the total interaction into a long-range part, generally. proportional to inverse powers of the intermolecular distance R, and a short-range repulsive part associated with exchange or overlap of the electronic distributions.These contributions have8 THE LENNARD-JONES LECTURE usually been evaluated separately, often by different techniques, and the results added to give a total A - B interaction potential A E A B = A E A B (long-range) + A E A B (short-range). (2) We shall refer to this technique as a " partitioned method." The second general approach, often described as the " supermolecule method," treats the interacting A * - B complex as a single large molecule; it then obtains the interaction potential A E A B by calculating the energy of AB and subtracting the ener- gies of separated A and B molecules: It is essential, of course, that the same theoretical method is used for all three species on the right-hand side of eqn (3).Examples of the application of both of these general techniques will, no doubt, be presented later in this meeting. Some remarks about their relative merits are therefore appropriate. THE PARTITIONED METHOD The long-range part of the partitioned potential (2) can usually be handled by methods which involve only properties of the isolated A and B systems or which avoid computations involving overlapping functions centred on A and B. If the unper- turbed wavefunctions for A and B are YA,'€'B (supposed not to overlap), then the interaction Hamiltonian VAB (the coulomb interaction between the electrons and nuclei of A and the electrons and nuclei of B) may be introduced as a perturbation. If this perturbation is applied to first order, the interaction energy is A E A B (long-range, first-order) = ( Y A Y B ] VAB [ Y A Y B ) .(4) This is the rigid electronic interaction potential, that is the interaction energy ensuing from the coulomb interaction of unperturbed electronic distributions of the isolated molecules A and B. The expression (4) can be evaluated directly, given the wave- functions YA and YB. Alternatively, if appropriate centres are chosen for A and B, a multipole expansion of VAB may be carried out, giving a series of terms in inverse powers of R, the distance between the molecular centres. For neutral molecules A and B (with the intercentre line along the z-axis), this takes the form (for axial molecules) A i E A B (long-range, first-order) = p&Bfi1(w)a-3 + [1UA@BfiZ(w) + pJ3@Af21(W)]R-4 + * * * ( 5 ) (6) where pA, pB and OA,OB are dipole and quadrupole moments of A and B.Here ~ J W ) = -2 cos 8, cos O2 + sin 8, sin 8, cos (9, - v2) . . . (0,,91),(02,92) being spherical polar coordinates of the axes of A and B relative to the AB line. The moments appearing in the long-range potential (5) may be computed by studies of the isolated molecules A and B or may be measured experimentally. In either case, this expansion provides a physically illuminating picture of the interaction at the largest distances. It was recognized at an early stage that the rigid electrostatic potential (5) failed to give any description of the long-range dispersion forces between spherical systems. To develop a more satisfactory theory, as originally shown by London: it is necessary to introduce the interaction potential VA, to second order. However, non-overlapJ .A . POPLE 9 between A and B wavefunctions may still be assumed. The additional contribution to the long-range potential is then by Rayleigh-Schrodinger perturbation theory. In this expression, Yi,Y!i are the wavefunction and energy for the ith excited state of A and similarly for B. The long-range second-order energy (7) can be further subdivided by separating the double summation The first of these three parts then corresponds to the interaction of the rigid electro- static distribution of A with the induced electronic density change in B; the second part has the converse interpretation. The third part of eqn (8) represents the coupling between the charge-density fluctuations in the two molecules; it represents the dis- persion interaction which exists even for spherical atoms.Each part of the second-order energy expression (8) may be further expanded in inverse powers of R. The leading term in the first part of expansion (9) (for neutral molecules) is then the dipole-induced-dipole interaction, proportional to P A c c B R - ~ where aB is the electronic polarizability of B. The leading term in the third part of eqn (8) is the dipole-dipole contribution to the dispersion interaction, proportional to Higher terms lead to the long-range expansion for spherical atoms AEAB (dispersion) = C6R-6 + C8R-8 + . . .. (9) For molecules, C,,c8 . . . will depend on relative orientations.All the parts of the first- and second-order contributions to the long-range inter- action can be handled by computations on wavefunctions for the isolated systems A and B. The situation is less satisfactory for the short-range part of the composite potential (2). This part depends on the overlap of the A and B wavefunctions. Al- though some progress can be made with an approximate theory involving only elec- tron den~ities,~ full ab initio computations require a study of the compound A - - B system. Some difficulty is encountered because the starting wavefunction YAY, is not fully antisymmetric with respect to all electrons. This can be overcome by appropriate modifications of perturbation t h e ~ r y . ~ However, the computation of the full interaction energy, even in first order, requires the valuation of Here d is a normalized antisymmetrizing operator, H A B is the complete Hamiltonian for A * B; H A , HB are the separate Hamiltonians for isolated A and B; YA,YB are still unmodified wavefunctions for the separate species.The evaluation of eqn (10) gives both the short- and long-range contribution inJirst order only. Thus it includes the long-range parts (4) or ( 5 ) but not the second-order long-range parts (7). A com- mon assumption is that these parts may be combined to give an appropriate full potential (1 1) In practical implementations of the partitioned approach, a choice has to be made for the ground-state wavefunctions YA,YB and for the excited wavefunctions Y&Yd A E A B (full) z A E A B (long + short, first-order) + AEAB (long, second-order).10 THE LENNARD-JONES LECTURE to be used in eqn (7), (10) and (1 1).A reasonable choice is to take ground-state single- determinant Hartree-Fock functions for YA,YB using a finite basis-set expansion for individual molecular orbitals and then construct excited functions Yi,Y$ using virtual orbitals from such a computation. An example of such a study is the (N2)2 potential recently obtained by Berns and van der Avoird.6 THE SUPERMOLECULAR METHOD In the supermolecular method the interaction between the molecules A and B is no longer treated as a perturbation. Rather, the wavefunction YAB for the compound A - B system is calculated for every required relative intermolecular configuration. The interaction energy is then At sufficiently large separations, the wavefunction YAB will become dYAYB where d is the antisymmetrizer.The interaction (12) then becomes identical with the first- order energy (10). If the intermolecular distance is so large that overlap is negligible, the antisymmetrizer has no effect and eventually (12) reduces to the first-order long- range expression (5) ; this is dominated by dipole-dipole interactions for neutral molecules. At shorter distances the supermolecule energy expression (12) takes some account of the repulsive intermolecular forces, since the compound wavefunction is always antisymmetrized and the full Hamiltonian automatically takes account of penetration effects. The supermolecule approach has the further advantage that it may also be used for extremely strong intermolecular interactions which lead to coalescence of AB into a single molecule with indistinguishable fragments.The dimerization of BH3 to give B2H6 with equivalent bridging hydrogens is an example. Deficiencies of the supermolecule approach are mainly associated with limitations on the type of wavefunction used. The great majority of supermolecule calculations in the literature are at the Hartree-Fock level. Thus YAB, YA and YB in eqn (12) are all single-determinant wavefunctions with appropriate optimized coefficients in a basis set expansion. Such wavefunctions will describe long-range multipole-multipole interactions insofar as the multipoles are themselves well described at the Hartree-Fock level. The interaction potential will include multipole-induced-multipole interac- tions, again insofar as the corresponding polarizabilities are well described at this level.However, no account is taken of dispersion forces, since these depend intrinsically on correlation between electrons in separate molecules, an effect completely neglected by the single-determinant wavefunction. Hartree-Fock supermolecule calculations are therefore likely to be most successful for systems where polar, electrostatic interactions dominate. Hydrogen bonding is an example. For the weaker interaction potentials which are dominated by dispersion forces, Hartree-Fock theory is inadequate. It may be noted at this point that the computational effort for a Hartree-Fock super-molecule calculation is comparable to that required for evaluation of the first- order interaction energy (10).Both computations require evaluation of the complete set of four-centre two-electron integrals over all basis functions. This part of a Hartree-Fock ab initio calculation usually dominates the later parts which process the integrals. It would clearly be advantageous to use the supermolecule method with a post Hartree-Fock technique which takes some account of electron correlation. However, not all such methods are suitable for studies of intermolecular force fields. They should have the property of size-con~istency,~ according to which the result of the calculation on the A - - B complex at infinite separation should be the sum of theJ . A . POPLE 1 1 energies calculated separately for A and B. If this condition is not satisfied, the inter- molecular potential calculation by eqn (12) will not tend to zero as the intermolecular distance tends to infinity.The Hartree-Fock technique itself is size-consistent when applied to closed-shell molecules. However, certain simple ways of handling electron correlation fail this test. For example, the method of configuration interaction involving all double sub- stitutions from single-determinant Hartree-Fock reference function is not size- consistent. This is because such a method fails to take account of" unlinked pairs '' of substitutions in the separated molecules A and B. Perhaps the simplest size-consistent post-Hartree-Fock procedure is Marller- Plesset perturbation theory8 carried out fully to any order. This theory handles electron correlation by treating the full Hamiltonian as a perturbation on the Fock Hamiltonian, for which the Hartree-Fock wavefunction is the exact solution.Thus, we introduce a perturbation parameter A and a perturbed Hamiltonian HA = I: + A(H - F) (1 3) where Pis the Fock Hamiltonian and H the full Hamiltonian. The energy correspond- ing to the Hamiltonian HA may then be expanded in powers of A En = E(O) + LE") + R2E'2' + A3E'3' + . . . (14) where explicit expressions for E(O), I?('), E(2) . . . may be obtained.8 In practice this series is terminated at some order n and the value of A then set equal to unity. The resulting energy is described as MPn. The MPl energy is identical to Hartree-Fock. Second-order theory (MP2) is the simplest order at which correlation is taken into account, all correlated electron pairs being treated independently at this level.Higher orders of Marller-Plesset (MP3 and MP4) take account of pair-pair interactions, orbital modification by correlation, correlated electron triads and unlinked pairs of pairs. Computer programs have now been developed for application of this theory up to full fourth order (MP4). Supermolecule methods based on Marller-Plesset theory, even at the MP2 level, should be effective techniques for studying intermolecular force fields. In addition to being size-consistent, they take account of electron correlation in the whole super- molecule A * * - B. The contribution of electron correlation to the intermolecular potential is then determined by subtracting the separate electron correlation energies for A and B.Thus, at the MP2 level, the contribution of electron correlation is AEAB (correlation) = E(2)(AB) - lP2)(A) - E(2)(B). (1 5 ) This is added directly to the Hartree-Fock supermolecule potential to give the overall MP2 force field. At sufficiently large distances it can be shown that the second-order energy expression (1 5 ) does include a dispersion-force term proportional to R-6. This term is, in fact, identical to that obtained from the partitioned long-range energy (7) provided that Hartree-Fock functions are used and provided that the energy denomin- ators in (7) are replaced by differences of Fock one-electron energies. It appears, therefore, that MP2 supermolecule studies (with a suitably chosen basis set) should provide a powerful method of generating intermolecular force fields which incorporate all the main long-range effects, including dispersion forces, as well as taking account of effects of electron correlation on the repulsive part of the potential surface.The higher-order MP3 and MP4 levels of theory could be used in a similar manner. Few Marller-Plesset supermolecule intermolecular force fields have yet been published. In the following sections we present some preliminary results in this12 THE LENNARD-JONES LECTURE direction. The procedure used is first to find the global minimum energy on the Hartree-Fock potential surface, a process greatly aided by recently developed analytic gradient techniques. The split-valence plus d-polarization basis set 6-3 lG* is used for these cal~ulations.~ This leads to a set of equilibrium geometries (optimized with respect to intra- and inter-molecular geometrical parameters) together with maximum intermolecular interaction energies (well depths).These Hartree-Fock structures (corresponding to the HF/6-31 G* theoretical model) are then used for single-point calculations with Marller-Plesset theory taken to fourth order in the space of single, double and quadruple substitutions from the Hartree-Fock determinant. This level (MP4SDQ) differs from full fourth-order theory by omission of triple substitutions which are more difficult to calculate. These single-point computations use the larger 6-31G** basis set which also contains polarization functions on hydrogen atoms.The final energies may be denoted by MP4SDQ/6-3 1 G**//HF/6-3 1 G*, where " / / " denotes " at the geometry of." They may be used to make some overall assessment of the role of electron correlation on binding energies. 3. INTERMOLECULAR BINDING BETWEEN FIRST-ROW HYDRIDES A systematic study of interaction potentials between closed-shell molecules should begin with the smallest and simplest such systems. These are H2 and the first-row simple hydrides XH,, where X goes from lithium to fluorine and m is the correspond- ing classical valence. We omit H2, where interaction energies tend to be small, and consider only the interactions involving the molecules LiH, BeH,, BH3, CHI, NH3, OHz and FH. Many of the complexes XH, - . . Y H , from this series have been examined at the HF/6-31G* level in joint work with D.J. DeFrees and P. V. R. Schleyer. The global minima have been determined leading to HF/6-3 1 G* equili- brium binding energies listed in table 1 . The corresponding structures are illustrated TABLE 1 .-HF/6-31G * BINDING ENERGIES OF XH, YH, (kJ rnol-') j 13 27 51 I 23 39 I 25 0 "Based on HF/6-31G* geometries. in fig. 1; detailed lengths and angles are available in the Carnegie-Mellon Quantum Chemistry Archive.lO The energies in table 1 may be partitioned into groups by the dotted lines shown. Interactions with methane are shown by lower level calculations to be generally small (<lo kJ mol-l) except for LiH. These will not be considered further. The com- plexes in the bottom right-hand corner (NH,OH,,FH with NH,,OH,,HF) are the hydrogen-bonded set with interaction energies in the range 13-51 kJ mol-I.AllJ . A . POPLE 13 1 (DZh) **..H*. H - Be '' Be - H * I . . - . H ,H . *. L i ' ' B e - H . . ' H .' .H .. .. ' Li' " B H %> 0 - H * * O p H H 16 (C,) F-H.* F, H FIG. 1 .-Structures of dimers involving first-row hydrides.14 THE LENNARD-JONES LECTURE involve single hydrogen bonds, the water dimer structure 16 being typical. It is known that the HF/6-31G* result is in excellent agreement with experiment l1 for the geometry of this species. However, there is evidence that the binding energy is too large because of basis set limitations. Smaller Hartree-Fock values of 15 and 21 kJ mol-1 have been reported in the literature using larger basis sets. In the top right-hand corner of table 1 are complexes representing donor-acceptor interactions between molecules having lone pairs of electrons (NH3, OH2, FH) on the one hand with electron-deficient molecules (LiH, BeH,, BH3) on the other.The interaction energies are considerably stronger (in the range 38-108 kJ mol-I). Finally the top left corner contains the interactions among the electron-deficient molecules (LiH, BeH,, BH3) themselves. These molecules may be considered as both electron acceptors and donors, the hydride end of bonds being electron-rich and thus capable of donating electrons towards the vacant orbital on the other molecule. For the three symmetrical dimers (LiH)2, (BeH2)2 and (BH3)2, complete coalescence to a hydrogen- bridged molecule is predicted. This is, of course, found experimentally for B2H6.The interaction energies here are even stronger, being in the range 85-207 kJ mol-l. TABLE 2.-MP4SDQ/6-3 1 G * * CORRELATION ENERGY CONTRIBUTIONS TO BINDING ENERGIES OF XH, YH, (kJ mol-l) LiH BeH2 BH3 NH3 OH2 FH LiH 6 23 56 BeH2 35 71 BH3 74 NH3 4 4 5 OH2 4 4 FH 5 a Based on HF/6-31 G* geometries. The single-point MP4SDQ/6-3 1 G**//HF/6-31 G* computations can be used to assess the role of electron correlation in modifying the intermolecular binding energy. Table 2 lists the correlation energy contributions at this level; table 3 gives the corres- ponding total binding energies. (The donor-acceptor complexes of NH,, OH2, FH with LiH, BeH,, BH, have not yet been examined at this level). The first conclusion to be drawn from table 2 is that electron correlation increases the strength of hydrogen bonds by a small and rather constant amount of ca.5 kJ mol-l. The contribution of correlation to the binding of electron-deficient complexes in the top left corner, on the TABLE 3.-MP4SDQ/6-31G** BINDING ENERGIES OF XH, YH, (kJ rno1-l) a LiH BeH2 BH3 NH3 OH2 FH LiH 199 191 264 BeH2 137 203 BH3 164 OH2 27 42 FH 30 NH3 17 31 54 Based on HF/6-31G* geometrics.J . A . POPLE 15 other hand, is quite large, particularly for compounds involving boron. The largest correlation effect is for diborane (BH3)2, which can reasonably be attributed to a transition from localized electron-pair bonds in 2BH3 to delocalized three-centre bonds in B2H6 where correlation effects are likely to be larger. The net result, noted in a previous study,12 is that electron correlation contributes nearly half of the total bind- ing of two BH3 molecules.The corresponding treatment of (LiH),, on the other hand, shows that the contribution of electron correlation for this dibridged species is quite small. This is perhaps to be expected as the binding in Li2H2 is a great deal more ionic than in B&&. The above results are discussed for the partial fourth-order (SDQ) Msller-Plesset theory. It is important to know how rapidly this series converges for the intermolecular contribution to the correlation energy. This can be answered, in part, by comparing MP2, MP3 and MP4SDQ contributions to some of the binding energies. These results are listed in table 4. They show that the second-order MP2 level of theory describes effects fairly well, with some small overestimation in all four of these cases.TABLE 4.<ORRELATION ENERGY CONTRIBUTIONS TO BINDING ENERGIES AT VARIOUS ORDERS OF M0LLER-PLESSET THEORY (kJ m0l-l) ~~ molecule HF MP2 MP3 MP4SDQ (BH312 89.5 173.5 167.5 163.7 (NH3)2 13.0 18.0 17.0 16.8 (OH,), 23 .O 28.6 27.1 27.2 WHIZ 25.0 31.2 29.3 29.8 _ _ _ _ ~ ~ ~ ~ ~ ~ a Using the 6-31G** basis and HF/6-31G* geometries. 4. INTERMOLECULAR BONDING INVOLVING ACETYLENE For a long time it has been recognized that the n-electrons of unsaturated hydro- carbons may act as electron donors in intermolecular bonding. This suggestion goes back originally to Dewar.13 There have been infrared spectroscopic studies of such complexes both in solution and in low-temperature matrices.Very recently, a ro- tational spectrum of the dimer involving acetylene and hydrogen chloride has been studied by rotational As a further example of the supermolecule approach to these problems? some recent work with Janet E. Del Bene on bonding to acetylene will be discussed. The experimental data on complexes between C2H2 and hydrogen halides have all been interpreted in terms of a " T-type " dimer. This has the proton of the proton- donor molecule directly above the middle of the n-bond of acetylene. These struc- tures (for HF and HCl) have been optimized at the HF/3-21G level, leading to the dimer geometries shown in fig. 2. Calculations of second derivatives of the energy with respect to nuclear coordinates confirm that these structures are indeed local minima in the potential surface.The total distance between the midpoint of the CC bond and the chlorine atom of hydrogen chloride is found to be 3.80 A, in reqsonable agreement with the experimental value of 3.699 A obtained by Legon et aZ.14 The energies of the hydrogen bonds in these systems may be evaluated by single- point calculations at these geometries and comparing with corresponding calculations on separated molecules. The level used is MP4SDQ/6-3 1 G*//HF/3-21 G. This gives equilibrium dissociation energies of 19 kJ mol-' for C2H2 - HF and 12 kJ mo1-I for C2H2 - - HCl. If electron correlation is not included (HF/6-31G*//HF/3-21G), the16 THE LENNARD- JONES LECTURE corresponding values are 17 and 9 kJ mol-'. Clearly, even if the proton acceptor is a non-polar unsaturated molecule such as acetylene, the hydrogen-bond energy is still dominated by the Hartree-Fock contribution, electron correlation corrections adding 2-3 kJ mol-l.An interesting point to investigate is the possible role of acetylene as a proton donor in hydrogen bonds. This is a possibility in view of the known high acidity of acetylenic hydrogen. In fact, a more complete search of the C2H2 HF surface does Y O 5 C H 2.36 , . . . . . , . 19 0.94 H- F H-C-C- 21 \ 1.05 C H 20 FIG. 2.-Structures of dimers involving acetylene. reveal a second potential minimum with the molecules attached H-CEC-H HF (actually distorted from the pure linear arrangement). However, the full correlated energy calculation yields a smaller binding energy than the T-type structure 19.Hydrogen fluoride is a relatively poor proton acceptor in hydrogen bonding. A more promising alternative is to consider hydrogen bonding between acetylene and ammonia in the C,, form 21. A preliminary study of this species indicates quite a strong bonding of 16 kJ mol-l. In fact ammonia is more strongly bound to acetylene in this configuration than it is in one with an NH bond acting as proton donor towards the n-electrons of C2H2. Species such as 21 have not, to my knowledge, yet been characterized by any experimental technique and represent a challenging theoretical prediction. 5. CONCLUSION The application of ab initio quantum mechanics to intermolecular binding has clearly come a long way in recent years. I believe that the supermolecule approach, using underlying techniques which make allowance for electron correlation, will become a standard procedure and will probably replace some features of the par- titioned approach. Given such a theoretical technique, it becomes possible to predict structures and binding energies for a large number of dimeric systems ranging from those with weak dispersion-type binding to those with extremely strong bonds leading essentially to new chemical bonds. In this sense, we are moving towards a unified theory of intra- and inter-molecular binding.J . A . POPLE 17 This research was supported by the National Science Foundation (grant no. CHE 8 1-0 1 06 1-0 1). J. 0. Hirschfelder, C. F. Curtiss and R. B. Bird, Molecular Theory of Gases and Liquids (Wiley, New York, 1974). F. London, 2. Phys. Chem., Teil By 1930,11, 222. R. G. Gordon and Y . S. Kim, J. Chem. Phys., 1972,56, 3122. D. N. Chipman, J. D. Bowman and J. 0. Hirschfelder, J. Chem. Phys., 1973,59,2830. R. M. Berns and A. van der Avoird, J. Chem. Phys., 1980,72, 6107. ' J. E. Lennard-Jones, Trans. Faraday Soc., 1929, 25, 668. ' J. A. Pople, J. S. Binkley and R. Seeger, Int. J. Quant. Chem. Symp., 1976, 10, 1. * C. Mraller and M. S . Plesset, Phys. Rev., 1934,56, 618. P. C. Hariharan and J. A. Pople, Theor. Chim. Acta, 1973, 28, 213. lo R. A, Whiteside, M. J. Frisch, J. S. Binkley, D. J. DeFrees, H. B. Schlegel, K. Raghavachari and J. A. Pople, Carnegie-Mellon Quantum Chemistry Archive (Carnegie-Mellon University, Pittsburgh, 1981). l1 T. R. Dyke and J. S. Muenter, J. Chem. Phys., 1974,70,2929. l2 R. Ahlrichs, Theor. Chim. Acta, 1974, 35, 59. l3 M. J. S . Dewar, J. Chem. Soc., 1946,406. l4 A. C. Legon, P. D. Aldrich and W. H. Flygare, J. Chem. Phys., 1981, 75, 625.

ISSN:0301-7249    

DOI:10.1039/DC9827300007

出版商:RSC    

年代:1982

数据来源: RSC

摘要:

Faraday Discuss. Chem. SOC., 1982, 73, 19-31 Intermolecular Perturbation Theory for Van der Waals Molecules BY ANTHONY J. STONE AND IAN C. HAYES Department of Theoretical Chemistry, University Chemical Laboratory, Lensfield Road, Cambridge CB2 1EW Received 1st December, 1981 We report an application to some Van der Waals systems of a perturbation analysis of the energy of interaction between the component molecules. The method used differs from the procedure reported previously, in that an expansion in powers of overlap is not required and errors from basis set superposition are corrected. The total interaction energy is obtained as a sum of terms, which can be identified as electrostatic, exchange, repulsion, charge transfer, efc. The individual terms are now less dependent on basis set than before (except that an improvement in basis set is reflected in a larger polarisation energy) and the totzl energies agree satisfactorily with supermolecule calculations in which the counterpoise method is used to correct for superposition error.1. INTRODUCTION Quantum-chemical studies of Van der Waals molecules are characterised by the fact that the interaction is very small when compared to the total energies involved. Thus the potential-energy surface needs to be calculated accurately with very sophisti- cated techniques if variational methods are to recover the very small energy differences of interest. However, the smallness of the interaction suggests that this is a problem very well suited to the application of perturbation theory, which can calculate these small energy differences without regard to the magnitude of the total energies.Addi- tionally, perturbation calculations of interaction energies automatically give a de- composition of the interaction energy, a feature not shared by variational schemes. Such decompositions break the interaction down into the various terms which chemists use when describing an interaction qualitatively, and this is a valuable aid to understanding and interpretation. Unfortunately, attempts to calculate intermolecular interactions using perturbation theory encounter well-known difficulties as soon as the molecules overlap significantly.' These may be called the ' exchange problem ' and the ' overlap problem'. The exchange problem arises because the perturbation, which describes interactions between particles on one molecule and particles on the other, is not symmetric with respect to all per- mutations of identical particles. One consequence is that there is no unique way to expand the interaction energy or the wavefunction in orders of magnitude of a perturbation parameter. The overlap problem arises because the natural expansion functions for the interacting system, namely products of wavefunctions for the separated molecules, do not form an orthogonal set, and therefore cannot be the eigenfunctions of any Hermitian ' unperturbed Hamiltonian'.Nevertheless we believe that the attractions of a perturbation description of inter- molecular interactions justify efforts to overcome these difficulties. One approach, suggested by Basilevsky and Berenfeld and implemented by Stone and Er~kine,~ adopts a self-consistent field description of the molecules as the starting-point.In20 INTERMOLECULAR PERTURBATION THEORY the unperturbed problem, the intermolecular integrals are omitted from the Fock and overlap matrices and, since the unperturbed Hamiltonian can be taken as a sum of one-electron Fock operators, it is symmetric with respect to electron interchanges. The overlap problem is handled by expanding in powers of the intermolecular overlap, so that the energy is obtained as a double series in perturbation and overlap. The expansion functions are Slater determinants constructed from orbitals for the combined system obtained by symmetrically orthogonalising the SCF orbitals for the separated systems. Early applications of this method gave encouraging result^.^ It has become apparent to us, however, that it has deficiencies.The first is that the ‘ charge-transfer ’ contribution to the second-order interaction energy, which involves excitations from an occupied orbital on one fragment to a virtual on another, is very sensitive to basis. This is a manifestation of basis-set superposition error.4 The second is that the expansion in powers of intermolecular overlap does not always converge adequately, and the errors so introduced distort the potential curve at short distances. Thirdly the symmetrical orthogonalisation introduces virtual components into the wave- function in zeroth order. This paper describes our attempts to overcome these difficulties.The method is given in the next section; section 3 is an extended discussion of the results for one system, and section 4 a comparison with two other systems. 2. METHOD The unperturbed and normalised wavefunctions for the two subsystems A and B with n A and n B electrons, respectively, are single Slater determinants composed, respectively, of n A and n B orthonormal spin-orbitals. The spin-orbitals of one sub- system are not orthogonal to the spin-orbitals of the other subsystem. These orbitals are obtained by the usual Hartree-Fock self-consistent field method in the Roothaan LCAO-MO formalism with the subsystems not interacting (that is, with all inter- molecular matrix elements set to zero, as if at infinite separation). Solution of the SCF equations also gives for each subsystem a set of virtual orbitals, orthogonal to the occupied orbitals of that subsystem but not to the occupied or virtual orbitals of the other.The zeroth-order approximation to the supersystem wavefunction is taken as the Slater determinant constructed using the nA occupied orbitals of A and the n g occupied orbitals of B. The energy to first order is the expectation value of the supersystem Hamiltonian over this zeroth-order wavefunction where the cofactors S(ab . . .;de . . .) of the overlap matrix appear because of the non- orthogonality of the orbitak6 In this expression and those to follow, summations are over spin-orbitals, and summations over i, j , k and I are restricted to occupied orbitals only. In the limit of large intersystem separation and zero overlap, the ground-state SCF wavefunction for the supersystem is this single determinant.To describe the effects of the interaction to higher order, we need further determinants in which one or more of the occupied orbitals is replaced by a virtual orbital.'^^ The corrections arise from single substitutions, which give the leading correction within the one-particleA . J . STONE AND I . C. HAYES 21 Hartree-Fock description, followed by double substitutions, which give the leading correction to that one-particle description, namely the correlation energy. Higher substitutions are less important. We might seek the supersystem wavefunction in a non-orthogonal group function (NOGF) formulation by solving the secular equation det ( H - ES) = 0 (2) where H i s the matrix of the full supersystem Hamiltonian over the set of determinants, and S is the overlap matrix. While such a variational approach may be tractable for small systems or when limited to single excitations," it will become rapidly intractable with increasing subsystem size or on considering double or higher substitutions.Thus a perturbational approach is adopted, and the secular determinant is expanded to obtain an estimate of the lowest eigenvalue where subscript p labels the determinants and 0 denotes the zeroth-order determinant. The summation over p includes all levels of excitation, but only single and double excitations will be considered. We now need matrix elements of one- and two-electron operators between deter- minants composed of non-orthogonal orbitals.The expression for such a matrix element involves cofactors of the matrix of overlap integrals between the two sets of orbitals appearing in the two determinants. To obtain these, we make use of the fact that we are only considering low orders of excitation. The overlap matrices whose cofactors are required differ in only one row (single excitations) or two rows (double excitations) from the ground-state overlap matrix, so we can proceed after inverting just one matrix by expanding cofactors of the substituted matrices in terms of the cofactors of the unsubstituted determinant. Note that no approximation is required. The expression (1) for the total (electronic) energy to first order becomes, when we replace the cofactors by their expression in terms of elements of the inverse of the overlap matrix where T = S-l is the inverse of the overlap matrix between occupied orbitals.The electronic energy to first order can be separated into four parts. These are (i) the zeroth-order electronic energy of the fragments ; (ii) the electrostatic energy, which includes electron-nuclear attraction and electron-electron repulsion, and which must be combined with the intersystem nuclear-nuclear repulsion to produce the overall electrostatic energy ; (iii) the exchange energy, given by the usual expression E = -(1/2) 2 2 (ijlji); iEA j e B and (iv) the repulsion energy, which is the remainder, and which is given by the expression The repulsion energy has an explicit dependence on overlap, in contrast to the ex- change energy, which has an implicit dependence uia the two-electron integrals involved.22 INTERMOLECULAR PERTURBATION THEORY The expression for the second-order energy due to single excitations is where o denotes an occupied orbital and v a virtual orbital.constructed using T as the MO basis density matrix P is a Fock matrix Q is defined by QOV == 2 svkTko; k and N is the normalisation of the substituted determinant (9) The single-excitation second-order energy is divided into two parts depending on whether the orbitals involved are on the same or opposite fragments; these are, respectively, polarisation and charge transfer. When double excitations are considered the expressions, although more compli- cated, still involve only the MO integrals and elements of S, T and Q.In practice only the contribution from two-electron integrals has been calculated; the contribu- tion from one-electron integrals would be absent but for the non-orthogonality of the orbitals. There are now five types of contribution, identified by the two excitations involved. (i) A double excitation within one fragment. This gives the intramolecular correlation for the subsystem but modified by the presence of the other fragment, via the non-orthogonality of the orbitals. We have not yet considered how the inter- action modifies this contribution. (ii) A single excitation within each fragment. This gives rise to the Hartree-Fock dispersion energy. But since it is calculated in a scheme which takes explicit note of the overlap of the orbitals, both the expression and the numerical value of this contribution are modified. (iii) A single excitation within one fragment together with an excitation from the same fragment to the opposite fragment.This is subject to basis extension error, because such contribu- tions would also be present if a calculation for the relevant subsystem were made in the presence of the orbitals but not the electrons or nuclei of the other subsystem. (iv) A single excitation within one fragment together with an excitation from the opposite fragment to the first fragment. This is an additional dispersion or charge- transfer correlation contribution. It is not subject to basis extension error in the manner of (iii) because it involves an excitation from an occupied orbital on each subsystem. (v) A double intermolecular excitation.This could be called a double charge-transfer or exchange-dispersion contribution. Finally we must consider corrections for basis superposition error. Consider a calculation in which the orbitals of both molecules are present but the nuclei and electrons of molecule A only. The energy correct to first order in this calculation is just the zeroth-order energy of A. Of the second-order single-excitation energy, there is a charge-transfer component but not a polarisation component. However, the " unperturbed " energy of a B orbital in this calculation is higher than in the original calculation, because it is not stabilised by attraction to the nuclei of B. It is estimated as the diagonal element of the Fock matrix which is obtained only if the orbitals of A are occupied.The reduced " charge-transfer " energy obtained in this way is ascribed to basis set extension, and is therefore subtracted from the charge-transfer contribution to the interaction energy.A . J . STONE AND I . C. HAYES 23 400 300 200 100 0 - -100 0 * 2 2 2 '9 I . x fi - - - - - Of the double-excitation second-order contributions, the dispersion and charge- transfer correlation contributions do not have a basis extension component, since they require an occupied orbital on each fragment; the double charge transfer will have a basis extension component but this has not been considered. I I 1 I I 6 7 8 9 3 . ARGON AND HYDROGEN FLUORIDE This system was studied with a basis set which is conventionally described as " split-valence." The basis set for argon was the MIDI-4 basis of Sakai et aZ.ll obtained from a minimal basis set by uncontracting the valence orbital.The basis set for fluorine was the standard Dunning contraction of (9s5p) to [3s2p] l2 and that for hydrogen the standard Dunning contraction of (4s) to [2s].12 The calculations were performed at the experimental HF distance of 1.73 bohr.* 1000 500 0 -5OC -loo( 6 7 8 9 R (Ar - * * F)/bohr FIG. 1.-Total interaction energies for (a) ArHF and (b) ArFH. SCF is the supermolecule interaction energy; Ghost is that energy corrected by the counterpoise method; BB is the energy obtained by lhe perturbation theory of Basilevsky and Berenfeld [ref. (2) and (3)] ; SH is the corrected perturbation energy of this work; SH1 is the uncorrected perturbation energy of this work; and SH2 is the alter- native method of correction (overcorrection) of this work.The cross in (a) marks the minimum obtained by Hutson and Howard [ref. (14)]. Calculations were performed at various Ar-F distances in the range 6-9 bohr in the two linear configurations ArHF alid ArFH. The various total energies, which are compared in fig. 1 for the two configurations, are as follows. The SCF energy is just the usual variational interaction energy, defined as the energy of the super- * 1 bohr "N 5.29 x lo-" m.24 INTERMOLECULAR PERTURBATION THEORY molecule minus the energies of the isolated fragments. This interaction energy is expected to be too strongly attractive, because it includes the basis set extension as part of the interaction. The Ghost curve is obtained by the usual method of correct- ing for this basis extension, using the counterpoise method of Boys and Bernardi.I3 For this energy, the energies of the isolated fragments are replaced by the energies of the fragments calculated using the basis of both fragments; these energies must be lower than the energies of the isolated fragments, so that the Ghost interaction energy is less negative than the SCF energy.The energy BB is that obtained using the SCF perturbation theory of Basilevsky and Berenfeld as implemented by Stone and E r ~ k i n e . ~ No attempt is made in this scheme to allow for basis set extension. The energy SH is that calculated with the perturbation scheme described above, with the charge-transfer energy corrected for basis set extension; this energy is the sum of the first-order energy and the second-order single-excitation energy, the double-excitation energy having been omitted so that the total is directly comparable with the SCF, Ghost and BB energies. The correction described above is just one way of correcting the perturbation energy for basis-set extension ; alternative methods include that of making no correction at all, which produces the curve called SH1, and that of making the correction with the virtual orbitals at their actual eigenvalues, rather than their effective eigenvalues, which produces the curve SH2.In the ArHF configuration, the SCF minimum of -0.001 460 hartree * at an.Ar-F distance of 6.7 bohr is reduced to a minimum of -0.000 280 hartree at 7.2 bohr on considering the Ghost correction.This shift is entirely in the direction expected as basis extension provides an attractive contribution which results in too deep a potential well at too short a separation. The SH curve has the same qualitative shape as these variational curves, with a minimum of -0.000 590 hartree at 7.0 bohr, and lies every- where between the SCF and Ghost curves. Hutson and Howard,14 by fitting an analytic potential surface to experimental data, find the well-depth to be 214 cm-' (0.000 975 hartree) at an Ar-F distance of 6.5 bohr. The alternative correction to the perturbation energy produces the curve SH2 which lies close to the Ghost curve. Now this alternative correction is obtained by considering virtual orbitals to lie at the energies at which they appear in their own fragment, which are lower than their effective energies as virtual orbitals of the other fragment, and is thus an overcorrec- tion.Then, since this correction is an overcorrection, the (variational) Ghost correction must itself be an overestimate of the true basis extension contribution to the interaction. The BB energy follows the SCF curve closely at separations greater than 7.5 bohr, but then turns rapidly with a minimum near to that of the Ghost curve, followed by a very rapidly rising repulsive wall. This change occurs when the estimate of the largest eigenvalue of the overlap matrix is 0.30 and causes the BB curve to have a qualitatively different shape, although the well depth and position agree with other curves.The ArFH configuration gives a shallower SCF well of -0.000 180 hartree at 6.5 bohr. The extension effect is the main cause of this well and the Gljost curve is repulsive. The SH curve now has different behaviour from the Ghost curve, giving a shallow minimum of -0.000 040 hartree at 7.0 bohr. The continued existence of a minimum in the SH curve is consistent with the variational ghost effect overestimating the true basis extension. The BB curve follows the SCF curve until it starts to diverge seriously at 7.0 bohr, where the overlap eigenvalue estimate is only 0.03, giving a very steep repulsive wall. This overlap is still rather small and this behaviour suggests that the divergence of the BB energy is not necessarily connected with large overlap.* 1 hartree % 2625.5 kJ mol-l.A . J . STONE AND I . C. HAYES 25 We now consider the SH energy decomposition. Fig. 2 shows the decompositions which have been plotted as follows. The electrostatic energy has been plotted as “ es ” but the two other first-order terms, exchange and repulsion, have been plotted 2000 - 1 1000- c Y, I 0, 3 g o - 1000 I I I I I 6 7 8 9 300 20 0 100 0 -1oc (b) 4 I I I I I I I I I I I I I I I I I 1 I I 6 7 a 9 R (Ar - F)/bohr FIG. 2.-Energy decomposition for (a) ArHF and (b) ArFH. es is the electrostatic energy; er is the sum of the exchange and repulsion energies; ct is the charge-transfer energy, corrected for basis set extension error; and pol is the polarisation energy. SH is the sum of these four terms, double is the sum of the double-excitation contributions.as their sum “ er ”. polarisation energy. “ ct ” is the corrected charge-transfer energy and “ pol ” is the “ SH ” is the sum of these contributions SH = es + er + ct + pol (11) and ‘‘ double ” is the sum of the second-order double-excitation contributions (dis- persion, charge-transfer correlation and double charge transfer). Thus the “ SH ” curve is the same as the curve called SH in fig. 1 . The electrostatic energy, attractive in both configurations, is much larger in ArHF than in ArFH. It is the least important of the first-order contributions. The electro- static energy is expected to be more attractive for ArHF because of the position of the hydrogen atom and the fluorine lone pairs. However, by displacing the electrostatic energy curves along the R-axis they may be brought almost into alignment.This may be interpreted as plotting against the value of the overlap determinant, or alternatively the displacement required could be used to estimate the effective interaction centre (the position of the dipole) of HF. Such an interpretation places the dipole 0.4 bohr from the fluorine atom.26 INTERMOLECULAR PERTURBATION THEORY The exchange and repulsion energies have very similar dependence on separation, although the former is negative and always smaller in magnitude than the latter, which is positive. Thus we have plotted only their sum, which is the major contribution to the first-order energy, and is sufficiently positive to make the total first-order energy positive. The separation dependence of this term is not quite as short range as an exponential dependence.The curves for the two configurations may be brought into alignment if they are replotted against separation from an effective centre of the HF molecule placed 0.4 bohr along the bond from the F atom. In the BB scheme, only the electrostatic energy is first order; exchange and exchange-repulsion are regarded as second-order effects. The electrostatic and exchange energies of the present scheme are the same as those of BB. The exchange- repulsion energy of the BB scheme has a behaviour different from that of the repulsion energy considered here. In particular, it is at least one order of magnitude larger, and it is of much longer range in ArHF than in ArFH.This seems to be happening at values of the overlap which are sufficiently small that we would not expect convergence difficulties and we are forced to accept that the BB scheme may be unreliable even in regions of small overlap. It is probably a consequence of the symmetrical orthogon- alisation used in the BB scheme. The second-order contributions, charge transfer and polarisation, turn the repul- sive curves obtained at first order into curves exhibiting minima and resembling the experimental surfaces. In both configurations charge transfer from Ar to HF gives greater stabilisation; this transfer is itself much larger in ArHF than in ArFH. This direction of charge transfer in ArHF is the direction of transfer which would be expected for an Ar - HF hydrogen bond.Ar to HF transfer is expected to be more important on the basis of energy eigenvalue differences; at 6.5 bohr separation, over 85% of the Ar to HF charge-transfer energy is accounted for by the HOMO-LUMO interaction. In ArFH, eigenvalue difference is not a sufficient explanation, the accep- tor orbital being not the LUMO but a higher unoccupied orbital of FH with more favourable overlap. Charge transfer between 0 orbitals is much more important than charge transfer between z orbitals. The polarisation interaction is the smallest of those considered; it is negligible for ArFH. Consideration of the orbital contri- butions shows that the stabilisation comes, as expected, from mixing of low-lying unoccupied orbitals with high-energy occupied orbitals. Note however that the polarisation contribution will be grossly underestimated because of the limited basis set.The same is true of the dispersion and other double-excitation contributions, which we do not discuss in detail. 4. ARGON AND HYDROGEN CHLORIDE; NEON AND HYDROGEN FLUORIDE The system Ar HCI was studied with a similar split-valence basis set. The argon and hydrogen bases were as used for ArHF, whilst the basis for chlorine was taken from ref. (1 1) with the valence orbital uncontracted. The calculations were performed at the experimental HCl distance of 2.41 bohr. Results are shown in fig. 3 and 4. The total energy curves are grossly similar to those of ArHF but the wells are less deep. The minima in ArHCl occur at between 8.0 and 8.8 bohr, depending on the particular calculation, whilst for ArClH they occur at 7.5 bohr for the SCF and SH1 curves.In the latter configuration the SH curve is almost purely repulsive (it has a very shallow minimum at large separation) and the Ghost curve is purely repulsive. In both configurations the alternative correction for basis set extension produces aA . J . STONE AND I . C . HAYES 27 curve SH2 which lies close to the Ghost curve, suggesting as before that the variational Ghost correction overestimates the basis extension effect. These results may be com- pared with the depth of 180 cm-I (0.000 822 hartree) at an Ar-Cl distance of 7.6 bohr determined by Hutson and Howard by fitting a potential surface to experimental data. The smaller well-depth in our calculation can be ascribed in part to under- estimation of the dispersion and polarisation energy.7 8 9 10 6 7 8 9 R (Ar * Cl)/bohr cross in (a) marks the minimum obtained by Hutson and Howard [ref. (15)]. FIG. 3.-Total interaction energies for (a) ArHCl and (6) ArClH. Notation as for fig. 1. The The first-order energy contributions for ArHCl and ArClH show the feature that they may be brought into superposition by being shifted along the separation axis. This time the shift required is dose to 1.2 bohr, corresponding to an effective inter- action centre placed ca. 0.6 bohr from the chlorine atom along the bond towards the hydrogen atom. The distance dependence of the first-order terms is the same with HC1 as with HF. The major charge-transfer contribution is from Ar to HCl. The LUMO of the combined system is a o* orbital of HC1 which lies considerably below the 4s orbital of Ar.The HOMO are the 7t orbitals of HC1 and, at lower energy, the 3p orbitals of Ar. The highest o orbital of HCl which could interact with the Ar 4s orbital lies at lower energy than all these, so that the HOMO-LUMO separation for charge transfer from Ar to HCI is 0.727 hartree compared with 1.421 hartree for charge transfer from HCl to Ar. Such considerations suggest that Ar to HCl transfer should be more important in both configurations, as indeed it is. Polarisation of both28 INTERMOLECULAR PERTURBATION THEORY I I I I 7 8 9 10 R (Ar * Cl)/ bohr FIG. 4.-Energy decomposition for (a) ArHCl and (b) ArCIH. Notation as for fig. 2. species is negligible in the ArClH configuration.In ArHCl, polarisation of Ar is less at separations above 7 bohr than in ArHF, but becomes larger at shorter separations; polarisation of HCl is larger than of HF at all separations. The long-range co- efficient of the contribution from polarisation of Ar in ArHCl is smaller than that in ArHF because the partial positive charge of the hydrogen nucleus is smaller. The system Ne HF was studied with a split-valence basis set, with the hydrogen and fluorine bases as before, and a neon basis comparable to the fluorine basis. The results are shown in fig. 5 and 6. The NeHF configuration was also studied with an extended basis contracted to [5s4p2d] on neon and'fluorine and [3s2p] on hydrogen. The results with the split-valence basis are similar to those with the argon systems.The minima are shallower than those of the argon-hydrogen-fluoride system by a factor of three in the NeHF configuration and by a factor of two in the NeFH con- figuration. The minima are also shifted to shorter separation by 0.5 bohr in both configurations. This shift is reflected in the three first-order energy contributions; these follow curves of the same shape as before but displaced along the separation axis by 0.7 bohr. The charge-transfer contributions are not shifted by as much as this, however. Charge transfer from hydrogen fluoride is similar in both configurations to that in the ArHF system. Charge transfer from neon to hydrogen fluoride is, however, considerably less than from argon. It is the reduction in this interaction which stops the minimum from moving in as far as might be expected from the change in the first-order energy.The polarisation energies are also correspondingly smaller. All the double-excitation contributions are reduced in importance. The dispersionA . J . STONE AND I . C. HAYES 29 30 0 a, e ki v o 2 8 5 c1 c 5 - 300 -600 100 50 0 - 5C I I I I 6 7 8 9 I ?\\ I \ I * I I I I 6 7 8 9 R (Ne F)/bohr FIG. 5.-Total interaction energies for (a) NeHF and (6) NeFH with the split-valence basis. Notation as for fig. 1. energy is expected to be reduced: the C, coefficient for NeHF will be smaller than that for ArHF. The components including charge transfer are naturally of lesser importance when the charge-transfer interaction itself is reduced. The contributions calculated with the extended basis set may be compared with those calculated with the split-valence basis set.All the first-order contributions are increased in magnitude. The better description of the diffuse regions of the molecules results in the electrostatic interaction being better reproduced, because the multipoles are more accurately represented ; whilst the diffuse tails on the wavefunctions result in larger exchange and repulsion energies. Because of the way in which the exchange and repulsion energies tend to vary with each other, and produce a net repulsion which acts against the electrostatic interaction, the overall first-order interaction energy is changed less than the individual components. The total first-order energy is in- creased (less attractive) with the extended basis.The uncorrected charge-transfer energies calculated with the extended basis are less than those from the smaller basis (in marked contrast to the situation with the BB energies, where the charge transfer, as well as the exchange-repulsion, is considerably larger with the extended basis). Comparing the corrected charge-transfer energies shows that the correction calculated for the smaller basis is not large enough, and the small basis overestimates the charge-transfer contribution. The charge transfer with the extended basis is also a contribution of longer range, consistent with the more diffuse wavefunctions. The polarisation energy is considerably underestimated by30 INTERMOLECULAR PERTURBATION THEORY 6 7 8 9 R (Ne 1 I I I _I 5 7 8 9 F)/bohr FIG.6.-Energy decomposition for (a) NeHF and (b) NeFH. Notation as for fig. 2. the smaller basis, owing to its lack of polarisation functions. The net result seems to be that the second-order energy is too large in the smaller basis set: the overestimate of the charge transfer exceeds the underestimate of the polarisation energy. The double-excitation second-order energy has not been calculated in the extended basis. The smaller basis set underestimates the first-order energy, which is a repulsive contribution at the separations considered, whilst it overestimates the second-order energy, which is an attractive contribution. Consequently the potential curve cal- culated with the smaller basis lies below that obtained with the extended basis. 5 .CONCLUSIONS The perturbation scheme which we have used has successfully overcome some of the deficiencies of our earlier scheme, namely the extreme sensitivity of some of the terms to changes in basis set, and the overestimation of the charge-transfer contribu- tion arising through basis extension error. There are various possible techniques for correcting the latter effect, and we have chosen one that seems sensible, but at present there seems to be no definitive procedure. It is also possible that modifications of the perturbation scheme, for example to use Epstein-Nesbet partitioning of the Hamil- tonian rather than Mlzrller-Ple~set,~~ would improve convergence. These are matters which await further investigation. The calculations reported here underestimate the polarisation and dispersion contributions to the energy, because the basis sets used lack the necessary polarisation functions.It is likely that these effects will be satisfactorily described when a goodA . J . STONE AND I . C. HAYES 31 enough basis is used. Otherwise we may conclude that charge transfer is important in describing inert-gas-hydrogen-halide (A HX) Van der Waals molecules, and that, as one might expect, it stabilises the A - HX configuration much more than the A * * * X H one, because the major contribution arises from transfer from the occupied inert-gasp, orbital to the antibonding XH CT* orbital, which has most of its amplitude on the hydrogen. The graphs of energy against separation for the terms other than charge transfer are approximately superimposable for the A HX and A XH configurations, if the distance is measured from the inert gas to an " inter- action centre " approximately a quarter of the way along the bond from X to H, and therefore these terms do not appear to favour one configuration rather than the other. I. C . H. acknowledges financial support from the S.E.R.C. P. R. Certain and L. W. Bruch, MTP International Review of Science: Phys. Chem. Ser. I , 1972,1, 113. M. V. Basilevsky and M. M. Berenfeld, Int. J. Quantum Chem., 1972,6,23; 555; 1974,8,467. A. J. Stone and R. W. Erskine, J. Am. Chem. Soc., 1980, 102,7185. N. S. Ostlund and D. L. Merrifield, Chem. Phys. Lett., 1976, 39, 612. C. C. J. Roothaan, Rev. Mod. Phys., 1951, 23, 69. R. McWeeny and B. T. Sutcliffe, in Methods of Molecular Quantum Mechanics (Academic Press, London, 1969), p. 51. K. Fukui and H. Fujimoto, Bull. Chem. Soc. Jpn, 1968, 41, 1989. E. L. Mehler, J. Chem. Phys., 1977, 67, 2728. lo E. L. Mehler, J. Chem. Phys., 1981, 74, 6298. Y. Sakai, H. Tatewaki and S. Huzinaga, J. Comput. Chern., 1981, 2, 100. l2 T. H. Dunning, J. Chem. Phys., 1970, 53, 2823. l3 S. F. Boys and F. Bernardi, MoI. Phys., 1971, 19, 553. l4 J. M. Hutson and B. J. Howard, MoI. Phys., in press. l5 J. M. Hutson and B. J. Howard, Mol. Phys., 1981, 43, 493. l6 P. Claverie, S. Diner and J. P. Malrieu. Irtt. J. Quantum Chem., 1967, 1, 751. ' J. N. Murrell, R. Randid and D. R. Williams, Proc. R. Soc. London, Ser. A, 1965, 284, 566.

