摘要:

FARADAY DISCUSSIONS OF THE CHEMICAL SOCIETY NO. 62 1977 Potential Energy Surfaces THE FARADAY DIVISION CHEMICAL SOCIETY LONDONFARADAY DISCUSSIONS OF THE CHEMICAL SOCIETY NO. 62 1977 Potential Energy Surfaces THE FARADAY DIVISION CHEMICAL SOCIETY LONDONA GENERAL DISCUSSION ON Potential Energy SurJaces Sth, 9th and 10th September, 1976 A GENERAL DISCUSSION on Potential Energy Surfaces was held in the School of Molecular Sciences at the University of Sussex on Sth, 9th and 10th September, 1976. The President of the Faraday Division, Professor D. H. Everett M.B.E., was in the chair: about 160 Fellows of the Faraday Division and visitors from overseas attended the meeting. Among the overseas visitors were: Prof. V. Aquilanti, Italy Miss B. A. Blackwell, Canada Mr. J.Bruinsma, The Netherlands Prof. D. L. Bunker, U.S.A. Dr. J. J. Burton, U.S.A. Mr. S . Carley, Canada Dr. hl. J. Coggiola, U.S.A. Prof. M. Daniels, U.S.A. Prof. D. H. W. den Boer, The Netherlands Dr. P. J. Derrick, Australia Prof. M. J. S. Dewar, U S A . Dr. A. Ding, West Germany Dr. D. A. Dixon, U.S.A. Mr. D. J. Douglas, Canada Dr. 1. R. Elsum, U.S.A. Dr. F. Engelke, West Germany Prof. K. Freed, U.S.A. Prof. T. F. George, U.S.A. Dr. D. W. Gough, Canada Mr. S . Goursaud, France Dr. K. D. Hansel, West Germany Prof. W. L. Hase, U S A . Dr. E. F. Hayes, U.S.A. Prof. D. R. Herschbach, U.S.A. Prof. D. V. S . Jain, India Dr. 0. Kafri, U.S.A. Dr. B. Katz, Israel Prof. W. Klemperer, U.S.A. Dr. A. Komornicki, U.S.A. Prof. K. Kuchitsu, Japan Dr. J. P. Kuntz, West Germany Prof. W.Kutzelnigg, West Germany Dr. U. T. Lamanna, Italy Dr. S . Leach, France Prof. R. J. Le Roy, Canada Dr. H. J. Loesch, West Germany Mrs C . M. Meerman van Benthem, Dr. V. Menendez, Spain Prof. W. H. Miller, U.S.A. Dr. D. N. Mitchell, Canada Dr. J. J. C. Mulder, The Netherlands Prof. E. A. Ogryzlo, Canada Dr. F. Pirani, Italy Prof. J. C . Polanyi, Canada Dr. L. L. Poulsen, Denmark Dr. K. Rasmussen, Denmark Prof. H. F. Schaefer, U.S.A. Dr. K. Shobatake, West Germany Mrs Sizun, France Dr. J. J. Sloan, Canada Dr. W. Thiel, West Germany Prof. J. Tomasi, Italy Prof. J. Troe, West Germany Dr. T. Valencich, U.S.A. Mr. J. J. Valentini, U.S.A. Mr. M. C. van Hemert, The Netherlands Prof. V. Vecchiocattivi, Italy Dr. J. C. Whitehead, U.S.A. Dr. P. E. S . Wormer, The Netherlands Prof.R. E. Wyatt, U.S.A. Dr. A. P. Zandee, The Netherlands TJie NetherlandsISBN: 0 85186 918 1 ISSN: 0301-7249 0 The Chemical Society and Contributors 1977 Printed in Great Britain by Richard Clay (The Chaucer Press), Ltd., Bungay, SuffolkCONTENTS Page 7 20- 29 40 47 59 67 77 92 110 127 138 169 179 185 197 210 Pot en tial Surfaces from Vibra t ion-ro tat ion Data by I. M. Mills Experimental Determination of Average and Equilibrium Structures of Polyatomic Gas Molecules by Diffraction and Spectroscopic Methods by K. Kuchitsu and K. Oyanagi Semi-classical Methods for Vibrational Energy Levels of triatomic molecules by N. C. Handy, S. M. Colwell and W. H. Miller Semi-classical Theory for Non-separable Systems by W. €3. Miller GENERAL DISCUSSION Potential Energy Surfaces for Ion-molecule Reactions by C.F. Bender, J. H. Meadows and H. F. Schaefer An Ab Initio Potential Surfacesfor the Reaction N+ + H, -+ NH+ + H by M. A. Gittins, D. M. Hirst and M. F. Guest Potential Energy Surfaces for Simple Chemical Reactions by G. G. Baht-Kurti and R. N. Yardley A Many-body Expansion of Polyatomic Potential Energy Surfaces: Applica- tion to H,, Systems by A. J. C. Varandas and J. N. Murrell Potential Energy Surface for Bond Exchange Among Three Hydrogen Molecules by D. A. Dixon, R. M. Stevens and D. R. Herschbach A New Empirical Potential Hypersurface for Bimoleczilar Reaction Systems by 0. Kafri and M. J. Berry GENERAL DISCUSSION DeterminiiTg Anisotropic Intermolecular Potentials for Van der Waals Molecules by R.J. Le Roy, J. S. Carley and J. E. Grabenstetter Tjie Rotational Spectroscopy of Van der Waals Molecules by W. Klemperer Q uan tum Chemical Calculation of In te rm o lecular Interact ion Po tent ials, Mainly of Van der Waals Type by W. Kutzelnigg Studies of the mechanisms of Some Organic Reactions and Photoreactions by Semi-empirical SCF MO Methods by M. J. S. Dewar Ethyl Radical Potential Energy Surface by C. S. Sloane and W. L. Hase6 CONTENTS 222 232 246 255 267 29 1 300 347 The Four-centre Reaction 12* -t F2 Studied by Laser-indirced Cherni- luminescence in Molecular Beams by F. Engelke, J. C. Whitehead and R. N. Zare Crossed Beam Studies of Endoergic Biinolecular Reactions: Production of Stable Trihalogen Radicals by J. J. Valentini, M. J. Coggiola and Y. T. Lee Radiative Transitions for Molecular Collisions in an Intense Laser Field by T. F. George, J.-M. Yuan, I. H. Zimmerman and J. R. Laing Alkali Atom-dimer Exchange Reactions: Na +- Rb2 by D. J. Mascord, P. A. Gorry and R. Grice The Reaction of F + H2 4 HF + H by J. C. Polanyi and J. L. Schreiber The Scattering of Hg(63P2) by CO, N2 and C 0 2 by J. Costello, M. A. D. Fluendy and K. P. Lawley GENERAL DISCUSSION Closing Remarks by H. C. Longuet-Higgins * The author to whom correspondence should be addressed is indicated by an asterisk after his name in the heading of each paper.