ISSN:0301-7249    

DOI:10.1039/DC9827300019

出版商:RSC    

年代:1982

数据来源: RSC

摘要:

Faraday Discuss. Chem. Soc., 1982, 73, 33-44 Mobility of the Monomers in the Van der Waals Molecule (N& Comparison with N2 Crystals BY AD VAN DER AVOIRD Institute of Theoretical Chemistry, University of Nijmegen, Toernooiveld, Nijmegen, The Netherlands Receiued 30th November, 1981 The Nz-N2 intermolecular potential has been obtained from ab initio calculations and represented in two analytic forms: a spherical expansion and a site-site potential (with different sites for each contribution to the potential). It is shown that, in the range of the Van der Waals minimum, the short-range exchange repulsion is the dominant anisotropic contribution, not the multipole-multipole interactions; this repulsion is mainly responsible for the (crossed) equilibrium configuration of the (N& dimer.Using this potential in lattice-dynamics calculations for solid N, (in the ordered u and y phases) with the harmonic and self-consistent phonon methods, yields generally very good agreement with experimental data : lattice structure, cohesion energy, translational phonon frequencies and their pressure dependence, and pressure dependence of the librational frequencies. The values of the librational frequencies and their temperature dependence are less well reproduced, however, especially going towards the orientational order-disorder, w-p phase transition; this is probably due to the larger amplitudes of the librations in the crystal and the failure of the self-consistent phonon model to deal with these. The (NJ2 dimer, for which we have made preliminary (rigid-rotor-harmonic- oscillator) calculations is floppier than the crystal. The barriers to internal rotation are rather low (20 and 40 cm-', dimer binding energy 125 cm-') and only one or two states in each internal-rotation direction correspond with " locked-in " N2 rotations (librations); the higher states will be (hindered) internal rotations.1. INTRODUCTION A Van der Waals molecule (dimer) is the smallest unit from a molecular crystal that is held together by the same forces that cause the cohesion of the crystal. Some- times (also for N2), this relation has been used to infer the equilibrium structure of Van der Waals molecules from the nearest-neighbour configurations in the crystal. This is too great a simplification, however, even when the interactions between the molecules are pairwise additive, since the surrounding of molecules in a crystal by several (nearest and further) neighbours will lead to optimum packing configurations which are generally different from the optimum dimer configuration.Still, knowing the structure and dynamical behaviour of the crystal can yield much information about the Van der Waals molecule and vice versa. The central role in such relations is, of course, played by the intermolecular potential. The nitrogen solid, as one of the simplest typical molecular crystals, has been the object of many experimental and theoretical studies [for reviews see ref. (1) and (2)]. (Hydrogen is even simpler, of course, but very atypical because of the small anisotropy in the intermolecular potential and the very large rotational constant of the H2 molecule, leading to almost free H, rotations in the solid.) Three crystal phases of solid N2 are known: the ordered a and y phases stable at low temperatures, the y phase at pressures above 3.5 kbar, and the orientationally disordered phase at temperatures above 35 K.' Many experimental data are available; of particular34 MOBILITY I N (N2)2 AND N, CRYSTALS interest here are the ~tructures,~ the cohesion energy,4 the phonon frequencies (at wavevector q = 0 from i.r.and Raman ~pectroscopy,~~~ for general q from inelastic neutron scattering )7 and the temperature and pressure (or volume) dependence of these properties (Gruneisen parameters, etc.). Several empirical N2-N2 model potentials have been proposed and it is generally believed that the calculation of the equilibrium structure and the phonon frequencies of the crystal provides a very good check on these model potentials. In particular the frequencies of the librational phonon modes in the ordered a and y phases and the conditions for the orientational order-disorder, a-p phase transition should be sensitive to the anisotropy (the orien- tational dependence) of the intermolecular potential.About the (N2)2 dimer much less is known. The only (experimental) study of this dimer (to our knowledge) has been made by Long et aL8 These authors describe the i.r. spectrum of (N2)2 in the region of the monomer stretch frequency (v,) at 2329.7 cm-l, measured at 77 K in the gas phase. This N2 stretch transition is i.r.-forbidden in the free monomer, but it becomes weakly allowed (and shifted to the red, but only by 0.1 cm-l) by the interactions in the dimer.(The same interactions lead to the collision-induced i.r. absorption from unbound N2 complexes, which is observed as a broad band in the spectrum underlying the discrete dimer peaks.) From the P and R (N,-N, end-over-end) rotational branches of this v1 transition it is concluded, via a model calculation that corrects for centrifugal distortion, that the N2-N, equilibrium distance Re is ca. 3.7 A. From the progression of the other observed side bands of the v1 transition it is deduced that the N2 monomers exhibit one libration at frequency v2 = 9.5 cm-l (a " locked-in " monomer rotation), while the higher monomer rotations in the dimer are just slightly hindered, because they have energies equal to or higher than the barrier to internal rotation.They lead to a perturbed monomer rotational S band. The barrier to internal rotation is estimated to lie at 15-30 cm-'. The equilibrium configuration of the (N2)2 dimer is not known, however, and the assignment of the spectrum is only tentative: it is based on the similarity with the N,-Ar and 0,-Ar spectra and some model calculations for the latter s y s t e m ~ . ~ ~ ' ~ From the same parallel, Long et aZ.* conclude that the (N2)2 dimer might have a T- shaped equilibrium configuration (0,-Ar and N,-Ar are found to be T-shaped), 9*10 which conclusion they support by looking at the crystal neighbour configurations and at the N2-quadrupole-N,-quadrupole interactions.(As we demonstrate below, these arguments are not conclusive, however.) In order to make theoretical predictions about the structure and dynamical properties of the N2 crystal and the (N,), Van der Waals molecule, one should know the intermolecular potential. One can try, nowadays, to extract this potential from ab initio calculations. The danger of fitting empirical potentials to the experimental data is, namely, that this fitting is usually not direct, but involves intermediate models for the dynamics. For instance, in fitting the N2-N2 potential to the phonon fre- quencies of the solid, it has been assumed that the lattice modes are harmonic. One then obtains " effective " model potentials (from the " effective " harmonic force constants) which may not describe other experimental properties well.Ab initio potentials do not have this drawback, but since the calculations are still difficult and the results contain inaccuracies, they must be checked by comparison with known experimental data. Berns and van der Avoird l 1 have calculated the N2-N, potential; Luty et a2.l2 have used it in lattice-dynamics calculations on solid a- and pN2 and directly compared the calculated properties of the crystal with measured data. Besides being a check on the calculated potential, this provides interesting information on the dynamical behaviour of solid N2 itself (e.g. about the anharmonicity of the trans- lational and librational motions of N2 in the crystal). These calculations on solid N2A .VAN DER AVOIRD 35 are briefly described in section 3. We have also started detailed calculations on the dynamics of (N,),, using the same ab initio potential (see section 2). Some prelimin- ary results are given in section 4. 2. N2-N2 POTENTIAL FROM ab initio CALCULATIONS The ab initio calculations leading to the N,-N, potential have been described in ref. (1 1) and (1 3). The following contributions have been included : (first-order) electrostatic multipole-multipole interactions, all R-', R- and R-9 terms; (second- order) induction, multipole-induced-multipole interactions, all R-* and R - lo terms ; (second-order) dispersion, induced-multipole-induced-multipole interactions, all Re6, R-8 and R-'O terms; (first-order) penetration and exchange effects due to overlap between the monomer wavefunctions.The induction terms appear to be negligibly small and, since these terms should provide the dominant three-body interactions 1 4 9 1 5 in a multimer (crystal), we expect the pairwise (molecule-molecule) potential to be a good approximation (estimated deviation from pairwise additivity mainly due to triple-dipole dispersion interactions and three-body exchange contributions : < 10% of the Van der Waals well depth). In order to express the potential and to formulate dynamical equations (section 4), we have to choose a dimer coordinate system. The geometry of the dimer is deter- mined by the vector R pointing from the centre-of-mass of molecule A to that of mole- cule B and by the vectors rA and rB defining the orientations of the monomer axes (the lengths of the vectors rA and rB are the monomer internuclear distances; these are assumed to be fixed, see below).These vectors can be expressed with all angles given either relative to an arbitrary space-fixed frame: R = (R, A) = (R, @', a,") rA = ( r A , cA) = ( r A , FA) rB = ( r B , $B) = (rJ3, @h, q&) or relative to a body-fixed frame attached to the dimer, e.g. with the z-axis lying along R and molecule B in the xz-plane, so that R = (R, 0 , O ) rA = (rA, OA, qA) rB (rB, eB, O). This body-fixed frame itself can be obtained from the space-fixed frame by rotations over three Euler angles Since the potential depends only on the internal dimer angles (OA, OB, qA), we can write, for rigid monomers: (a, P, r) = (as, OS, 4%)- V(R, $A, $B, R) V(R, eA, eB, ?A)- (1) Two different analytic representations of the ab initio potential have been given." (a) A site-site potential (i.e.a generalized atom-atom potential with the force centres shifted away from the nuclei), fitted to 36 interaction-energy values, calculated for six values of R (3 < R / A < 4.4) and six combinations of angles ( 8 ~ , BB, qA). This site-site potential has the form36 MOBILITY IN (N2)2 AND N2 CRYSTALS where r l , are the distances between sites on the molecules A and B. The positions of these sites along the N-N axes are optimized for each term in the potential (2) separately, and the parameters qi, A , B, C are determined by separate fits of the three different terms to the corresponding ab initio contributions.(b) A spherical expansion: V A B = ( 4 ~ ) ~ ' ~ 2 vLA,LB,L(R) A=A,LB,L(PA,YIB,ff) (3) LA,LB,L with the angular functions (in terms of spherical harmonics, YL,M, and 3-j coefficients) : in space-fixed and body-fixed coordinates, respectively. The first term V ~ , ~ , ~ ( R ) is just the isotropic potential, while terms with LA # 0 and/or LB # 0 are anisotropic con- tributions. It appears that, for N2-N2, terms up to L A = LB = 4 inclusive are important. The coefficients in eqn (3) can be written as uLA,LB,L(R) = o%,LB,L(R) + u E , L B , L ( R ) . The long-range contributions have been obtained directly from the multipole expanded electrostatic and dispersion energies.13 The short-range coefficients have been calculated I1 at R = 3 A from ab initio results at 105 different angle combinations (OA, OB, qA) by numerical integration (Gauss-Legendre and Gauss-Chebyshev quadrature), while they have been assumed to vary with distance as exp(-a - bR - cR2).All parameters are given in ref. (1 1). Looking at the orientational (OA, OB, qA) dependence of the ab initio potential (in both representations) we observe the following remarkable features (see fig. 1). The multipole-multipole interactions (led by the quadrupole term) are indeed most attrac- tive for a T-shaped dimer (0, = 90', OB = qA = OO), but the minimum for a shifted parallel dimer (0, = 8, z 45", qA = 0') is equally deep. The anisotropy in the dis- persion interactions, although these are always attractive, is just as important as the anisotropy in the multipole-multipole interactions, however.The dispersion term would favour a h e a r geometry (0, = 6 , = qA = oo), for given distance R, the long- range terms together would favour the T-shape or shifted parallel configurations. However, at the distance R = 4 A, which is about the equilibrium distance in the iso- tropic potential, the dominating anisotropic effect is caused by the short-range overlap effects (mainly exchange repulsion). So the exchange repulsion is most important in determining the dimer equilibrium configuration (as has been found empirically by Henderson and Ewing 9910 for the N,-Ar and 0,-Ar dimers, which are T-shaped). In the case of (N2)2, however, the requirement of minimum exchange repulsion favours the crossed structure (0, = 8, = qA = 90') or the parallel one (OA = OB = go", qA = 0').The quadrupole-quadrupole interaction is more repulsive in the latter case, so that the minimum in the potential occurs for the crossed configuration (see fig. 2). 3 . STRUCTURE AND LATTICE DYNAMICS OF SOLID N2 Performing lattice summations with the ab initio N2-N, potential in its site-site representation, Luty et a1.12 have calculated the crystal equilibrium structure (i.e.A . VAN DER AVOIRD 37 optimized the lattice constants within the given space group Pa3 for cc nitrogen with 4 molecules in the unit cell and P4,lmnm for y nitrogen with 2 molecules in the unit cell), by minimizing the total internal energy (or, in the self-consistent phonon model, the Gibbs free energy at given temperature and pressure). Nearest-neighbour pairs in the crystal correspond to dimer configurations: 8A = 90°, 0, = 35" and qA = 55" in the a phase (fixed by symmetry) and OA = 90", 0 , = 42" and qA = 90" in the y phase.So the nearest-neighbour configurations, which apparently lead to a minimum in the total lattice energy, do not correspond to the energy minimum for each in- 3.0 2.5 2 .o 1.5 1 .o 0.5 rl I - E 0.0 M 2 -0.5 -1.0 -1.5 -2.0 -2.5 r- i I....- FIG. 1 .-Orientational dependence of different long-range (multipole) and short-range (exchange and penetration) contributions to the Nz-N2 potential at R = 4 A; (-) spherical expansion (3) and (- - -) site-site (2) representations of the potential." L dividual nearest-neighbour pair. The dimer binding energy for the angles found in the solid is ca.75% of the maximum dimer binding energy at 0, = 0 , = vA = 90". After optimizing the structure, lattice dynamics calculations have been carried out l2 by the harmonic l6 and self-consistent phonon l7 models. The self-consistent phonon method had originally been developed for light noble-gas (i.e. atomic)38 MOBILITY I N (N2)2 AND N2 CRYSTALS -0.5 r( I - 0 * E s - 4 -1.0 Q -1.5 -2.0 I /f I I FIG. 2.-Orientational dependence of the Van der Waals well depth, A E m i n , and equilibrium distance, Rmln, in N2-N2; @ab inirio results, ( - ) spherical expansion (3) and ( - - - ) site-site (2) represen- tations." crystals (He in particular) with a large zero-point vibrational energy." It corrects the lattice modes for the effects of anharmonicity in the potential by using effective force constants which are derived by minimizing the first-order expression for the Gibbs (or Helmholtz) free energy of the crystal (i.e.the canonical ensemble average of the " exact " anharmonic potential over harmonic oscillator states). This minimiz- ation leads to an expression for the effective force constants : the second derivatives of the potential, Fourier transformed for a given wavevector q, averaged over the mole- cular displacements. Also this average is a canonical ensemble mean value, i.e. one has to take quantum-mechanical expectation values (for all wavevectors q) weighted by Boltzmann factors (for a given temperature T ) ; in practice, this is performed via the displacement-displacement correlation function. Since this correlation function depends on the lattice vibration functions (the phonon eigenvectors) and thereby on the effective force constants again, the calculation has to be carried out self-con- sistently.In each self-consistent phonon cycle also the structure of the crystal has been optimized l2 by minimizing the free energy; after convergence this results in the average lattice structure at given temperature and pressure. Also the phonon fre- quencies, and all thermodynamic properties derived via the statistical partition func- tion, become functions of T and p . Wasiutynski l8 has generalized the self-consistentA . VAN DER AVOIRD 39 phonon formalism to molecular crystals, by dealing explicitly with the librational phonon modes, making the assumption that these modes have small amplitudes, however.Luty et aZ.12 have used the computer program written by him. Results for the c( phase, where most experimental data are available, are shown in table 1 ; the y phase results are similar. The increase of the phonon frequencies (at q = 0) with pressure can be observed in fig. 3. It appears that the calculated results TABLE l.--cc-Nz CRYSTAL DATA AT ZERO PRESSURE AND T = 15 K calculated harmonic self-consistent experimental phonon iat t ice constant/A cohesion energy / kJ mo1-1 6.43 5.61 1 (at T = 0 K, / cm-' 536 including zero-point / K 772 motions) (0,090) phonon frequencics/cm - 42.4 52.9 77.7 A , 52.8 translational T, 52.6 vibrations 1:: 58.9 78.8 34.9 46.4 59.1 64.4 72.3 37.1 39.2 77.6 58.1 61 .O M ( d a , nla, 0) mixed (44 44 n/a) librations 23 5.763 6.48 540 778 41.1 50.7 73.7 49.2 49.0 54.8 73.3 32.7 43.8 55.8 60.4 67.6 34.7 36.5 72.3 55.2 58.4 5.644 6.92 577 83 1 32.3 36.3 59.7 46.8 48.4 54.0 69.4 27.8 37.9 46.8 54.9 62.5 33.9 34.7 68.6 43.6 47.2 Results from ref.(12), harmonic values with potential B (best site-site potential), self-consistent Results from ref. phonon corrections with potential A (slightly simplified site-site potential). (4) and (7). [lattice constants, cohesion energy, phonon frequencies and their pressure dependence (or volume dependence: Griineisen parameters)] are in good agreement with the measured data. This is the more satisfactory if one considers that no adjustments of the potential to the experimental data have been made. [The usual (semi-)empirical potentials are fitted to these data.] Looking in particular at the dynamical properties, one observes that the phonon frequencies which correspond to translational vibr- ations of the N2 molecules in the crystal are in almost perfect agreement with the neutron scattering results (only 2 cm-l too high on the average); the correction for anharmonicity by the self-consistent phonon method is essential to reach this agree-40 MOBILITY I N (N2)2 A N D N2 CRYSTALS 140 12c 100 3 80 9 \ 3 60 4 c 20 g-phase 1 y-phase 0 2 4 6 0 p ikbar FIG.3.-Pressure dependence of the phonon frequencies (for q = 0) in solid N2, CI and y phases, calculated l2 by the self-consistent phonon method at T -- 12 K. Solid lines: librational modes; dashed lines : translational modes. ment (the harmonic frequencies are 6 cm-I too high).For the pure librational modes the agreement is less good; these are too high by 12 cm-l, on the average, although the self-consistent phonon corrections work in the right direction (harmonic frequencies are 14.5 cm-l too high). These corrections are too small, however, which must probably be ascribed to the fact that the self-consistent phonon method,17 in particular its generalization to molecular crystals,18 does not sufficiently correct for anharmoni- city in the case of the librations, which actually have rather large amplitudes (ca. 15') even at low temperature. At higher temperatures close to the ct--p phase transition [remember that in the p phase the librations pass into (hindered) rotations], the self- consistent phonon method fails even more in following the observed softening of the librations (see fig.4 for a typical librational mode). Of course, it may also be that the orientational dependence of the potential is less accurate than its R-dependence (which should be almost perfect, judged by the values of the translational mode frequencies), but the agreement of several calculated properties (e.g. the Griineisen parameters, also for the librational modes) and the theoretical failure of the self- consistent phonon method to account for larger amplitude librations, point to the conclusion that it is the dynamical (self-consistent phonon) model which must be refined in the first place, at least for the librations. We expect that the calculationsA . VAN DER AVOIRD 41 0.0 - 1 .o r( I E 2 3 -4 -2 .o 0 10 20 30 40 TIK FIG.4.-Temperature dependence of the Eg librational frequency l2 in a-Nz; A u = u ( T ) - c o ( T = O ) . on the (N2)2 dimer, where we apply a dynamical model that correctly deals with large- amplitude librations and hindered rotations, using the full (anharmonic) anisotropic potential, will throw more light on this problem. 4. STRUCTURE AND DYNAMICS OF (N2)2 As we have seen in section 2, the ab initio calculations predict that the equilibrium configuration of the (N2)2 dimer is a crossed one (6, = OB = q, = 90") with Re = 3.5 A and binding energy AE = 1.5 kJ mo1-I = 125 cm-' = 180 K. The barriers to internal rotation are rather low: ca. 20 cm-' for a rotation over qA through the parallel structure (with 0, = 8, = 90"; yA = 0" and almost the same Re = 3.6 A); ca.40 cm-' for a rotation over 0, or 8, through the T-shaped structure (0, = 90"; OB = qA = 0"; Re = 4.2 A). These results are in good agreement with the quan- tities obtained from the i.r. spectrum:* Re z 3.7 8, and a barrier to internal rotation of 15-30 cm-' (in one direction), even though the equilibrium configuration is different from the T-shaped structure predicted by Long et aL8 According to our calculations a T-shaped structure would lead to a much larger equilibrium distance: Re = 4.2 A. Knowing the complete anisotropic N2-N2 potential [see section 2, the spherical expansion (3) is especially convenient here], we can solve the dynamical problem for the dimer, assuming that the N2 molecules can be considered as rigid rotors (length ro). Actually the N2 stretching frequency (2329.7 crn-') is so much higher than the dimer vibrational frequencies (see below) that this approximation should hold very well.The hamiltonian for the dimer then becomes, in space-fixed coordinates:42 MOBILITY IN (N2)2 AND N2 CRYSTALS where 1, jA and j, are the angular momentum operators associated with the angles R, and 3,, respectively, and pd = m and p = m/2 are the dimer and monomer reduced masses (m is the nitrogen nuclear mass). One can solve for the bound states of this hamiltonian by the close-coupling secular-equation method, formulated for atom- diatom systems by Le Roy et d . 1 9 For a diatom-diatom dimer one has to use a basis : The expressions in parentheses ( . . . ) are Clebsch-Gordan coefficients.The radial part of the problem, i.e. the generation of the (numerical) basis functions xn,(R) from a pseudo-diatomic problem with the potential V(R), which may be the isotropic poten- tial U , , ~ , ~ ( R ) , and the calculation of the radial matrix elements, is just the same as for the atom-diatom problem l9 [using the spherical expansion (3) for the anisotropic potential]. The angular matrix elements lead to generalized Percival-Seaton 2o coefficients, which contain 9-j and 69 symbols. It is more convenient to use the body-fixed coordinate system, however, and to transform the hamiltonian (5) into the form where the angular momentum vectors J, jA and jB are now associated with the body- fixed * angles (u,p), (eA,vA) and (OB,qB) (see section 2).J is the overall angular momentum; one should remember that the components of J with respect to the body- fixed frame obey the converse of the usual angular-momentum commutation relations. For this hamiltonian (7) one can use the basis where Di,MI are symmetric top eigenfunctions. The angular matrix elements with the potential (3) are simpler (although they still contain a 9-j symbol). In order to get an initial idea about the positions of the vibrational levels in the (N2)2 dimer, we have solved for the eigenstates of the body-fixed hamiltonian (7) in the rigid-rotor-harmonic-oscillator approximation. Neglecting the rotation-vibration coupling terms J*jA and J*j, and assuming infinitesimal displacements from the equilibrium coordinates Re, SAe, BBe, (vA - qB)e, the body-fixed hamiltonian (7) easily yields the expressions for the Wilson G matrix 21 in terms of internal-displace- ment coordinates R, OA, BB and (qA - vB).[This G matrix can also be derived by the methods of ref. (21), using the Eckart conditions, but this is much more elaborate.] The force-constant matrix F, which contains all second derivatives of the potential (3) with respect to R, BA, OB and (pA - qB), has been derived using formulae from ref. (22). Instead of the spherical potential (3), we have also used the site-site potential (2) and derived the expressions for the F and G matrices in terms of the atom-atom * Actually, this coordinate system is not completely body-fixed. The third Euler angle (7 = pi, section 2), which should rotate molecule B into the xz plane, is not used.Instead, one uses pB as an “internal” angle. One must realize that the motions with pA = pB are in fact overall rotations, however, while the internal angle is qA - pe. This distinction is usually not made explicitly in the literature.A . VAN DER AVOIRD 43 distances rt j . [Note that these are different from the site-site distances; the methods of ref. (21) can be generalized to deal with this problem, however.] The results of both rigid-rotor-harmonic-oscillator calculations are listed in table 2. TABLE 2.-HARMONIC FREQUENCIES FOR (N& Equilibrium configuration OA = OB = qA - qB = 90" (DZd symmetry). v/cm - normal symmetry v/cm-' spherical coordinate site-site potential (2) potential (3) (Re = 3.6A) (Re = 3 .5 4 ~ ~~~ ~ r A + rB A1 2329.7 a r A - rB BZ 2329.7 R A1 38.9 39.2 9A9 9 B E 27.7 22.1 '?A - '?B Bl 15.2 13.9 a Experimental monomer value,* assumed to be unchanged for dimer. Although we do not expect the rigid-rotor-harmonic-oscillator model to be valid for a floppy molecule such as N2-N,, we can still draw the following conclusions. (Henderson and Ewing9Sl0 have shown in their calculations for N,-Ar and 0,-Ar that the harmonic-oscillator model is not too bad an approximation for the lowest levels.) In the first place, it is reassuring to observe that the two different analytic representations of the ab irzitiu potential produce frequencies which are reasonably close to each other. These frequencies are clearly lower than the frequencies of the translational and librational phonon modes in the N2 crystal (table l), so we may con- clude that the Van der Waals molecule (N2), is floppier than solid N,.Just as for the crystal (cf. table I), we expect that the anharmonic effects will further lower the frequencies in the dimer. For the internal rotations in the pdirection (barrier z 20 cm-l) we can expect just one or two librational states (" locked-in " N2 rotations): the ground state at ca. 7 cm-l from the bottom of the well and possibly another state near the top of the barrier (fundamental transition frequency 5 14 cm-'), whereas all the higher states are (hindered) internal rotations. For the internal rotations in the OA and 8, directions (barrier ca. 40 cm-') we expect a similar picture with higher frequencies: zero-point energy ca.12 cm-l, fundamental transition frequency 5 25 cm-l. The bending/internal-rotation coordinates S,, OB will probably mix rather strongly with the stretching coordinate R, because of the significant shift in Re when going from the crossed equilibrium structure to the saddle point in the T-shaped configuration. Generally, the picture obtained from the calculations agrees well with the type of internal motions deduced from the (N2)2 i.r. spectrum by Long et a1.' A more detailed interpretation of the spectrum can be made when we have solved the complete dynamical problem in the (N2), dimer in the close-coupling secular-equation formalism. Looking at the dimer properties for non-zero temperature will also be useful for understanding the dynamics of the solid.At very low temperatures only the libra- tional states in the dimer will be populated; at higher temperatures an increasing fraction of dimers will become orientationally disordered with (hindered) internal N2 rotations, The transition should resemble the a-p phase transition in the solid, although it will probably occur at lower temperature.44 MOBILITY I N (N2)2 AND N2 CRYSTALS I thank Drs Fred Mulder, Rut Berns and Tadeusz Luty for their contributions to the calculation of the N2-N2 potential and to the lattice dynamics calculations on solid N2, Maarten Claessens and The0 van der Lee for their assistance with the rigid-rotor- harmonic-oscillator calculations on (N2)2 using the site-site potential, and Drs Paul Wormer, Jonathan Tennyson, Brian Sutcliffe and Wim Briels for useful discussions. T. A. Scott, Phys. Rep. C, 1976, 27, 89. J. C. Raich and N. S. Gillis, J. Chem. Phys., 1977, 66, 846. R. L. Mills and A. F. Schuch, Phys. Rev. Lett., 1969, 23, 1154; A. F. Schuch and R. L. Mills, J. Chem. Phys., 1970,52, 6000. B. C. Kohin, J. Chem. Phys., 1960, 33, 882. M. M. Thiery and D. Fabre, Mol. Phys., 1976, 32, 257. F. Fondere, J. Obriot, Ph. Marteau, M. Allavena and H. Chakroun, J. Chem. Phys., 1981, 74, 2675. C. A. Long, G . Henderson and G, E. Ewing, Chem. Phys., 1973,2,485, G. Henderson and G. E. Ewing, Mol. Phys., 1974, 27, 903. ’ I. K. Kjemsand G. Dolling, Phys. Rev. B, 1975, 11, 1639. lo G. Henderson and G. E. Ewing, J. Chem. Phys., 1973, 59, 2280. l1 R. M. Berns and A. van der Avoird, J. Chem. Phys., 1980,72, 6107. l2 T. Luty, A. van der Avoird and R. M. Berns, J. Chem. Phys., 1980,73,5305. l3 F. Mulder, G. van Dijk and A. van der Avoird, Mol. Phys., 1980, 39, 407. l4 A. Bayer, A. Karpfen and P. Schuster, Chem. Phys. Lett., 1979, 67, 369. l5 E. Clementi, W. Kolos, G. C. Lie and G. Ranghino, Int. J. Quantum Chem., 1980, 17, 377. l6 G. Venkataraman and V. S . Sahni, Rev. Mod. Phys., 1970, 42, 409. ” N. R. Werthamer, in Rare-gas Solids, ed. M. L. Klein and J. A. Venables (Academic Press, l8 T. Wasiutynski, Phys. Stat. Solidi, 1976, B76, 175. l9 R. J. Le Roy and J. S. Carley, Adv. Chem. Phys., 1980,42, 353. 2o I. C. Percival and M. J Seaton, Proc. Camb. Philos. Soc., 1957, 53, 654 21 E. B. Wilson, J. C . Decius and P. C. Cross, Molecular Vibrations (McGraw Hill, New York, 22 D. A. Varshalovich, A. N. Moskalev and V. K. Chersonski, Quantum Theory of Angular London, 1976), vol. I, p. 265. 1955), Momentum (Nauka, Leningrad, 1975).

ISSN:0301-7249    

DOI:10.1039/DC9827300033

出版商:RSC    

年代:1982

数据来源: RSC

摘要:

Faraday Discuss. Chem. Sue., 1982, 73, 45-62 An Intermolecular Potential-energy Surface for (HF)2 BY ANDREW E. BARTON AND BRIAN J. HOWARD Physical Chemistry Laboratory, Oxford University, South Parks Road, Oxford OX1 3QZ Received 18th January, 1982 Molecular-beam spectroscopic data on (HF);? are used to determine the HF * * * HF intermolecular potential-energy surface. The method used in based on the BOARS approximation of Holmgren et al. (J. Chern. Phys., 1977, 67,4414), which is extended to diatom-diatom weakly bound complexes. Molecular constants are calculated for several proposed potential-energy surfaces and compared with experiment. The optimised potential surface is shown to be similar to those derived from ab initio calculations. It is believed accurately to represent the shape of the true surface in the region of the potential minimum and along the path of the tunnelling motion; further information is required before an accurate well depth may be obtained.The molecular-beam radiofrequency and microwave spectra of atom-diatom Van der Waals complexes 1*2 have provided valuable information on the intermolecular potential-energy surface between rare-gas atoms and hydrogen halide molecule^.^ In such complexes the potential surface is basically a function of just two variables, the distance R between the atom'and the centre of mass of the diatom and 0, the angle between the intermolecular axis R and the diatom bond axis; the experimental data are insensitive to the dependence of the potential on the bond length of the diatom, principally because of the high frequency of the vibrational motion of the hydrogen halide compared to that of the other motions within the complex.In this paper we extend the work to diatom-diatom complexes and determine a potential surface for the weakly hydrogen-bonded complex (HF),. In this case the intermolecular potential is a function of four internal coordinates (neglecting the HF bond lengths). This doubling of the degrees of freedom These are defined in fig. 1 . FIG. 1.-Coordinate system for (HF),. adds greatly to the complexity of the potential function to be used and it has been necessary to base our potential on the results of earlier investigations discussed below. Hydrogen bonding is a widely studied interaction both experimentally and the~retically.~ Much of the information about the nature of the binding is obscured, however, because the majority of the studies have been performed in the condensed phases where solvent and lattice effects also exist.In the gas phase relatively few46 P . E . SURFACE FOR (HF)2 studies exist, for example ref. (5)-(8); these are largely restricted to a determination of the structure and an analysis of the harmonic force constants. However, more information is available for (HF),. The microwave and radiofrequency spectra were obtained by Dyke et al.,9 who showed that the dimer was an asymmetric top with an unusual tunnelling motion between the two equivalent hydrogen-bonded structures More recently detailed spectroscopic information on both the K = 0 and K = 1 states of the complex has been obtained lo and these data will be used to determine the potential-energy surface ( K is the angular momentum about the intermolecular axis).In addition a non-linear hydrogen bond is suggested from a crude analysis of the data. Smith has obtained an analysis of the dissociation energy of the dimer (25 4 kJ mol-l) from the temperature dependence of the infrared spectrum, but this estimate may well be too high, an observation which has been made for atom- diatom c~mplexes.'~~'~ Most of the early studies concentrated on the determination of the equilibrium structure of the dimer and the binding energy. A list of these studies is given by P0p1e.l~ Many of these studies have also assumed a linear hydrogen bond for the complex [e.g. ref. (15)], but a full geometry optimisation has indicated the presence of a non-linear hydrogen bond.16 These studies have provided a wide range of equilibrium geometries (0, = 30-80"; 0, = 0-30"; R = 2.5-2.8 A) and dissociation energies (12-33 kJ mol-l).This is largely the outcome of not using Hartree-Fock quality wavefunctions and is a consequence of basis-set superposition errors." Van Duijneveldt has shown that the range of values given above may be significantly reduced by a careful consideration of these errors.'' The effects of electron correlation are also ~ignificant.'~ A decomposition analysis of the binding energy by Morokuma 2o has shown that the multipole (electrostatic) contribution to the energy dominates that of exchange and charge-transfer effects and is largely responsible for the equilibrium angular geometry of the complex.21 Other workers have found similar results using high-quality basis sets.22 Several potential-energy surfaces have been derived from fits to these ab initio calculations.About fifty points are required for even the simplest characterisation of the surface, but the inclusion of more points can be prohibitively expensive especially for calculations with larger basis sets. Yarkony et aZ.23 have mapped out a surface from calculations using a (9s 5p ld/4s lp) basis set and have fitted the points to several simple functional forms of the surface. These ab initio calculations have been used by other workers in studies of the liquid-phase24 and vibrational relax- a t i ~ n . , ~ Both sets of studies have yielded quantities calculated from the potential surface which agree tolerably well with experiment.Other potential surfaces have been derived from different sets of calculations,26-28 but the experimental consequences have not been fully analysed. In this paper we shall use the surface of Alexander and De Pristo 29 derived from Yarkony's ab initio calculation as a starting point for the (HF), potential surface. Many ab initio calculations of (HF), have been undertaken. MOLECULAR-BEAM SPECTRA In order to make use of the spectroscopic constants and to make an accurate comparison between calculated and experimental quantities, it is necessary to reviewA . E . BARTON A N D B . J . HOWARD 47 briefly the spectroscopic parameters derived for (HF),. This information is different from that available for atom-diatom c o m p l e x e ~ .~ * ~ ~ First (HF)2 exhibits the tunnel- ling motion, mentioned earlier, which splits all rotational levels. In addition, spectra for (HF), have been observed in states with non-zero angular momentum, K, about the intermolecular axis. Because a change in K is a major perturbation to the system and results in a significant change in the molecular constants, states of different K are best treated independently. For each value of K, the rotational energy levels are accurately represented by the expression 'F 3{v, + S,[J(J + 1) - K21) (2) where v represents the tunnelling frequency and the symbols 5 label the symmetry of the tunnelling ~ t a t e s . ~ The parameter 6 represents the J-dependent centrifugal distortion of the tunnelling frequency, or alternatively the difference in rotational constants in the two tunnelling states; it is small compared to the tunnelling frequency.(B + C)/2 is the measured rotational constant of the near-prolate asymmetric top and DJ its centrifugal distortion parameter. Separate values of (B + C)/2, v and DJ for each K have been defined since the distortion of these para- meters upon changing the angular momentum K is not negligible 31 and cannot be treated as a perturbation, see below. To eqn (2) we must also add the effects of asymmetry doubling in states with K st 0. Thus for K = 1 we should add 1 B - c * AE' = & [ (7-) J(J + 1) + U2(J + 1)2 . (3) The resultant spectroscopic parameters for (HF)2 are given in table 1.In addition to the rotational parameters mentioned above, Stark measurements have provided the values of the electric dipole in the K = 0 and K = 1 states: paK=' = 2.988 63 (10) D and pUK=l = 2.836 84 (20) D.* These quantities are predominantly the expectation values of the sum of the vector components of the dipole moments of the two HF molecules along the a-inertial axis, i.e. pu,,[(cos 0,) + (cos O,)]. Because of the large contribution of the induced moments and possible charge transfer, care must be taken when attempting to obtain useful structural information from these quantities. However, the contributions to the induced moments from the dipole and quadrupole moments of the neighbouring molecule are calculated to be fairly constant over a fairly large range of angular geometry.For example for O1 = 45-65' and 62 = 0-20', we estimate an induced moment of 0.30 & 0.05 D. This yields a value of (cos 8,) + (cos 0,) of ca. 1.47 (3). The constancy of @induced with respect to change in angular geometry suggests that Ap = paK=O - ,uUK=l is due almost entirely to changes in the vector components of the permanent moments of H F resulting from the change in K. A simple harmonic model suggests that an increase of K from 0 to 1 produces an increase in 0, and O2 of ca. 5 O and IApinducedI < 0.01 D. The sources of information on the shape of the potential surface are as follows: (a) Rotation constant (B + C)/2. This is dominated by the state moment of inertia of the dimer and is a reliable measure of the equilibrium intermolecular distance Re.(b) Dipok moment p. This is determined by the angular geometry of the complex. * D % 3.3356 x C m.48 P . E . SURFACE FOR (HF);! Providing proper account is taken of induction and charge-transfer contributions this provides information on the equilibrium values of O1 and 02. (c) Centrifugal distortion of the dipole moment Ap = ,uo - pr. As mentioned earlier this is dominated by the changes in angular coordinates O1 and O2 resulting from the large centrifugal forces owing to rotation about the a-axis. This provides information on the angular curvature of the potential near the minimum. TABLE 1 .-CALCULATED AND OBSERVED SPECTROSCOPIC CONSTANTS FOR (HF)2 this work Alexander O experimental 6494.9 6532 37 ( 1 7 ) 0.0595 0.061 2 96.2 91.4 - 0.658 (+0.005) (-3.9) 1.068 2.6173 0.1516 0.3714 (- 2.5) 680 1.2 6822 21 0.0665 0.0662 95.5 94.6 - 0.393 (-3.2) 1.362 3.0472 0.4041 (-2.1) (- 0.059) 6494.963 (36) 6528.907 (30) 33.944 (47) 0.059 5(1) I' - 95.207 6(14) 91.189 6(18) 0.002 393 (12) 0.658 697 (3) -4.033 (14) -4.523 8(10) 1.064 286 8(4) 2.988 6(1) 0.151 8(2) a Ref.(29). pseudo CT term. Ref. (10). All values in MHz unless otherwise stated. * ca. 0.60 D is due to the P ( K = 0 ) - p ( ~ = 1). Observed - calculated. (d) Centrgugal distortion constants Dj and DjK. Dj is a measure of how easily the hydrogen bond may be stretched by centrifugal forces and is thus largely deter- mined by the radial curvature of the potential near the minimum. is dominated by the change in intermolecular distance occurring when O1 and O2 are increased owing to the large centrifugal forces resulting from rotation about the a-axis.This provides information on the coupling of the bending and stretching degrees of freedom. (e) TunneZling splittings vo and vl. Quantum-mechanical tunnelling is very sensi- tive to the height, width and shape of the barrier penetrated. Since these quantities are measured in both K = 0 and 1, it is expected that the shape of the potential along the path for tunnelling and in particular the barrier height can be accurately determined. For a known intermolecular distance this is determined by the equilibrium angular structure and to some extent the amplitude of the bending motions. This is expected to reinforce the information mentioned above on the equilibrium geometry and the bending force constants.In all a total of nine separate pieces of information can be used to determine the potential. Most of the information is on the shape of the potential around the ( g ) The asymmetry splitting (B - C).A . E . BARTON A N D B . J . HOWARD 49 minimum in the region sampled by the zero-point vibrational motion. ation is available on the absolute depth of the potential well. Little inform- THEORY The approach used is similar to that used for atom-diatom c o m p l e ~ e s . ~ ~ . ~ ~ ~ We also make the approximation that there is no coupling between the low-frequency vibrations and rotations of the dimer and the stretching vibrations of the diatoms, which are at a very much higher frequency. This in effect treats the diatomic mole- cules as rigid rotors.All molecular constants and potential parameters should be considered as averages over the zero-point vibrational motion of the diatom molecules. HAMILTONIAN Following Dyke et ~ 1 . ~ we write the Hamiltonian for a diatom-diatom complex such as (HF), in terms of a body-fixed axis system: where p is the reduced mass of the complex, which is m H F / 2 ; b, and b, are the rotational constants of the diatoms in the dimer, which will differ little from b H F , the rotational constant of free HF in its ground vibrational state; J i s the total angular momentum of the complex and j, and j2 are the angular momenta of the two HF molecules measured with respect to the molecule-fixed axis (the z-axis is aligned along the inter- molecular vector R).The terms in eqn (4) represent, respectively, the end-over-end rotational kinetic energy of the complex, the vibrational kinetic energy associated with the stretching of the hydrogen bond and finally that associated with the bending of the hydrogen bond. The last term is the intermolecular potential expressed as a function of the internal coordinates defined in fig. 1. Since the potential must be invariant to a complete rotation of an HF molecule by 27c, it is convenient to expand the potential as a sum of products of spherical har- monics which form a suitable complete set of angular functions. Thus v ( R A 0 2 9 ~ ) = 2 V1,grn(R)C1rn(@l~l)C1* - m ( 0 2 ~ 2 ) ( 5 ) Il'm where Clrn(@,9) is the reduced spherical harmonic [4n/(21 + l)]% Yim(@,y) and V,,t,(R) is a suitable function of the intermolecular distance.Polar coordinates 9, and v2 have been included explicitly but V is only a function of 9 = 9, - q2. Because of the symmetry of the potential to tunnelling (equivalent to permuting the two mole- cules) eqn (5) can be ~implified.~ The most general intermolecular potential is then V(R,@17@2,~) = 2 2 v11',(R)(Eclrn(@191)C~'-rn(@2~2) + Ci - r n ( @ 1 9 1 ) c i f m ( @ 2 ~ 2 ) I l'<I m a 0 The form of the function Vrlprn(R) will be discussed later. Eqn (4) may be expanded and conveniently rewritten as50 P . E. SURFACE FOR (HF)2 where b(R) = bHF + h2/2pR2. The only good quantum number of the system is the total angular momentum J, but for a complex like (HF)2 with a strongly anisotropic angular potential the projection K of J along the intermolecular axis R is also nearly conserved.CALCULATION OF SPECTROSCOPIC PARAMETERS The solution of the Schrodinger equation in many variables is often most con- veniently obtained by the secular determinant method 35 using a set of product basis functions which span the space of each of the variables involved. However, for a Hamiltonian like that in eqn (7) with three rotational and four internal vibrational variables this can prove computationally very expensive. A far more efficient pro- cedure is to extend the BOARS (Born-Oppenheimer angular radial separation) method of Holmgren et a1.32933 used for atom-diatom complexes, to the case of diatom-diatom complexes. In this we seek an approximate vibrational wave function in which the bending and stretching degrees of freedom are separated in the following manner W b s ( R , 0 l d % , d = Vb(0190299 ;R)xbs(R) (8) where the (Vb(01,02,$9 ;R)} are bending wavefunctions depending parametrically on R.This is analogous to the Born-Oppenheimer separation of electronic and nuclear degrees of freedom in normal m01ecules.~~ Because the molecular-energy levels are represented as a power series in J(J + 1) and because the spectroscopic observables are J-independent quantities, it is convenient to deal initially with a problem that is independent of J and to calculate all J-dependence by perturbation theory. As mentioned earlier, rotation about the intermolecular axis may not be treated as a perturbation and the corresponding quantum number K will be included explicitly.We now define the bending wavefunctions q$(0,,0,,p;R) to be eigenfunctions of the J-independent fixed-R Hamiltonian, Hb, i.e. In the absence of the potential, the solution to the above equation will be essentially that of two free rotors whose wavefunctions are spherical harmonics. It is thus reasonable to expand the bending wavefunction as a sum of products of spherical harmonics VW1,02,V $1 = c CfXR) YJlkl(elY1) YJ2k2(02V2) (10) i where K = kl + k2 and where the symbol j is used to represent all possible quantum numbers jlklJ2k2. The appearance of the coordinates q1 and y2 and not just 9 = q1 - q2 is analogous to the use of redundant angular coordinates in treating electronic angular momenta in linear molecules 37 but this need not worry us as long as we ensure that J, = j l z + j 2 , everywhere. The number of angular basis functions required is very much greater than for the corresponding atom-diatom case,3o mainly because of the increased number of degrees of freedom.However, the number of basis functions used in the problem may be substantially reduced by taking into account the symmetry of the system under the tunnelling motion. Dyke et aL9 have shown that a proper consideration of the per- mutation-inversion symmetry group of (HF)2 leads to an approximate identification ofA . E. BARTON AND B . J . HOWARD 51 the simultaneous permutation-inversion operator Pr2 with the symmetry of the tunnelling motion. Under this operator we obtain (1 1) pi% Y j k ( O l q l ) - yjk(7t - @29q2) = (- l ) j e k yjk(82q2) and similarly for Y j k ( 6 2 q 2 ) .This operator also affects the rotational coordinates. If spherical-top basis functions 9$@K (a’py) are used,34 the effect of the permutation- inversion operator is As a result the symmetrised vibrational functions which multiply 9 $ K ( ~ p y ) are qFf(@,,82,yl;R) = 2 C,Kf (R>E YJlk1(@191) Yj2k2(022032) .i L!z (- 1 ) j 1 - j 2 Y j z k 2 ( 4 9 1 ) Y J l k l ( Q 2 Y 2 ) I (13) where The bending eqn (9) is now solved independently for each symmetry & and for each value of K ; only values of K > 0 need be considered since reflection symmetry ensures degeneracy of &K. For each value of K we have kl + k, = K, and a com- plete set of functions is obtained with kl > k2. Thus kl is allowed to take values from K/2 if K is even and ( K + 1)/2 if K is odd; k, = K - k,. For each value of these quantum numbers ji can take any value 3 Ikil.It is usually necessary to include values of j 2 up to 10 to obtain reasonable convergence of the eigenvalues of the Hamiltonian matrix. The expressions for the matrix elements are given in Appendix 1. Solving eqn (9) at a series of values of R gives a set of effective potential functions Ut*(R) for radial motion. The radial part of the wavefunction zbs(R) in eqn (8) is then given, within the adiabatic appro~imation,~~ by the solution of the one-dimen- sional Schrodinger equation represents the symmetry of the wavefunction under Pf2. (14) ti2 a2 I] - 2m + W ( R ) + rf*(R)]&* (R) = Ei%fs*(R). The term zf*(R) takes into account, to first order, the dependence of t$*(81,02,q;R) on R, and is given by where J dQ represents integration over the angular variables 01,02,q and C&* is the first derivative of C&* with respect to R.A simple method for determining r(R) from the derivatives of the potential was presented in ref. (30) but, because of the large number of basis functions used, it proved expensive here and numerical differenti- ation of C&* was used. The effect of operating the total Hamiltonian in eqn (7) on the zeroth-order vibrational wavefunctions is given by52 P . E. SURFACE FOR (HF)2 where the matrix elements of the non-adiabatic perturbation Zna are defined as a function of R by the expression The effects of Zna and the J-dependence in the Hamiltonian may be included using perturbation theory.This J-dependence in the Hamiltonian may be split into a rotational part (1 8) " J 2 z r o t = and a Coriolis part Xna has only matrix elements off-diagonal in bending state. It contributes to the vibrational energy in second and higher orders. CALCULATION OF SPECTROSCOPIC CONSTANTS For a rotationless molecule, the principal molecular quantity derived from molecular-beam spectra of ( HF)2 is the tunnelling frequency, which corresponds to the difference in energy of the pair of adjacent vibrational levels of different permu- tation symmetry Pt2. However, compared with the well depth or the separation of other vibrational levels, the tunnelling frequency is a very small quantity so that in order to fit this parameter accurately an extremely large basis set of angular basis functions is required.Convergence to at least 0.01 cm-' was sought and for this values of kl up to 7 had to be included. The highest corresponding values o f j , was ca. 12 for k , z K/2, diminishing t o j , = 8 for ki = 7. This produced a Hamiltonian matrix of dimension 391 x 391 to be diagonalised, and it is this diagonalisation that is the major contributor to time of computing molecular parameters for a given potential surface. Initial calculations used smaller matrices of dimension 256 x 256 where a convergence to within 0.05 cm-l was obtained. In these initial fits to obtain a potential surface, a radial grid of 11 points was used (2.29-3.29 A), and eqn (14) was solved 38 over a smooth radial curve of 1000 points obtained by cubic-spline interpolation.A finer radial grid of 49 points (2.29-3.49 A) allowed convergence of the rotational constant to ca. 0.1 MHz; for the coarse grid errors were ca. 1 MHz. A smaller number of radial points is required for (HF)2 than for atom-diatom systems, since the well is an order of magnitude deeper and nearly harmonic over the range of the zero-point motion in the ground state. In addition the zeroth-order Born-Oppenheimer wavefunction is a much better approxim- ation tp the total wavefunction than for the atom-diatom cases; this is shown by the absence of peaking in the adiabatic and non-adiabatic correction terms as a function of R. For example z(R) in the ground state reaches a maximum of 1.9 cm-l, whereas for Ar-DCl, z(R) has a sharp peak of height 40 cm-l for the Neilsen-Gordon surface [see ref.(33), fig. 3(a)]. The small value of z(R) for (HF)2 can be attributed to the smooth slow change of the bending wavefunction q$+ with respect to R. As a result Lagrange n-point interpolation formulae 39 could be used to calculate the radial derivatives C,K,*.A . E. BARTON AND B . J . HOWARD 53 In calculating the vibrational energies and tunnelling frequencies the only neglected term in the Hamiltonian is the non-adiabatic correction Z,,, which couples together different bending states. This term can be included by perturbation theory but because of the accuracy of the original Born-Oppenheimer separation such effects provide a negligible correction ((0.01 cm-’) to the tunnelling frequency and were usually neglected.The other spectroscopic parameters were obtained as expectation values of the appropriate operator over the zeroth-order wavefunctions. Where necessary cor- rections were calculated using second-order perturbation theory. The expressions for these observables are the same as in the atom-diatom system except that the states IubK) are replaced throughout by lubK & l}. The rotational constant +(B + C), is the coefficient of J(J + 1) in the rotational energy. The dominant contribution comes from the expectation value of Ifrot ; also there are corrections in second order from the Coriolis coupling of states of K differing by one, and from the non-adiabatic coupling of states. Thus for K = 0 A large number of excited bending and stretching vibrational states are required in the above summation, and as a result the calculation can prove computationally expensive. Instead we have used the theory of Epstein 40 for calculating second-order corrections to energies within the Born-Oppenheimer approximation.If there is a perturbation between different bending states, then treating R as a constant, there is a second-order correction to the energy in state b : where x i b , ( R ) = $V)b*ZrV)bvds2. Then averaging over the stretching vibrational wavefunction gives the second-order correction to the energy J%’ = Sxbs(R)E~”(R)xbS(R)dR. (23) This approximation is obeyed well when the separation of bending curves is large compared with the spacing of stretching vibrational levels. For the lowest level the approximation used should lead to a slight overestimate of the interaction energy, since integrals involving a possibly larger energy denominator, (Eubl - Eooo) have been replaced by a smaller R-dependent denominator Ub(R) - Uo(R).However, since the separation of the curves Ub(R) and Uo(R) vary little with R, the errors in the approxim- ation are probably < 5%. The asymmetry splitting in the K = 1 state is given by a similar coupling of the K = 1 and K = 0 states by the Coriolis Hamiltonian. Thus where I& is the hybrid state 2-*()K = 1 ) & (K= - 1)) and it is the positive combin- ation that is displaced by the Coriolis interaction. Again the summation is simplified by the use of the Epstein approximation and results for the optimised potential are shown in table 3.54 P .E . SURFACE FOR (HF)2 TABLE 2.-POTENTIAL PARAMETERS ~~ ~ this work Alexander 29 this work Alexander 29 Ao/cm-' Bo/A- ' Al/cm- ' BJfi-' A2/cm- ' B2/A - A,/cm- B3/A - A4/cm-' B,/A- 671.7 (10.1) a 4.04 (0.03) 3.5 151 .O (0.3) 1.8 537.0 2.1 0.0 -30.0 (9.0) - 629.6 3.78 3.59 54.07 1.89 2.08 192.3 535.1 224.9 1.49 0.0 - 79.08 - 0.898 0.0 35.29 - 3.855 0.0 44.3 773.31 767.73 358.1 3 393.03 165.85 0.0 79.0 (6.3) 0.0 Uncertainties quoted as one standard deviation. The centrifugal distortion constant D, is the coefficient of J2(J + 1)2 in the expres- sion for the energy. This can be determined by perturbation theory, but it was found convenient to solve the radial eqn (14) adding Zrot to the radial potential in the form of a centrifugal barrier h'J(J+ 1)/2puR2.The equation was solved for a range of values of J (0-10). The eigenvalues were fitted to a power series in J(J + 1). The TABLE 3.-vALUES OF SPECTROSCOPIC PARAMETERS ( K = 0 STATES) FOR (HF)2 + state - state energies/cm- zeroth-order - 1086.9452 non-adiabatic -0.0169 total - 1086.9621 tunnelling frequency/cm- rotational constants/MHz zero th-order 6594.99 non-adiabatic 0.14 total 6496.82 P1 : zeroth-order DJ : zeroth-order/kHz 61.2 coriolis -98.31 (B - C)/MHz b = O b - 8.44 b = 1-5 99.61 b = 6-9 5.02 total 96.19 - 1086.2871 -0.0166 - 1086.3037 0.6584 (&0.005 cm-') 6594.39 0.13 - 101.56 6492.96 (+5 MHz) 1.4332 (~0.001) 61.2 (-I1 kHz) -9.55 98.96 2.02 91.43 ( - I 5 MHz) ' P I is (000 - IP(cosOl) + (cos0,)lOOO + >, see text. Contributions from different bending channels, b, of K = 0 state.constant term corresponded to the vibrational energy ; the coefficient of J(J + 1) gave the zeroth-order rotational constant B and the coefficient of J2(J + 1)' gave D,. In reality there are further higher-order Coriolis corrections to DJ, but these are much smaller than the experimental errors and can be neglected. The last remaining term to be calculated is (cos8,) + (cos~,) in the K = 0 and K = 1 states. In a tunnelling molecule the expectation values of these quantities areA . E . BARTON AND B . J . HOWARD 55 identically zero and the measured dipole moment is really a transition moment between the + and - tunnelling states. Thus (25) (cOs&) + ( ~ 0 ~ 0 2 ) = Spoxoo(~0~4 + cos&)poxoodRdR. Higher-order corrections are not warranted.PARAMETERISATION OF THE POTENTIAL SURFACE It is known that at long range the forces between two HF molecules are dominated by the classical electrostatic forces. The work of Morokuma and coworkers 20*21 suggests that this is still an important contribution to the intermolecular interaction in the region of the potential minimum. To this must be added at least some angle- dependent repulsion. However, since the exchange repulsion depends upon the over- lap of the charge distribution of the two molecules and since the contours of the electron density in an HF molecule do not deviate greatly from ~pherical,~' it is reasonable to assume that the basic features of the repulsive potential can be represen- ted by the first few terms in the series in eqn (6).Thus the potential was assumed to take the form V(R,8,,8,,p) = A,exp [- Bo(R - 2.79)] - A,exp[-&(R - 2.79)] + A,exp[-B,(R - 2.79)]{~0~8, - COS~,} + A2exp[-B2(R - 2.79)](2 cos& cos0,) + Velect. (26) The repulsive terms were assumed to vary exponentially with distance. Also a reference distance of 2.79 A, a first guess at the intermolecular separation, was used in these calculations so that the pre-exponential terms A possessed physically meaningful values. It was found necessary, like Alexander,29 to use two exponential terms to represent the dominant isotropic interaction. The classical electrostatic interaction between two linear molecules was expanded as an inverse power series in intermolecular distance. All terms up to R-5 were included.Written in terms of the permanent multipole moments of HF, Velect can then be written as56 P. E . SURFACE FOR (HF), D 42 and Q = 2.36 (3) D A.43 No value is known for the octupole moment but it was felt necessary to include the dipole-octupole term in the potential since it is formally of the same magnitude as the quadrupole-quadrupole term. We shall, however, treat the octupole moment as an adjustable parameter to add flexibility to the attractive part of the potential. First it assumes a very limited parameterisation of the repulsion. Secondly higher-order electrostatic forces make a significant, perhaps lo%, contribution to the energy in the region of the minimum. However, these higher terms vary rapidly with angle and their contribution to experimental observables appears to be insignificant for all except the tunnelling frequency, and here the dipole-octupole term may give the required flexibility.Thirdly no account has been taken of the induction and dispersion forces. The former are calculated to be much smaller than the neglected electrostatic contribution to the interaction, and the dispersion interaction is expected to be dominated by the angle- independent isotropic contribution. This has little affect on molecular constants except the well-depth, and can be mimicked in the region of the potential minimum by the two isotropic exponential terms in the potential. Finally no attempt has been made to represent valence and charge-transfer contributions to the interaction. Although these terms are probably not small, the energy-partition technique of Morokuma suggests that they have little angle dependence and can be largely included in the isotropic term in the potential.The potential in eqn (26) is similar to that derived by Alexander and DePristo from ab initio calculations. They wrote the potential 29 in a coupled representation suitable for scattering calculations. However, transforming to a decoupled representation as we have used (see Appendix 2), their potential can be written in the form A number of criticisms can be made of this assumed potential. V(R,O1,Bl,p) = Aoexp[--Bo(R - 2.79)] - A,exp[--B3(R - 2.79)] + {A,exp[-B,(R - 2.79)] - A,exp[-B4(R - 2.79)]} (cos0, - cos0,) - (A,exp[-B,(R - 2.79)] - A,exp[-B,(R - 2.79)]} (2cos0, cos0, - sine, sine, cosp) + {[A6(2.79/R) - A,]exp[--Bti(R 2.79)I)f(COSQi, COSe2) + Veiect (28) where f(cose,, case,) = cosel (3cos2e2 - 1) - cos0, (3c0s281 - 11, the angle dependence of the leading dipole-quadrupole term.Alexander neglected the quad- rupole-quadrupole and dipole-octupole terms as well as the higher-order contributions to the electrostatic potential. These potential parameters are listed in table 2. RESULTS AND DISCUSSION The values for the spectroscopic parameters calculated from the optimised and Alexander potentials are given in table 1 ; these can be readily compared with the experimental values. A limited parameterisation of the potential permits a quite accurate fit to the experimental quantities, and we believe that this indicates that the shape of the potential in the region of the minimum has been accurately determined, In addition the inclusion of the theoretical electrostatic interaction means that the potential is asymptotically correct at long range. The slightly large calculated values of (B + C)/2 indicate an over-short equilibrium separation.The slightly The Alexander potential also does remarkably well.A . E . BARTON AND B . J . HOWARD 57 180 120 0 . 3 60 0 60 120 180 W" FIG. 2.-Contour plot of the optimised potential for (HF)2 showing the tunnelling path. Contours are at 100 cm-' intervals above the minimum. The centre-of-mass separation is Re (table 4) and p = 180" (fig. 1). The positions of the potential minima are indicated by dots. 180 120 60 0 60 120 180 @,lo FIG. 3.-Contour plot of the Alexander potential.Contours are the same as for fig. 2, and the centre-of-mass separation is again fixed at Re (table 4) and @ = 180".58 P. E. SURFACE FOR (HF)2 incorrect values determined for the tunnelling frequency and the dipole moment indicate some errors in the angle dependent part of the potential. In fig. 2 and 3 may be compared contour plots of the optimised and Alexander potentials as a function of 0, and O2 at the equilibrium intermolecular distance. The equilibrium angular structures are 8, = 67", 82 = 12" and 8, = 56", 8, = 17", respectively, and it is this FIG. 4.-Radial potential for the optimised surface for (HF),. v, = 180" and &,Oz are fixed at the values given in table 4. The contributions to the potential are as follows: Vmm = isotropic inter- action, vmd = term involving cos 0, - cos &, vdd = term involving cos O1 cos &, and V,,,, = long range multipole attraction. change in 61 that is basically responsible for the large value of the dipole moment calculated for the Alexander potential.The slightly more linear geometry ensures a longer path for the tunnelling motion. This, combined with a higher barrier, is largely responsible for the low calculated value of the tunnelling frequency for the K = 0 state. In order to obtain a good fit to the K-dependence of the dipole moment it was necessary to include a term of the form of the dipole-octupole interaction in the attractive potential. Note that at Ro = 2.79 A, the value of p/Ri of 79 cm-I is equivalent to an octupole moment of 1.45 D A2.This is very close to the ab initio values of Cade and Huo 44 (I .87 D A2) and of Maillard and Silvi 45 (1.78 D A2 at the SCF level and 1.699 D A2 from a CI calculation). Thus this term in the attractiveA . E. BARTON AND B . J . HOWARD 59 potential appears to possess approximately the physical meaning we have attributed to it. Any discrepancy between the observed and expected values of the dipole- octupole parameter is probably owing to the effects of suppressed higher multipole terms. Regarding the form of the calculated potential it may be seen that the repulsive forces are dominated by the isotropic (angle-independent) terms and this appears to support our view of a nearly spherical HF molecule. This is demonstrated in fig. 4, where the radial dependence of the potential at the equilibrium values of el, B2 and 9 200c l0OC 3 I n C 2 LL u -loo( -200( 1 I I 1 I I I 2 4 6 RIA FIG.5.-Radial potential for the Alexander surface for (HF)2. O1 and 0, are fixed at 50" and 15" respectively and (p = 180". The contributions to the curve are labelled in the same way as in fig. 4. is decomposed into its various components. The results for the Alexander potential are very similar except there is a slightly deeper well, see fig. 5. Another important observation is the large amount of zero-point motion possessed by the complex. As well as the bending motion corresponding to tunnelling between the two equivalent hydrogen-bonded configurations, the stretching motion has a large effect on the spectroscopic parameters. For example the equilibrium inter- molecular distance, Re, differs greatly from the effective distance in the ground vib- rational state.For the optimised potential we obtain Re = 2.675 A, which should be compared with the value of 2.79 A derived by Dyke et aL9 from a simple analysis of60 P. E . SURFACE FOR (HF)2 the rotational constant. Since this latter value corresponds to some average inter- molecular distance in the ground vibrational state it can be seen that care must be taken when comparing with the results of ab initio calculations, which should corres- pond to the minimum of the potential. To facilitate a direct comparison between our optimised potential and the results of other calculations we have listed the harmonic force constants V,, = a2V/ax8y in table 4.In general the agreement is remarkably good. The differences in V,, occur TABLE 4.-HARMONIC FORCE CONSTANTS AND EQUILIBRIUM STRUCTURES FOR (HF), this work Alexar der Lischka a Curtiss (6-3 1 G * *) 67.5 12 2.67 5 1705.7 (4.87) 0.188 0.074 0.107 - 0.066 0.01 7 -0.010 0.004 5 118.5 2.70 302 (0.86) 55.5 17 2.56 1998.5 (5.71) 0.200 0.069 0.100 - 0.061 0.027 -0.010 0.009 2 110 2.56 382 (1.09) 56 7 2.80 1330 (3.8) 0.13 0.048 0.15 0.008 2 0.000 01 5 123 2.80 -350 (1) - 0.054 -0.008 4 64 8 2.78 1645 (4.7) 0.20 0.079 0.17 - 0.000 12 lls7p2d/6slp SCF calculation; ref. (28). This is the angle relative to the centres-of-mass of Values Vee, Vvv; units of mdyn A. VRe; units of mdyn. elbar = 180 - elbar; bar = tunnelling barrier geometry. h Z classical the HF units. For the angle relative to the F-F axis subtract 0.9" for 0, and add 0.9" for 0,.in brackets in kcal mol-'. barrier height. Units of mdyn A-1 (1 dyn = 10-5N). because of the large fractional differences in the equilibrium values of 0,; as 0, tends to zero, Vvv must of necessity also tend to zero. There is also quite good agreement in the values of the well depth. We unfortunately have no way of accurately assessing our values. However, assuming we have correctly included the long-range electro- static interaction and from an estimate of the neglected terms (dispersion and induction forces) we expect our value to be accurate to &20%, giving a value of 20.5 & 4 kJ mol-l. Unfortunately little or no information is available from molecular-beam data on the changes in the average bond lengths of the HF units that occur on dimer form- ation.Estimates from ab initio calculations suggest that the change in bond length is small (0.002-0.004 A).28 Spectroscopic data are also available for (DF),, but these were not included in our least-squares fits. This is because the tunnelling frequency is only 0.053 cm-l in K = 0 states, beyond the accuracy of our calculations. In addition because of the smaller value for the rotational constant of DF a larger number of rotational basis functions are required to obtain eigenfunctions of similar quality to those of (HF),. As a consequence a prohibitively large amount of computer time would be required to fit both sets of data, and it is believed that very little additional information on the potential surface would be obtained.A .E . BARTON A N D B . J . HOWARD 61 In conclusion we have shown that the Born-Oppenheimer separation method may be extended to diatom-diatom systems but with a large increase in computation time. The molecular-beam data on (HF)2 contain sufficient information to determine a potential-energy surface which should be accurate around the potential minimum. Finally the derived potential surface almost certainly confirms a non-linear hydrogen bond. APPENDIX 1 MATRIX ELEMENTS OF HAMILTONIAN APPENDIX 2 TRANSFORMATION BETWEEN COUPLED A N D DECOUPLED EXPRESSIONS FOR THE INTERMOLECULAR POTENTIAL Alexander writes his potential function as a triple sum over spherical harmonics to facilitate semiclassical scattering calculations 46 as V(R) = 2 A'1'2'(R) 2 (4m1l2m2IW YV? Y12m2 Y& 11121 rn x Cl1kC'L-k since we use the expansion v(&) 2 V l 1 t 2 k C l l k C l 2 - k I,[& then V and A can be related by the equation and the values of v l 1 1 2 k may then be calculated.62 P .E . SURFACE FOR (HF)Z [ K. C. Jackson, P. R. R. Langridge-Smith and B. J. Howard, Mol. Phys., 1980, 39, 817. ’ A. E. Barton, T. J. Henderson, P. R. R. Langridge-Smith and B. J. Howard, Chem. Phys., 1980, 45,429. J. M. Hutson and B. J. Howard, Mol. Phys., 1981, 43, 439. M. D. Joesten and L. J. Schaad, Hydrogen Bonding (Marcel Dekker, New York, 1974); P. Schuster, in The Hydrogen Bond, ed. P. Schuster, G. Zundel and C. Sandorfy (North-Holland, Amsterdam, 1976). K. C. Janda, J. M. Steed, S. E. Novick and W. Klemperer, J.Chem. Phys., 1977, 67, 5162. T. R. Dyke K. M. Mack and J. S. Muenter, J. Chem. Phys., 1977,66,498. D. J. Millen, J. Mol. Struct., 1978,45, 1 ; A. C. Legon, D. J. Millen and 0. Schrems, J. Chem. Soc., Faraday Trans. 2, 1979, 3, 592. P. D. Aldrich, A. C . Legon and W. H. Flygare, J. Chem. Phys., 1981,75, 2126. T. R. Dyke, B. J. Howard and W. Klemperer, J. Chem. Phys., 1972, 56, 2442. lo B. J. Howard, T. R. Dyke and W. Klemperer, to be published. l1 D. F. Smith, J. Mol. Spectrosc., 1959, 3, 473. l’ W. Klernperer, Ber. Bunsenges. Phys. Chem., 1974, 78, 128. l3 A. W. Miziolek and G. C. Pimentel, J. Chem. Phys., 1976,65, 4462; 66, 3840. l4 J. D. Dill, L. C. Allen, W. C. Topp and J. A. Pople, J. Am. Chem. Soc., 1975, 97, 7220. l5 G. H. F. Dierkson and W. P. Kramer, Chem.Phys. Lett., 1970, 6, 419. l6 L. A. Curtiss and J. A. Pople, J. Mol. Spectrosc., 1976,61, 1 ; P. N. Swepston, S. Colby, H. L. l7 S. F. Boys and F. Bernardi, Mol. Phys., 1970, 19, 558. l9 H. Lischka, J. Am. Chem. SOC., 1974,96, 4761. ’O S. Yamabe and K. Morokuma, J. Am. Chem. SOC., 1975, 97, 4458. 21 H. Umeyama and K. Morokuma, J. Am. Chem. Soc., 1977,99, 1316, ’’ P. Kollman, J. McKelvey, A. Johansson and S . Rottenberg, J. Am. Chem. Soc., 1975,97, 955; P. Kollman, J. Am. Chem. Soc., 1977, 99, 4875; G. LeRoy, G. Louterman-Leloup and P. Ruelle, Bull. SOC. Chim. Belg., 1976,85, 393; S. Nagase and T. Fueno, meor. Chim. Acta, 1974, 35, 217. 23 D. R. Yarkony, S . V. O’Neill, H. F. Schaeffer 111, C. P. Baskin and C. F. Bender, J. Chem. Phys., 1974,60, 855. 24 M. L. Klein, I. R. McDonald and S. F. O’Shea, J. Chem. PhyJ., 1978,69, 63; M. L. Klein and I. R. McDonald, J. Chem. Phys., 1979, 71, 298. 25 L. L. Poulsen, G. D. Billing and J. 1. Steinfeld, J. Chem. Phys., 1978, 68, 5121 ; G. D. Billing and L. L. Poulsen, J. Chem. Phys., 1978,68, 5128; Chem. Phys., 1979,36,271. 26 W. L. Jorgensen, J. Am. Chem. Soc., 1978,100, 7824; W. L. Jorgensen and M. E. Cournoyer, J. Am. Chem. Soc., 1978,100,4972; W. L. Jorgensen, J. Chem. Phys., 1979, 70, 5888. 27 G. A. Parker, R. L. Snow and R. T. Pack, Chem. Phys. Lett., 1975,33, 399. 28 H. Lischka, Chem. Phys. Lett., 1979, 66, 108. 29 M. H. Alexander and A. E. DePristo, J. Chem. Phys., 1976,65, 5009. 30 J. M. Hutson and B. J. Howard, Mol. Phys., 1980, 41, 1123. 31 K. Yamada and M. Winnewisser, J. Mol, Spectrosc., 1977, 68, 307. 32 S. L. Holmgren, M. Waldman and W. Klemperer, J. Chem. Phys., 1978, 69, 1661. 33 S. L. Holmgren, M. Waldman and W. Klemperer, J. Chem. Phys., 1977, 67, 4414. 34 D. M. Brink and G. R. Satchler, Angular Momentum (Clarendon Press, London, 1968); A. R. 35 R. J. LeRoy and J. S. Carley, Adu. Chem. Phys., 1980, 42, 353. 36 M. Born and K. Huang, Dynamical Theory of Crystal Lattices (Oxford Unversity Press, 37 J. T. Hougen, J . Chem. Phys., 1962, 36, 519; J. K. G. Watson, Mol. Phys., 1970, 19, 465. 38 J. W. Cooley, Math. Comp., 1961, 13, 363. 39 J. Singer, Elements of Numerical Analysis (Academic Press, London, 1964), chap. 7. 40 S. T, Epstein, J. Chem. Phys., 1976, 65, 5526. 41 R. F. W. Bader, I. Keaveny and P. E. Cade, J. Chem. Phys., 1967,47, 3381. 42 J. S. Muenter, J. Chem. Phys., 1972, 56, 5409. 43 F. H. DeLeeuw and A. Dymanus, J. Mol. Spectrosc., 1973, 48, 427. 44 P. E. Cade and W. Huo, J. Chem. Phys., 1967, 47, 614. 45 D. Maillard and B. Silvi, Mol. Phys., 1980, 40, 933. 46 M. H. Alexander and A. E. DePristo, J. Chem. Phys., 1977, 66, 2616. Sellers and L. Schafer, Chem. Phys. Lett., 1980, 72, 364. T. P. Green and F. B. van Duijneveldt, unpublished work. Edmonds, Angular Momentum in Quantum Mechanics (Princeton University Press, 1957). London, 1954).