ISSN:0301-7249    

DOI:10.1039/DC9776200001

出版商:RSC    

年代:1977

数据来源: RSC

摘要:

GENERAL DISCUSSIONS OF THE FARADAY SOCIETY 35 1 Date Subject Volume 1962 1962 1963 1963 1964 1964 1965 1965 1966 1966 1967 1967 1968 1968 1969 1969 1970 1970 1971 1971 1972 1972 1973 1973 1974 1974 1975 1975 1976 1977 Inelastic Collisions of Atoms and Simple Molecules High Resolution Nuclear Magnetic Resonance The Structure of Electronically-Excited Species in the Gas-Phase Fundamental Processes in Radiation Chemistry 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 Molecules in Excited States The Photoelectron Spectroscopy of Molecules Molecular Beam Scattering Intermediates in Electrochemical Reactions Gels and Gelling Processes Photo-effects in Adsorbed Species Physical Adsorbtion in Condensed Phases Electron Spectroscopy of Solids and Surfaces Precipitation Potential Energy Surfaces Oxidation For current availability of Disciission volumes, see back cover.33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62GENERAL DISCUSSIONS OF THE FARADAY SOCIETY 35 1 Date Subject Volume 1962 1962 1963 1963 1964 1964 1965 1965 1966 1966 1967 1967 1968 1968 1969 1969 1970 1970 1971 1971 1972 1972 1973 1973 1974 1974 1975 1975 1976 1977 Inelastic Collisions of Atoms and Simple Molecules High Resolution Nuclear Magnetic Resonance The Structure of Electronically-Excited Species in the Gas-Phase Fundamental Processes in Radiation Chemistry 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 Molecules in Excited States The Photoelectron Spectroscopy of Molecules Molecular Beam Scattering Intermediates in Electrochemical Reactions Gels and Gelling Processes Photo-effects in Adsorbed Species Physical Adsorbtion in Condensed Phases Electron Spectroscopy of Solids and Surfaces Precipitation Potential Energy Surfaces Oxidation For current availability of Disciission volumes, see back cover.33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62

ISSN:0301-7249    

DOI:10.1039/DC97762BX003

出版商:RSC    

年代:1977

数据来源: RSC

摘要:

Potential Surfaces from Vibration-rotation Data BY IAN M. MILLS Department of Chemistry, University of Reading, Reading, Berkshire RG6 2AD Received 24th May, 1976 The problems of inverting experimental information obtained from vibration-rotation spectro- scopy to determine the potential energy surface of a molecule are discussed, both in relation to semi- rigid molecules like HCN, NOz, H,C=O, etc., and in relation to non-rigid or floppy molecules with large amplitude vibrations like HCNO, C302, and small ring molecules. Although standard methods exist for making the necessary calculations in the former case, they are complex, and they require an abundance of precise data on the spectrum that is rarely available. In the case of floppy molecules there are often data available over many excited states of the large amplitude vibration, but there are difficulties in knowing the precise form of the large amplitude coordinate(s), and in allowing for the vibrational averaging effects of the other modes.In both cases difficulties arise from the curvilinear nature of the vibrational paths which are not adequately handled by our present theories. The object of this Discussion is to bring together workers in the area of molecular physics and molecular chemistry with a common interest in determining potential energy functions. One source of information on the potential energy function (p.e.f.) of a molecule is vibration and vibration-rotation spectroscopy. This is a field which has made considerable progress in the last fifteen years, but it has also become a specialist subject which may not be easy to follow for workers in related fields.My object in this paper is to bring non-specialist readers up to date with develop- ments in this field, and with the strengths and weaknesses of this approach to determining the p.e.f. of a molecule. The prominent weakness of the approach is well known, but it must be stated again at the outset: it is that vibrational spectroscopy only provides information on the p.e.f. close to the minimum, because experimental data are rarely available for more than the first few quanta in each normal mode. Excitation energies of perhaps a few thousand cm-1 from the minimum (i.e., about 0.5 eV E 18 millihartree, or -50 kJ mol-l) represent the upper limit of information from vibrational spectro- scopy, and often the limit is less than 1000 cm-1 (0.12 eV w 4.2 millihartree, or -12 kJ mol-l).This is only 10% of a typical bond dissociation energy. Thus, vibrational and vibration-rotation spectroscopy give detailed information on the shape of the p.e.f. for only a rather smalzpocket in the surface around the minimum, or equilibrium configuration. The detail and precision of the information on the p.e.f. derives from the study of spectra at high resolution, in the gas phase, showing rotational structure and vibration-rotation interaction effects. Both experimentally and theoretically this subject has made great advances. The experimental techniques which have proved particularly fruitful are the developments in conventional high-resolution infrared and Raman spectroscopy, the development of fourier-transform spectrometers based on a Michelson interferometer, the developments in millimetre wave experimental8 POTENTIAL SURFACES FROM VIBRATION-ROTATION DATA techniques, and the application of lasers to high-resolution spectroscopy (for example double-resonance experiments, and laser-Stark and laser-Zeeman spectroscopy).Resolution of 0.02 cm-I is not uncommon today in infrared spectro~copy,l-~ and modern interferometers achieve 0.005 cm-l throughout the mid-infrared region 4 9 5 ; laser experiments can achieve effective resolution of a few MHz (lo4 cm-1).6-8 Gas phase Raman vibration-rotation spectra are now available with resolution of the order 0.25 and pure rotational spectra with much higher resolution.lo Milli- metre wave spectroscopy based on microwave techniques is available in a number of laboratories up to 400 GHz, and, in at least one case, spectra have been observed up to 1000 GHz (30 cm-') with the usual microwave precision and resolution (of less than 1 MHz).ll The analysis of vibration-rotation spectra, and the inversion of the data to give a molecular potential energy function, involve the theory of vibration-rotation spectro- scopy in polyatomic molecules, the theory of force constant calculations, and the application of least-squares refinement techniques to obtain the p.e.f.which gives the best fit to the experimental data. Since I have recently attempted to review these topics in other publication~,'~-'~ and have there given references to other review articles, I shall not go into the detail of the analysis in this paper.I shall discuss some examples in sufficient detail to show where the difficulties arise, first for the usual case of semi-rigid molecules, and then for the particular case of molecules with low- frequency large-amplitude vibration(s) where it is often the case that detailed experi- mental data can be obtained over many excited vibrational states. It is in this last field that the work of my own laboratory has been concentrated in the last few years. 