ISSN:0301-7249    

DOI:10.1039/DC9827300045

出版商:RSC    

年代:1982

数据来源: RSC

摘要:

Faraday Discuss. Chem. SOC., 1982, 73, 63-70 Molecular-beam Studies of Van der Waals Complexes of Atmospheric Interest BY JOHN s. MUENTER, ROBERT L. DELEON AND AKIMICHI YOKOZEKI Department of Chemistry, University of Rochester, Rochester, New York 14627, U.S.A. Received 9th December, 1981 The radio-frequency and microwave spectra of the Ar.03 and Ar.S02 Van der Waals complexes have been analysed with the conventional first-order centrifugal distortion Hamiltonian plus inversion and centrifugal distortion of the inversion frequency. The centrifugal distortion constants were then used to obtain a force field describing the Van der Waals vibrations. This procedure worked quite well for Ar.03, generating vibrational properties appropriate for a semi-rigid complex. Ar-S02, however, cannot be described by such a harmonic model.Ar.S02 has a very isotropic Van der Waals potential where stretching and bending vibrations are strongly coupled. In addition, a qualitative study of many SO2-containing bimolecular complexes was made. It was possible to categorize these complexes as having strong, moderate or weak interactions. The relatively weak, but long-range Van der Waals interaction can be of great importance in many different aspects of atmospheric studies. The complex chemistry and dynamics of the upper atmosphere exist in a low-pressure, low-temperature medium where Van der Waals complexes can exert maximum influence. The stability of the ozone layer is a prime example of an upper atmosphere condition which can be affected by weak interactions. Many aspects of atmospheric pollution also relate to long-range intermolecular potentials.The work presented here involves Van der Waals complexes containing a wide variety of molecules of atmospheric importance. The emphasis is placed on species containing ozone and sulphur dioxide. Two quite different experimental approaches were used. The first involves high-resolution radio-frequency and microwave spec- troscopy on Ar-0, and Ar-SO,. These molecular-beam electric resonance studies provide detailed structural and electronic properties of the complexes and initial reports of this work have been published.lP2 The emphasis here will be on vibrational information obtained from centrifugal distortion analysis. These molecules provide examples of both the usefulness and limitations of obtaining potential function information from centrifugal distortion data.The second set of experiments were qualitative studies of the production and characterization of bimolecular complexes containing SOz. A wide range of both organic and inorganic partners for SO2 complexation were examined. The relative strengths of the weak interactions were estimated by observing the competitive nature of Van der Waals molecule formation. Some qualitative structural properties were obtained from electric deflection experiments. EXPERIMENTAL The apparatus and operating conditions used for studying Ar-03 and Ar-SO, have been previously described.'P2 Briefly, a few percent of O3 or SO, in 2.5 atm of Ar were ex-64 ATMOSPHERIC VAN DER WAALS COMPLEXES panded through a 25 pm diameter nozzle.The molecular beam formed in this way passed through an electric resonance spectrometer and into a mass-spectrometer detector. The observed linewidth for transitions up to 18 GHz was 1.5 kHz f.w.h.m. Electric dipole moment components were obtained from Stark-effect measurements. Transition assign- ments were made with extensive radio-frequency-microwave double-resonance experiments. The same apparatus was used in the qualitative study of S02-containing bimolecular complexes. In this work, however, only the molecular-beam source and mass-spectrometer detector were fully utilized. The electrostatic quadrupole focusing fields of the intervening spectrometer were used in some cases to observe electric deflection. To completely characterize the production of bimolecular species, SO,-X, it would be desirable to vary the following parameters: identity of X, concentration of X, concentration of SO2, rare gas, source pressure, source temperature, nozzle diameter and species observed. Such a multiple dimension study would be extraordinarily time consuming and several of these variables were fixed to define a set of standard operating conditions.Room tem- perature and 25 pm diameter nozzles driven by 2.5 atm of Ar were employed for most measurements. SO, and X concentrations were varied by powers of two over an 8-1 % range. A total of 18 partner molecules, X, were used and the following weakly bound species formed in the expansion were investigated: Ar,, (S0J2, Xz, AraSO,, Ar-X and S02.X.In many cases trimolecular and larger complexes were also formed. In those cases where S02.X was formed in quantities to suggest future spectroscopic study, more extensive studies have been made. Equal concentrations of SO, and partner X were initially used. RESULTS AND DISCUSSION INTERNAL DYNAMICS OF Ar.0, AND Ar-SO, The previously published work on Ar-03 and Ar-SO, presented the experimental data, structural information, electric dipole moment components and, quite sur- prisingly, inversion frequencies. The structural properties of these two complexes are extremely similar. The geometries of both exhibit perpendicular structures with the Ar atom located above the triatomic molecule's plane, over the centre-of-mass of SO, or 03. The separation between the atomic and molecular subunits of both com- plexes is approximately the sum of the Van der Waals radii involved.Both molecules have very small dipole moments along the direction, R, connecting Ar with the molecular centre-of-mass (very nearly the a inertial axis) and much larger moments perpendicular to this axis. The spectroscopic, structural and dipole moment data for Ar.0, and Ar-SO, are summarized in table 1 with the coordinates defined in fig. 1. TABLE 1 .-SEMI-RIGID PROPERTIES OF Ar.03 AND Ar-S02 property Ar.0, Ar50, AIMHz BIMHz CIMHz A/MHZ RIB+ 4" do PbID AID cr for fit/MHz 12 222.78 1946.64 1715.39 462.87 3.416 78 0 0.101 0.0 0.464 0.94 9 146.40 1485.21 1 3 1 8.64 975.12 3.675 80 0 0.267 0.0 1.465 1.75J . S . MUENTER, R . L . DELEON A N D A . YOKOZEKI 65 FIG.1.-Coordinates for Ar-03 or Ar.S02. R is the distance from Ar to the centre-of-mass of the molecule portion, 19 is the angle between R and the C2 axis of the molecules and q~ is the torsional angle about C2. 6' = 0 when R and C2 are coincident. q~ = 0 when the complex has C, symmetry. The existence of a high-frequency tunnelling motion was not initially recognized in Ar.0, because nuclear spin statistics require half of each inversion doublet to be absent. Thus, inversion splittings cannot be directly observed. However, a more extensive set of transitions was available for Ar-SO, which clearly showed that all of the c-type transitions were rotation-inversion, i.e. these transitions involved a rota- tional energy plus or minus an inversion energy. With this information, additional Ar-0, transitions were observed confirming inversion in both molecules.The presence of inversion and the observed spin statistics requires the torsional angle p to be zero in both cases. The first indication that the internal dynamics of the two complexes were quite different, in spite of the similar structures, was given by the inversion frequencies. Both frequencies (975 MHz for Ar-SO, and 460 MHz for Ar-0,) are extremely high since heavy atoms must be moving large distances. Since the reduced mass must be larger for Ar-SO, and it also exhibits the greater inversion frequency, Ar-SO, must have a more isotropic Van der Waals potential than Ar-0,. It was felt that this qualitative difference made a more quantitative study of potential functions very desirable.While there could be no hope of gaining sufficient information to do the kind of detailed calculations which have been applied to rare-gas-hydrogen-halide molecule^,^ a simple centrifugal distortion analysis in the spirit of the Ar-CO, study appeared very attractive. A large amount of information is available in the sense that the experimental transition frequencies are known with uncertainties of < 1 kHz, but a semi-rigid model containing 3 rotational constants and inversion frequency can only fit the data with standard deviations of 1-2 MHz. The difficulty, of course, is that there are 3 low-frequency vibrations associated with the Van der Waals bond and the force constant matrix used need not be particularly diagonal. The experimental frequencies were first fitted to a Hamiltonian containing the usual 3 rotational constants, the 5 independent first-order centrifugal distortion constants as defined by W a t ~ o n , ~ and an inversion energy which lowered even K-l and raised odd K- I rotational levels.2 For Ar-0,, the addition of centrifugal distortion improved the data fit only by a factor of 5 relative to the rigid-rotor plus inversion analysis.For Ar-SO,, there was virtually no improvement when centrifugal distortion was included, even though the number of parameters being adjusted increased from 4 to 9. This is not surprising since a significant vibration-rotation interaction associated with the large-amplitude inversion motion can be expected. This problem has been treated in detail by Lide in the study of amides and by Butcher and Costain ' for cyclo- pentene. Assuming that the rotational constants for the upper and lower inversion levels are the same, the vibration-rotation interaction manifests itself as a centrifugal66 ATMOSPHERIC VAN DER WAALS COMPLEXES distortion of the inversion frequency.The interaction arises from rotation per- pendicular to the near prolate axis and will be 6 9 7 equal to a[J(J + 1) - (Pi)], where the constant a is to be determined from the data fit. In the symmetric top limit this is, of course, equal to the J(J + 1) - K 2 term which is well known for ammonia.8-* Including the parameter a in the semi-rigid analysis, i.e. fitting A , B, C, A and a, brought no improvement, but adding centrifugal distortion dramatically improved the fit.For Ar-0,, the standard deviation went from 0.9 MHz to 20 kHz and from 1.8 MHz to 70 kHz for ArmSO,. Care was taken to ensure that the constants obtained were statistically significant. While the residuals generated by this model are still much larger than experimental uncertainties, further extensions are clearly unjustified and the present values must be regarded as being effective molecular constants. The problem of extracting Van der Waals potential function information from these constants is non-trivial. The simplest approach is to obtain z values from the centrifugal constants and relate these to a force field. Fortunately, some reasonable assumptions greatly simplify the problem and make it possible to obtain 6 z values from 5 Watson constants.These complexes belong to the C, point group and have 4 vibrations of A' symmetry and 2 A" modes. The symmetric vibrations include the symmetric stretch and bend of SO2 or O3 at relatively high frequencies, and the low- frequency Van der Waals vibrations corresponding to stretching of the coordinate R and the bending of 8. The two A" vibrations are the high-frequency asymmetric stretch of the molecular subunit and bending of the Van der Waals coordinate 9. Assuming negligible interaction between the high-frequency molecular motions and the Van der Waals bond vibrations, and using the known l o * l l vibration properties of the isolated molecules, leaves four force constants to be determined. In this case, the Wilson F matrix for A' modes becomes two separate 2 x 2 blocks with the low- frequency block containing unknown F R R , F R 0 and Foe elements.The A" matrix is diagonal with only Fvv to be determined. Six z were obtained from the 5 empirical distortion constants by noting that z,,,, is independent of the low-frequency vibrations for a geometry with 8 = 90". Both complexes have structures with 8 quite close to 90" and the following conclusions were carefully checked and found to be completely insensitive to any reasonable variation in the specific value used for z,,,,. Thus, z,,,, was calculated from the known SO2 or O3 properties and the remaining 5 z were calculated from the 5 Watson constants. The net result is that the 4 required F matrix elements are just overdetermined and the force-field matrix can be fitted to the distortion constants.Table 2 lists the molecular constants obtained from the data fit and table 3 gives the F matrix elements and the corresponding vibrational frequen- cies. The vibrational frequencies in table 3 described two very diffcrent Van der Waals complexes. Ar-03 is well described as a semi-rigid rotor. The 3 low-frequency vibrational modes are determined with reasonable accuracy and are of the magnitude expected for a Van der Waals complex. The values listed in table 3 are stable with respect to altering details of the calculation procedure. The off-diagonal element connecting the stretch and 8 bend is small and it is realistic to consider these two motions separately. While it is difficult to estimate uncertainties for the Ar.0, vibrational frequencies in table 3, it is felt that these frequencies provide a good description of the Ar.0, Van der Waals bond vibrations.Ar-S02, however, is completely different from Ar.0,. With the exception of FpP, the force constants are extremely small. The relatively large value off-diagonal * It was not possible to separate from the inversion frequency an interaction proportional to angular momentum parallel to the prolate axis (the KZ term in NHJ because transitions with > 2 could not be observed.J . S. MUENTER, R . L. DELEON AND A . YOKOZEKI 67 TABLE 2.-cENTRIFUGAL DISTORTION RESULTS FOR h . 0 3 AND k's02' 12 225.16(4) 1 947.3 3 (6) 1715.37(6) 460.91 (4) 122(5) -40 - 94(2) -61(2) - 77(2) - 1 lO(115) - 1175(105) 19 9146.15(5) 1485.63(2) 13 18.28(3) 978.12(6) 517(7) -10 -25(3) - 60(4) -51(2) - 105(50) - 792(29) 72 Uncertainties are one standard deviation for fitting 15 transitions for Ar.03 and 20 transitions zlabB 5 racrBB + 2raBaB. for Ar.S02.This value calculated from molecule properties, see text. TABLE 3.-vIBRATIONAL PROPERTIES OF Ar.0, AND Ar.S02 property Ar.0, Ar.S02 FRR/mdyn A-' 0.0067 0.0001 Feelmdyn A- ' 0.025 0.0006 F,e/mdyn A-' - 0.0002 0.0007 F,,/mdyn A-' 0.0073 0.0025 v,/cm - 50 0 v&m - 26 6 v,/cm - 29 17 FR, element and effectively zero value for FRR clearly indicate that the inherent harmonic approximation of this treatment will not work for Ar-SO,. The concep- tual separation of a Van der Waals stretch and 0 bend is not possible. Only the q~ vibration has any meaning in this Ar-SO, analysis because of its A" representation. Qualitatively, not only is the Ar-SO, potential more isotropic, but the proper choice of coordinates to describe the complex is not obvious.In any description, the Van der Waals bond vibrations will be of very low frequency. The remaining information concerns the inversion. For Ar-O, it is reasonable to assume a barrier shape and then estimate its height. Assuming the inversion co- ordinate is 0, which corresponds to 0, rotating about its a axis, and using a Dennison and Uhlenbeck potential l2 produces a barrier height of 50 4 30 cm-l. The large uncertainty is due primarily to uncertainties in the equilibrium structure and not the shape of the potential. Since the Ar.SO, inversion path cannot be postulated, a similar estimate is not realistic in this case.Finally, there is the inversion distortion constant, a. In NH,, perpendicular rotation increases the inversion frequency as predicted by the distortion increasing the height of the pyramid. Similarly, in Ar.03 the inversion frequency increases with perpendicular rotation but a simple physical interpretation is not obvious. Little can be said about the sign difference for a in Ar-SO,.68 ATMOSPHERIC VAN DER WAALS COMPLEXES BIMOLECULAR COMPLEXES CONTAINING so, Bimolecular Van der Waals molecules are obviously much more important to atmospheric considerations than atom-molecule complexes. To find out which bi- molecular species might be subjects for detailed spectroscopic studies, a broad survey was undertaken. Table 4 gives the binding partners investigated, listed in order of TABLE 4.-SO2 BINDING PARTNERS inorganic organic a Listed in order of increasing molecular weight.increasing molecular weight for inorganic and organic molecules. The inorganic species were chosen primarily on the basis of availability and the organic substances are generally the simplest representatives of several large classes of molecules. The primary goal was to identify which complexes could be made in abundance, but it was hoped that a qualitative measure of relative interaction energies could also be obtained. Most significant are the facts that the mechanism of complex formation in the nozzle expansion is complicated and poorly understood and that the molecular-beam detector does not measure complex concentrations because of unknown fragmentation during ionization. The most effective approach found was to study the competitive aspects of Van der Waals molecule formation.Thus, as well as monitoring the intensity of the S02-X signal the ratios of S02*X to Ar,, S02.X to X2 and S02.X to (SO,), were measured. The decrease of Ar, with SO,-X production relates both to the consumption of Ar, in formation of more strongly bound species and to the increasing temperature of the beam. (SO,), is relatively strongly bound and effective competition of partner X for the available SO, indicates a relatively strong S02.X interaction. Similarly, where X, is easily produced the effective competition of SO2.X for X indicates a strong inter- action. While no single criterion could be used to judge the interaction strength, three categories of bond strengths could be established from the S02.X intensity and the three above ratios.As listed in table 5, molecules could be labelled as having a strong, moderate or weak interaction with SOz. Those molecules in the strong category either showed significant magnitude for each of the intensities or ratios, or exhibited overwhelming behaviour with respect to at least one of the criteria. Those molecules in the weak category neither generated significant S02.X signals nor com- peted effectively with the formation of other weakly bound species. Obviously, the remaining molecules were placed in the moderate group. Many difficulties stand in the way of this latter goal.J . S . MUENTER, R .L . DELEON A N D A . YOKOZEKI 69 TABLE 5.-cATEGORIES OF so2 BINDING strong moderate weak H20 OCS SO2 CS2 C2H2 CZH4 CHlO CH3Cl (CH3)zCO N20 COZ NO2 C2H6 CO N2 NO 0 2 CH4 The most satisfying result obtained is that one's chemical intuition about which factors contribute to strong complexation is apparently correct. Those molecules which are very polar and very polarizable give strong interactions while small, low- polarizability molecules with small permanent moments interact weakly with SO,. The role of electric polarizability is clearly established in both the CH3CH,, CH2=CH2, C H r C H and CO,, OCS, CS, series. The importance of dipole moment can be seen in comparing CH4 with CH3Cl or CO with H2C0. Electric deflection studies clearly indicate that (SO,), has a non-symmetric polar structure.SO,.HC--CH exhibited abnormally small deflection which could arise from a structure in which the SO, moment is perpendicular to the A axis of the complex. Finally, all of the molecules in the strong classification are potential candidates for detailed spectroscopic work. Preliminary projects of this nature are in progress for (SO,), and S02-H20. CONCLUSIONS While the static or geometric properties of Ar.0, and Ar-SO, appear quite similar, the dynamical behaviour of these two complexes is very different. Several aspects of AreSO, would suggest that it is more strongly bound than Ar.0,. SO, is both more polar and more polarizable than O3 and the distance R in Ar.SO2 is shorter than the sum of the Van der Waals radii, while R in Ar.0, is slightly longer than indicated by contacting Van der Waals spheres.Yet Ar-0, is clearly very much more rigid than ArSO,. This is just another reminder that there is no direct correlation between bond strength and rigidity. The second part of this work presents a very qualitative study of many complexes. By observing the ability of S02-X formation to compete with the formation of other Van der Waals molecules, an approximate categorizing of bond strengths was achieved. The results obtained are in agreement with intuitive ideas concerning what affects Van der Waals bond strengths. This work has been supported by the United States Department of Energy Con- tract DE-AC02-77EV04321. R. L. DeLeon, K. M. Mack and J. S. Muenter, J. Chem. Phys., 1979,71,4487. ' R. L. DeLeon, A. Yokozeki and J. S. Muenter, J. Chem. Phys., 1980, 73, 2044. S. L. Holmgren, M. Waldman and W. A. Klemperer, J. Chem. Phys., 1977, 67, 4414; J. M. Hutson and B. J. Howard, Mol. Phys., 1980, 41, 1123. J. M. Steed, T. A. Dixon and W. A. Klemperer, J. Chem. Phys., 1979, 70,4095. J. K. G. Watson, J . Chern. Phys., 1966, 45, 1360; 1968,48, 181. D. R. Lide, J. Mol. Spectrosc., 1962, 8, 142. ' S. S. Butcher and C. C. Costain, J. Mol. Spectrosc., 1965, 15, 40.70 ATMOSPHERIC VAN DER WAALS COMPLEXES C. H. Townes and A. L. Schawlow, Microwave Spectroscopy (McGraw-Hill, New York, 1955), p. 308. D. Kivelson and E. B. Wilson, J. Chem. Phys., 1952, 20, 1575. lo L. Pierce, J. Chem. Phys., 1956, 24, 139. '' W. Gordy and R. L. Cook, Microwaue Spectroscopy (Wiley-Interscience, New York, 1970). '* D. M. Dennison and G . E. Uhlenbeck, Phys. Reu., 1932,41, 31 3.

ISSN:0301-7249    

DOI:10.1039/DC9827300063

出版商:RSC    

年代:1982

数据来源: RSC

摘要:

Faraday Discuss. Chern. Soc., 1982, 73, 71-87 Determination of Properties of Hydrogen-bonded Dimers by Rotational Spectroscopy and a Classfication of Dimer Geometries BY A. C. LEGON AND D. J. MILLEN Christopher Ingold Laboratories, Department of Chemistry, University College London, 20 Gordon Street, London WC1 H OAJ Received 21st December, 1981 Rotational spectroscopy is a rich source of information about the molecular geometry, the potential-energy function and the electric-charge distribution of simple hydrogen-bonded dimers in the gas phase. Techniques for the observation of such spectra are now available and have led to a considerable amount of information for a wide range of dimers. We outline two techniques that have been used and then review the various molecular properties that can be derived from the observed spectroscopic constants.We indicate how vibrational ground-state rotational and centrifugal distortion constants can lead to information about the molecular geometry and potential-energy function and how observations of the Stark effect and nuclear quadrupole hyperfine structure allow conclusions about charge redistribution on dimer formation. We also show how important inform- ation can be obtained by the study of rotational spectra of dimers in vibrationally excited states. Two specific examples, HCN * HF and N2 * * HF, are examined in detail before a general discussion of results for a number of dimer species is presented in which geometries, force constants and dissociation energies are compared systematically.We show that, although the observed geometries correspond to broad potential energy minima, it is nevertheless possible to propose a simple rule which accounts for the preferred equilibrium conformation. Rotational spectroscopy is a rich source of information about the molecular properties of simple hydrogen-bonded dimers B - - HA. Moreover, since the spectra are obtained at low pressure, the molecular properties so determined refer to the isolated dimer unencumbered by lattice or solvent interactions. The molecular properties that can be obtained in this way include (i) the principal parameters charac- terizing the one-dimensional radial potent ial-energy function of the dimer, namely the geometry (re), the force constant (.fD) and the dissociation energy (De), (ii) details of the hydrogen-bond bending potential-energy function and (iii) information about the electric-charge distribution in the dimer, namely the electric dipole moment (‘p) and the electric field of gradients (a2 V/az2) at certain nuclei. In this paper we discuss briefly two experimental methods for detecting rotational spectra of species B - - HA, with special emphasis on the advantages and limitations, and consider the use of each technique by detailed reference to (HCN,HF) and (N,,HF).We then discuss the results for four types of simple dimer: (a) species B * * HX in which the acceptor atom carries only one non-bonding electron pair, (h) species B * HX in which the acceptor atom carries two non-bonding pairs, (c) species B . - * HX in which the acceptor atom has three non-bonding pairs and (d) species B - * - HX in which the acceptor molecule B has a 7c bond but no non-bonding pairs.Finally, we propose a simple rule which accounts for the observed geometries.72 ROTATIONAL SPECTROSCOPY OF HYDROGEN-BONDED DIMERS EXPERIMENTAL In this paper we discuss results derived from rotational spectra that have been obtained mainly by two techniques. The first of these involves conventional Stark modulation micro- wave spectroscopy of dimers in binary gas mixtures at equilibrium at temperatures 7200 K and pressures of ca. 50 mTorr. *' The second technique involves Fourier-transform micro- wave spectroscopy of a pulse of gas mixture (B and HA diluted in, say, argon) expanded supersonically from a nozzle into an evacuated Fabry-Perot cavity. The microwave pulse induces a macroscopic polarization in the gas when it is in collisionless expansion, It is required that the half-life, T2, for the decay of the macroscopic polarization is much greater than that (z,) for decay of the microwave pulse within the Fabry-Perot cavity.Accordingly, when the polarized dimers in the cold gas ( T FZ 5 K) begin to emit at some rotational tran- sition frequency and the detector is opened, the microwave pulse has dissipated and only the molecular emission survives to be detected. Details of the theory and operation of this technique have been de~cribed.~.~ Although equilibrium rotational spectroscopy is restricted mainly to moderately strongly hydrogen-bonded dimers and has moderate resolution, it has the significant advantage that rotational spectra are obtained in vibrationally excited states of the low-lying hydrogen-bond modes as well as in the vibrational ground state.The pulsed-nozzle, Fourier-transform method has a very high sensitivity to molecular dimers and a high resolution because of the low effective temperature of the gas pulse and because the molecular emission occurs while the gas is in collisionless expansion. Very weakly bound molecular dimers can be investigated with this instrument. A con- comitant disadvantage of the low effective temperature is, however, that spectra in the vibrational ground state only are observed. The equilibrium method, through the Stark effect, readily furnishes electric dipole moments of dimers, while the high resolution of the pulsed nozzle, Fourier-transform method allows the investigation of nuclear quadrupole and nuclear-spin-nuclear-spin coupling effects.Results obtained by other groups using molecular-beam electric resonance spectroscopy will also be discussed. The two techniques outlined above are complementary. RESULTS AND DISCUSSION (i) EQUILIBRIUM ROTATIONAL SPECTROSCOPY; HCN 9 HF The many molecular properties that can be determined from the rotational spectrum of a hydrogen-bonded dimer are well illustrated by the example of the linear complex HCN - - - HF.' Fig. 1 shows the J = 5t4 transition of this species in the ground state and various excited vibrational states, observed using the technique of equili- brium rotational spectroscopy. The strongest transition (at 35 908.35 MHz) is as- signed to the vibrational ground state and the remainder to vibrational satellites associated with the low-lying vibrational modes.On formation of a linear hydrogen- bonded dimer, the loss of three degrees of translational freedom and two degrees of rotational freedom results in three new vibrational modes, of which two are doubly degenerate and all are expected to have a considerably smaller energy spacing than any of the monomer modes. The approximate form of these new modes for HCN * - H F is shown in fig. 2. The doubly degenerate bending modes are denoted by va and vB, in order of increasing energy, and the stretching mode by v D . Tn fig. 1 the progressions in the low-frequency bending mode vs are identified using the convenient notation (va, uB'ti), where I is the vibrational angular momentum quan- tum number.An explicit label for the mode vB is not included in this notation because states with uB > 0 are not sufficiently populated at the experimental temperature. * 1 Torr = (101 325/760) Pa.A . C . LEGON AND D. J . MILLEN (0,001 73 I l l l r l l l l l l l r l l l r l I l l l l l l l l l L 3 8.0 37- 5 37.0 36.5 36.0 35.5 frequency/GHz FIG. 1 .-J = 5+4 transition of HCN - * HF in the vibrational ground state (0,OO) and in vibrationally excited states (u,,upl !). .... % vB ....- FIG. 2.-Diagrammatic representation of the low-frequency normal modes of HCN * * HF.74 ROTATIONAL SPECTROSCOPY OF HYDROGEN-BONDED DIMERS There is much information to be derived from the spectrum in fig.1. First, the ground-state transition frequency, taken with those of other J + 1 t J transitions, leads to accurate rotational constants for this and other isotopic species of the dimer. Under the well-established assumption of unchanged monomer geometries on dimer formation, the set of ro(N * * F) distances shown in table 1 results. The agreement TABLE VA VALUES OF ro(N - - - F) FOR HCN - - - HF isotopic species ro(N - - - F)/A between the values obtained from different isotopic species is remarkably good. We note that deuterium substitution leaves ro(N - - F) sensibly unchanged. A similar result has been obtained for CH3CN * HF.4a Secondly, the intensities of vibrational satellites relative to that of the ground state give the vibrational separation up = l+O as 91 20 cm-I and u, = It0 as 197 & 15 cm-’ in the two lowest-energy modes of the molecule.By combining these values with those of the analogous high-frequency modes vB and v,, obtained from infrared spectroscopy, 4b stretching and bending quadratic force constants have been obtained. From a model involving motion essentially along the dissociation coordin- ate of the dimer, the force constantf, is found to have a value of 26.3 N m-l. For the angle bending force constants we use the result obtained from a similar investigation 4b of CH3CN * HF, from which it was established, with the aid of the centrifugal distortion constant DJK, thatfo, (see fig. 3 for a definition of internal coordinates in HCN * * HF) is effectively zero. A similar assumption in the present case leads to the set of values f v = 3.7 x and fe = 6.3 x J for H”C14N.. .H19F. FIG. 3.-Internal coordinates used to describe bending of the HCN * * - HF molecule. Thirdly, while relative intensities lead to force constants, absolute intensities of rotational transitions in the equilibrium gas mixture of HCN and H F lead to the dissociation energy of the dimer.’ For a rotational transition originating from a state characterised by the rotational and vibrational quantum numbers J and u, the absolute (or integrated) intensity is given by I = (8n3n,,J/3ckT)Ipij]2v~ (1) where nu, J is the number density of molecules in the state u,J. Thus, if ,u is known (see below), can be determined directly. If the partition function of each of the molecules M participating in the dimer formation H C N + H F = H C N * * * H F (2) no,o(HCN - HF)/no,o(HCN)no,o(HF) = (h2/2npkT)3/2exp(D,/RT) (3) is known, values of E*,~(M) are then readily obtained and lead through the relationshipA .C . LEGON AND D. J . MILLEN 75 to the zero-point dissociation energy Do. The results of such determinations are Do = 18.9 Fourthly, the Stark effect, which can be distinguished in fig. 1 by its phase difference of n, provides a direct route to the determination of the electric dipole moment p of the dimer and its enhancement Ap over the sum of the monomer moments of this linear species. The mean value obtained from a number of measurements gives p = 5.612 & 0.01 D * and Ap = 0.80 D.6 Finally, it is in principle possible in a given rotational transition to resolve the nuclear quadrupole hyperfine structure arising from the I4N nucleus.The change in the 14N nuclear quadrupole coupling constant on dimer formation leads to additional information about the electric-charge redistribution that accompanies dimer form- ation. While results are not yet available for HCN - - - HF, these changes have been determined for other dimers, for example N2 - - HF,7 and will be discussed below. 1.1 kJ mo1-I and D, = 26.1 & 1.6 kJ mol-'. (ii) PULSED-NOZZLE, FOURIER-TRANSFORM SPECTROSCOPY: Nz * HF The dimer formed between molecular nitrogen and hydrogen fluoride is more weakly bound (see below) than that in which hydrogen cyanide is the acceptor molecule. Consequently, in order to observe its rotational spectrum, recourse to the pulsed-nozzle, Fourier-transform technique is appropriate.The spectrum is characteristic of a linear molecule but, because this technique detects only transitions in the vibrational ground state, the arguments used to establish the linear equilibrium geometry for HCN * HF are not available here. There is, however, considerable J -L____L___y -1.4 -1.2 -1.0 -+- frequency /MHz FIG. 4.-J = It0 transition in the vibrational ground state of 14N "N HF showing 14N nuclear quadrupole and H, I9F nuclear-spin-nuclear-spin hyperfine structure. Reconstructed from observed spectroscopic constants with frequencies offset from vo = 6355.4038 MHz. but less direct evidence for a similar geometry for N2 * * HF. Each rotational transition carries a complicated hyperfine structure arising from nuclear quadrupole coupling of the two I4N nuclei and the nuclear-spin-nuclear-spin coupling of H and 19F.This structure simplifies appreciably for the species l4NI5N - - HF, as shown for the J = It0 transition in fig. 4, in which is reproduced a computer simulation based * 1 D M 3.3356 x C m.76 ROTATIONAL SPECTROSCOPY OF HYDROGEN-BONDED DIMERS on the experimentally determined spectroscopic constants. Each separate component in fig. 4 has been resolved, thus illustrating the very high resolution of the pulsed- nozzle technique. Analysis of the hyperfine structure gives the nuclear quadrupole coupling constants x,,(l) and x,,(2) as displayed in table 2. The labelling of the N TABLE 2.-sPECTROSCOPIC CONSTANTS OF Nz * * HF 14N14N * * * HF 3195.3534 17.2 - 4.91 (14) - 4.75(14) 107(6) 15N14N * * * HF 3 11 6.4771 16.0 - -4.697(2) 1 lO(3) l4NISN - - - HF 3177.7361 17.1 - 4.978(3) - 115(5) lSN1'N * * HF 3101.1081 16.0 - - 105(1) ' S, is the H, 19F nuclear-spin-nuclear-spin coupling constant; see ref.(7) for details. atoms is such that N(2) is that involved in the weak binding. The difference AX,, = x,,(l) - x,,(2) = -0.286 MHz is significant and can be shown to be a lower limit to the equilibrium value of this quantity. Taking Ax,, as the equilibrium value allows us to discuss the electrical changes that occur in the N2 molecule on dimer formation. Two electrical effects make contributions to Ax,,: first, the presence of the HF molecule gives rise to different electric field gradients, and therefore nuclear quadrupole coupling constants, at N(l) and N(2) and, secondly, the electron polarization of the nitrogen molecule by the HF molecule has a similar result.As the electric multipole moment expansion of HF and the dimer geometry are known, the first of these contributions can be calculated (including the effects of Sternheimer shielding at the N nuclei) and hence the difference in the I4N nuclear quadrupole coupling constants caused by polarization of N2 by HF, Ax:, = -0.33 MHz, can be estimated. An interpretation of Ax:, is possible in terms of a simple valence-bond model. The valence-bond structures assumed to contribute to the N2 * HF dimer ground state are shown in table 3. Using the Townes-Dailey model for nuclear quadrupole TABLE 3 .-14N-NUCLEAR QUADRUPOLE COUPLING CONSTANTS, X/MHZ, FOR VALENCE- BOND STRUCTURES OF N2 * * ' HF valence-bond structure X U ) a X(2) Ax ~~~ ~ ~~~~~~~ (I) N=N * * H-F -5 -5 0 (11) k=N * * * H-F -12.5 0 - 12.5 (111) N=6 - - * H-F 0 - 12.5 f12.5 x values calculated for N, N+ and N - using the Townes-Dailey model [see C.H. Townes and A. L. Schawlow, Microwme Spectroscopy (McGraw-Hill, New York, 1955), p. 2391. coupling constants allows the values x (1) and x (2) shown for each structure in table 3 to be estimated. The final assumption is that dimer formation stabilizes structure I1 relative to structure 111. Since transfer of 1 electron from N(l) to N(2) in I to give I1 results in Ax = -12.5 MHz, the observed value corresponds to the analogous transfer of ca. 0.03e as HF takes up its equilibrium position.A.C . LEGON A N D D . J . MILLEN 77 (iii) A DISCUSSION OF DIMER PROPERTIES BASED ON A SIMPLE ELECTRON-PAIR MODEL FOR THE ACCEPTOR MOLECULE Now that a number of simple hydrogen-bonded dimer geometries is available from rotational spectroscopy, it is possible to present a discussion of dimer properties with the aid of a classification based on an electron-pair (n and n type) model for the acceptor molecule B. Each of the four classes (a) one non-bonding pair, (b) two non- bonding pairs, (c) three non-bonding pairs and (d) n-bonding pairs will be discussed in turn. (a) ONE NON-BONDING ELECTRON PAIR ON ACCEPTOR ATOM The dimers HCN - H F and N2. - HF, both of which are linear species and have been discussed above, are isoelectronic.We can understand the preferred geometry if the H atom in HF seeks the non-bonding pair on an N atom in each case, with the HF molecule lying along the supposed axis of the conventionally viewed non- bonding pair. In HCN there is only one non-bonding pair while in NZ, of course, there are two equivalent pairs. A third molecule forming a dimer in the isoelectronic series is carbon monoxide, which has two inequivalent axial non-bonding pairs. If, as is observed in the above examples, the HF molecule prefers to bind to a non-bonding rather than a n-bonding pair, the question arises as to which non-bonding pair is the better acceptor. Mixtures of CO and HF have been examined using the pulsed-nozzle Fourier- transform spectrometer and exhibit rotational spectra characteristic of a linear mole- ~ u l e .~ . ~ The J = 1-0 transition attributed to a species (C0,HF) is shown in fig. 5. frequency/MHz FIG. 5.-J = It0 transition in the vibrational ground state of l6OI2C * * * H19F showing H, 19F nuclear spin-nuclear-spin hyperfine structure. Each component is split further into a doublet as a result of the phenomenon of Doppler doubling. Frequencies are offset from 6127 MHz. There are four hyperfine components arising from the effects of H, 19F nuclear-spin- nuclear-spin splitting and each of these is further split into a doublet by a spectro- meter artefact. Spectroscopic constants have been obtained for five isotopic species, which allows five values for the distance r,(C F) to be calculated under the usual assumption that monomer geometries survive dimer formation.Only if the atoms78 ROTATIONAL SPECTROSCOPY OF HYDROGEN-BONDED DIMERS are assumed to be in the order 0-C * - HF does the internally consistent set of values displayed in table 4 result. This demonstrates unambiguously that the hydro- gen atom is bound to the carbon atom. TABLE 4.--r,(C - - - F) DISTANCES FOR OC * . - HF isotopic species ro(C * * * F)/A a Linear model, CO and HF bond lengths assumed unchanged on dimer formation. In the isoelectronic series B - HF, where B = HCN, N2 or CO, it is important to obtain and compare the strengths of the intermolecular binding as measured by D, andf,. While these are directly available for B = HCN from equilibrium rotational spectroscopy [see section (i)], they must be obtained less directly for B = N2 and CO.The method used relies on the pseudodiatomic approximation for B - - HF and the assumption that the hydrogen-bond stretching potential function is of the Lennard- Jones type.2 Thus the centrifugal distortion constant DJ is related to fa and D, as follows: and Dj = 4BZ/o: (4) D, = for2/72. (5) TABLE 5.-cOMPARISON OFf& Go and D, DETERMINED BY TWO METHODS FOR HCN - - * HF 27 200 24.8 26 1 9 7 i 15 26.1 f 1.6 diatomic approximation a experimental values L. W. Buxton, E. J. Campbell, M. R. Keenan, A. C . Legon and W. H. Flygare, unpublished bA. C. Legon, D. J. Millen and S. C. Rogers, Proc. R. SOC. London, Ser. A , 1980, 370, results. 213. Table 5 compares values so calculated for HCN - * HF with the more directly deter- mined values.The agreement in table 5 is sufficiently good to allow use of this method to make the comparison within the isoelectronic series shown in table 6. We TABLE 6.-COMPARISON OF BINDING STRENGTHS IN THE ISOELECTRONIC SERIES B a * * HF (B = HCN, CO, N,)A . C. LEGON AND D . J . MILLEN 79 note that, in this series, we generate the next member by moving a proton from the extreme left-hand nucleus in the dimer to the adjacent nucleus. Evidently, in proceed- ing from HCN to N2, the nitrogen atom thus becomes a worse acceptor. In the next step, we therefore assume that the oxygen atom becomes a worse acceptor than the carbon atom. Consequently, the most stable dimer of HF and CO has the atomic order OC Moreover, the carbon atom in CO has become a better acceptor than the nitrogen atom in NZ.It is also of interest to compare the strengths of binding calculated on the basis of the pseudodiatomic model within the series (HCN, HX), where X = F, C1 and Br, and within the corresponding series (OC, HX), for which the results are recorded in table 7.8-13 The order F > C1 > Br in each case is as expected. HF. TABLE 7.-cOMPARISONS OF THE BINDING STRENGTHS IN THE SERIES HCN ’ ’ HX AND OC - . HX WHERE X = F, C1, Br HC4N * - * HF 27 200 24.8 HCi4N * - H3’Cl 11.2 111 14.3 HC15N - - * H79Br 8.5 85 12.0 OC * Hi9F 10.8 125 11.8 oc . - - ~ 3 5 ~ 1 4.5 69 6.8 OC - * H79Br 3.3 52 5.6 In summary, it is found for a number of hydrogen-bonded dimers in the series B * - - HX (where B = HCN, N2, CO and X = F, C1, Br) that the geometries are all linear even though binding energy varies by as much as a factor of five.Thus, in all cases the H-X direction coincides with that of a non-bonding pair. This conclusion has also been shown to apply to RCN - * - HF, where R = CH3,4 (CHJ3C l4 and NC,I5 and to HCN - HCN.16*17 (b) TWO NON-BONDING ELECTRON PAIRS ON ACCEPTOR ATOM In order to test more severely the conclusion of the preceding paragraph it is necessary to investigate the geometries and properties of dimers in which the acceptor atom in B has more than one non-bonding pair of electrons. A convenient starting point is provided by molecules B in which an oxygen atom is the acceptor. The simplest possible hydrogen-bonded dimer in which an oxygen atom is the acceptor is (H20, HF). The rotational spectrum of this species has been investigated in an equilibrium gas mixture of water and hydrogen fluoride by using a Stark- modulation microwave s p e c t r ~ m e t e r .~ ~ * ~ ~ The pattern of rotational transitions is characteristic of that of a very nearly prolate rotor with a 3 : 1 nuclear spin statistical weighting of intensities appropriate to a molecule in which a pair of equivalent protons is exchanged by the operation C;. Moreover, the magnitude of the observed rotational constants accords with a dimer model in which H20 is the proton acceptor and HF is the proton donor. These facts exclude all geometries for (H,O,HF) except the C,, planar model and the C, non-planar form with a barrier to inversion of the configur- ation at the oxygen atom sufficiently low that the vibrational wavefunctions can be classified according to the C2, symmetry point group.It is crucial now to distinguish experimentally between the C, and Cfv equilibrium conformations. The equilibrium conformation of H20 * - HF has been established unambiguously80 ROTATIONAL SPECTROSCOPY OF HYDROGEN-BONDED DIMERS by a careful analysis 2o of the rotational spectra in excited vibrational states that are associated with the low-energy hydrogen-bond modes, the form of each of which is shown schematically in fig. 6. Vibrational satellites in the rotational spectrum have **.*4$ .... Q 1 % FIG. 6.-Diagrammatic representation of the low-frequency normal modes of H20 * * HF. been identified for a number of states, including (1 ,O,O), (2,0,0), (0,l ,O) and (0,2,0) where the notation (us(,), up(i), u,) is used.Three types of information gained from the satellite spectra have been employed, First, the variation of the rotational con- stants with Fig. 7 shows how (B + C), varies with and ug(i, will be considered.A . C. LEGON A N D D . J . MILLEN 81 each of ~ g ( ~ ) and vg(i). We merely note for the moment the striking difference in behaviour. Secondly, the vibrational spacings v = It0 and 2+l shown in table 8 TABLE 8 .-VIBRATIONAL SEPARATIONS FOR THE LOW-FREQUENCY HYDROGEN-BOND MODES AND VS(i) OF HzO * - HF vibrational separation/cm- mode v = It0 v = 2 4 VB(0) 64f 10 267 d= 35 VB(i) 157f 10 330 f 30 have been obtained from relative intensity measurements of the appropriate vibrational satellites for states associated with each of the modes vpt0) and V/j(i).We note that, while vB(i) exhibits effectively harmonic behaviour, the order of the spacing in vg(o) is 1 +O < 24- 1. A similar contrast has already been noted in the variation of (B + C), with v , where the mode Vpti) shows familiar behaviour but that of vg(o) is irregular. These results for v ~ ( ~ ) are characteristic of vibrational states governed by a double- minimum potential-energy function that has a low barrier, examples of which are known from the study of puckering modes in small ring molecules. The qualitative conclusion must be therefore that the equilibrium conformation of H20 . HF is that with C, symmetry. By using the usual one-dimensional approximation, we can determine the equili- brium value of the out-of-plane angle p (as defined in fig.8) simultaneously with a FIG. 8.-Out-of-plane angle p in HzO HF. determinatipn of the quantitative form of the potential function Y(p). We take the optimum potential-energy function Y(q) to be that which best reproduces the observed vibrational spacings in V B ( ~ ) and the observed variation of ( B + C), with V B ( ~ ) . The result is numerically : V(V)/cm-l = 328p4 - 406p2 (6) and graphically as shown in fig. 9, in which vibrational energy levels are also included. We have also used the observed variation of the electric dipole moment p with vg(o) as an additional constraint. A curvilinear model for the motion of the hydrogen atoms was assumed in order to obtain V(p) from Y(z), where z is a dimensionless reduced coordinate, in terms of which such calculations are conveniently made. It can be seen from fig.9 that the equilibrium value of 9 is 46" and that the barrier height is 126 cm-'. The value of p is not far removed from the value of approxim- ately half the regular tetrahedral angle expected if the four electron pairs (two bonding, two non-bonding) were tetrahedrally disposed about the central oxygen atom and if, at equilibrium, the HF molecule were to lie along the supposed axis of a non-bonding pair, as conventionally envisaged. We note that if the HF molecule is considered bound to a given non-bonding pair, the other such pair would exert a cooperative82 ROTATIONAL SPECTROSCOPY OF HYDROGEN-BONDED DIMERS effect on its binding in a manner that reduces both the barrier at the planar conforma- tion and the equilibrium value of p.The fact that a low barrier is observed and a consequent large vibrational amplitude occurs for H20 - * - HF even in the zero-point state draws attention to the need for caution in evaluating geometrical parameters from ground-state rotational constants in molecules of this type. Now that the non-bonding pair approach is found to be in accord with the I I I I I I I -80 - 4 0 0 4 0 8 0 V1° in fig. 8. FIG. 9.-Variation of the potential energy V(V) with p in H20 HF. The angle rp is defined equilibrium geometry of the dimer H,O - * HF, it is important to examine the con- figuration at oxygen in hydrogen-bonded dimers in which the angle between the non- bonding pair axes is expected to differ significantly from that in water.Itjs commonly assumed that for the molecules oxiran and oxetan the decreased angle COC is accom- panied by an increased angle between the non-bonded pair axes. The three molecules B = H20, (CH&O and (CH2)20 thus form a series in which the angle / \ pro- gressively decreases (104'3 1 ', 91 "44' and 61 "38') while presumably the angle between the non-bonded pairs increases correspondingly. Rotational spectra have accordingly been observed for dimers formed by hydrogen fluoride with each of oxiran 21 and oxetan.22 Ground-state rotational constants have been evaluated and used with the assumption of unchanged monomer geometries on dimer formation to determine the quantities r,(O - . F) and p in each case. Fig. 10 compares these two quantities for B - - HF where B = H20, (CH2)30 and (CH2),0.0 The striking result is that q increases in a manner that parallels the decrease in the/ \ angle noted above, a result which is consistent with the simple non-bonded pair approach. A further result which fits the general pattern is that for H20 * * HOH, where a value of p =58 & 6" (not far removed from half-tetrahedral) has been reported.23 Two other gas-phase dimers involving hydrogen bonding to oxygen should be men- 0A . C . LEGON A N D D . J . MILLEN 83 FIG. 10.-Comparison tioned. For N,O * of uo(O * F) and v, values in B HF, where B = H20, (CH2)30 and (CHAO. H F the angle between the N,O and HF axes has been reported 24 to be 47" (which is approximately the value expected from the simple model) while for C 0 2 - - - H F a linear geometry has been reported.2s (C) THREE NON-BONDING ELECTRON PAIRS ON ACCEPTOR ATOM Only two examples have been reported in which the acceptor atom in B has three non-bonded pairs of electrons.Both of these examples involve hydrogen bonding to the F atom in HF; one is (HF),, in which HF is also the proton donor and the other is HF - Both species have been investigated by Klemperer et al. using the molecular-beam electric-resonance technique. A detailed study of the HF dimer and its deuterated species 26 establishes a non-linear model for the geometry in which the F F distance is 2.7 & 0.05 8, and the HF unit acting as the acceptor is bent ca. 70" from the F - * * F axis. A similar non-linear geometry 27 holds for H F - * - HCl.These observations can be interpreted in terms of the non-bonding pair model if it is assumed that the three non-bonding pairs and the bonding pair in HF are disposed tetrahedrally about the fluorine nucleus. Then if the axis of the HF donor molecule coincides with the axis of one non-bonding pair an angle corresponding to that reported (ca. 70") would result. HCl, in which HCl is the proton donor. (d) "C-BONDING ELECTRON PAIRS WITH NON-BONDING PAIRS In the isoelectronic series B - - HF (where B = HCN, N,, CO) discussed above, the acceptor molecule has n-bonding electron pairs as well as non-bonding pairs. In view of the linear geometry established in each case, the proton of H F evidently prefers to seek the axis of the non-bonding pair rather than the region of high n- electron density. It is, therefore, of interest to enquire into the geometry of the dimer in which the isoelectronic acceptor molecule B is acetylene, since this has no non-84 ROTATIONAL SPECTROSCOPY OF HYDROGEN-BONDED DIMERS bonding valence electrons.Recently the rotational spectrum of (HC=CH, HC1) in its vibrational ground state has been detected by the technique of pulsed-nozzle, Fourier-transform microwave spectroscopy.28 The rotational constants have been interpreted unambiguously in terms of the T-shaped geometry for the dimer shown diagrammatically in fig. 1 l(a). The HC1 molecule lies along the C2 axis of acetylene, its hydrogen atom pointing at the centre of the C=C bond. The distance from the mid-point of C r C to C1 is obtained under the usual assumption of unchanged monomer geometry.This is the first example of a hydrogen bond to a n-bond detec- ted in the gas phase. A similar investigation of (ethylene, HCl) 29 leads to the geometry shown in fig. ll(b). Again the HCl molecule lies along the C2 axis that is in free ethylene per- C FIG. 11 .-Dimer geometries for B * HCl, where B is (a) acetylene, (b) ethylene and (c) cyclopropane. pendicular to the molecular plane. This result shows conclusively that the proton in HC1 seeks the region of maximum electron density offered by the n-bond, which is along the axis in question. Cyclopropane is well-known to behave in some ways like an unsaturated hydro- carbon. It is of interest that it forms a dimer with HCl of the geometry shown in fig. 11(~).3'-~~ The HCl axis coincides with a median of the cyclopropane equililateral triangle and the H atom points at the centre of a C-C bond.This result accords with the Coulson-Moffitt model of cyclopropane, which predicts a region of highA . C . LEGON AND D. J . MILLEN 85 electron density on the median but displaced away from the ring, and which for the purposes of hydrogen bonding allows cyclopropane to be viewed, at the edge of interest, as a distorted ethylene molecule. Finally, by use of the pseudodiatomic model [eqn (4) and ( 5 ) above] it is possible to compare the quantities (fo and D,) that define the strength of intermolecular bind- ing in a number of molecules B - - - HCl. These quantities are recorded for the TABLE COMPARISON OF BINDING STRENGTHS IN THE SERIES B - - HCl Ar 1.2 32 1.5 oc 4.5 69 6.8 H2C=CH2 6.6 84 7.5 HC=CH 6.9 88 7.7 (CH2)3 8.7 87 11.5 HCN 11.2 1 1 1 14.3 a Ref.(33). series B = Ar,33 OC, HC=CH, H2C=CH2, cyclopropane and HCN in table 9. The increase in these quantities from B = Ar to B = HCN is understandable in terms of the expected electron-donor ability of B. C 0 N C L U S I 0 N S An important conclusion about the preferred equilibrium geometries can be drawn from the results obtained for the variety of simple hydrogen-bonded dimers discussed in the preceding section. Care must be taken in drawing general conclusions about geometry because, as already noted for H20 - - * HF, distortion from equilibrium by bending the hydrogen bond costs little in energy. In case it might be thought that, because of the double-minimum potential-energy function, the ease of hydrogen-bond bending in H20 * HF is unique, we now present an analysis which shows that bend- ing of this type is relatively easy even when it is governed by a single-minimum potential-energy function.This is followed by a simple rule in terms of which the preferred equilibrium geometries can be understood, even though the energetic stability of the observed conformer is so small. It is possible to examine the energetics of angular distortion of the hydrogen bond in CH,CN * HF for each of the angles shown in fig. 3. Values of the energy AE required to produce angular distortions of A0 and A9 in the range 0-30" are given in table 10. The calculations are based on the harmonic force constants [given in ref.(4)]. Although it has not been established that the force field is harmonic over that range, the nearly uniform spacings observed in the satellite progression for several quanta of the bending mode v p suggest that for this mode at least the harmonic assumption may well be a good approximation. Table 10 also gives distortion ener- gies as a percentage of the dissociation energy based on the value of D, for HCN - - - HF, since that for CH,CN - * HF is not available. The value of D, for the latter is probably somewhat larger and so percentages in table 10 are likely to be a little over- estimated. It is seen that distortion from linearity at N is relatively easy to bring about ; even a distortion of A8 = 30" reduces the hydrogen-bond energy by only 15%.By contrast distortion away from linearity at H leads to a more rapid reduction in binding energy. Thus for Arp = 30" the result is a reduction in binding energy of86 ROTATIONAL SPECTROSCOPY OF HYDROGEN-BONDED DIMERS TABLE ~~.-HYDROGEN-BOND DISTORTION ENERGIES FOR CH,CN * - HF distortion A9 AP angle/" AElkJ mol-' AEID, AEIkJ mol-' AEID, 10 0.4 2% 1.5 6% 20 1.7 7% 5.9 22 % 30 3.8 15% 13.8 53% 53%. These findings can readily be interpreted in terms of repulsion energy between N and F that rises rapidly as H moves away from the N * * - F line. This is in contrast to the situation in which the hydrogen bond remains linear (A0 > 0, Av) = 0), when resistance to bending is small. Although large changes in the angle 0 (see fig. 3) can be made at the cost of very little energy, a simple rule can be proposed which summarises the equilibrium geo- metries for the species B - * - HX discussed in this paper and which can be assumed to apply to gas-phase, hydrogen-bonded dimers generally : THE RULE The gas-phase geometry of a dimer B - HX can be obtained in terms of the non- (i) the axis of the HX molecule coincides with the supposed axis of a non-bonding bonding and n-bonding electron pairs on B as follows: pair as conventionally envisaged, or, if B has no non-bonding electron pairs but has n-bonding pairs, (ii) the axis of the HX molecule intersects the internuclear axis of the atoms form- ing the n-bond and is perpendicular to the plane of symmetry of the n-orbital.Rule (i) is definitive when B has both non-bonding and n-bonding pairs.The investigations reported here that use the pulsed-nozzle, Fourier-transform microwave spectrometer were carried out in collaboration with the late W. H. Flygare while one of us (A. C . L.) was on sabbatical leave at the University of Illinois in 1980. We thank L. C . Willoughby for obtaining the spectrum reproduced in fig. 1 . A research grant from the S.R.C. is gratefully acknowledged. A. C. Legon, D. J. Millen and S. C . Rogers, Proc. R. SOC. London, Ser. A, 1980, 370, 213. T. J. Balle, E. J. Campbell, M. R. Keenan, and W. H. Flygare, J. Chem. Phys., 1980, 72, 922. T. J. Balle and W.H. Flygare, Rev. Sci. Instrum., 1981, 52, 33. (a) J. W. Bevan, A. C. Legon, D. J. Millen and S. C. Rogers, Proc. R. Soc. London, Sev A, 1980, 370,238. A. C. Legon, D.J. Millen, P. J. Mjoberg and S. C. Rogers, Chem. Phys. Lett., 1978,55, 157. A. C. Legon, D. J. Millen and S. C. Rogers, J. Mol. Spectrosc., 1978, 70, 209. P. D. Soper, A. C . Legon, W. G. Read and W. H. Flygare, J. Chern. Phys., 1982, 76, 292. * A. C. Legon, P. D. Soper, M. R. Keenan, T. K. Minton, T. J. Balle and W. H. Flygare, J. Chew Phys., 1980, 73, 583. A. C. Legon, P. D. Soper and W. H. Flygare, J. Chem. Phys., 1981, 77, 4944. lo A. C. Legon, E. J. Campbell and W. H. Flygare, J. Chem. Phys., 1982, in press. l1 E. J. Campbell, A. C. Legon and W. H. Flygare, J. Chem. Phys., 1982, in press. l2 P. D. Soper, A. C . Legon and W. H. Flygare, J. Chem. Phys., 1981,74,2138. l3 M. R. Keenan, T. K. Minton, A. C . Legon, T. J. Balle and W. H. Flygare, Proc. Natl Acad. l4 A. S. Georgiou, A. C. Legon and D. J. Millen, Proc. R. SOC. London, Ser. A, 1980, 370, 257. (b) R. K. Thomas, Proc. R. SOC. London, Ser. A, 1971,325, 133. Sci. USA, 1980,77, 5583.A . C . LEGON AND D . J . MILLEN 87 l5 A. C. Legon, P. D. Soper and W. H. Flygare, J. Chem. Phys., 1981, 74, 4936. l6 A. C. Legon, D. J. Millen and P. J. Mjoberg, Chem. Phys. Lett., 1977, 47, 589. l7 L. W. Buxton, E. J. Campbell and W. H. Flygare, Chem. Ph-ys., 1981,56, 399. la J. W. Bevan, A. C. Legon, D. J. Millen and S. C. Rogers, Chem. Commun., 1975, 341. l9 J. W. Bevan, Z . Kisiel, A. C . Legon, D. J. Millen and S. C . Rogers, Proc. R. SOC. London, Ser. A , 2o Z . Kisiel, A. C. Legon and D. J. Millen, Proc. R. SOC. London, Ser. A , in press. 21 A. S. Georgiou, A. C . Legon and D. J. Millen, Proc. R. Soc. London, Ser. A, 1981,373, 511. 22 A. S. Georgiou, A. C. Legon and D. J. Millen, J. Mol. Struct., 1980, 69, 69. 23 T. R' Dyke, K. M. Mack and J. S. Muenter, J . Chem. Phys., 1977, 66, 498. 24 C. H. Joyner, T. A. Dixon, F. A. Baiocchi and W. Klemperer, J . Chem. Phys., 1981, 74, 6550. 25 F. A. Baiocchi, T. A. Dixon, C. H. Joyner and W. Klemperer, J. Chem. Phys., 1981, 74, 6544. 26 T. R. Dyke, B. J. Howard and W. Klemperer, J. Chern. Phys., 1972,56,2442. '' K. C. Janda, J. M. Steed, S. E. Novick and W. Klemperer, J. Chern. Phys., 1977, 67, 5162. A. C. Legon, P. D. Aldrich and W. H. Flygare, J. Chem. Phys., 1981,75, 625. 29 P. D. Aldrich, A. C . Legon and W. H. Flygare, J. Chem. Phys., 1981, 75, 2126. 30 A. C. Legon, P. D. Aldrich and W. H. Flygare, J . Am. Chem. SOC., 1980, 102, 7584. 31 A. C. Legon, P. D. Aldrich and W. H. Flygare, J. Am. Chem. SOC., in press. 32 L. W. Buxton, P. D. Aldrich, J. A. Shea, A. C. Legon and W. H. Flygare, J. Chem. Phys., 1981, 75, 2681. 33 Calculated from results given by S. E. Novick, P. Davies, S. J. Harris and W. Klemperer, J. Chem. Phys., 1973, 59, 2273 and S. E. Novick, K. C. Janda, S. L. Holmgren, M. Waldman and W. Klemperer, J. Chem. Phys., 1976, 65, 11 14. 1980,372, 441.

ISSN:0301-7249    

DOI:10.1039/DC9827300071

出版商:RSC    

年代:1982

数据来源: RSC

摘要:

Faraday Discuss. Chem. SOC., 1982, 73, 89-108 Infrared Spectra of Hydrogen-Rare-gas Van der Waals Molecules BY A. ROBERT W. MCKELLAR Herzberg Institute of Astrophysics, National Research Council of Canada, Ottawa, Ontario, Canada KIA OR6 Received 17th December, 198 1 Previously measured spectra of the Van der Waals molecules formed from H2 or Dz molecules and Ar, Kr or Xe atoms have been used to derive detailed potential surfaces for these systems. The success of the analyses has prompted a new program to obtain improved high-resolution spectra which may serve as a basis for refined potentials. The first results of these new studies are reported here. The spectra, which lie in the 2-4 pm wavelength region of the hydrogen stretching vibration, are observed in absorption through a long path in a mixture of hydrogen and the rare gas at low temperature. The new results were obtained with a path of 220 m in a 5.5 m multiple-traversal cell at 77 K for H2-Ar, HD-Ar and D2-Ar, and at 97 K for H2-Kr.A 2 m vacuum infrared grating spectrometer was used with electronic signal averaging to record the weak spectra. The resulting spectral linewidths of ca. 0.1 cm-l represent an improvement of from 3 to 6 times over the previous spectra, with the result that much greater detail is revealed in the more crowded regions of the spectra, Furthermore, fully resolved spectra were observed in the Q(0) regions of HD-Ar and Dz-Ar which were previously too weak to be studied in detail. All the spectra studied here are subject to pre- dissociation because the upper states of the observed transitions lie at 3000-5000 cm-l, far above the 40-70 cm-' binding energy of the Van der Waals bond.Predissociation is clearly observed as line broadening in the HD-Ar S(0) region, and also detected for one line of Hz-Kr in the S(0) region. In other regions, the predissociation linewidths must be less than ca. 0.1 cm-'. These observations are of interest since predissociation in Van der Waals molecules is currently an area of considerable theoretical activity and there are relatively few experimental measurements. 1 . INTRODUCTION A discrete infrared spectrum due to the H2-Ar Van der Waals molecule was first observed in 1965 by Kudian et aZ.,l shortly after the discovery of an analogous spectrum due to the (HJ2 dimer. Further studies resulted in similar spectra due to H2-Kr and H2-Xe,3 and aIso in rather more diffuse spectra due to H2-N, and H,-C0.4 These investigations were extended to considerably higher resolution and to the species containing D2 by McKellar and Welsh,5 whose spectra showed direct evidence of the effects of anisotropy in the hydrogen-molecule-rare-gas-atom intermolecular potential functions.Similar experiments were also performed on H2-Ne, D,-Ne and HD-Ar.7 Subsequently there has been a considerable amount of theoretical work 8-14 performed to analyse these high-resolution results and to extract potential-energy surfaces from them. The combination of the spectroscopic data with data from various molecular-beam scattering experiments has thus resulted in increasingly precise determinations of the detailed forms of the hydrogen-rare-gas interaction poten- t i a l ~ .~ ~ J ~ The great success of these studies suggests that new experimental data on the spectra would be very worthwhile as a basis for refined analyses of the inter- molecular potentials. The present paper describes the first results of a program to obtain such improved high-resolution spectra for the hydrogen-rare-gas systems. Spectra of the molecules90 HYDROGEN-RARE-GAS SPECTRA H,-Ar, HD-Ar, D,-Ar and H,-Kr have been measured with the resolution improved by a factor of from 3 to 6 compared with the best previous experiment^,^ higher resolution results in much greater detail being uncovered in the more crowded regions of the spectra. Furthermore, fully resolved spectra have been obtained in spectral regions [Q(O) in HD-Ar and D,-Ar] which were previously too weak to study in detail, and definite evidence has been seen of the effects of predissociation on the spectra: such observations furnish an additional sensitive test of intermolecular potentials.The spectra studied here occur in the region of the middle infrared (A = 2.0-3.5 pm), and they accompany the vibration-rotation transitions in the fundamental bands (u = l+O) of the hydrogen molecules: HZ, HD and D,. They are observed by relying on the very small equilibrium concentrations of the Van der Waals species, and using long absorption paths (> 10 m), moderate gas pressures (0.05-5 atm *) and low temperatures (in the region of the boiling point of the rare gas).If a high-resolu- tion spectrometer is used, the linewidths in the spectra are generally limited by pres- sure broadening. Higher-resolution spectra thus require lower density (pressure) and hence longer pathlength and/or increased sensitivity to weak absorption features. Since the equilibrium concentration of Van der Waals molecules varies approximately as the density squared, the variation in the required pathlength/sensitivity is much stronger than for a normal (monomer) molecule (indeed, for a normal molecule in the pressure-broadened regime, the peak absorption of an isolated line is constant with density). In the present work, the pathlength, sensitivity and instrumental resolution are all increased relative to earlier studies.'- An example of the improve- ment is shown in fig.1, which shows part of the spectrum of the H,-Ar molecule observed with the conditions of ref. (l), (3) and (9, and the present study. As the sample density is further reduced, the resolution in the spectra improves until the linewidths become limited by Doppler broadening or by the finite lifetime of the Van der Waals molecules in their upper states. All the spectra studied here are subject to such predissociation effects, because the upper states lie at energies of 3000- 5000 crn-', which are far above the 40-70 cm-l binding energies of the Van der Waals bonds. However, the predissociation lifetimes may vary greatly with the particular species being studied and with the various quantum states of each species, and in most cases the lifetime-limited linewidths have not been reached experimentally.In one sense, the spectra studied in this paper constitute a component of the well- known collision-induced spectrum l6 of hydrogen. Short-lived binary collisions between, say, H, molecules and Ar atoms in an H, + Ar mixture give rise to charac- teristically broad (ca. 100 cm-l) collision-induced absorption lines, while the few H2 molecules that happen at any moment to be bound to Ar atoms by the weak Van der Waals forces give rise to much sharper spectral lines arising from transitions between bound states of the Van der Waals molecule. In both cases, the mechanism for absorption is the dipole moment induced by the proximity of the H, to the Ar. The overall appearance of the fundamental band with these two contributions may be seen, for example, in fig. 1 of ref.(3) or (7), or fig. 2 of ref. (6). Electric dipole vibration-rotation transitions are, of course, forbidden for isolated H, and DZ, and very weak for HD. Even weaker electric quadrupole transitions occur in all the hydrogen species. A hydrogen-rare-gas molecule is best understood starting with the approxiniation that the hydrogen molecule within the Van der Waals complex is completely free to vibrate and rotate. This assumption is a very good one; even in solid hydrogen, rotation and vibration remain remarkably unperturbed. The energy levels of H,-Ar * 1 atm = 101 325 Pa.A . R . W. MCKELLAR 91 L t- I I 1 I I 4480 4490 4500 4510 4520 wavenumber Icm- FIG. 1,-Spectrum of the H2-Ar Van der Waals molecule accompanying the S(0) transition of H2 under a range of experimental conditions.Traces (a), (b), (c) and ( d ) correspond, respectively, to the results of ref. (l), (3) and ( 5 ) and the present work; the temperature, density and absorption path for each trace are as follows (1 amagat is the density of a gas sample at 1 atm pressure and 0 "C temper- ature): (a) 99 K, 9.6 amagat, 13 m; (b) 91 K, 4.0 amagat, 48 m; (c) 86 K, 1.2 amagat, 165 m; ( d ) 77 K 0.3 amagat, 220 m. The vertical (absorption) scale is not common for the four traces. are then given simply by the sum of two diatomic molecule energies, that of the free H, molecule (quantum numbers v and j ) and that of the pseudodiatomic H,-Ar molecule (quantum numbers n and Z for vibration and rotation).The resulting spectrum consists of small, closely spaced Van der Waals bands centred around each of the widely spaced H2 vibration-rotation transitions. The angular dependence of the induced dipole moment leads to rotational selection rules of Aj = 0, &2 and AZ = & 1, &3 (with only AZ = & l allowed when j = O+O). At the low temperatures used in the experiments, only the Q(0) ( j = O+-0), Q(l) ( l t l ) , S(0) ( 2 ~ 0 ) and S(1) ( 3 t l ) transitions are observed for species containing H,, and only Q(0) and S(0) for those containing HD or DZ. The free vibration and rotation of the hydrogen within the Van der Waals molecule are in fact slightly hindered by the dependence of the intermolecular potential on the hydrogen internuclear distance and by the anisotropy of the potential with respect to hydrogen rotation.The vibrational perturbation is manifested as a slight shift (<0.05%) in the free hydrogen vibrational frequency. The anisotropy couples the angular momenta j and I to form the resultant total rotational angular momentum J.92 HYDROGEN-RARE-GAS SPECTRA Each rotational level denoted by I is split into 25' + 1 (or 21 + 1 if I < j ) sublevels labelled by different values of J. The resulting transitions of the Van der Waals molecule are thus shifted or split, though these effects are only apparent in the spectra a t fairly high resolution. 2. EXPERIMENTAL The results presented here were obtained using a new low-temperature multiple-traversal absorption cell. The cell has a base pathlength of 5.5 m, and was generally used with 40 traversals, giving a total path of 220 m [cf.165 m used in ref. (5)]. It is constructed of stain- less steel, and consists of an innermost tube containing the sample gas and multiple-reflection l7 mirrors, a surrounding space which may be filled with liquid coolant, and an outermost vacuum jacket for thermal insulation. The inner diameter of the sample tube is 12 cm and the outer diameter of the cell is 22 cm. Experiments on the molecular species containing Ar were performed with liquid-nitrogen coolant at 77 K, whereas those involving H,-Kr were performed at ca. 97 K, which was obtained by maintaining an overpressure of ca. 5 atm on the liquid N2 by means of a pressure-relief valve. Spectra were recorded with a 2 m vacuum grating spectrometer,18J9 using a 72 line mm-' grating (in the 7th to 11 th orders), a cooled PbS detector and a carbon-rod continuum source.The instrumental resolution was generally in the range of 0.06 to 0.10 cm-l. The optical path from source to cell and cell to spectrometer was evacuated to minimize the effects of strong atmospheric water-vapour absorption in the 2.5-3.3 pm region. In order to achieve the best possible sensitivity to weak absorption lines, a signal averag- ing technique was used to achieve long integration times (5-30 h) over the relatively limited (ca. 40 cm- ') spectral regions surrounding each hydrogen transition frequency. For this purpose, the spectrometer was operated in a very fine scanning mode achieved by moving its output (" camera ") mirror on a 2 m long pivot arm by a precision lead screw.Provision for this very slow scan mode had been incorporated in the original spectrometer design by Douglas,18 but it was never used because the normal (grating rotation) scanning mode proved satisfactory. By driving the fine scan mode lead screw with a digitally controlled '' stepless " l9 stepping motor, an extremely reproducible scan of any limited wavelength region could be achieved. This mechanical stability made it possible to use a digital signal averager (Nicolet model 535) to accumulate many successive spectral scans without any loss in resolution due to lack of coincidence between scans. Typically, 30 scans of 2048 points and 25 min duration were accumulated during an overnight run; the resulting noise levels were as low as 0.1 % of the continuum.It was thus possible to operate with low sample pressures that gave peak absorptions of only a few percent in the Van der Waals transitions and still obtain good quality spectra. This fact and the increased pathlength were responsible for the improved resolution of the present results. The Nicolet signal averager was inter- faced to a desk-top computer and plotter (Hewlett-Packard models 9825 A and 7225 A) which enabled spectra to be smoothed, calibrated, measured and plotted after each run. The figures presented in this paper result directly from these plots; an example of the raw output of the signal averager is shown in fig. 2. The slope and curvature of the background in this spectrum, which is exaggerated by the suppression of zero on the vertical scale, is due to the variation of the grating, source and detector sensitivities as a function of wavelength, and to the spectral band pass of the spectrometer's order-sorting prism.The appearance of the same spectrum after calibration, smoothing and background flattening is shown below in section 3 ( 4 . Wavelength Cali bration was accomplished using visible neon emission lines recorded in high grating orders before or after each run. The absolute calibration was usually checked using absorption standards recorded simultaneously with the Van der Waals spectra. These standards included the v3 band 'O of CH4 [present as an impurity in the D2-Ar Q(0) spectra], the 2-0 band 21 of CO (introduced at low pressure into the spectrometer tank), the v3 band 24 of H20 (residual water vapour present in the spectrometer vacuum tank), and the hydrogen quadrupole transitions themselves.The frequencies assumed for the free hydrogen tran- sitions are listed in table 1. Le Roy and Van Kranendonk ' point out that there is a problemA . R . W . MCKELLAR 93 70 0 4 0 0 800 1200 1600 2000 channel number FIG. 2.-Raw spectrum of the H,-Kr molecule in the H2 S(0) region. The sample temperature was 97 K, and the total pressure was 108 Torr in an approximately equal mixture of para-Hz and Kr. The spectrum shown is the signal-averager output resulting from the sum of 25 scans of duration 32 min each. The same spectrum after processing is shown later in fig. 11. TABLE 1 .-ASSUMED TRANSITION FREQUENCIES (IN cm- l) FOR FREE HYDROGEN MOLECULES transition H2 a HD Dz Qi(W 4161.166 3632.152 2993.614 Qd 1) 41 55.254 SdO) 4497.8 3 8 3887.681 3166.359 4712.902 a From a fit (unpublished) to available 0-0 and 1-0 band data; fitted values from ref.(22); fitted values from ref. (23). introduced by the practice of calibrating relative to the hydrogen quadrupole lines since these lines are subject to fairly substantial pressure shifts. They also show that this difficulty can be circumvented since the pressure shifts depend on the very potential parameters that are extracted from the Van der Waals spectra. At any rate, the uncertainties introduced by pressure shifts are very small (t0.01 cm-') in the present work because of the low pressures used. Para-H2 and ortho-D, were prepared by the usual technique of liquefying the hydrogen at ca.20 K in the presence of a chrome alumina catalyst. An ortho-H2 separation apparatus such as used previously was not available for the present study, so normal H2 (75OA ortho) was used to obtain spectra in the Q(l) and S(l) regions for H,-Ar and H2-Kr. 3 . RESULTS (a) H2-Ar The spectrum of H2-Ar in the H, Q-branch region is shown in fig. 3 for para-H2 and in fig. 4 for normal H2. These and successive spectra are shown in terms of percentage transmission, rather than absorption coefficient as in earlier papers. The94 100 98 96 HYDROGEN-RARE-GAS SPECTRA -- -- 10 9c 100 95 ~ I I I 1 ** I I I 4155 41 60 4165 wavenumber/cm - FIG. 3.-Spectrum of H2-Ar accompanying the Q(0) transition of H2. The temperature was 77 K and the sample pressures were 350 Torr (upper trace) and 100 Torr (lower trace) in an approximately equal mixture of pura-Hz and Ar.Present in the lower trace, and denoted by asterisks, are lines of the 2-0 band of CO included for wavelength calibration (the 3 weaker lines are due to 13C160). The vertical arrow indicates the Q(0) frequency of free H2 (there is no line at this position); note the shift between this position and the " centre " of the H2-Ar spectrum. para - H, - I I I 4140 4150 41 60 4170 wavenumber/cm- FIG. 4.-Spectrum of H,-Ar accompanying the Q(1) transition of H2 in a normal H2 + Ar mixture at 79 Torr and 77 K. The short upper trace is the H2-Ar Q(0) spectrum (fig. 3) with intensity scaled to match that due to the 25 % para-H2 present in normal H2.The vertical arrow indicates the Q(l) quadrupale transition of free H t .A . R. W . MCKELLAR 95 para-H, spectrum, which is entirely due to the Q(0) ( j = 060) transition, consists of simple P (AZ = - 1) and R (AZ = + 1) branches and is shown for two different sample pressures in fig. 3. The lower trace has some CO 2-0 band lines superimposed for calibration purposes. The normal H2 spectrum is mostly that of the Q(l) transition ( j = l t l ) , but as indicated above the main trace (fig. 4) it also contains weak lines of Q(0) due to the 25%para-H, in normal H,. In each figure, the transition frequencies of free H2 are marked by arrows, and in fig. 4 this coincides with the quadrupole Q(l) transition arising from the unbound H2 in the absorption cell [there is no Q(0) quad- rupole transition].The small shift (ca. 1 cm-') between the apparent centre of the H,-Ar band and the free H, frequency is evident in each case. The S(0) ( j = 2 t 0 ) spectrum of H,-Ar is shown in fig. 1, and a more detailed view of the central part of the spectrum is shown in fig. 5. The upper trace in fig. 5, wavenumberlcm- FIG. 5.-Central portion of the H2-Ar spectrum accompanying the S(0) transition of H2 in a para- H2 + Ar mixture at 77 K: (a) 60, (b) 153 Torr. The whole spectrum is illustrated in fig. 1. The vertical lines below the lower trace indicate lines for which measured positions are listed in table 2. taken with a gas pressure of 153 Torr (20.3 kPa), already shows more detail than the best previous spectrum; the lower trace, taken with 60 Torr, shows almost complete resolution of the H,Ar P- and R-branch lines.The irregular pattern of these lines (cf. fig. 3) is a direct result of anisotropy in the H,-Ar potential. Even more irregular- ity is evident in the S(l) ( j = 3 t l ) region, shown in fig. 6 ; here each line of the N (AZ = -3) and T (A1 = +3) branches may be split by anisotropy into up to 6 com- ponents. The splittings are especially evident in the T branch (lower trace of fig. 6) and they constitute a very sensitive measure of anisotropy which was not previously utilized 599912 because the lines are not well re~olved.~ Many weaker and blended features, especially in the S(1) spectrum, are not listed here Measured positions of the well resolved lines of H,-Ar are given in table 2.96 9 5.- HYDROGEN-RARE-GAS SPECTRA I I I I I I I I (further details may be obtained on request from the author).The assignments of I and J quantum numbers given in table 2 are straightforward except for the S(0) P and R branches and the S( 1) T branch, where they were obtained from calculated spectra.*v9 For blended features, table 2 either gives no assignment, or only that for 1. The absolute calibration of the Q(0) wavenumbers is relative to CO lines 21 (fig. 3) and those of Q(I), S(0) and S(1) are relative to the respective quadrupole lines, with the frequencies assumed in table 1. The uncertainty of these measurements is <0.02 cm-I. The accuracy is thus improved by a factor of ca. 2 compared with previous measurement^,^ but this improvement is unfortunately not as great as the improvement in resolution.The explanation of this discrepancy lies in the signal-averaging tech- nique used here: variations in coincidence between different scans of less than ca. 0.05 cm-I would not significantly affect the resolution in the final averaged spectrum, but they would deteriorate the line-position measurements. Because of such varia- tions, the measurements were not reproducible to the level that one would normally hope for (ca. 1/10 of a linewidth, or better than 0.01 cm-l). (b) HD-Ar The spectrum of HD-Ar was previously studied with an absorption path of 66 m : the resulting rotational structure of the Van der Waals molecule was essentially un- resolved in the Q(0,I) and S(1) regions and fairly well resolved in the S(0) region.The present experiment, with a 220 m path, results in a well resolved Q(0) spectrum but an only marginally improved S(0) spectrum. The Q(0) spectrum is shown in fig. 7; in this region there are two lines due to residual water vapour, and one ofA . R. W . MCKELLAR 97 TABLE 2.-MEASURED POSITIONS OF LINES DUE TO THE H2-Ar MOLECULE ACCOMPANYING THE FUNDAMENTAL BAND OF H2 assignment assignment assignment vlcm - I LJ vlcm-' 4.J vjcm - I 1,J vlcm-' j = 3 t l assignment 4 J 4153.932 41 54.760 41 55.740 4156.776 41 57.880 41 59.001 41 61.256 41 62.3 3 7 41 63.400 4164.417 4165.365 4166.215 41 66.8 17 5,5 t 6,6 4,4445 3,3 4-4,4 2,2+3,3 1,1+2,2 0,o t 1,l 1,l +-0,0 2,2 t l , 1 3,3 t 2 , 2 4,44-3,3 $5 t 4 , 4 6,6+5,5 7,7 4-6,6 4478.1 50 4480.496 448 3.3 63 4486.51 3 4489.847 4490.49 3 449 1.042 4491.863 4492.828 4493.740 4495.134 4497.838 4498.366 4499.125 4499.306 4499.860 4500.261 4500.45 6 4501 -006 450 1.272 4501.522 4502.036 4502.155 4502.48 6 4502.915 4503.320 4505.81 5 450 8,940 451 1.825 4514.304 4,6 t 7 , 7 3,5 t 6 , 6 2,4 t 5 3 1,3 t 4 , 4 0,2 +3,3 1,2+2,2 quad.2,l t l , l 3,l t 2 , 2 2,2 4- 1,l 3,2 4-2,2 4,2 +3,3 3,3+-2,2 4,3 4-3,3 5,3 +4,4 4,4+3,3 5,4 +4,4 6,4 +- 5,5 3,l t O , O 6,5 +- 5,5 6,6 4-53 4 , 2 t l , l 5,3 t 2 , 2 6,4t3,3 7,5t4,4 41 35.96 4138.315 4141.177 4 1 44.28 5 4147.525 4147.982 41 48 327 4 1 49.806 4150.846 4151.938 41 55.254 4 156.370 41 57.453 4 1 58.476 41 59.43 1 41 60.300 4160.825 4164.028 41 67.098 41 69.951 4172.372 4,5 t 7 . 6 3,44-6,5 2,3 t 5,4 1,2+4,3 0,1+3,2 5 t 6 4 t 5 3 +-4 2 t 3 14-2 quad.24-1 3 + 3 44-3 54-4 6 t 5 3,2+0,1 4,3 t 1,2 5,4+2,3 6,5 t 3 , 4 7,6 +4,5 4693.38 5 4695.769 4698.685 4701.860 4705.258 4709.230 471 2.902 4713.415 47 19.945 4720.340 4722.066 4723.545 4724.172 4726.467 4727.067 4728.978 4729.550 4 t 7 3 t 6 2+5 14-4 Oe3 quad." 4,l t l , l 4 , 2 t l , l 4,l +1,0 5,3 +2,2 5-2 6,4 4-3,3 6+3 7,5 t 4 , 4 7 t 4 ' " quad." indicates quadrupole transitions due to free H2 in the absorption cell. these obscures the I = I t 2 HD-Ar transition. The H20 lines were minimized by prolonged pumping on the spectrometer vacuum tank. The S(0) spectrum is shown in fig. 8, and the measured line positions for HD-Ar are listed in table 3. The upper trace in fig. 8 is the spectrum observed after the absorption cell was filled with HD but before argon was added.The R(2) dipole ( j = 3 t 2 ) and S(0) quadrupole ( j = 2 t 0 ) lines of HD are prominent in this background spectrum, and other weak lines are also present: some of these are due to residual H20 in the spectrometer tank, and others due to an impurity (possibly C2H2) in the HD sample. The broad, weak features in the background (ca. 3880, 3885 and 3889 cm-l) may be due to the (HD)2 Van der Waals dimer, which has not previously been studied [CL (H2)2 and (D2)2 spectra 2*25]. When argon is added to the HD (middle trace of fig. 8), the resulting spectrum shows prominent HD-Ar features, but the lines are less sharp than H,-Ar98 HYDROGEN-RARE-GAS SPECTRA 9 5 1 ! : : : : ! : : ; : ! : : : 3625 3630 3635 wavenumber/cm- FIG.7.-Spectrum of HD-Ar accompanying the Q(0) transition of HD in an HD + Ar mixture at 184 Torr and 77 K. Two H20 lines, due to residual water vapour in the spectrometer vacuum tank, are indicated. The vertical arrow indicates the Q(0) transition frequency of free HD (there is no line at this position). 100 90 n x O 100 3 2 b C -g C Y 90 -164 + 138 Torr Y H D + A r I HD S,(O) quadrupole - HD R , [ 2 ) dipole 3870 388 0 389 0 3900 wavenumberlcm- FIG. 8.-Spectrum of HD-Ar accompaning the S(0) transition of HD. The top trace shows the absorption due to the HD sample alone; it shows the prominent R(2) and S(0) transitions as well as weak lines due to residual H20 and another impurity. When Ar is added (middle trace) the spectrum due to HD-Ar appears, but when the total pressure is lowered (bottom trace) this spectrum does not become significantly sharper (see text).A .R. W. MCKELLAR 99 TABLE 3.-MEASURED POSITIONS OF LINES DUE TO THE HD-Ar MOLECULE ACCOMPANYING THE FUNDAMENTAL BAND OF HD Ql(0)j = O t O S , ( O ) j = 24-0 assignment assignment v/crn-' 1, J v/cm-' 1,J 3625.41 3625.978 3 626.65 3 3627.370 3628.130 3628.348 3628.887 3629.644 3630.47 3632.02 3632.838 3633.590 3634.333 3635.055 3635.752 3636.388 3636.966 3637.42 7,7 t 8,8 6,6 4-7,7 5,5 +6,6 4,44-5,5 3,3 +4,4 H20 a 2,24-3,3 H20 O,O+l,l 1,l t O , O 2 , 2 t l , l 3,3 t 2 , 2 4 , 4 t 3 , 3 5,5 t 4 , 4 6,64-5,5 7,7 c 6 , 6 8,8 4-7,7 9,9 4- 8,8 3869.28 3871.13 3873.09 3874.354 3875.26 3877.51 3887.68 1 3895.89 3 8 98 -04 3900.08 3901.89 3903.53 3904.73 6,8 t 9 , 9 5,7 4-8,8 4,6 +7,7 R(2) dipole ' 3,5 +6,6 2,4 t 5,5 S(0) quad.' 5,3 +2,2 6 , 4 t 3 , 3 7,5 t 4 , 4 8,6 c 5 , 5 9,7 4-6,6 10.8 +7,7 ' Line due to residual HzO in spectrometer vacuum tank, used to calibrate this band.24 * Thir The R(2) dipole and S(0) quadrupole lines H 2 0 line is blended with the l , l t 2 , 2 line of HD-Ar.of free HD were used for absolute calibration of this band. under similar conditions. When the total pressure is lower (bottom trace of fig. S), the HD-Ar lines do not become significantly sharper. This behaviour indicates that the region of predissociation-limited linewidth has been reached for HD-Ar S(O), as suspected previ~usly.~ Predissociation is expected to be more rapid in Van der Waals species containing HD than in those containing H, or D2 because of the much greater effective anisotropy in the intermolecular potential, which results from the non-coincidence of the centre of mass and geometric centre in HD.The selection rule A j = 0,1,2, . . , applies to predissociation in HD-Ar, whereas in H,Ar or D2-Ar, the possible channels for predissociation are limited by A j = 0,2,4 . . . In contrast to S(O), it was found that the HD-Ar Q(0) spectrum continued to sharpen as the pressure was lowered. In this case, predissociation of the upper state (u = 1, j = 0) must involve a change in u, and this process is expected to be much slower than the Au = 0 processes possible from the upper state ( u = 1, j = 2) of S(0). The observed linewidths are discussed in more detail in section 3(e). When the previous HD-Ar S(0) measurements were compared with calculations 11 based on the potential function derived from H,-Ar and D,-Ar, relatively large discrepancies were found." These were attributed to a particular sensitivity of HD- Ar energy levels to different portions of the potential.However, it was later realized l4 that the HD-Ar calculations 11*i2 were not valid, in that a larger basis set is required in this case and that shifts due to coupling with the continuum (also responsible for the observed predissociation) cannot be ignored. Such computational difficulties, together with the experimental difficulties in accurately measuring the positions of the100 HYDROGEN-RARE-GAS SPECTRA 100- 99 98 9 7 - 3 L G ." v) *E EJ 100- l-l + broadened lines, render the HD-Ar S(0) spectrum less useful than originally hoped.'J1 However, these problems are considerably less serious for the newly observed HD-Ar Q(0) spectrum. (c) D,-Ar Spectra of D2-Ar accompanying the Q(0) and S(0) transitions of D2 are shown in fig.9, and the measured line positions are listed in table 4. The upper trace in fig. 9 -- -. c H4 I 1 I 2990 2995 300C g5t t I I 1 I I I I 31 50 3160 3170 31 80 wavenumber/cm - FIG. 9.-Spectra of D2-Ar accompanying the Q(0) and S(0) transitions of D2 in an ortho-D2 + Ar mixture at 164 Torr (upper trace), 106 Torr (lower trace), and 77 K. The upper [Q(O)] trace shows absorption due to CH4 impurity; note the shift between the Q(0) frequency of free D2 (marked by the vertical arrow) and the apparent band origin " yo" of the Dz-Ar pattern.The background curvature and relatively poor signal-to-noise ratio of the lower [S(O)] trace are due to absorption by ice on the cell windows (see text). represents the first observation of the resolved Q(0) spectrum for D2-Ar; in previous work this band was too weak and obscured by impurity absorptions to be observed. Some absorption due to an impurity [the P(2) and P(3) lines of the v3 band 2o of CH,] remain in the Q(0) region, and these were used to help fix the absolute wavenumber scale. The observed absorption in these lines corresponds to a methane impurity ofA . R . W . MCKELLAR 101 only ca. 0.5 ppm in the ultra-high-purity (99.9995%) argon sample used. In pre- liminary experiments with high-purity (99.995 %) argon, the CH4 absorption was more than ten times stronger.TABLE 4.-MEASURED POSITIONS OF LINES DUE TO THE D2-Ar MOLECULE ACCOMPANYING THE FUNDAMENTAL BAND OF Dz Ql(0) j = 0 +O S , ( O ) j = 2 c o vlcm - assignment vlcm - 1 assignment 1,J LJ 2987.040 2987.485 2987.988 2988.549 2989.08 a 2989.690 2990.297 2990.904 299 1.542 2994.042 2994.659 2995.287 2995.83 5 2996.402 2996.952 2997.475 2997.955 2998.375 9.9 +10,10 8,8 c9.9 7,7 t 8,8 6,6 +7,7 5,5 t 6 , 6 4,4 + 5,5 3,3+4,4 2,2 +3,3 1,l t 2 , 2 2,2+3,3 3,3 +4,4 4,4 4- 5,5 5,5 +6,6 6,6 t 7 , 7 7,7 +8,8 8,8+9,9 9,9 t10,lO 10,lO + 1 1,ll 3 148.025 3 149.246 3 150.692 3 152.290 3153.995 3 155.768 3 1 57.585 3 159.472 3 161.35 3166.359 3 170.29 3 172.159 3 173.986 3 175.725 3 177.387 3 1 78.964 3 180.41 8 3 18 1.676 3 182.64 8,10+ll,ll 7,9+10,10 6,8+9,9 5,7+8,8 4,6+7,7 3,5+6,6 2,4+5,5 1,3 4-4,4 0,2+3,3 4,24-3,3 5,3 +4,4 6,4+5,5 7,5+6,6 8,6+7,7 9,7 +- 8,8 10,8t9,9 11,9+10,10 12,104-11,ll S(0) quad.* This line is blended with a line due to CH4 impurity (see fig. 9). This is the S(0) quadrupole transition of free DZ. While recording the D,-Ar spectra it was found that the intensity of the infrared radiation transmitted through the absorption cell fell considerably over the period of a few days required to complete an experiment. This effect is believed to be due to the accumulation of a thin layer of ice (H,O) on the side of the (cold) cell windows facing the insulating vacuum space. The ice absorption was especially severe in the D, S(0) region, which lies quite near the peak of the 3.1 pm ice band.The ultimate signal-to-noise ratio obtained for S(0) was limited by this effect, and the spectral background in this region was also affected with the result that the lower trace in fig. 9 has not been " flattened " very successfully. ( d ) H,-Kr Spectra of the H,-Kr molecule are shown in figs. 10-12, and measured line positions are listed in table 5. Most of the comments made regarding H,-Ar in section 2(a) also apply to these H,-Kr results. Analysis * of the previous H,-Kr measurements suggested that they were less satisfactory than those for H,-Ar and H,-Xe, and it is anticipated that the new data in table 5 represent a correspondingly greater improve-102 '00-i, 95- HYDROGEN-RARE-GAS SPECTRA n 90.- x ._ 100- W m v) -g 3 c) 9 5 - 4 155 4 160 4 165 wavenumberlcm- FIG.10.-Spectrum of H2-Kr accompanying the Q(0) transition of Hz in a para-H2 + Kr mixture at 103 Torr and 97 K. The asterisks mark absorption lines of CO present as an impurity in the infrared source chamber. The arrow marks the transition frequency of the free H z molecule; note the shift (ca. 1.6 cm-') between this and the apparent centre of the H2-Kr pattern. r I I I I 4480 4490 4 500 41 50 1 I I 1 1 1 4490 4495 4500 wavenumberlcm- FIG. 11.-Spectrum of Hz-Ar accompanying the S(0) transition of H2 in a para-Hz + Kr mixture at 108 Torr (upper trace) or 69 Tom (lower trace) and 97 K. The arrows mark the S(0) quadrupole transition of free Hz, and the vertical lines indicate features for which measured positions are given in table 5.A . R . W .MCKELLAR e v, .M .- g 100.- c e +a 95.. 103 I I I 1 I I I I I 4 1 4 0 4 150 4160 4170 I I I I ment. In particular, the new Q(l) measurements differ considerably from those of McKellar and Welsh,5 thus confirming the suspicion of Le Roy and Van Kranendonk * that there were serious measurement errors in this region. (e) LINEWIDTH MEASUREMENTS The measured linewidths (full width at half maximum) of typical lines in a number of bands of the H2-Ar and HD-Ar molecules are shown as a function of density in fig. 13(a). The behaviour of widths for D,-Ar and H2-Kr was similar to that shown in fig. 13(a), but the HD-Ar S(0) spectrum was rather different, and is considered separately below. The pressure broadening measurements have been corrected for the effects of instrumental resolution, but they are only accurate to ca.10 or 20%. In particular, the small differences between the different bands in fig. 13(a) are only marginally significant, and may partly be due to the different hydrogen/argon mixing ratios used in each case. From these data, the pressure broadening coefficient for lines of H,-Ar, HD-Ar and D2-Ar appears to be ca. 0.16 cm-’ amagat-l* at 77 K, which is equivalent to 22 MHz Torr-’. As mentioned above, the widths in the HD-Ar S(0) spectrum were found to be limited by predissociation at the pressures used here. The measured widths of re- solved N and T lines (see fig. 8) are plotted in fig. 13(b) as a function of Z‘, the quantum number for end-over-end rotation in the upper state (u = 1, j = 2) of the transition. Note that different J’ substates are involved for N- and T-branch lines, and there may thus be two points for each I’ (this actually occurs only for I’ = 5).The un- * See caption to fig. 1.1 04 HYDROGEN-RARE-GAS SPECTRA TABLE s.-MEASURED POSITIONS OF LINES DUE TO THE H2-Kr MOLECULE ACCOMPANYING THE FUNDAMENTAL BAND OF H2 Ql(0)j = O+-0 S , ( O ) j = 24-0 Ql(l)j = 1-1 S,(l)j = 3 t l v/cm-' assignment v/cm-' assignment v/cm-' assignment v/cm- assignment 1,J LJ 4J 1,J 41 52.207 4152.767 4 1 53.540 41 54.421 4155.365 41 56.361 41 57.390 4158.435 4 1 60.5 1 7 4161.553 4162.551 4163.544 41 64.502 4 1 65.394 41 66.226 41 66.940 7,7 + 8,8 6,6 4-7,7 5,5 t 6 , 6 4,4 t5,5 3,3 t 4 , 4 2,2 +3,3 1,1+2,2 0,0+1,1 1,l t 0 , o 2,2 t 1,l 3,3 t 2 , 2 4,4t3,3 5 3 +4,4 6,6t5,5 7,7 +6,6 8,8t7,7 4475.3 8 8 4477.675 4480.326 448 3.1 96 448 6.220 4489.3 64 4489.886 4490.093 4490.68 5 4490.891 4491.617 4491.779 4492.766 4493.678 4496.462 4497.321 4497.838 4499.027 4501.882 4502.192 4502.739 4503.039 4503.561 4503.758 4504.200 4507.244 45 12.864 451 5.325 45 17.27 5,7 4-83 4,6t7,7 3,5 t 6 , 6 2,4 t 5,5 1,3 t 4 , 4 0.2 t 3 , 3 6,7 t 7 , 7 6,6 t 7 , 7 5,6 t 6,6 5,5 t 6 , 6 4,5 t 5 , 5 4 , 4 4 3 2,o t 1,l 2 , 1 t l , l quad.6,5 t 6 , 6 7,6 t 6 , 6 7,7 c6,6 4,2+1,l 5,3 t 2 , 2 7,5 +4,4 8,6t5,5 9,7 t 6 , 6 4133.351 4135.630 4138.239 4141.027 4143.915 4146.156 41 46.793 4147.5 13 4 1 48.390 4 149.326 4 1 50.300 41 51.285 41 52.234 41 54.510 41 55.254 4155.520 4156.530 4157.542 4158.500 41 59.405 4163.052 4 1 65.944 41 68.739 4171.398 4173.772 5,6 t 8,7 4,5 t 7 , 6 3,5 t 6 , 5 1,2 4-4.3 2,3 +.5,4 0,1+3,2 quad. 4,3 t l , 2 5,44-2,3 6,5t3,4 7,6t4,5 8,7t5,6 4689.03 4690.734 4693.082 4695.775 4698.688 4701.758 4704.972 4712.902 4720.920 4721.663 4722.260 4724.626 4725.266 4727.390 4728.027 4729.866 4730,537 4732,535 6 t 9 5 t 8 44-7 3 t 6 2 t 5 14-4 O t 3 quad. 5,2 t 2 , 2 5,3 t 2 , 2 5,2 t 2 , 1 6,4 t 3,3 6,3 t 3 , 2 7,5 +4,4 7,4 t 4 , 3 8,6t5,5 8,5 t 5 , 4 certainties in the measurements of fig. 13(b) are rather large (10-2073, but the striking trend of narrower widths at higher I' in the T branch is quite definite. It may easily be observed in the spectrum itself (fig. 8), and was in fact noted by Weiss 26 in the earlier results of M~Kellar.~ The open points in fig. 13(b) are calculated widths from the recent work of Corey and Le Roy:27 they reproduce the magnitude and 2'- dependence of the observed widths quite well, though they underestimate all the widths by ca. 20%.There are no calculated widths for 1' = 8 and 9 because these levels lie above the u = 1, j = 2 energy asymptote and are thus subject to direct rotational predissociation. In spite of this, they are the narrowest lines in fig. 13(b). T(7), with 1' = 10, is not plotted in the figure, but it is visible in fig. 8; it definitely shows broadening due to direct rotational predissociation. The calculations of Corey and Le Roy 27 predict that the sharpest levels for HD- Ar with u = 1, j = 2 are those with I' = J', and particularly I = J = 2 with a width of 0.08 cm-'. These particular levels are not involved in N or T transitions, but they do participate in P and R transitions.Intensity calculations for H2-Ar suggest thatA . R. W. MCKELLAR 0 . 0 . 8 - 0.6- 105 I I I 0.5 1 .o 1.5 A A A A o A A A n - A 0 d e n s i t y (amagat) A (b) - a 0 0 0 0.4 r( I E E 4 3 5 f 2 3 . ? c .- I 0.2 0.41 0 0 1 3 5 7 9 I’ FIG. 13.-Measured linewidths (full width at half maximum) for lines in the spectra of hydrogen-rare- gas Van der Waals molecules, corrected (approximately) for instrumental resolution. The upper graph (a) shows pressure-broadening measurements [O, H2-Ar Ql(0); 0, H,-Ar S1(0); +, H2-Ar Ql(l); A , HD-Ar Ql(0)]. The lower graph (6) shows widths of N-branch (A) and T-branch (0) lines in the HD-Ar Sl(0) spectrum as a function of l’, the quantum number for end-over-end rotation in the upper state.These widths are due to predissociation, and they are compared with calculations (open symbols) of Corey and Le Roy.27 Note: in (b) the solid triangles should be moved one unit to the right. P-branch transitions with 1’ = J’ are considerably stronger than the corresponding R- branch lines. Thus one might expect to see a series of relatively sharp lines in the HD-Ar S(0) spectrum P branch. This is indeed the case, as can be seen in fig. 8 where there are 5 or 6 sharper (but unresolved) lines in the 3880-3885 cm-’ region. This observation and the comparison in fig. 13(b) offer strong support for the reli- ability of the secular equation approach used by Corey and Le Corey and Le Roy have also calculated predissociation widths for H2-Ar and D,-Ar S(0) spectra. Their predicted widths generally fall in the range from 0.005 to 0.05 cm-I, which is narrower than can be detected experimentally in the present study.There is, however, one exceptional level predicted to be considerably broader : 1 = 2, J = 0 with a predicted width of 0.123 cm-l in H,-Ar. It turns out that this level is only involved in one possible line of the S(0) spectrum, the R-branch transition with 1,J = 2,0+1,1. Unfortunately, in H,Ar this transition is predicted ‘ 9 ’ to lie106 HYDROGEN-RARE-GAS SPECTRA directly underneath the H, S(0) quadrupole line (see fig. 5) and thus its width cannot easily be checked. And in D2-Ar, the central part of the S(0) spectrum was not sufficiently well-resolved to observe this transition clearly.However, all is not lost: assuming that the relative predissociation lifetimes for H2-Kr will be similar to H2-Ar and D2-Ar, we can turn to the S(0) spectrum of this molecule, in fig. 1 1 . An expanded view of the relevant region is shown in fig. 14. At a total pressure of 69 Torr, the H, l:!;!;!;!;!:!:!;.: 4 4 9 2 L 4 9 4 4 4 9 6 4 4 9 8 4 500 wavenumberlcm- FIG. 14.-Expanded view of the H2-Kr S(0) spectrum shown in fig. 11. The upper trace is calculated using the line positions and intensities of Dunker and G ~ r d o n . ~ Note the line at 4496.46 cm-I whose greater width (0.144 cm-') is believed to be due to predissociation. S(0) quadrupole line has a width of 0.083 cm-' (including instrumental resolution), and most isolated H2-Kr lines have widths of ca.0.10 cm-l. However, one H,-Kr line has a measured width of 0.144 cm-l, as shown in fig. 14, and this line is indeed the Z,J = 2 , 0 t l , l transition, which terminates on the level (2,O) expected 27 to be short- lived. Moreover, this line (at 4496.46 cm-') is predicted 'p9 to be unblended and well isolated, and its observed peak intensity is less than predi~ted,~ just as expected for a line broadened by predissociation. The contribution to the width of this line due to predissociation may be estimated to be ca. 0.1 1 cm-l, which is remarkably close to the value of 0.123 cm-' predicted by Corey and Le Roy 27 for the same transi- tion in H2-Ar. Beswick and Requena 28 have predicted predissociation widths for H,-Ne, H2-Ar and H2-Kr in a few chosen (I,J) levels with j = 2, 4 and 6.Their calculations are for u = 0 rather than v = 1 as required here, but a comparison is still of interest. For H,-Kr with j = 2, I = 2 and J = 0, their calculated width (f.w.h.m.) is 0.022 cm-', which is considerably smaller than the experimental value reported here. 4. DISCUSSION In the above discussion of predissociation, only brief mention was made of the " direct " rotational predissociation process involving levels located above the asymp-A . R. W . MCKELLAR 107 tote of the potential surface and bound only by the centrifugal energy barrier in the potential for larger I values. These effects were well observed in previous ~ p e c t r a , ~ and the relevant widths have been well calculated.8 An interesting aspect of the present work is that transitions involving these quasibound levels tend to " disappear " as the pressure is reduced and the resolution increased, because they do not continue to sharpen below a certain pressure.This effect is apparent at the outer edges of the N and T branches in fig. 1, and may also be seen by comparing many of the present spectra with their earlier eq~ivalents.~. Further work on Van der Waals spectra accompanying the fundamental bands of molecular hydrogen is planned to improve the measurement accuracy of the present results, and to extend these high-resolution studies to other species: HD-Kr, D,-Kr; H,-Xe, HD-Xe, D,-Xe; D,-Ne; (H,),; (D2),; (HD),. Similar spectra may also be studied in other spectral regions. The hydrogen overtone (u = 2 ~ 0 ) spectra in the 1-2 pm region are expected to be ca.25 times weaker than those studied here, but they should yield considerably more information on the dependence of the potential surfaces on. the internal stretching motion of the hydrogen. Van der Waals spectra will also accompany the pure rotational (u = Ot-0) hydrogen transitions in the 20-70 pm far-infrared region. A study of these spectra should yield more precise informa- tion on the anisotropy of the potential for u = 0. This is because the ( u = l+O) spectra studied to date tend l4 to give information on anisotropy for u = 1, since the j = 2 level which supplies a majority of this information is only observed for u = 1. At much lower frequencies, it is also possible to observe pure hyperfine spectra of H,-Ne, H,-Ar and H,-Kr with 21 = 0 and j = 1, as demonstrated recently by Waaijer 29 using molecular-beam magnetic resonance; these results may be used to impose further constraints on parameters of the intermolecular potentials.In summary, new high-resolution spectra of the H,-Ar, HD-Ar, D,-Ar and H,-Kr molecules have been obtained in the regions of the hydrogen fundamental bands. Compared with previous work, the spectral resolution is improved by a factor of 3-6 and the measurement accuracy by a factor of 2 or more. Fully resolved Q(0) spectra for HD-Ar and D,-Ar are reported for the first time. Predissociation broadening due to Au = 0, Aj = 1,2 processes is observed for certain transitions of HD-Ar and H2-Kr. These observations are especially valuable because, in spite of the considerable recent interest in predissociation processes in Van der Waals mole- cules, there are very few experimental measurements involving systems where the relevant potential surfaces are known.I am grateful to J. L. Hunt for providingpara-hydrogen catalyst, to J. G. Potter for construction of the slow-scan drive, to C . A. Harris and J. W. C. Johns for experi- mental assistance and to I. K. M. Strathy for initial assembly of the 5.5 m cell. Special thanks are due to J. R. Austin for invaluable assistance in computer programming and data collection. A. K. Kudian, H. L. Welsh and A. Watanabe, J. Chem. Phys., 1965,43, 3397. A. Watanabe and H. L. Welsh, Phys. Rev. Lett., 1964, 13, 810. A. K. Kudian and H. L. Welsh, Can. J. Phys., 1971, 49, 230. A. K. Kudian, H. L. Welsh and A. Watanabe, J. Chem. Phys., 1967, 47, 1553. A. R. W. McKellar and H. L. Welsh, J . Chem. Phys., 1971, 55, 595. A. R. W. McKellar and H. L. Welsh, Can. J . Phys., 1972,50, 1458. A. R. W. McKellar, J. Chem. Phys., 1974, 61, 4636. R. J. Le Roy and J. Van Kranendonk, J . Chem. Phys., 1974,61,4750; and the Supplement to this work : Chemical Physics Research Report CP-22 (University of Waterloo, Canada, 1974). A. M. Dunker and R. G . Gordon, J . Chem. PhyJ., 1978, 68, 700.108 HYDROGEN-RARE-GAS SPECTRA lo A. N. Petelin, Opt. Spectrosc. (Engl. Transl.), 1975, 38, 21. l1 H. Kreek and R. J. Le Roy, J. Chem. Phys., 1975, 63, 338. It R. J. Le Roy, J. S. Carley and J. E. Grabenstetter, Faraday Discuss. Chem. SOC., 1977, 62, 169. l3 J. S. Carley, Faraday Discuss. Chem. SOC., 1977, 62, 303. I4 R. J. Le Roy and J. S. Carley, Adv. Chem. Phys., 1980, 42, 353. l5 H. Thuis, S. Stolte and J. Reuss, Comments Atom. Mol. Phys., 1979, 8, 123. l6 H. L. Welsh, in Spectroscopy, M.T.P. International Reviews of Science, PhysicaZ Chemistry (Butterworths, London, 1972), vol. 3, p. 33. J. U. White, J. Opt. Soc. Am., 1942, 32, 285. A. E. Douglas and D. Sharma, J. Chem. Phys., 1953, 21, 448. l9 J. W. C. Johns, A. R. W. McKellar and D. Weitz, J. Mol. Spectrosc., 1974,51, 539. 2o A. S. Pine, J. Opt. Soc. Am., 1976, 66, 97. 22 A. R. W. McKellar, W. Goetz and D. A. Ramsay, Astrophys. J., 1976, 207, 663. 23 A. R. W. McKellar and T. Oka, Can. J. Phys., 1978,56, 1315. 24 C. Camy-Peyret, J. M. Flaud, G. Guelachvili and C. Amiot, M d , Phys., 1973, 26, 825. 25 A. R. W. McKellar and H. L. Welsh, Can. J. Phys., 1974,52, 1082. 26 S. Weiss, J. Chem. Phys., 1977, 67, 3840. 27 G. C Corey, MSc. Thesis (University of Waterloo, Waterloo, Canada, 1980); G. C. Corey and R. J. Le Roy, unpublished work, 1981. 28 J. A. Beswick and A. Requena, J. Chem. Phys., 1980,72,3018; see also: J. A. Beswick and A. Requena, J. Chem. Phys., 1980,73,4347. 29 M. Waaijer, Thesis (Katholieke Universiteit, Nijmegen, The Netherlands, 1981). G. Guelachvili, Opt. Commun., 1973, 8, 171.

ISSN:0301-7249    

DOI:10.1039/DC9827300089

出版商:RSC    

年代:1982

数据来源: RSC

摘要:

GENERAL DISCUSSION Dr. P. G. Burton (University of Wollongong) said: In supporting his error esti- mate of 5 1 kcal mol-' in the values he quoted for the strength of the Van der Waals interactions between the first-row hydrides, Prof. Pople only referred to the approach to convergence, through several orders of perturbation theory, of the estimate of the electron correlation contribution to these interaction strengths. However, an independent source of uncertainty in these quantities is the systematic errors stemming from the limited basis sets employed. Even with the largest 6-31G** bases used, errors in the individual molecule multipole moments at the (SCF and) CI level will surely affect the computed interaction strengths to a greater degree than A1 kcal mol-'. In addition, since such small basis sets are incapable of accounting for a large fraction (ca.30%) of even the valence correlation energy of the molecules studied, no matter what order of perturbation theory is used, there is a systematic error in the computed total energies of the molecules which is two orders of magnitude greater than the claimed precision. Prof. Pople has presented no evidence that this large systematic error is unimportant in the comparison of the Van der Waals complex and separated molecule energies, nor any evidence as to the magnitude of the basis set superposition artefact which is involved in these comparisons. Prof. J. A. Pople (Carnegie Mellon University, Pittsburgh) said : An error estimate of & 1 kcal mol-' would be appropriate only for the weaker interactions between first-row hydrides.The larger interaction energies between systems such as BH3 may indeed be subject to greater error for reasons of basis-set limitation, as you indicate. Prof. J. Jortner (Tel-Aviv University) said : The charge-transfer contribution to intermolecular Van der Waals bonding may be amenable to experimental detection by the observation of the enhancement of the infrared intensity of one of the components. Intermolecular charge flow is expected to exhibit a vibronic-type contribution to the infrared intensity.' This mechanism is well documented for " charge-transfer " complexes in condensed phases. Such a charge-transfer correlation effect implies a relation between the intensity enhancement and an energetic shift of the fundamental frequency towards lower energies.R. S. Mulliken and R. Pearson, Molecular Complexes (Wiley-Interscience, New York, 1969). Prof. V. Magnasco (University of Genoa) said: 1 fully agree with the philosophy inspiring Dr. Stone's paper. To make progress in the understanding of intermolecular interactions (possibly including chemical reactions) we ought to resort to some kind of perturbation approach, where knowledge of the properties of the isolated molecules would be of great value, as Prof. Buckingham said in the course of this meeting. The proper treatment of the small overlap existing between atoms or molecules at relatively large internuclear separations is essential to give a correct description of the intermolecular interaction and its breakdown into components of different orders.On similar premises, we have recently developed a perturbation treatment which is second order in the intermolecular potential and infinite order in the intermolecular overlap. The method, which makes use of the whole set of bonding and antibonding orbitals of the separate systems, allows for a full account of the non-orthogonality existing in second order between dzflerent interacting partners, giving results which110 GENERAL DISCUSSION are easily analysable into components with a direct physical meaning. In this context, our previous work on first-order interactions has been improved by allowing for polarisation and delocalisation (charge transfer) described in terms of single excitations, and applied both to the study of the orientation dependence of the intermolecular inter- actions in the short to medium range ' and to the problem of rotational barrier in ethane.3 Further work is in progress mostly on intramolecular interactions.The overall results are close to those obtained from conventional MO-SCF calculations. Electron correlation might be included as well in terms of double excitations. Secondly, turning to Dr. Stone's paper, I would like to make a few comments. which I believe can be of rather general interest. (i) In decomposing its first-order energy into components, Dr. Stone does not mention explicitly the A-correction term to the interaction energy, which is zero if v0 is an eigenfunction of the unperturbed Hamiltonian Ho. Now, although it is well known that this term is small for Hartree-Fock wavefunctions of the separate mole- cules, giving a correction which vanishes at least as the fourth power in the intermole- cular ~ v e r l a p , ~ * ~ its importance has been seen to increase considerably for the heavier rare-gas atoms and at larger di~tances.~ A possible answer to that might be in the fact that Dr.Stone defines the exchange repulsion energy [eqn (6)] implicitzy as the difference between the total (electronic) energy to first order [eqn (4)] and the electro- static (or Coulombic) energy. The A-correction might then well be absorbed into the exchange term. However, we would expect a sensible loss in accuracy in obtaining such differences when the quantities involved are almost of the same order of magnitude, and a lack of physical transparency in the definition of the exchange component.Starting from the same zeroth-order wavefunction in the form of a single deter- minant of non-orthogonal spin-orbitals {vi"), we have given an expIicit expression for the first-order exchange energy among several overlapping charge distributions in terms of the one-electron density matrices occ pR(0011 ;l') = 2 V/i"(l)V/i"*(l'), I fdxlpA(OO1 1 ;1) = NA (1) A all i j pf(0011;l') = - 2 2 vio(l)Aijvjo*(l'), Jdx1pf(00/1;1) = 0 ( 2 ) where eqn (2) is the overlap density matrix and Aij = (SM-l)i,j with M = 1 + S the matrix of the unperturbed spin-orbital basis. If we denote by the induction potential due to nuclei and unperturbed Hartree-Fock wavefunction of the separate molecules electron density on B, for the first-order exchange is The first term in eqn (4) is the true two-electron exchange, which survives in the case of no overlap, and coincides with eqn ( 5 ) of Dr.Stone's paper in the case of twoGENERAL DISCUSSION 111 molecules. The second term gives the interaction of the overlap density (2) in the Coulomb-exchange field provided by nuclei and unperturbed electron density of all other groups. The last term gives the Coulomb-exchange interaction of the overlap densities among themselves. Since eqn (4) describes the penetration of charge clouds due to their overlap, the exchange component of the first-order interaction energy has also been called by us first-order penetration. We can notice that all terms in eqn (4) do arise from either charge overlap or non-local exchange potentials, and that all non-additiuity of intermolecular forces in first order stems from the last two terms in (ii) The use of basis sets of insufficient size (i.e.below Hartree-Fock) in yo is expected to introduce errors into the Coulombic and exchange overlap components of the first-order interaction. Such errors are known to be rather large for the heavier rare gases, even using two-term SCF wavefunctions based upon Slater orbital^.^ (iii) Use of orbital energy differences in the calculation of excitation energies [eqn (7)], being tantamount to assuming an uncoupled procedure where the details of the many-electron nature of the wavefunction are lost, is expected to underestimate second- order eqn (4) * V. Magnasco and G.F. Musso, Chem. Phys. Lett., 1981, 84, 575. V. Magnasco, Atti Accad. Ligure Sci. Lett., 1981, 38, 3 and references therein. G. F. Musso and V. Magnasco, J . Chem. SOC., Faraday Trans. 2, 1982,78, 1609. J. N. Murrell and A. J. C . Varandas, Mol. Phys., 1975, 30, 223. A. Conway and J. N. Murrell, Mol. Phys., 1972, 23, 1143; 1974, 27, 873. V. Magnasco, Mol. Phys., 1979,37, 73. ' V. Magnasco and G. Roncallo, Chem. Phys. Lett., 1981, 79, 125. Dr. A. J. Stone (Cambridge University) said: The calculations reported in the paper do not use big enough basis sets to give useful values for the dispersion and polarization energies. We have since done calculations on Ar * * - HCI using an [8~4p2d/3~2p/7~5p2d] basis, and find that the minimum appears at about the distance found by Hutson and Howard, but that the well is ca.10% deeper. The largest attractive contribution by far is the dispersion term, but there is also a substantial contribution from the " charge-transfer correlation " term, which arises from double excitations of the form AOCCBOCC-+AVirtAVirt. The single-excitation charge-transfer term is smaller, and the polarization term larger, than for the smaller basis. Similar results were obtained for Ar - We have done some calculations using Epstein-Nesbet energy denominators instead of the Moller-Plesset denominators described in the paper. The com- putational labour is much increased, and it is not clear whether the results are better. It must be remembered that in this calculation a more negative energy (which the Epstein-Nesbet formalism does give) is not necessarily a better result, because there is no variational principle for the energy diflerence that we seek.The A-correction mentioned by Dr. Magnasco arises when the first-order energy is partitioned in a certain way. Our calculation uses a different partitioning scheme, merely separating out the electrostatic and exchange terms. The total first-order energy is the same in both schemes. We do not believe that our scheme leads to any significant loss of numerical accuracy. HF using an [8~4p2d/3~2p/5~4p2d] basis. Dr. P. G. Burton (University of Wollungong) said: My first comment relates to the relative importance of higher than double excitations, compared to single and double excitations, in contributing to the Van der Waals interactions. While I agree that the higher substitutions are less important than the single and double excitations, and so the intermolecular perturbation theory should concentrate first on the latter, our own112 GENERAL DISCUSSION results on very small systems (e.g.He-He and H2-H,) indicate that these higher excitations are directly responsible for a significant fraction of the total Van der Waals interaci ion. Recently we have directly compared the Van der Waals well depths, computed with and without these higher excitations, for the He-He and H2-H, systems using large scale CI supermolecule calculations and moderately large orbital basis sets (ca. 40 functions per electron pair). While we do not yet have full CI for these four-electron systems with these basis sets, we have variational PNO-CI (limited to double excit- ations) and the non-variational but size-consistent CEPA2-PNO result^.^ The pair natural orbitals (PNO) used in these calculations prediagonalize the C1 expansions for these systems and lead to very compact CI wavefunctions, with minimal loss of accuracy (ca.0.7% of E,,,, when a selection threshold of The CEPA2-PNO calculations allow, in an approximate but systematic way, for the important independent pairs of double excitations which are required to simultan- eously correlate both electron pairs of these systems, i.e. to provide for statistically independent simultaneous correlation of both subsystems. In the He-He system we find that the more limited PNO-CI accounts for only 90% of the Van der Waals well depth, while in H2-H2 the higher-order excitations are even more important and account for 18% of the computed well depth (PNO-CI = 82% of CEPA2-PNO well depth). In the light of these results it seems inadvisable to exclude higher-order excitations from consideration as important contributors to the Van der Waals inter- action of systems with a great many more electrons.My second comment relates to the treatment of the basis set superposition correc- tions in the intermolecular perturbation theory for Van der Waals molecules. Al- though Boys and Bernardi introduced the " function counterpoise " correction in the context of correlated wavefunctions for intermolecular force calculation^,^ it has been most widely applied at the SCF level of wavefunction computation.The magnitude of the basis-set extension or superposition error is quite different (and typically smaller) at the SCF than at the (limited or) full CI level, simply because the less constrained correlation Ansatz greatly increases the opportunity for an arbi- trary basis function to contribute to the description of a molecular system. When a molecule is represented using a given basis set, the maximum extent of the basis set extension error that is possible in an intermolecular force supermolecule calculation is given at the SCF level by the difference between the computed SCF energy and the corresponding Hartree-Fock limit energy for that molecule. With a given limited or full CI Ansatz, the corresponding upper limit to the superposition error is given by the difference between the computed CI energy and the exact non-relativistic energy for the molecule. We can illustrate these ideas numerically by considering a basis set developed for our recent H,-H, study, where we have the advantage of the precise knowledge of the exact energies of H2 at either the SCF or CI level.The partially optimized basis designed for H, consists of 39 independent basis functions, with characteristics defined by the quantities EsCF = - 1.133 40 E h (- 1.133 63 E J , E,, = - 1.172 45 Eh (- 1.174 47 Eh) and Q2" = 0.4516 (0.4574). The values in brackets are those for the Kolos and Wolniewicz wavefunctions for H2 at the same bond distance (we make these comparisons at re, although ro was used for the intermolecular force compu- tations).The error at the SCF level by restricting the H, basis to these 39 functions is seen to 0.23 x Eh (441 cm-I), an order of magnitude larger. E h (882 cm-I). In practice the actual superposition corrections we found with this basis Eh is used). Eh (50 cm-'), while the error at the CI level is 2.02 x Thus the upper limit to the CI superposition error for H,-H, is 4.04 xGENERAL DISCUSSION 113 for each H2 in the H2-H2 calculations \\*ere greatest for shortest intermolecular dis- tances (midbond to midbond) considered (R = 3.0 au) and even here in the worst case geometry (the T configuration) the CI superposition correction was less than 0.097 x low3 E h (21.2 compared to the total interaction energy for this same geometry of 43 230 X E h (9487.9 cm-I). The corresponding correction for this geometry, at R = 6.5 au near the Van der Waals minimum for the isotropic potential, was 0.009 X E h (1.95 cm-I).Although the actual superposition corrections are seen to be orders of magnitude less than the theoretical upper limit for these corrections, these corrections nevertheless enter at the SCF level and CI level in a similar ratio to that suggested by the respective limiting values, over the whole range of the potential. The reason that the actual superposition corrections are such a small fraction of the theoretical maximum value of the corrections is that at intermolecular distances near the Van der Waals minimum, the overlap of the basis functions of one molecule with the electron distribution of the other molecule is so small.How large the theoretical maximal SCF and CI superposition errors are in a particular supermolecule calculation depends on (a) how close to a full C1 the chosen CI Ansatz represents and (b) the basis set employed. The very small values for the superposition corrections that we have quoted above arise only because the H2 basis set employed accounts for 96.3% of the correlation energy of this molecule; typically nothing like this approach to full correlation is achieved with conventional basis sets (2-5 basis functions per electron), certainly not for systems as large as ArHF. Given the expectation that the SCF superposition error in a given basis might be expected to be an order of magnitude less than the corresponding error at the CI level, the fact that Stone and Hayes have quoted SCF superposition corrections in their ArHF and ArHC1 studies which are, even at the SCF level, comparable to the well depth in the vicinity of the Van der Waals minimum, implies that the corresponding CI superposition corrections should be an order of magnitude greater than the well depth with the basis sets employed.P. G. Burton, J. Chem. Phys., 1979,70,3112; Chem. Phys. Lett., 1981,82,335; and unpublished results. R. Ahlrichs, H. Lischka, V. Staemmler and W. Kutzelnigg, J, Chem. Phys., 1975, 62, 1975; (6) R. Ahlrichs, Cumput. Phys. Commun., 1979, 17, 31 ; (c) S. Koch and W. Kutzelnigg, Theor. Chim. Acta, 1981, 59, 387. S. F. Boys and F. Bernardi, Mol. Phys., 1970, 19, 553. N. S. Ostlund and D. L. Merrifield, Chem. Phys. Lett., 1976, 39, 612.P. G. Burton, P. D. Gray and U. E. Senff, Mol. Phys., to be published, * P. G. Burton and U, E. Senff, J. Chem. Phys., 1982, 76, 6073. Dr. A. J. Stone (Cambridge University) said: The basis-extension error is certainly larger for the double-excitation terms, which correspond to the electron-correlation part of the interaction energy, than for the single-excitation terms. The magnitude of the error in the latter case is of the order of 0.5 x E h for Ar - - HF in the region of the minimum, as can be seen from fig. 1 of our paper. The double-excitation terms, however, separate very conveniently, some of them being free of basis extension error entirely, and some being wholly basis extension error, at least to zeroth order in overlap. The former include the dispersion energy, arising from excitations of the form AoccBocc-+AvirtBvirt, and the “ charge-transfer correlation” AoccBocc+AvirtAvirt- These contribute cn.-1.0 x El, and -0.7 x loa3 Eh, respectively, in the region of the minimum of Ar - - - HF. The “ extension-correlation ” AOCCAOCC+ AvirtBvirt is an example of a term which is entirely attributable to basis extension error, to zeroth order in overlap, and its magnitude at the minimum for Ar - * - HF is -20.5 x Eh. Naturally it is not included in the total interaction energy. It is evident that, as Dr. Burton suggests, the basis-extension effect is very much larger for114 GENERAL DISCUSSION the correlation terms than for the single excitations; but it seems to be a relatively straightforward matter to isolate it and correct for it within the perturbation form- alism.Dr. J. M. Hutson (University of Waterloo) said : Douketis and Scoles have recently attempted to model intermolecular forces in the Ar - * * HF system using a semi- empirical Hartree-Fock + dispersion (HFD) model. In calculations of this type it is not clear what point to use as the origin of the dispersion forces, but the results depend critically on the point chosen; in particular, there is no reason to suppose that the H F centre of mass is the appropriate origin, although this assumption has fre- quently been made in the past. Drs. Stone and Hayes have found that the electrostatic and exchange/repulsion energy curves for Ar - - - HF and Ar - * - FH are similar if the intermolecular distance is measured to a point 0.4 au along the F-H bond.Is there any evidence for similar behaviour in the induction and dispersion energies ? C. Douketis and G. Scoles, 1982, unpublished work. Dr. A. J. Stone (Cambridge University) said: We have not examined the induc- tion and dispersion terms yet for Ar - - FH in a large basis. Nor have we investi- gated the interaction potential for non-linear geometries. Consequently the evidence for an effective interaction centre 0.4 bohr along the bond is at present very slight. Our present belief is that attempts to model intermolecular forces in terms of a single interaction site for each molecule are misguided. A distributed multipole analysis of HF shows that the electrostatic forces are very much better described in terms of a two-site or three-site model, and a suitable two-site model gives a good fit to the data of Hutson and Howard for Ar - * - HC1.2 A distributed multipole model also accounts well for the dielectric second virial coefficient of HCl, which a one-centre multipole expansion completely fails to A.J. Stone, Chem. Phys. Lett., 1982, 83, 233. C. G. J o s h and A. J. Stone, 1981, unpublished work. ' A. J. Stone and S. L. Price, 1982, unpublished work. Prof. J. Jortner (Tel-Aviv University) said : An interesting relation may exist between the thermal excitation of intermolecular vibrational modes in (N2)n clusters and phase transitions in solid N2. Thermal excitation of intermolecular vibrations in the cluster will result in orientational disorder, which is reminiscent of the K- (ordered)-+P(disordered) phase transition in the solid.A cardinal question is: What is the smallest size of the (N2),* cluster expected to exhibit a solid-state-like phase transition? Obviously, the dimer (n = 2) is too small, and solid-state-like behaviour is expected to set in for larger values of n. In this context, the calculation of the thermodynamic properties of these clusters will be of considerable interest. In alluding to the intriguing question concerning the onset of " solid-state proper- ties '' in finite clusters it is imperative to distinguish between local properties of clusters, e.g. vibrational excitations of intramolecular vibrations, electronic excitations of a guest molecule and global (collective) properties, such as phase transitions.These local and global solid-state features will set in at different coordination numbers (size) of the clwter. Prof. A. van der Avoird (University of Nijmegen) said : Concerning the question at which cluster size Van der Waals molecules begin to acquire solid-state properties, I would like to quote a recent paper by Etters et a2.l who have calculated thermodynamicGENERAL DISCUSSION 115 properties of (CO,), clusters for 2 2 n 2 13 using a classical Monte Carlo scheme. It is demonstrated in this paper that surprisingly small clusters, even dimers, already show rather well defined orientational order-disorder " phase " transitions and melt- ing. E. D. Etters, K. Flurchick, R. P. Pan and V. Chandrasekharan, J. Chem. Phys., 1981, 75, 929. Prof. W. Klemperer (Haruard University) said: We have attempted to study the structure of (N,), by molecular-beam methods.We found, however, that the electric dipole moment was below the limit of detectability by deflection methods.' We should like to call attention to the observation of several rotational resonances for the perhaps closely related isoelectric system (CO),. This study is at present incomplete. We believe it is possible to extend this work, and thus we suggest that it would be valuable to have detailed studies of the inter- molecular potential for (CO), as have been discussed for the N2 dimer. S. E. Novick, P. B. Davies, T. R. Dyke and W. Klemperer, J . Am. Chem. Soc., 1973,95, 8547. P. A. Vanden Bout, J. M. Steed, L. S. Bernstein and W. Klemperer, Astuophys.J, 1978,234,503. Prof. Ph. Brechignac (University of Paris, Orsay) said : Coming back to the N2 dimer and to the similarity between the two molecules N, and CO, I wish to point out that such similarity arguments should be used with much caution. Indeed, we measured a few years ago by infrared double resonance the propensity rules for the rotational-energy transfer in gaseous CO, which are governed by the anisotropy of the CO-CO potential. The results were found to be very different from what would be inferred from a N2-N, surface. In the latter case odd-AJ collisional transitions are symmetry-forbidden, and because of the quadrupole moment (1.5 D A) of the N2 molecule the rotational relaxation is dominated by A J = 2 changes. The CO molecule has a quadrupole moment very close (2.2 D A) to that of N, and a fairly small dipole moment (0.1 D) so that similar behaviour would be predicted by a first Born-approximation kind of calculation from the long-range multipolar anisotropic interaction.However, the experimental findings are, very unexpectedly, that 2/3 of the rotationally inelastic transitions proceed by A J = 1 , while 1/3 proceed by A J = 2 and < 10% by A J >, 3. The only way in which we can understand these results is if the short-range anisotropy is essential for the outcome of inelastic collisions. The CO-CO potential should have a rather large both P,- and P,-shaped short-range anisotropy to be responsible for such selection rules. The P,-like part is necessary to account for the efficiency of A J = 1 transitions, while the P,-like part has to allow for some cancellation of the A J = 2 Q-Q contribution by destructive inter- ference during the dynamics.In conclusion, it seems impossible to predict from simple arguments the rotational-energy-transfer behaviour of such weakly polar molecules. Of course any structural information on the CO dimer would be very interesting in order to see whether it confirms the double-resonance results. Prof. A. van der Avoird (University of Nijmegen) said: I do believe that CO molecules behave rather differently from N,, mainly due to the importance of P, terms in the short-range anisotropic interaction with other collision partners. Support for the importance of such terms can be found in a recent calculation of the CO-H, interaction potential, followed by a spherical expansion [cf.eqn (3) of our paper]. It appears that, among the anisotropic vLA,LB,L(R) terms in the potential (where LA refers to CO and LB to H,), vl,o,l is smaller than v , , ~ , ~ but considerably larger than u , , ~ , ~ , at least in the short range. At the same time it should be remembered that the116 GENERAL DISCUSSION short-range contribution to u2,2,4 is substantial. (This term does not just contain the qua drupole-quadrupole interaction .) M. C. van Hemert, Thesis (University of Leiden, 1981). Dr. R. Altman, Dr. M. Marshall and Prof. W. Klernperer (Haruard University) said: It is well known that the isoelectronic species N2 and CO are quite similar. This similarity is useful in illustrating the likely complexity in charge distributions of weakly bound complexes.The geometric structure of CO and N2 complexes is extremely similar. The dipole polarizability of CO and N, are also very similar. The dipole moments of complexes involving these species are however quite different. We illustrate this point below. co 1.95 2.60 2.886 0.592 0.482 2.41 0 1.5178 0.388 NZ COIN2 1.76 1.1 1 2.38 1.09 2.875 1 .oo 0.35 0.35 1.38 2.41 3 1 .oo 1.2497 0.2525 1.53 We note that the ratios of induced moments do not scale simply with polarizability nor are they constant from one complex to another. In this sense the electrical properties appear to reflect the complexity of bonding in these Van der Waals acceptor- donor complexes. Prof. A. D. Buckingharn (Cambridge University) said: Prof. Klemperer has given us interesting results for the electric dipole moments of BF3C0, BF3N2, HClCO and HClN,.He concluded, from a comparison of the dipole moments of the correspond- ing complexes of CO and N2, that the induced dipoles are not proportional to the polarizabilities. However, the electric field at the CO and N2 molecules is far from uniform. The absence of a centre of symmetry in CO means it has a dipole-quad- rupole polarizability A that will produce a dipole in a field gradient. The induced dipole linear in the field is Apa = aa&p + +Aa@&y + . where AaE, is the gradient of the electric field. The tensor Aaay is responsible for rotational Raman scattering by compressed CH4, for in a tetrahedron aaa is isotropic but AXy2 # 0. In the non-uniform field near an atom or molecule in an optical field, the rotating CH4 molecule has a fluctuat- ing induced dipole proportional to A,,, which couples its rotation to the radiation field.2*3 The tensor Aapy may be considered to describe the asymmetric distribution of the polarizability within the molecule ; it vanishes in centrosymmetric molecules.The dipole induced in CO by a field gradient can be evaluated since its A-tensor has been determined by ab initio cornp~tation.~ For a linear structure of the type +CO, the ratio of the dipoles proportional to A and to a isGENERAL DISCUSSION 117 where n = 3 if a dipole and 4 if a quadrupole, produces the field. For CO, Azzz/azz = -0.474 A where the z axis is from C to 0.4 Thus if R = 3 8, ApA/Apa = 0.237 for a dipole and 0.31 6 for a quadrupole.Thus the simple model Ap = a Eis not adequate. However, the general long-range form Ap = a * E + +A*AE + * . * + $p: E2 +Y&Y i E3 + . . . may be appropriate. Non-linear contributions to Ap, determined by the hyper- polarizabilities p,y etc. probably contribute only a few percent of Ap. A. D. Buckingham, Ado. Chem. Phys., 1967, 12, 107. A. D. Buckingham and G. C. Tabisz, Mol. Phys., 1978, 36, 583. D. P. Shelton and G. C. Tabisz, Mol. Phys., 1980,40, 299. R. D. Amos, Chem. Phys. Lett., 1980,70,613. Miss S. H. Ling, Mr. R. R. Miledi and Dr. M. Rigby (Queen Elizabeth College, London) (communicated). Dr. Barker has pointed out the value of empiricism in the de- velopment of intermoleqular potentials. We have attempted to establish an accurate potential for nitrogen using procedures which have proved reliable when applied to the inert gases.Our approach has been to use results from quantum-mechanical calculations in the regions for which they are believed to be reliable, at short and long range. We believe that the behaviour in the well region cannot yet be obtained with sufficient accuracy in this way, and have used comparisons with thermophysical properties to establish the shape of the potential well. The form of the short-range potential was studied by carrying out ab initio cal- culations for the dimer (N2)2 in several relative orientations and at a number of separ- ations. The results were in good general agreement with those recently published by other worker~,l-~ and have been fitted to a simple analytic function based on a diatomic site-site interaction model.The long-range function was based on the results of Mulder et aZ.,4 which includes several electrostatic and dispersion contributions. Electron-cloud overlap effects on the quadrupole-quadrupole energy were taken into account using the procedure of Ng et aL5 The total potential function was represented as a sum of these two interactions, with the long-range functions modified in the well region by a variety of cutoff functions similar to those employed in the HFD potentials for the rare gases.6 Various damping functions have been studied, and the most suitable form determined on the basis of second virial coefficients and 0 K lattice properties. Allowances were made for the contribution to the sublimation energy due to non-additive triple-dipole contribution^,^ and quantum corrections to the virial coefficients were estimated on the basis of approximate potential models.Calculations of dilute-gas shear viscosity were then carried out, using the Mon- chick-Mason scheme, for some of the better potentials. No single potential was found which was capable of completely reconciling the lattice properties, virial coefficients and viscosities. This may reflect the inadequacy of the Monchick-Mason procedure. For our best potential functions, the maximum attractive energy for the N2 dimer was found for the crossed structure, in agreement with the findings of Berns and van der Avoird.8 However, our potentials differed in several respects from those of these workers, and the unweighted angle averaged potential from our models had deeper wells and less steep repulsion than their potentials.F. H. Ree and N. W. Winter, J. Chem. Phys., 1980,73,322. B. Jonsson, G. Karlstrom and S. Romano, J. Chem. Phys., 1981, 74, 2896. F. Mulder, G. van Dijk and A. van der Avoird, Mol. Phys., 1980,39,407. K-C. Ng, W. Meath and A. R. Allnatt, Mol. Phys., 1979, 38, 449. * D. G. Bounds, A. Hinchcliffe and C. J. Spicer, Mol, Phys., 1981,42,73.118 GENERAL DISCUSSION R. Ahlrichs, R. Penco and G. Scoles, Chem. Phys., 1976,19,119. R. M. Berns and A. van der Avoird, J. Chem. Phys., 1980,72, 6107. ' P. Monson and M. Rigby, Mol. Phys., 1980, 39, 1163; 1981,42, 249. Dr. J. Tennyson and Prof. A. van der Avoird (University qf Nijmegen) said: In the last few months we have performed fully variational ro-vibrational calculations on the nitrogen dimer, (N2)2, utilizing a secular equation method based on the close-coupling approach.We have formulated the problem in body-fixed coordinates using the Hamiltonian given by eqn (7) of our paper. As in eqn (8) the basis set is constructed from R - 'xn(R>g42 d e A , vA,& ,PB)D~, M A a A O ) (1) (2) with y$k".jB = 2 yjA,m(oA,qA) y j B , k - m(eB,qB)(ikljA,m ; j B , k - m). m Following recent work on atom-diatom systems the radial functions were written in terms of associated Laguerre polynomials (3) (4) Xn(R) == p+ NnaY(a+1)/2e-Y/2La "01) Y = A exp[--P(R -Re)] where the parameters A , p and Re (and hence a) are optimized variationally. paper, the integrals over the angular coordinates can be performed analytically With this basis and the spherically fitted dimer potential, eqn (3) and (4) of our <gj';$' 1 vAB(R,'A,oB,vA - qB)Iq$kzJB) @A'pA'@B'IB A B 2 uL L(R)(-l).i--k+jA+jB z= bkkl A B, LA, LB, L x [j'(j' + 1)jA(jA + 1)jtXjE; + 1)L(L + l ) L A ( L A + 1)LB(& + l)j(j + l)jA ( j A + 1 ) j d j B + I)]+ For J (the total angular momentum) = 0 use of all basis functions with n 5 4 and jA,jB I 9 leads to a secular problem of dimension 3350.This can be greatly simplified, however, by use of the permutation and inversion symmetry of the system. Schematically, functions of the form (6) cDpi(j),p(jA) = 2 - y w j k , j B a & g j i : j A ) ; j A + j B even and two-vectors describe a basis for the irreducible representations when J = 0, where p ( i ) is the parity of i.This allows the problem to be solved in ten blocks, and means that for J = 0 no secular matrix larger than 475 x 475 needs to be diagonalised. It is possible to use the local symmetry of basis functions (6) and (7) about the equilibrium structure (6, = o B = 90", v, = qA - qB = 90" or 270") to make a corres- pondence between the present results and the fundamental vibrational levels given by the harmonic-oscillator rigid-rotor model. Table 1 makes such a comparison. TheGENERAL DISCUSSION 119 TABLE 1 .-" FUNDAMENTAL " FREQUENCIES OBTAINED WITH THE FULL CALCULATIONS AND THE HARMONIC OSCILLATOR MODEL, USING THE SPHERICAL POTENTIAL OF BERNS AND VAN DER AVOIRD coordinate symmetry frequenciedcm - DZd SJ 0 Ci harmonic full R A1 A: 39.2 33.2 8,,8B E E+ 22.1 14.2 v B1 B: 13.9 8.I zero-point energy 48.7 47.1 anharmonicity in the potential causes the harmonic-oscillator model to overestimate the fundamental frequencies by up to 40%. The zero-point energy, however, is well represented, suggesting that the ground state is fairly harmonic in character. Besides these vibrationally excited levels there is one other low lying level (sym- metry B y ) which lies only 2.4 cm-' above the ground state. This is due to the splitting of the ground state by tunnelling through the low barrier, 25.5 cm-l, in the 9 co- ordinate which separates the two equivalent equilibrium structures at 9 = 90" and 270". Fig. 1 shows cuts through various wavefunctions for UA = OB = 90". The 360 q / o 180 0 5.0 360 180 0 7.0 9.0 5.0 7.0 360 180 d" 0 5 .O 7.0 9.0 RlG 9 .o FIG. 1.-Amplitude of the wavefunctions of the lowest states of A:, B; and B t symmetry with J = 0.The cuts are with OA = 8, = 90". Solid and dashed contours represent regions of positive and negative amplitude, respectively.120 GENERAL DISCUSSION A,+ and B; states lie well below the barrier; the Bf state is the highest state " local- ised " in the q coordinate. This is in good qualitative agreement with the analysis by Long et aL2 of their infrared spectrum. Higher excitations should become increasingly free-rotor like. This trend is difficult to detect due to coupling with vibrations in other coordinates and in particular the effect of the slightly larger barrier in the 8 directions. Many of the observed free internal rotor levels seem to be resonances in the continuum which would be difficult to reproduce with a secular equation method.Fig. 2 shows the ground state and those states that we identify with the R and 180 180 O,/" 90 - 0 I I 0 5.0 7.0 9.0 5.0 7.0 9 .o 180 90 @I O 0 5.0 7.0 9.0 Ria0 FIG, 2.-Amplitude of the ground-state, stretch fundamental and (@A,&) fundamental wavefunctions with J = 0. The cuts are for p = OB = 90". Solid and dashed contours represent regions of positive and negative amplitude, respectively. (8,,8,) fundamentals. In terms of the harmonic-oscillator model the stretching fundamental is strongly mixed with a state deriving from an overtone in the (OA,O,) bending coordinate. Mixing such as this would be difficult to simulate within the BOARS approximation used by Barton and Howard for (HF),.With the basis outlined above, we have obtained all the bound states with J = 0 and 1. The J = 1 states were calculated neglecting Coriolis interactions, making k a good quantum number. This approximation has been tested by performing full calcu- lations for certain symmetry blocks. The effect of the Coriolis interactions is found to be small.GENERAL DISCUSSION 121 The nitrogen dimer can be regarded as a collision complex made up of two N2 monomers each having either ortho or para nuclear spin symmetry. This symmetry is fully reflected by our symmetry adapted basis functions (6) and (7). Conservation of this symmetry means that for ortho-para complexes the J = I , k = 1 state is lower than the J = 0 state.The full set of results is currently being prepared for publi- cation. J. Tennyson and B. T. Sutcliffe, J. Chem. Phys., 1982, Oct. 15th. C. A. Long, G. Henderson and G E. Ewing, Chem. Phys., 1973,2,485. Prof. R. J. Le Roy (University of Waterloo) said: It is very striking to note that the calculations described by Barton and Howard and by Tennyson and van der Avoird effectively involve the accurate solution of systems of up to several hundred coupled differential equations using, respectively, the corrected Born-Oppenheimer method of Hutson and Howard or the secular equation method of Grabenstetter and LeRoy [see ref. (4) for example]. I wonder if these authors might have any comments on the relative merits of these two schemes for such very large scale applications? A.E. Barton and B. J . Howard, Faraday Discuss. Chem. SOC., 1982, 73, 45. J. Tennyson and A. van der Avoird, Favaduy Discuss. Chem. SOC., 1982, 73, 118. J. M. Hutson and B. J. Howard, Mol. Phys., 1980,41, 1123. R. J. LeRoy and J. S . Carley, Ado. Chem. Phys., 1980,42, 353. Dr. J. Tennyson (University of Nijmegen) said: I would like to make three points in reply to Prof. LeRoy. First the BOARS method used by Barton and Howard for (HF), represents only an approximation to the secular equation method with the same angular basis functions (channels) and a saturated radial basis set. The methods thus do not yield the same level of accuracy, and it is clear from our results on the nitrogen dimer * that certain couplings (Fermi resonances) which we find significant are neglected in the BOARS approximation.Secondly, the bottleneck in the secular equation method is diagonalisation, even for the diatom-diatom problem where the potential matrix elements involve sums over 9 - j symbols. The current generation of vector processors can greatly speed up this step, meaning the method is suitable for more taxing problems. Finally, I note that both Hutson and Howard and ourselves used methods formu- lated in body-fixed coordinates, unlike the method of Grabenstetter and LeRoy. Besides being computationally simpler than the equivalent space-fixed formulation, body-fixed coordinates allow the simplifying approximation of neglecting the Coriolis interactions. We have found this approximation to be surprisingly A.E. Barton and B. J . Howard, Faraday Discuss. Chem. Soc., 1982,73,45. J. Tennyson and A. van der Avoird, J. Chem. Phys., 1982, Dec. 1st. R. J. LeRoy and J. S. Carley, Adu. Chem. Phys., 1980,42,353. J. Tennyson and B. T. Sutcliffe, J . Chem. Phys., 1982, Oct. 15th. Dr. B. J. Howard (Oxford University) said: Although the BOARS method presented by us uses a Born-Oppenheimer separation of angular and radial variables as a first approximation, it is not true that the final result is any less accurate than the close coupling and secular equation methods. All the non-adiabatic couplings between channels (including Coriolis interactions) are included by perturbation theory using the correction Born-Oppen heimer (CBO) method of Hutson and Howard.' The advantage of the CBO method is that it leads to a considerable saving in computing time over the other methods.It is also capable of dealing with near degeneracies and Fermi resonance if Brillouin-Wigner rather than Rayleigh-Schrodinger perturbation theory is used.2122 GENERAL DISCUSSION J. M. Hutson and B, J. Howard, Mol. Phys., 1980,41, 1113 and 1123. J. G. Frey and B. J. Howard, Chem. Phys., submitted for publication. , Prof. R. J. Le Roy (University of Waterloo) said: The absence of short-range 9- dependent terms in the potential energy surface of Barton and Howard reflects the lack of sensitivity of the available spectroscopic data to such terms. However, our studies of the H,-inert-gas systems suggest that level broadening due to internal- rotational predissociation is relatively much more sensitive to the short-range potential anisotropy than are level spacings and expectation values of the type analysed by Barton and Howard.Measurements of predissociation lifetimes should therefore be a sensitive probe of the short-range potential anisotropies. A. E. Barton and B. J. Howard, Faraday Discuss. Chem. SOC., 1982, 73,45. Prof. J. M. Lisy (University of Illinois) said: The gas-phase vibrational spectra of hydrogen-bonded systems can now be measured using vibrational predissociation Such information will be quite useful in testing new intermolecular potentials. The vibrational predissociation spectrum of (HF)2 contains a combin- ation band involving intramolecular and intermolecular modes. This would be con- sistent with a hydrogen-bond vibrational energy of ca.90 cm-’. It would be useful to compare this value with the harmonic vibrational frequencies of the (HF)2 potential surface of Barton and H ~ w a r d . ~ J. M. Lisy, A, Tramer, M. F. Vernon and Y. T. Lee, J. Chem. Phys., 1981,75,4733. Lee, Faraday Discuss. Chem. SOC., 1982,73, 387. A. E. Barton and B. J. Howard, Faraday Discuss. Chem. SOC., 1982,73,45. ’ M. F. Vernon, J. M. Lisy, D. J. Krajinovich, A. Tramer, H. S. Kwok, Y. R. Shen and Y. T. Dr. B. J. Howard (Oxford University) said: The potential-energy surface for (HF)2 determined in the present paper leads to vibrational frequencies considerably larger than those estimated from an analysis of the predissociation s p e c t r ~ m . ~ ’ ~ Within the harmonic approximation, the frequency for the stretching of the hydrogen bond is calculated to be 178 cm-’ while the symmetric and antisymmetric bends are deter- mined to 337 and 520 cm-l, respectively.The remaining torsional vibrational frequency is of a similar magnitude but is poorly determined. Because of large an- harmonicity effects, especially in the bending motions, the above vibrational frequen- cies are too large. Far better estimates of the vibrational intervals are 148, 160 and 304 cm-l, respectively ; these values are obtained directly from the first-order Born- Oppenheimer separation calculations outlined in the paper.l A. E. Barton and B. J. Howard, Faraday Discuss. Chem. SOC., 1982,73,45. M. F. Vernon, J. M. Lisy, D. J. Krajinovich, A. Tramer, H. S. Kwok, Y. R. Shen and Y.T. Lee Faraday Discuss. Chem. SOC., 1982, 73, 387. ’ J. M. Lisy, A. Tramer, M. F. Vernon and Y. T. Lee, J. Chem. Phys., 1981,75,4733. Prof. G. E. Ewing (Indiana University) said: I feel a need to emphasize that molecu- lar beam experiments on Van der Waals molecules have offered a rather restrictive pic- ture of the intermolecular potential surface. The complexes, produced by supersonic nozzles, reside in energy levels near the minimum of this surface, and radiofrequency and microwave transitions do not interrogate energy levels far from this minimum. While these experiments do provide valuable information on the equilibrium geo- metries of Van der Waals molecules, details of the intermolecular surface essential to understanding other properties are missing. Consider the case of (HF), for example.Here the elegant analysis by Barton and Howard of the pioneering molecular beam experiments of Dyke et al., reveal theGENERAL DISCUSSION 123 semi-rigid geometry of (HF)2 locked into its bent configuration by the strong aniso- tropic hydrogen bond. It is the details of the intermolecular surface minimum which dictate this structure. However, there is another view of this surface which is revealed by quite different experiments. Imagine two HF molecules in high rotational states which are the products of predissociation of vibrationally excited (HF)2.3 Or in another type of experiment, vibrationally excited H F molecules collide and transfer their energy into rotational motion^.^ In the limit of exceedingly high rotational states the anisotropic intermolecular forces will be averaged out for the product molecules of these experiments.The two HF molecules will appear to each other as, essentially, isoelectronic Ne atoms. The reality of this effective surface is revealed by the isotropic interaction term for HF + H F (the result of ab initio calculations) which appears in fig. 4 or 5 of the paper by Barton and Howard.' The effective isotropic hard-sphere diameter [i.e. where Vmm(R) = 01 for HF + HF is 2.9 A, remarkably close to that for Ne + Ne with CT = 2.8 A.5 In order to understand vibrational-energy- transfer experiments we therefore need the details of the intermolecular potential surface at two extremes: near the minimum and at high energies above the dissoci- ational limit of the Van der Waals bond.The effective shape of the potential surface for the vibrational relaxation experiments is discussed Of course there is only one intermolecular potential surface for a Van der Waals molecule, and many types of experiments are required to map it completely. For- tunately we are provided with a great variety of these experiments at this meeting. A. E. Barton and B. J. Howard, Faraday Discuss. Chem. Suc., 1982,73,45. T. R. Dyke, B. J. Howard and W. Klemperer, J. Chem. Phys., 1972,67, 2442. M. F. Vernon, J. M. Lisy, D. J. Krajnovich, A. Tramer, H-S. Kwok, Y . Ron Shen and Y. T. Lee, Faraday Discuss. Chem. Suc., 1982, 73, 387. 0. D. Krough and G. C. Pimentel, J. Chem. Phys., 1977,67, 2993. J. 0. Hirschfelder, C. F. Curtiss and R. B. Bird, The Molecular Theory of Gases and Liquids (Wiley, New York, 1954).L. L. Poulson, G . D. Billing and J. I. Steinfeld, J. Chem. Phys., 1978,68, 5121. ' G . Ewing, Chem. Phys., 1981, 64,411. Dr. A. Tramer (University of Paris, Orsay) said: Infrared intensities for X-H stretching vibrations yield important information about charge distribution in hydro- gen bonded complexes, but it seems of interest to extend this study to overtone spectra: in a number of hydrogen-bonded systems in solution the enhancement of the X-H fundamental intensity is accompanied by a drastic decrease in the oscillator strength for the Av = 2 transition. This peculiar behaviour has not been explained: it may be due to the effects of " electric anharmonicity " [higher terms in the p =f(Q) development] cancelling those of " mechanic " anharmonicity.The estimation of ap/aQ and of a2p/i3Q2 (where Q is the normal coordinate of the X-H stretching mode of the hydrogen-bonded complex) from an ab initio treatment would be of great interest. Exact experimental data for infrared intensities of funda- mentals and overtones in isolated Van der Waals molecules are also needed. Mr. J. T. Brobjer (University of Sussex) said : In their paper Barton and Howard have used a multipole expansion to represent the attractive (essentially the electrostatic) part of their potential, It must, however, be questioned whether the multipole expansion is accurate at these relatively short distances, and how many terms in the expansion are needed for a certain level of accuracy. By using a Morokuma-type partitioning technique it is possible to calculate the exa:t electrostatic energy between two monomer wavefunctions, and this can be compared with the multipole expansion using the multipole moments of the monomer wavefunctions.An alternative method124 GENERAL DISCUSSION of calculating the electrostatic energy is to use a point-charge representation of the monomer charge distribution. Prof. Murrell and I have constructed a set of point- charge models which are constructed so as to reproduce the monomer multipole moments. The simplest model for H F is of two charges positioned on the nuclei whose magnitude is determined to give the dipole moment (either the experimental moment or that of the above-mentioned wavefunction depending on the mode’s inten- ded use).The most accurate model we have constructed for HF has three point charges and it reproduces all multipole moments up to and including the hexadecapole moment. We have shown that these multipole-fitted point-charge models give a more accurate electrostatic energy than an equivalent multipole expansion. The multipole expansion for the HF dimer with dipole-dipole, diple-quadrupole and quadrupole-quadrupole moments is in error by ca. 50% at R = 2.67 A. The point- charge model which fits dipole and quadrupole moments is in error by ca. 15% at the same distance. At the experimentally determined intermolecular distance the best point-charge model gives a dimer structure in agreement with experimental findings, A more detailed account of our results for (HF)2 is avai1able.l ’ J.T. Brobjer and J. N. Murrell, J. Chem. SOC., Faraday Trans. 2, 1982,78, 1853, Prof. J. S. Muenter (Uniuersity of Rochester) said: In reply to Stephen Berry’s informally-posed question of whether the isotropy of the Ar-SO, potential might arise from the size of the sulphur atom, we present fig. 3. Indeed, the van der Waals radius FIG. 3.-Schematic representation of Ar*S02. of sulphur dominates the sulphur dioxide shape. However, this view of SO2 does not address the large dipole and quadrupole moments of SO2 which contribute to the anisotropy of the complex. Ian Mills pointed out, privately, that the Ar-SO, force constant data in table 3 of our paper were inconsistent, with the off-diagonal element being too large. This problem arose from the extreme correlation in the data analysis. This is another indication of the isotropic potential of Ar-SO,.The absence of this problem in Ar-SO, is consistent with its semi-rigid nature. Prof. A. W. Castleman Jr, Dr. B. D. Kay, Dr. F. J. Schelling and Dr. R. Sievert (Uniuersity of Colorado) (communicated). For a number of years we have been actively engaged in a laboratory investigation of the properties, energetics of form- ation, stabilities and binding energies, and structures of both Van der Waals and ion clusters. Some of the results have potential significance to problems in the atmo-GENERAL DISCUSSION 125 spheric sciences. As part of the work on neutral Van der Waals species we have examined the formation dynamics and have carried out electrostatic deflection measurements for a variety of substances, including multiple clusters of SO2 and H20, SO3 and H20, and clusters of sulphuric acid, Other relevant studies have been made on clusters of up to forty water molecules, mixed clusters of nitric acid and water, and the nitric-acid/water/ammonia system.Investigations of pure water clusters have indicated, among other things, the enhanced structural stability of particular cluster species and the absence of a permanent dipole moment for water clusters (H20),, with 3 < n < 9, both effects indicative of a probable cyclic structure. Table 2 shows TABLE Z?.-FOCUSING OF MOLECULES IN ELECTROSTATIC FIELDS. EVIDENCE OF PERMANENT DIPOLE MOMENT focus do not focus (H20),(3 d n d 9) (NH3), (3 d PI d 6) (HN03)(H,0),(2 G n d 10) (CH30H), (3 d n < 17) (C2H50H),(3 f n d 9) (CD30D), (3 < n d 7) some of the species whose dipole moments have been examined to date.Experi- ments conducted with nitric acid are intended to probe solvation effects and the process of ion-pair formation upon the coclustering of an electrolyte with successively more water molecules.' A similar mechanism of atmospheric aerosol formation has recently been postulated which involves the reaction of a hydrated acidic cation with an appropriate hydrated basic anion following an ion-ion recombination.2 Inform- ation gained from these experiments are relevant to both atmospheric processes and in contributing to our knowledge of solvation and formation of the bulk liquid phase. Recent molecular-orbital calculations indicate that the complex H20(S02) should be fairly strongly bound (ca.10 kcal mol-') with a large dipole moment (ca. 7 D) and have a Lewis acid-base pair structure. Recently, there has been evidence that the kinetics of certain bimolecular reactions may be altered when the reactants are in the presence of another molecule to which one of them can become cl~stered.~ This can possibly have major implications for atmospheric chemistry. In order to account for the observed rapid rate of aerosol formation via the photo-oxidation of SO2 in the presence of H20, Reiss and coworkers postulated the existence of clusters of these specie^.^ In order to confirm the existence of a stable gas-phase H20(S02) adduct, as well as to examine the possibility of existence of higher (H20),(S02), clusters, electrostatic deflection experiments were conducted on cluster species produced by nozzle expansion of a mixture of SO2 and H20.Our results indicate that the H20(S02) species exists and is very polar, as evidenced by its strong refocusing in a quadrupole field. At higher partial pressures of SO2, the126 GENERAL DISCUSSION ion intensity detected as (H2O.SO2)+ did not refocus, strongly suggesting that multi- channel cluster fragmentation may play a dominant role in the ionization of the more highly-clustered species, as it does in atomic cluster^.^ A dominant ionization channel may be (H20)m32(S02)n>2 + e+(H,O-SO,)+ + neutral fragments + 2e. That is, the (H,O.SO,) + ion detected in the high-partial-pressure experiments may arise predominantly from non-polar higher clusters present in the neutral beam and not from the polar (H20.S02) entity.Similar studies currently in progress are intended to compare the sulphur-trioxide-water cluster with its isomer, sulphuric acid. Preliminary evidence indicates the existence of a stable H20(S03) adduct. Recent calculations predict a structure for this species for which the oxygen of the water is bound to the sulphur of the SO3 group. Studies of sulphuric acid clusters have similarly shown that while the monomer possesses a large dipole moment and can be refocused in electrostatic fields, the dimer is non-polar and does not refocus. This suggests that the complex has a head-to-tail configuration, the individual molecular dipole moments cancelling.The presence of significant dimerization among sulphuric acid molecules might retard the rate of nucleation to the condensed (aerosol) state, as the kinetics of clustering is dominated by long-range dipole-dipole and dipole-induced-dipole forces. B. D. Kay, V. Hermann and A. W. Castleman Jr, Chem. Phys. Lett., 1981,80,469. 1980, 283, 55. P. M. Holland and A. W. Castleman, Jr., J. Photochem., 1981, 16, 347. E. Hamilton Jr and C. Naleway, J. Phys. Chem., 1976, 80, 2037. D. Marvin and H. Reiss, J. Chem. Phys., 1971, 69, 1897. B. D. Kay, Ph.D. Thesis (University of Colorado, 1982) and references therein. (a) N. Lee,Ph.D. Thesis (Yale University, 1976); (6) A. Hermann, E. Schumacher and L. Waste, J. Chem. Phys., 1978,68, 2327. * (a) E. E. Ferguson, personal communication; (b) F. Arnold and P.Fabian, Nature (London), Dr. F. A. Baiocchi and Prof. W. Klemperer (Haruard University) said: In comment- ing upon the very important work presented by Drs. Legon and Millen, we wish to make a few points. The linearity of the structure of C0,HF is well established in our view. We note that the microwave and radio spectrum of both hydride and deuteride were investigated. In addition the complex SCOHF was shown to have a similar structure. More recently Robert Altman and Mark Marshall have completed a structural study of C02HCl (and DCl). This system is also linear. In our view the difference between C0,HF and N20HF remains an intriguing problem. The structure of hydrogen-bonded complexes with oxygen-containing systems appears to be a rich field.We have recently completed radiofrequency and micro- wave spectroscopy of the hydrogen fluoride-formaldehyde system. The structure of the complex is a rigid planar geometry, shown below. The OH bond length is short, 1.74 A. The angle between the CO and HF units is 71", a surprising angle in view of the large dipole moment of both units, since the dipole-dipole interaction is almost zero for this orientation. The electric-dipole-GENERAL DISCUSSION 127 moment components are large, namely pa = 3.75 D and f i b = 1.40 D. This n-bonded complex complements the complexes discussed by Legon and Millen. Finally we wish to point that there is a considerable arbitrariness in fitting Lennard- Jones potentials to weak bonds with the hope of extracting binding energies.We note, since we have also done what Legon and Millen have done, that everyone when fitting hydrides uses the heavy-atom distances in the fitting procedure even though it is the hydrogen that is the actual bonding atom. When molecules involving heavy atoms are considered then the actual bonding (nearest) atoms distance is used in fitting the two parameters of the Lennard-Jones potential. In this sense the present procedure is arbitrary. Mr. A. J. B. Cruickshank (University ofBrisrol) said : Dr. Legon’s paper presents not only zero-point dissociation energies, but also effectively complete specifications of the vibrational modes characterising each hydrogen-bonded dimer. Since the geometry and the intermolecular distances are known, it should be possible to set up the rotation- vibration partition function for each dimer species.Indeed Dr. Legon refers to the partition functions for the dimer and the participating molecules. Is it not possible therefore, given the effective enthalpy of formation, -Do, to calculate the standard Gibbs energy of formation and the association constant, and so the equilibrium con- centration of hydrogen-bonded dimers? Could one thereby make an estimate of the contribution made by these dimers to the gross thermodynamic properties of the gas mixture ? Dr. A. C. Legon and Prof. D. J. Millen (Uniuersity College, London) said: We thank Prof. Klemperer for his comments about the geometry of CO, - - HF and for his presentation of the recently determined structure of H,C=O - - - HF.We note that the latter result fits well into the simple classication scheme presented in our paper. In connection with the use of Lennard-Jones potential-energy functions to describe the radial potential-energy variation in the dimers discussed in our paper, we recognise its parametric nature and limitations. However, as pointed out in introducing the paper, it is a simple function which allows a value of the dissociation energy D, to be determined for such dimers from the ground-state spectroscopic constants alone under the pseudodiatomic approximation. In addition to this advantage, the Lennard-Jones potential, in the one case where comparison with a directly measured value of D, is possible (HCN * * * HF in table 5 of our paper), leads to good agreement and, more- over, the changes in D, along series such as HCN - - * HX and OC - - - HX where X = F, C1, Br (see table 7) are qualitatively in agreement with chemical expectations.In reply to Prof. Berry’s informally-posed question about the isomer CO * - - HF, a careful search was made for its rotational spectrum by using the pulsed-nozzle, Fourier-transform technique but without success. If, as appears likely, this isomer were even 20 cm-l higher in wavenumber than OC - - HF it probably would not be detected at the effective temperature of ca. 3 K characteristic of the technique. Mr Cruickshank is of course correct in suggesting that, in the case of HCN - * * HF, sufficient information exists to calculate the thermodynamic functions, and in particular the equilibrium constant, for the reaction since in this case a proper value of Do and the values of the quantities necessary to calculate the partition function are available.This is not true in other cases where either information about Do and/or the bending vibrations is lacking. We plan to128 GENERAL DISCUSSION calculate the thermodynamic functions for the association reaction involving HCN and H F. Dr. J. T. Brobjer (University of Sussex) said: I would like to point out that it is possible to predict the lowest-energy geometry for most dimers by using the inform- ation contained in the multipole moments of the monomers. It is relatively well known that for polar molecules it is largely the electrostatic part of the potential that determines the dimer structure. With our multipole-fitted point- charge models we have calculated the equilibrium structure (for any given &,) and our results are in close agreement with experiment, as shown in table 3.TABLE 3 .-COMPARISON OF EXPERIMENTAL DIMER CONFIGURATIONS WITH THOSE PREDICTED BY OUR MULTIPLE-FITTED POINT-CHARGE MODELS ~~ ~ expt.a p c b expt. p.c. expt. p.c. expt. p.c. R' 5.27 5.27 6.4 6.4 6.69 6.57 8.28 8.28 81 - - 6 0 -6 -5 -7 0 0 82 60-70 66 50 62 79.5 91 0 0 HCN - * HF HCN - - * HCl H2 * * * HF N2 * * * HF expt. p.c. expt. p.c. expt. p.c. expt. p.c. R 6.32 6.32 R R R R R R 0, 0 0 0 0 90 90 0 0 8 2 0 0 0 0 0 0 0 0 expt. p.c. expt. p.c. 5 R 5.62 5.63 - 0 -0.7 -4 - 81 02 58 73 46 52 Experimental; ' point charge; ' R expressed in atomic units between centre of mass (note that the point-charge models give 19~ and O2 but not R).Note that a lone-pair model in its simplest form would fail to account for the differences in angles for (HF),, HF * - - HCI, HCl HF and (HCl),. Dr. A. C. Legon and Prof. D. J, Millen (University College, London) said: Dr. Brobjer's proposal is interesting. We note, however, that the multipole-fitted point- charge model requires, first, A,, either from experiment or from reliable ab initio calculation, in which cases the geometry is likely to have been established, and secondly, several terms in the multipole expansion of the electric charge distribution, which are generally not known except for the simplest molecules. We believe that the lone-pair model in its simplest form does not make a sig- nificantly worse prediction of the angles than the point-charge model in the cases of (HF), and HF - - .HC1 (for which experimental angles are known) and in the case ofGENERAL DISCUSSION 129 HCl - - HF we are not aware that experimental values are available. Finally, we take the view that it is the experimental angle that allows us to diagnose the lone-pair direction most directly and thus the result for (HCI), is the best experimental evidence available for a localised orbital description of the HC1 monomer. Dr. B. J. Howard, Dr. C. M. Western and Mr. P. D. Mills (Oxford University) said: We wish to raise the question of the nature of the Van der Waals bond. Traditionally the attractive forces involved have been considered as the natural extension of the long-range electrostatic, induction and dispersion forces to the region of the potential minimum.It has, however, also been shown that the structure of some Van der Waals molecules correlates strongly with a HOMO-LUMO picture of the interaction and, since there must be some orbital overlap between the molecules in this region, that this leads to the possibility of an incipient chemical bond. More recently, electric quad- rupole coupling constants have been measured for quadrupolar rare-gas nuclei in Kr- HC1 and Xe-HF.3 The observed field gradients at the nuclei can be accounted for by an induced electric quadrupole in the rare-gas electron distribution by the neigh- bouring hydrogen halide. It is concluded that there is no need to invoke theories of charge transfer to explain the Van der Waals interactions in these species.We have investigated the radiofrequency and microwave spectra of the Van der Waals molecule Ar-NO using the technique of molecular-beam electric resonance spectroscopy. The molecule is an ideal probe of electronic rearrangement in that it contains the open-shell molecule NO which possesses both unpaired electron spin and unquenched orbital angular momen tum. NO also possesses suitable unoccupied orbitals for possible HOMO-LUMO charge transfer. In addition the electron distribution can be monitored via the large nuclear hyperfine interactions due to the nitrogen nucleus. Analysis of the rotational spectrum shows the complex to be T-shaped as predicted by molecular-beam scattering data and electron-gas calculations of the inter- molecular potential.However, unlike conventional open-shell asymmetric-top molecules, the orbital angular momentum is only partially quenched, the spin-orbit coupling being significantly larger than the barrier to free orbital motion of the un- paired electron. We determine the vibrationally averaged structure to be near T- shaped with an argon to NO centre-of-mass distance of 3.65 A and with this inter- molecular axis at an angle of 95" to the NO internuclear axis. The barrier to free orbital motions as measured by the difference in energy between the in-plane and out- of-plane z* orbital on NO is ca. 3 cm-l, much less than the total bond energy of ca. 100 ~ m - ' . ~ In addition, the nitrogen nuclear hyperfine structure can be explained by projecting the known hyperfine constants of NO on to the inertial axes of Ar-NO.The observed hyperfine splittings are reproduced to within 1% and we conclude that there is little electron rearrangement. This result together with the unusually small barrier to orbital motion further support the view that the individual molecules retain their identity in a Van der Waals molecule and that there is no need to invoke incipient chemical bonding. S. J. Harris, K. C . Janda, S. E. Novick and W. Klemperer, J. Chem. Phys., 1975,63,881. M. R. Keenan, L. W. Buxton, E. J. Campbell, T. J . Balle and W. H. Flygare, J. Chem.Phys., 1980,73, 3523. L. W. Buxton, E. J. Campbell, M. R. Keenan, T. J. Balle and W. H. Flygare, Chem. Phys., 1981, 54, 173. H. H. W. Thuis, P1z.D. Thesis (Nijmegen, 1979).G. C. Nielson, G. A. Parker and R. T. Pack, J. Chem. Phys., 1977,66,1396. Prof. A. D. Buckingham (Cambridge Uniuersity) said: Dr. Howard raised the matter130 GENERAL DISCUSSION of electron delocalization in Van der Waals molecules containing the nitric oxide radical. He reported that he saw no evidence for electron transfer in the Van der Waals molecule. A very sensitive test of electron delocalization is provided by measurements of the unpaired electron density at a nucleus in the neighbour. This can be achieved by observations of hyperfine splittings that involve a nuclear spin of the diamagnetic neighbouring molecule.' Delocalization can also be detected by measuring the effect of a paramagnetic buffer-gas on the n.m.r. resonance f r e q ~ e n c y .~ ~ ~ Thus for 129Xe ( I = 5) the large downfield shifts proportional to the pressure of 0, and of NO have been interpreted in terms of a contact shift resulting from the over- lap of the 5s atomic orbitals of Xe with the unpaired orbitals of the O2 or NO. Al- though the intermolecular contact shift was readily measured in the case of Xe in O2 and NO, its magnitude is reduced by the smallness of the overlap of the 5s orbital of Xe with the unpaired orbitals of O2 or NO because of " shielding " by the 5p electrons. The effect would presumably be larger in Hg in O2 or NO (199Hg has I = + and an abundance of 16.86%). G . W. Canters, Corvaja and E. de Boer, J . Chern. Phys., 1971,54,3026. C. J. Jameson, A. K. Jameson and S . M. Cohen, Mol. Phys., 1975, 29, 1919.A. D. Buckingham and P. A. Kollman, Mol. Phys., 1972, 23, 65. ' C. J. Jameson and A. K. Jameson, Mol. Phys., 1971, 20, 957. Prof. Z. H. Zhu (Chengdu University, China) and Prof. J. N. Murrell (University of Sussex) said: The dimer of HCN has been examined by microwave spectroscopy and found to have a linear structure.l This contrasts with the structure of the hydrogen halide dimers which, although having a linear hydrogen bond, are non-linear overall. and from thermodynamic measurements an enthalpy of dimerization of -3.3 kcal mol-l was ~ b t a i n e d . ~ Theoretical calculations by Johansson et al.5 have been made on the collinear dimer using SCF theory and an STO-3G basis (there had earlier been calculations with the CNDO method).6 Retaining the monomer geometry they obtained a distance RCN = 3.2 for the hydrogen-bond bridge and a stabilization energy of 3.7 kcal mol-l.They also calculated the energy of a side-by-side (rectangular) structure but found no stable form. In contrast the CNDO calculations found this side-by-side structure to be more stable than the collinear.6 We have re-examined the dimer structure using a larger basis and made a more extensive examination of the potential energy surface. Calculations were first made at The existence of the HCN dimer in the gas phase was postulated as early as 1929 1.053 1.142 2.233 1.059 1.145 H- C-N -----H-C--N FIG. 4.-Calculated energy and geometry of (HCN)'. E = - 185.8056 au; AE = 4.559 kcal mol-' the 6-31 G level using the Gaussian 76 program to find the optimum geometry of the collinear dimer; the results are shown in fig.4. The CN distance of 3.292 A is only a little longer than the value found by Johansson and co-workers but we predict a short- ening by 0.1A of the CN bond length in the H-bond donor molecule. We have con- firmed (fig. 5 ) that this structure is stable to a bending displacement which retains the linear hydrogen bond. Calculations were then made at the geometry found above using the 6-311G** basis (a triple valence basis plus polarization functions on all atoms).8 After correc- tion for basis set superposition error this gave a binding energy of 4.56 kcal mo1-I. With this basis the dipole moment of the HCN monomer is calculated to be 3.21 DGENERAL DISCUSSION 131 -185.54 - 1 85.56 - -1 85.58 - -185.60 - -185.62 - -185.64 -185.66 FIG.5.-Change in energy of (HCN)2 on distortion from the linear structure. which is slightly greater than the experimental value of 2.98 D.9 If we take the major contribution to the dimer energy as being the dipole-dipole energy, then we can make an empirical correction to the binding energy and arrive at the value 3.93 kcal mol-l. It is surprising that the previous ab initio calculations on the side-by-side dimer did not find a stable structure as the dipole-dipole energy is stabilizing for this configur- ,,.,j 70.0 6 5 . 0 &- 1.053 1.144 H - C - N g; I 78.49' I l'e I N - C J H €=-185.8017 a u A€=2.110 kcal mol-' 3.25 3.50 3.75 4.00 4.25 4.50 RIA FIG. 6.-Contours of the side-by-side dimer calculated with the 6-31G basis. Contour a = -185.66015 &, intervals 5 x lo-" Eh.132 GENERAL DISCUSSION ation. We suspected that this was because there was insufficient geometrical flexi- bility in the previous work and to confirm this we have obtained contours on a two- variable surface for the geometry shown in fig.6. With fixed monomer dimensions we found at the 6-31G level the geometry shown. At this geometry, using the 6--311G** basis and superposition correction we obtained a binding energy of 2.11 kcal mol-’. After the empirical dipole moment correction this reduces to 1.82 kcal mo1-’. By symmetry, the side-by-side dimer could not be detected by microwave but it may contribute to non-ideal gas behaviour at high temperatures. A. C. Legon, D. J. Millen and P. J. Mjoberg, Chem. Phys.Lett., 1977,47,589. T. R. Dyke, B. J. Howard and W. Klemperer, J . Chem. Phys., 1972,56, 2442. H. Sinosaki and R. Hara, Tech. Repts. Tohoku Imp. Univ., 1929, 8, 19. W. F. Giauque and R. A. Ruehrwein, J. Am. Chem. SOC., 1939,61,2626. A. Johansson, P. Kollman and S. Rothenberg, Theor. Chim. Acta, 1972,26,97. J . R. HoyIand and L. B. Kier, Theor. Chim. Acta, 1969, 15, 1. ’ W. J. Hehre, R. Ditchfield and J. A. Pople, J. Chem. Phys., 1972, 56, 2257. * R. Krishnan, J. S. Binkley, R. Seeger and J. A. Pople, J . Chem. Phys., 1980,72, 650. Stand., no. 10 (1967). R. D. Nelson, Jr., D. R. Lide, Jr., and A. A. Maryatt, Nut/ Stand. Ref. Data Ser., Nut1 Bur. Dr. J. M. Hutson (Uniuersity of Waterloo) said : Dr. McKellar has observed infrared lines of HD-Ar broadened by predissociation, and has compared the measured widths with preliminary calculations by Corey and LeRoy (see fig.13 of McKellar’s paper). However, the calculations were performed using the secular equation (SE) method for calculating widths, and this has since been found to be unreliable.2 Prof. LeRoy and I have therefore repeated the width calculations using the close-coupling method and the BC3 potential of Carley and L ~ R o ~ . ~ The basis set included all channels with HD rotational quantum number up t o j = 3, and the coupled equations were solved 0.0 1 3 5 7 9 I‘ FIG. 7.-Comparison of observed and calculated widths for S1(0) lines of HD-Ar broadened by predissociation. A, A, N-branch, J’ = I’ + 2; 0 , 0, T-branch J‘ = I’ - 2 (closed symbols, observed; open symbols, calculated).GENERAL DISCUSSION 133 using the method of DeVogelaere.Level energies and widths were calculated by fitting the S-matrix eigenphase sum to a Breit-Wigner formula with a quadratic background term.4 The calculated level widths are compared with the experimental widths in fig. 7: note that states with I’ > 7 can also predissociate by tunnelling through the centrifugal barrier, and were not considered here. The agreement between the observed and calculated level widths is very good. This confirms that the observed broadening is due to predissociation by internal rotation, and is additional evidence for the validity of the Carley-LeRoy BC3 potential for H,-Ar. J. E. Grabenstetter and R. J. LeRoy, Chem. Phys., 1979, 42, 41. R. J. LeRoy and J. S. Carley, Adu. Chem. Phys., 1980,42, 353. A. U. Hazi, Phys. Rev. A , 1979,19,920; C . J. Ashton, M. S. Child and J. M. Hutson, J. Chem. Phys., 1982, to be published. * R. J. LeRoy, G. C. Corey and J. M. Hutson, Faraday Discuss. Chem. Sac., 1982,73, 339. Dr. G. G. Baht-Kurti and Mr. I. F. Kidd ( University of Bristol) said : We would like photon energy/lO-’ cm-I FIG. 8.-Total photofragmentation cross section for the process Ar-H2(v = 0 , j = 0) ”y, [Ar-Hz(u = 0 , j = 2)]* _j Ar + H,.134 GENERAL DISCUSSION to report that we are in the process of performing calculations of photofragmentation cross-sections for the H,-Ar van der Waals molecule, which is one of the systems examined experimentally by Dr. McKellar. The theory underlying our calculations has been described in previous publications.' -3 The cross-sections which we calculate are of the form where yJMt is the original H,-Ar bound wavefunction, ~,Y-(~~W) is a scattering wave- function in which k specifies the direction of the Ar-H, relative motion and vimj are the H2 vibrational and rotational quantum numbers. The cross-section given above is integrated over all scattering directions, summed over final mj and averaged over initial Mi quantum numbers. Further details of the theory are available from pub- lished paper~.I-~ To illustrate the type of results we are currently able to obtain we present in fig. 8 and 9 below two different types of cross-section. In fig. 8 we show a total photofragmentation cross-section (i.e. summed over all -12 -10 - 8 - 6 - 4 -2 0 2 4 6 8 10 12 photon energy/10-3 cm-' hv FIG. 9.-Partial photofragmentation cross section for the process Ar-H2(u = 0,j = 0) [Ar-H2(v = 1 , j = 4)]* - Ar + H2(u = 1 , j = 0).GENERAL DISCUSSION 135 10 a 6 4 d I 2 2 X x .3 U ?v 2 w c -; E ._ M ." - 1 - 1 - 1 - 1 1 -10 -8 -6, - 4 - 2 0 2 4 6 8 1 real S matrix x lo-' FIG. 10.-Argand diagram of diagonal SI1 matrix element over energy range corresponding to fig. 8. final states of H,) centred around a photon energy of 360.2031 em-'. The processes corresponds to the excitation of a metastable rotationally excited state of the H, within the Ar-H, Van der Waals molecule Ar-H2(yZi Lv_ [Ar-H, v = o j = 2 ] * --+ Ar + H,. ground state Fig. 9 shows a partial photofragmentation cross-section corresponding to the produc- tion of H2 in its u = 1, j = 0 state. This cross-section is centred around the higher photon energy of 5285.0002 cm-I and corresponds to the processes Ar-H2(3Zt)hY- [Ar-H,(Yzi)]* ---+ Ar + H2('jZ$). ground state Of particular interest is the fact that the lineshape of this absorption cross-section is not symmetric. This can be interpreted as originating from a Fano-type interaction between the metastable bound state and the c o n t i n ~ u m . ~ In fig. 10 we plot an Argand diagram of a diagonal element of the S matrix which arises in our calculations over the energy range corresponding to that of fig. 8. As may readily be seen, the phase of136 GENERAL DISCUSSION the S matrix element changes by 2n over this small energy range, thus displaying a characteristic resonance behaviour. M. Shapiro, J. Chem. Phys., 1972,56, 2582. G. G. Balint-Kurti and M. Shapiro, Chem. Phys., 1981, 61, 137. U. Fano, Phys. Rev., 1961,124, 1866. * I. F. Kidd, G . G. Baht-Kurti and M. Shapiro, Faraday Discuss. Chem. SOC., 1981, 71, 287.

ISSN:0301-7249    

DOI:10.1039/DC9827300109

出版商:RSC    

年代:1982

数据来源: RSC

摘要:

Faraday Discuss. Chem. SOC., 1982, 73, 137-151 Spectroscopy and Photophysics of Organic Clusters BY DONALD H. LEVY, CHRISTOPHER A. HAYNAM AND DONALD V. BRUMRAUGH * James Franck Institute and Department of Chemistry, University of Chicago, Chicago, Illinois 60637, U.S.A. Received 1 1 th December, 198 1 A supersonic free jet expansion has been used to prepare small clusters of aromatic organic com- pounds, and their structure and energy-transfer properties have been studied by laser-induced fluores- cence. In the first, the two monomer units were side by side in a planar configuration. In the second, the two molecules were in a T-shaped configuration. In the T-shaped geometry the two rings are not equivalent, and spectra resulting from excitation of one or the other ring could be distinguished.A mixed-gas expansion of tetrazine and benzene in helium produced a mixed tetrazine-benzene dimer, a trimer consisting of two tetrazines and one benzene, and bands that were attributed to larger clusters. The benzene-tetrazine dimer had a stacked, parallel-plate geometry, while the T2B trimer had a benzene stacked parallel over a planar tetrazine dimer. Dimers of dimethyl tetrazine were found to be more tightly bound in the excited electronic state. Excitation of the 6 d ring mode in the dimethyl tetrazine dimer led to intra- molecular vibrational relaxation into the cluster modes. Dissociation was observed following excit- ation of the 6aX transition of the tetrazine-benzene dimer. The dimer of s-tetrazine was found to have two observable conformations. In recent years the technique of laser-induced fluorescence spectroscopy has emerged as a powerful tool for the study of Van der Waals molecules.’ While it lacks the extreme resolution of electric or magnetic resonance methods, it is very sensitive and relatively easy to apply.Moreover, analysis of the electronic spectrum is the only way to obtain information about electronically excited states of Van der Waals molecules. In addition to examining the structure of electronically excited states, laser-induced fluorescence has also allowed the study of Van der Waals photochemistry and photophysics. Until the studies described in this paper, all work in our laboratory was directed toward Van der Waals molecules composed of a single chemically bound molecule, the substrate, surrounded by one or more rare-gas atoms or first-row diatomic mole- cules Van der Waals bound to the substrate.The substrate was the chromophore and absorbed in the visible or non-vacuum U.V. where there were easily available excitation lasers. Also the molecule had to fluoresce with some finite quantum yield so as to be detectable by fluorescence techniques. The Van der Waals atoms or molecules had much larger electronic excitation energies, and only their ground electronic states seem to be important in describing the structure and dynamics of these species. We have now begun the study of small clusters of aromatic organic molecules where there is potentially more than one chromophore in the molecule. Prior to this work there were no spectroscopic data on the structure of such clusters.More- over, there were no data on how energy transfer either within a monomer unit or from * Present address: Eastman Kodak Company, Kodak Park, Rochester, New York 14650, U.S.A.138 ORGANIC CLUSTERS one monomer to another was influenced by the weak bonds of the cluster. Therefore this appears to be a fruitful area of study. This paper is a report of laser-induced fluorescence studies of clusters containing s-tetrazine or substituted tetrazines as the chromophore. A preliminary report of work on benzene clusters has been published.' The clusters are formed in a supersonic free jet and are small polymers of a single organic monomer or mixed polymers of more than one organic monomer, e.g. tetrazine and benzene.In favourable cases we are able to resolve rotational structure in the fluorescence excitation spectrum and in this way probe the structure of the cluster. Analysis of the dispersed fluorescence spectrum following excitation to a well-defined excited-state level provides both struc- tural information about the ground electronic state as well as dynamical information about the redistribution of energy between the time of excitation and the time of emission. EXPERIMENTAL The experimental apparatus is similar to that which we have used to study the spectroscopy of rare-gas-substrate Van der Wads molec~les.~ The clusters are prepared in a mixed-gas superonic expansion consisting of a small amount of the molecule of interest seeded into a carrier gas, usually helium.When two monomeric units are to be combined in a cluster, seeded gas mixtures of the individual monomers in helium are prepared, and the two seeded mixtures are then mixed with each other prior to expansion through the nozzle. Mixtures are prepared by passing the carrier gas over a solid or liquid sample of the seed molecule, and the concentration of the seed in the mixed gas is controlled by varying the temperature of the mixing chamber thus varying the vapour pressure of the seed. In this way we can independently control the total pressure and the individual concentrations of the seed molecules. Typical operating conditions would be seed gas in helium at a total pressure of 5 atm expanded through a 100 pm diameter nozzle. It should be understood that syn- thesis of a particular cluster may require expansion conditions which are quite different from these typical values.Fluorescence was excited by an argon-ion laser pumped tunable dye laser. Spectra were taken using three different dye-laser configurations depending on the required resolution and scan length. Long-range survey scans were made using a low-resolution laser containing only a birefringent filter as a tuning element. These spectra had a resolution of ca. 1 cm-I and had resolved vibrational structure but no resolved rotational structure. They could, however, cover a range of several hundred cm-'. High-resolution spectra were taken with a single-mode laser which could be hopped from mode to mode but could not be scanned smoothly between modes. In this case the effective resolution was the mode spacing which was on the order of 0.01 cm-', and the length of a single scan was ca.4 cm-'. In most cases these conditions allowed us to scan a complete vibrational band while resolving rotational structure. Finally, in a few cases, we were able to use a single-mode ring laser which could be scanned between modes. This laser had an effective resolution of ca. 0.003 cm-' but could only scan a 1 cm-' interval and thus required several individual scans to cover a vibrational band. Fluorescence excitation spectra were taken by scanning the dye-laser wavelength and collecting as much fluorescent light as possible at all frequencies. Light was collected by an achromatic camera lens and focused to an image on a slit which could be used to spatially select the fluorescent light.The Doppler width of the spectrum could be reduced by narrow- ing the slit, and the width of the slit was chosen in each case to match the Doppler width to the resolution of the laser. The fluorescence was detected by a cooled photomultiplier behind the slit operating in a photon-counting mode. Dispersed fluorescence spectra were taken by tuning the laser to an absorption frequency of the cluster, keeping the laser frequency fixed, and then dispersing the fluorescent light with a 1 m monochromator with a dispersion of 4 8, mm-'. The fluorescent light was collected and imaged onto the monochromator slit by a camera lens and periscope.D . H . LEVY, C . A . HAYNAM AND D . V . BRUMBAUGH 139 Most components of the experimental apparatus have been described in more detail elsew here.3 RESULTS AND DISCUSSION A .TETRAZINE Before describing the spectra of tetrazine clusters, it is helpful to examine the fluorescence excitation spectrum of the monomer shown in fig. 1. Tetrazine itself is a near symmetric oblate top ( K = 0.74), the near symmetric-top axis being the out- of-plane axis. The visible transition is a n * t n transition that is out-of-plane polarized, and therefore the transition moment is parallel to the symmetric-top axis. The ground and excited electronic states have almost the same geometry and rotational constants, and this produces the typical parallel band structure seen in fig. 1 . The He <,,-> I I I I , L . , I I 1 I " " . " " ' -98 -84 -70 -56 -42 -28 -14 Yo 14 28 42 56 70 frequency/GHz FIG.1 .-High-resolution fluorescence excitation spectrum of tetrazine, He-tetrazine and He2- tetrazine. The bottom trace shows the composite spectrum of the three molecules. band has a strong central Q branch consisting of many overlapped lines and evenly spaced P and R branches going off to the low- and high-frequency sides of the Q branch. The slight asymmetry of the molecule produces a slight splitting of the individual K components of a given J transition, but the dominant feature of the140 ORGANIC CLUSTERS spectrum is the regular spacing between adjacent P- and R-branch lines. To a first approximation this spacing is given by the average rotational constant B = i(l3 + C) given by the moments of inertia about the in-plane axes.Also shown in fig. 1 are the fluorescence excitation spectra of two Van der Waals molecules, He-tetrazine and He,-tetrazine.' The helium atoms sit above and below the plane of the tetrazine ring and their additional mass contributes to the moments of inertia about the in-plane axes and thus reduces B. However, the Van der Waals molecules are still near symmetric tops, the ground and excited electronic state geometries are still similar, and the transition is still parallel polarized. Therefore the overall appearance of the spectrum is the same, only the scale is changed due to the smaller rotational constant. B. TETRAZINE DIMERS In fig. 2 we show the low-resolution fluorescence excitation spectrum of a mixture 1 D I I I I I I I I 18300 18200 18100 18000 v/cm-' FIG.2.-Low-resolution fluorescence excitation spectrum of tetrazine in helium at (a) T, = 0 "C and (6) T, = 24 "C, where T, is the temperature of the tetrazine sample container. Increasing T, produces a higher concentration of tetrazine. Features assigned to the tetrazine dimer are marked D. The upper and middle ( x 30) traces are normalized to produce the same intensity at the strongest feature. The lower ( x 1) trace is identical to the middle trace but is taken at 1/30 the sensitivity to keep all features on scale. of tetrazine in helium. The quantity T, is the temperature of the mixing chamber containing the tetrazine sample, and therefore the lower trace is taken at a higher concentration of tetrazine, all other experimental parameters being held constant. Both spectra have been renormalized to have the same intensity in the off-scale tetrazine monomer origin band at 18 128 cm-l, and we note the three features marked D increase in intensity relative to the monomer origin as the concentration is raised.We assign these three features to the tetrazine dimer. The high-resolution fluorescence excitation spectra of the two dimer bands that are red-shifted with respect to the monomer origin are shown in fig. 3. The overallD . H . LEVY, C. A . HAYNAM A N D D. V . BRUMBAUGH 141 pattern of the rotational structure is characteristic of a perpendicular transition (transition moment perpendicular to the symmetric-top axis) of a near symmetric top with similar ground- and excited-state geometries. The overall pattern of these spectra rule out certain geometries.In the first place, the overall extent of the spectrum would tend to rule out a trimer or higher polymer as being responsible for these bands. Moreover, a stacked structure with two parallel plates on top of each vlcm-' FIG. 3.-High-resolution fluorescence excitation spectra of the perpendicular tetrazine dimer bands. The geometries responsible for the two spectra are shown in the figure. (a) vo = 18 089 cm-', (b) vo = 18 103 cm-'. other would be a near symmetric top with the common out-of-plane axis being the symmetry axis, and this would produce a parallel-type transition. This would be true regardless of the interplate separation, and therefore the dimer producing the spectra in fig. 3 cannot have a stacked parallel plate geometry. The spacing between the large Q-branch peaks in fig.3 is largely determined by the rotational constant A - B. The spacing of the Q branches in the 18 103 cm-l band is surprisingly large for a molecule as big as the dimer, and this greatly restricts the possible geometries that are compatible with the spectrum. The only geometries that reproduce the large Q-branch splittings require an A-inertial axis that lies in the planes of both monomer units passing through or near the centre of both monomers. Moreoever, the extensive structure in the four central Q-branch features involving K = 1 levels is produced by the slight asymmetry and disappears in a true symmetric top. A geometry that has two perpendicular rings (fig. 4) with the A-axis lying in the plane of both rings is too symmetric to produce the observed K = 1 splitting6 The geometry that gives the best fit to the spectrum has the two rings lying in the same plane with a separation of 5.6 8, between their centres.The spectrum is rela- tively insensitive to the rotation of the two rings about their own out-of-plane axes, and we are still trying to determine how well we can measure these geometric para- meters. Note that we see only a single transition arising from the side-by-side planar geometry, which implies that the orientation of the rings about their out-of-plane axes produces an overall geometry in which the two rings are symmetrically equiva-142 ORGANIC CLUSTERS c h -1 VO v/cm - 1 FIG. 4.-Synthesized high-resolution spectrum of the tetrazine dimer in a twisted geometry. This geometry is obtained from the planar geometry shown in the lower trace of fig.3 by rotating one ring 90" about a line passing through the centres of the two rings. lent.* One such geometry that would produce the observed spectrum and that would have two equivalent rings is shown in fig. 5. This structure allows partial hydrogen bonding between the hydrogens on one ring with the nitrogens on the other ring. Such a structure not only reproduces the spectrum and preserves the equivalence of the rings, but it explains the forces that hold the dimer in a planar configuration. The band centred at 18 089 cm-l is also a perpendicular transition but the smaller spacing between the Q branches indicates that it is produced by quite a different geometry.The best fit that we have been able to obtain is produced by a T-shaped geometry with the centres of the two rings separated by 4.37 A. This geometry has been proposed for the ground state of the benzene dime^-,^ and it is the geometry of nearest neighbours in the tetrazine crystal * where the centre of the nearest-neighbour rings is 4.40 A. In a T-shaped configuration the two rings are not symmetrically equivalent, and therefore each electronic transition should produce two bands corresponding to excitation of the two inequivalent rings. A T-shaped dimer of tetrazine is a near prolate symmetric top with the symmetry axis being the line drawn between the centre of the two rings. Since the z*+n transition is polarized out-of-plane, excitation of the ring that is parallel to the symmetry axis (the upright of the T) will produce a perpendicularly polarized rotational structure (AK = & l), whereas excitation of the ring that is perpendicular to the symmetry axis (the crosspiece of the T) will produce a parallel polarized rotational structure (AK = 0).The band at 18 278 cm-' is parallel polarized (fig. 6 ) and its rotational structure leads to the same inter-ring * The helium Van der Waals molecule formed by binding a single helium atom above the plane of one ring makes the two rings unequivalent. In this species we observe two transitions separated by 0.2 cm-'.D . H. LEVY, C . A. HAYNAM AND D. V. BRUMBAUGH 143 I I I I ~ I I I I ~ I I I I ~ I I ~ - 1 UO 1 v/cm-' FIG. 5.-Synthesized high-resolution spectrum of the tetrazine dimer in the planar geometry shown in the lower trace of fig.3. -0.2 -0.1 Z'O 0.1 0.2 0.3 v,'cm-' FIG. 6.-High-resolution fluorescence excitation spectrum of the parallel component of the T-shaped tetrazine dimer. vo == 18 278 cm-', R = 4.369 -+ 0.005. distance as that of the 18 089 cm-I perpendicular band. We therefore assign the 18 278 cm-' and 18 089 cm-' bands as the parallel and perpendicular components arising from a T-shaped geometry. c. MIXED TETRAZINE-BENZENE CLUSTERS In fig. 7 we show the fluorescence excitation spectrum that is produced when a mixture of tetrazine and benzene seeded into helium carrier gas is expanded in a144 M D ( a ) M ORGANIC CLUSTERS M n I D M, /"0 :M . /xlo supersonic free jet. For comparison, the spectrum of tetrazine in helium is reproduced in the lower trace.In the spectral region shown in the figure, three new bands (marked with asterisks) appear upon the addition of benzene, and these are assigned to mixed clusters of tetrazine and benzene. The band at 18 260 cm-l, shown in fig. 8, has rotational structure associated with I I I I 1 - 2 -1 L'o 1 v/cm - FIG. &--(a) High-resolution fluorescence excitation spectrum of the stacked parallel plate mixed dimer of tetrazine and benzene. (b) Also shown is the spectrum of the He-tetrazine-benzene Van der Waals molecule. vo = 18 260 cm-'.D . H . LEVY, C . A . HAYNAM A N D D . V . BRUMBAUGH 145 a parallel-type transition, and we beliebe that it is produced by a dimer containing one molecule of benzene and one of tetrazine.The geometry that best fits this band is a structure with the two rings stacked and parallel to each other. A second blue-shifted band can be seen in fig. 8 which also has a parallel rotational structure. We assign this to a Van der Waals molecule consisting of a helium atom bound to the exposed tetrazine face of the tetrazine-benzene dimer. There are several bands to the red of the 18 260 cm-l band that are due to higher clusters of benzene and tetrazine. Two of these are marked in fig. 7, but there are other weaker bands that are observable at higher sensitivity and under different expansion conditions. We are in the process of analysing their rotational structure when this is possible, but the overall width of the rotational profiles suggests that all of these bands are due to clusters larger than the dimer. For example, the rotational structure of the 18 190 cm-I band shown in fig. 9 is due to a B-type transition and "0 1 2 vlcrn-' 3 FIG.9.-(a) High-resolution fluorescence excitation spectrum of the B-type transition produced by the mixed (tetra~ine)~-benzene trimer. The geometry responsible for this band has the two tetrazine rings planar and side-by-side, as in the lower trace of fig. 3, with the benzene stacked above one of the tetrazinesand parallel to the tetrazine ring. Also shown are the spectra of two helium Van der Waals molecules of this trimer : (b) He-(tetra~ine)~-benzene and (c) He2-(tetrazine)2-benzene. vo = 18 190cm-'. can be reproduced assuming a trimer consisting of two tetrazine molecules side by side in a plane with a benzene monomer stacked above the plane of one of the tetrazine rings and parallel to that ring.Also seen in this figure are two bands assigned to Van der Waals molecules containing one and two helium atoms bound to the (tetrazine),-benzene trimer. D . MILD EXCIMERS-DIMETHYL TETRAZINE DIMERS All of the cluster spectra described above have the common characteristic that in the fluorescence excitation spectrum the bands assigned to the clusters are only slightly shifted from those of the monomer. The monomer-cluster shift is a measure of the difference in binding energy of the cluster in its ground and excited electronic state, a red shift corresponding to a deeper well in the excited electronic state. There- fore the small spectra1 shift implies that the potential surface describing the cluster146 ORGANIC CLUSTERS binding does not change very much upon electronic excitation. The fact that the dominant spectral features are single vibronic bands rather than long, intense vibra- tional progressions supports this interpretation that the potential surface is relatively insensitive to the electronic state. A small spectral shift with no extensive activity in the low-frequency vibrational modes has been the rule for rare-gas-substrate Van der Waals molecules as well as for the molecular clusters described above.The fluorescence excitation spectrum of dimethyl tetrazine (DMT) is shown in fig. 10. Once again, the temperatures that label the spectra are the temperatures of x1 XI0 { l I l , ] , 1 , 1 , 1 , 1 , 1 17500 17000 v/cm - FIG, 10.-Low-resolution fluorescence excitation spectrum of dimethyl tetrazine (DMT) in helium.Temperatures noted to the right of each trace are the temperatures of the sample container holding the DMT. Higher temperatures correspond to higher concentrations of DMT in the gas mixture. Features assigned to the DMT monomer are marked M. Other features are assigned to the DMT dimer. The lower three traces have been normalized to a common intensity for the strongest feature. the DMT reservoir, and therefore higher temperatures correspond to higher con- centrations of the seed gas. The lower three spectra are normalized to a common intensity for the DMT dimer origin band at 17 496 cm-', and features that grow in relation to the monomer as the concentration is raised are assigned to the DMT dimer.The spectrum of the DMT dimer is qualitatively different from that of the tetrazine dimer. The largest wavelength feature of the DMT dimer spectrum is red shifted by 548 cm-l with respect to the monomer origin, and a fairly harmonic progression of five members with we = 79 cm-l is observed. The red shift indicates that the excited electronic state has a significantly larger binding energy than the ground electronic state, and this would be expected to produce vibrational activity in the dimer stretching mode, as is observed. The ability of aromatic hydrocarbons to form excimers in solution is well-known,1° and there is some similarity between the DMT gas-phase dimer spectrum and the spectra of aromatic hydrocarbon excimers.Excimers are more tightly bound in their excited states by several thousand cm-l, and the fluorescence is very strongly red shifted from the absorption. The influence of the excimer state has been invoked to interpret the supersonic-jet spectrum of the benzene dimer,2 although the initialD . H . L E V Y , C. A . HAYNAM AND D. V . RRUMBAUGH 147 absorption in that molecule has been assigned to a less tightly bound T-shaped dimer. The supersonic-jet spectrum of the DMT dimer is similar to that of a hydrocarbon excimer in that it is red shifted, but the shift is different in magnitude from that expected from a hydrocarbon excimer. The change in binding energy of the DMT dimer upon electronic excitation is about an order of magnitude less than that ex- pected for a hydrocarbon excimer, and for this reason we have called these states mild excimers.We do not know of any observation of an 7c"t-n excimer in solution, and the spectral resolution available in solution is probably too low to allow the observation of a mild excimer. The only other mild excimer that we have observed is a mixed cluster of DMT and tetrazine. The fluorescence excitation spectrum of this molecule is shown in fig. 1 I . * 1 17300 17200 17100 vlcrn-' FIG. 11 .-Low-resolution fluorescence excitation spectrum of (a) tetrazine, (b) dimethyl tetrazine and (c) a mixture of tetrazine and dimethyl tetrazine. The spectrum of the DMT-tetrazine cluster has a very long vibrational progression with an intensity that peaks in the middle of the progression and dies off to both the red and the blue.One member of the progression is overlapped by the DMT dimer feature near 17 100 crn-l, and at least two weaker members of the progression can be seen to the red of 17 100 cm-l. Because of the intensity fall-off, we cannot be sure that the longest wavelength member of the progression that we observe is the origin of the transition, but if it is the origin, the excited electronic state is more tightly bound than the ground state by ca. 450 crn-'. E. RELAXATION AND PHOTOCHEMISTRY When two monomer units are formed into a dimer, the six translational and six rotational degrees of freedom of the separated monomers become three translational and three rotational degrees of freedom of the cluster plus six new vibrational modes which we will call cluster modes.Because the cluster binding is so weak, the cluster modes have low frequencies, and therefore even at moderate excitation energies the148 ORGANIC CLUSTERS density of vibrational states can be large. We should expect to see intramolecular relaxation effects produced by the large density of vibrational states, and ultimately the flow of energy into the stretching mode should dissociate the cluster. These effects have been observed in both chemically bound molecules ''J~ and Van der Waals m~lecules,'~ and we have recently started to study such effects in molecular clusters. In fig. 12 we can see the effect of a large density of states on the fluorescence I 0 I I I 4 1 1 2 vlcm-' I FIG.12.-High-resolution fluorescence excitation spectrum of (a) the origin and (b) 6aA bands of the dimethyl-tetrazine dimer. The instrumental resolution was the same for both spectra. (c) The assignment of the (DMT),-He Van der Waals molecule is uncertain. excitation spectrum of the DMT dimer. The lower trace shows the rotational structure of the origin band of the dimer. The spectrum, although complex and at this time unanalysed, is sharp and reasonably well resolved. The upper trace shows the rotational envelope of the 6a; transition taken with the same instrumental resolu- tion. The band is clearly broadened, and we observe only a band contour with no resolved single lines. The V6n mode in the DMT is an in-plane elongation of the ring and has a frequency of 517 cm-' in both the monomer and the dimer.Because of the high density of cluster mode states, even at the relatively small excitation energy of 517 cm-', there are a large number of isoenergetic states that can mix with the 6a' level. These states can be described by a wavefunction that has all of the monomer modes in their zero-point levels (or perhaps has some of the low-frequency monomer modes partially excited) but has varying degrees of excitation in the six cluster modes. The mixing of the 6a' level with the various isoenergetic cluster levels is often referred to as intramolecular vibrational re1axati0n.l~ It should be mentioned that whether or not there is a time-dependent relaxation depends on whether the excitation of the collection of isoenergetic states is coherent or incoherent.The effect of a large density of states may be clearly seen in fig. 13, which shows the dispersed fluorescence spectra of the DMT monomer (lower trace) and the DMT dimer. Both of these spectra are taken by tuning the exciting laser to the 6a; absorp- tion frequency, keeping the excitation frequency fixed, and dispersing the emitted light, The feature marked with an asterisk in both spectra is at the laser frequency and is a combination of resonance fluorescence and scattered laser light, The monomer emission spectrum, while complex, consists of well-resolved singleD . H . LEVY, C. A . HAYNAM A N D D. V . BRUMBAUGH 149 * 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 I I I I I 18 000 17500 17000 16500 vlcm-' FIG. 13.-Dispersed fluorescence spectra obtained by exciting (a) the dimethyl tetrazine (DMT) mono- mer 6aJ transition and (6) the DMT dimer 6a; transition.The excitation frequencies are marked *. features. On the other hand, the dimer spectrum has some weak sharp features just to the red of the exciting frequency but is dominated by a few broad features several hundred cm-I wide. When relaxation occurs, a large number of cluster states are populated and the observed emission spectrum is the superposition of the emission spectra from each of these states. If the potential surface did not change at all upon electronic excitation, only Av = 0 emission transitions would be allowed, they would all occur at the same frequency, and the emission spectrum would be narrow. Be- cause there is a significant change in the frequency of the stretching mode and pre- sumably some small change in the other five cluster modes, the emission spectra from the several states are not identical and the composite spectrum is broad.The coarse structure in the dimer spectrum is due to the activity of the V6n mode leading to a 6a: progression in the emission spectrum. The appearance of the weak sharp features could arise in either of two ways. If the initial excitation were coherent, the initial state would be equivalent to the single 6af level and would therefore produce a sharp emission spectrum. In time this coherent superposition would dephase into the collection of relaxed states and would produce a broad emission spectrum. If the dephasing time were short compared with the fluorescence lifetime of a few ns, the emission would be broad; if it were long, the emission would be sharp.If the dephasing time was similar to the fluorescence life- time, the spectrum would have both broad and sharp features and the relative intensi- ties would be a measure of the ratio of the two lifetimes. A second, and probably more likely, explanation of the appearance of both sharp and broad features would presume that the excitation was incoherent. In this case the individual excited-state wavefunctions would be linear combinations of the 6a' state with no excitation in the cluster modes and the 6a0 level with excitation in the cluster modes. Therefore each individual state would have some transition prob- ability for emission to a few de-excited (or weakly excited) ground-state cluster levels (sharp spectrum) and some transition probability for emission to many excited ground- state cluster levels (broad spectrum).The ratio of sharp to broad intensity in the150 ORGANIC CLUSTERS emission spectrum would be a measure of the fraction of the 6a' basis function that appeared in each of the excited eigenstates. If this is the mechanism involved here, the appearance of sharp structure requires that the number of cluster levels not be too large so that after mixing each eigenstate still has some significant fraction of the 6a' state. This second mechanism is now thought to be responsible for the appearance of sharp and broad structure in the emission spectra of jet-cooled alkyl benzenes.12 A final phenomenon which we should consider is the possibility of photochemistry, the transfer of enough energy from the initially excited ring vibration to the stretching cluster mode to allow the cluster bond to break.In the spectra just described, the 6ai excitation of the DMT dimer, this process is energetically forbidden. In fig. 10 we can see a dimer feature at 17 465 cm-', just to the red of the monomer origin. This is the 6ai band of the dimer and has built on it the stretching progression of the dimer. Because the 6ai feature is to the red of the monomer, we know that the excitation energy of the 6a' level above the zero-point level of the excited state is less than the difference in binding energy between the ground and excited electronic states. There- fore the 6a' vibrational energy is certainly less than the excited-state binding energy itself, and even if all of this energy flows into the stretching coordinate it will be in- sufficient to break the bond. We have looked for photochemistry in the DMT dimer by exciting up to 6a3, but all emission seems to come from the bound dimer, not from the monomer fragment that would be produced by a photochemical reaction.This failure to break the bond could be an energetic constraint if 6a3 contained less than the binding energy. Even if 6a3 were above the energy threshold, the presence of the other non-dissociating modes could lengthen the dissociation lifetime so that it was much longer than the fluorescence lifetime. Very recently we have looked for photochemistry in the tetrazine-benzene dimer.In this case v,, is a higher frequency (703 cm-' as opposed to 517 cm-l in DMT), and since it does not form even a mild excimer, the binding energy is probably lower. We find that excitation to 6a' produces only relaxation but that excitation to 6a2 pro- duces photochemistry. We are currently trying to determine if the lack of photo- chemistry at 6a' is due to insufficient energy or whether it is due to an insufficiently rapid rate. This material is based upon work supported by the National Science Foundation under grant CHE-7825555, and by the donors of The Petroleum Research Fund ad- ministered by the American Chemical Society. C. A. H. was supported by the Fannie and John Hertz Foundation. D. H. Levy, Annu. Rev. Phys. Chem., 1980, 31, 197; in Photoselective Chemistry, Adv. Chem., Phys., ed. J. Jortner, R. D. Levine and S. A. Rice (Wiley-Interscience, New York, 1981) vol. 47, part I, pp. 323-362. P. R. R. Langridge-Smith, D. V. Brumbaugh, C. A. Haynam and D. H. Levy, J. Phys. Chem., 1981, 85,3742. W. Sharfin, K. E. Johnson, L. Wharton, and D. H. Levy, J. Chern. Phys., 1979,71, 1292; R. E. Smalley, D. H. Levy and L. Wharton, J. Chem. Phys., 1976, 64, 3266. R. E. Smalley, L. Wharton, D. H. Levy and D. W. Chandler, J. Mol. Spectrosc., 1977, 66, 375. R. E. Smalley, L. Wharton, D. H. Levy and D. W. Chandler, J. Chern. Phys., 1978,68,2487. All synthetic spectra were produced using a spectral simulation program originally written by L. Pierce, Notre Dame University. ' K. C . Janda, J. C. Hemminger, J. S. Winn, S. E. Novick, S. J. Harris and W. Klemperer, J. Chem. Phys., 1975,63,1419; J. M. Steed, T. A. Dixon and W. Klemperer, J. Chem, Phys., 1979 70 4940. D. V. Brumbaugh, C . A. Haynam and D. H. Levy, J. Chem. Phys., 1980,73, 5380. * F. Bertinotti, G . Giacomello and A. M. Liquori, Acta Crystallogr., 1956, 9, 510.D . H . LEVY, C. A . HAYNAM AND D. V . BRUMBAUGH 151 lo J. B. Birks, Rep. Prhgr. Phys., 1975, 38, 903. l1 J. B. Hopkins, D. E. Powers and R. E. Smalley, J . Chem. Phys., 1979,71,3886; 1980,72,2905; 1980, 72, 5039; J. B. Hopkins, D. E. Powers, S. Mukamel and R. E. Smalley, J. Chem. Phys., 1980,72, 5049; J. B. Hopkins, D. E. Powers and R. E. Smalley, J. Chem.Phys., 1980, 73, 683. l2 P. S. H, Fitch, C. A. Haynam and D. H. Levy, J. Chem. Phys., 1981,74, 6612. l3 K. E. Johnson, W, Sharfin and D. H. Levy, J. Chem. Phys., 1981, 74, 163; J. E. Kenny, T. D. Russell and D. H. Levy,’J. Chem. Phys., 1980,73,3607; J. E. Kenny, K. E. Johnson, W. Sharfin and D. H. Levy, J . Chem. Phys., 1980, 72, 1109.

ISSN:0301-7249    

DOI:10.1039/DC9827300137

出版商:RSC    

年代:1982

数据来源: RSC