1. SEMI-RIGID MOLECULES By " semi-rigid " I mean molecules without any exceptionally low-frequency large-amplitude vibrations; for the sake of argument " low-frequency " means a wavenumber below about 300 cm-? In this case, the p.e.f.rises rapidly for dis- placements in all directions from the minimum in the potential energy surface. The limitation of vibrational spectroscopy to relatively small displacements from the minimum naturally suggests the representation of the p.e.f. as a power series expansion in displacement coordinates from equilibrium : We may thus talk of the quadratic, cubic, and quartic force field, etc. Most analyses are based on a p.e.f. defined in this way, and vibration-rotation theory is founded on the assumption that this series converges reasonably rapidly, perhaps by a factor of 10 for successive terms in (1) for displacements appropriate to one-quantum excitation. Attempts to determine the coefficients in (1) are described as force constant calculations. The choice of coordinates in (1) requires care.A first requirement seems to be that they should be geometrically defined internal coordinates (so that the force constants are unique and invariant to isotopic substitution); the usual choice is to take valence bond stretching and angle bending coordinates. However, another factor in the choice may be that certain definitions give more rapid convergence than others, a point to which I return later. Also the choice of geometrically defined coordinates leads to a non-linear transformation into normal coordinates, which complicates anharmonic force constant calculations, as also discussed later.IAN M. MILLS 9 The formulation of (1) as a Taylor series ensures that each force constant is equal to the appropriate derivative of Y at equilibrium.For example Equations of this kind allow us to visualise cubic force constants as the rate of change of quadratic constants with respect to one of the coordinates, and thus give a physical interpretation of the higher order constants. Indeed, the signs and magnitudes of cubic and quartic constants can sometimes be estimated by arguments of this kind. 1.1 HARMONIC FORCE FIELD CALCULATIONS The problem in harmonic force field calculations may be described in the following way. The observed vibration frequencies of the fundamentals are essentially the diagonal force constants in the normal coordinates, and the interaction force con- stants in the normal coordinates are by definition zero. Thus the problem is to determine the form of the normaZ coordinates, in terms of which the harmonic force field is known as soon as the vibration frequencies are known.The forms of the normal coordinates are usually expressed in terms of a basis of internal valence coordinates Ri through the L matrix: R = LQ, that is, Ri = ELiQ,. (3) r Thus the elements of the L matrix, L: = (aRJaQr)const Qs etc. give the proportional contributions of the Ri to a unit displacement in Qp Alternatively they may be expressed in a basis of 3N Cartesian displacements 6x subject to the Eckart conditions through the AL matrix: SX = ALQ = M-lBt(L-')+Q. (4) This provides a pictorial representation of the normal coordinates in terms of Cartesian displacements. To determine the p.e.f. in a representation of geometrically defined internal coordinates, as in eqn (I), we construct the transformation F = (L-1)ffi-l (5) 6) (ii) (iii) where A is a diagonal matrix with elements Ar = 47c2c2u),2. The elements of the F matrix are then the quadratic (or harmonic) force constants in eqn (1).The experimental data most sensitive to the forms of the normal coordinates are: Isotopic frequency shifts, because these depend on the amplitude of motion at the substituted atom; centrifugal distortion effects, which are related to the dependence of the moments and products of inertia on displacements along the normal co- ordinates, ar(aB) = (a&B/aQ,) ; coriolis resonance effects, which are related to the forms of the normal co- ordinates through the zeta constants [r,s(a), which give a measure of the vibrational angular momentum generated by a pair of normal coordinates Qr and Qs about the cc axis.It is worth noting that vibration-rotation interactions, which provide data of types (ii) and (iii), provide important information on the p.e.f.; it is for this reason that high resolution gas-phase spectra are so valuable. The calculation of harmonic force-fields from spectroscopic data is usually made by a non-linear least-squares calculation, in which a trial force field is refined to10 POTENTIAL SURFACES FROM VIBRATION-ROTATION DATA achieve agreement between the calculated and observed data. The calculation is complicated by the variety of different types of data involved, by the relatively large number of parameters which have to be simultaneously refined, and by the effects of anharmonicity, for which it is usually difficult to make precise corrections. The uncertainties in the resulting force constants are almost invariably correlated, so that at the end of the day it is hard to assess how reliably the p.e.f.has been determined- even for specialist readers. When there are more than two or three vibrations of the same symmetry species one must always be suspicious of the reliability, i.e., the uniqueness, of a harmonic force field calculation, and it is generally accepted that interaction force constants (FJ # j ) are less reliably determined than diagonal force constants. Despite these difficulties, well-determined harmonic force fields are now available for many simple molecules. In this category I would put most triatomic molecules, many linear four or five-atomic molecules, many simple C,, molecules (XY, or XYZ3), many simple organic molecules like formaldehyde, ethylene, allene, and even benzene, and many simple inorganic molecules for which isotopic shifts associated with substitution at a central metal atom have been studied.Also, chemically related series of molecules have often been studied together, of which the classic example is the series of saturated hydrocarbon molecules; in such cases apparently reliable harmonic force fields have been determined by constraining force constants to remain unchanged between chemically similar groups in different molecules. Many more force constant calculations have been reported in which constraints have been imposed in order to obtain a unique solution for the p.e.f.Lack of data make this unavoidable for larger molecules, but it also makes the reliability of the results even more difficult to assess. Harmonic force field calculations have recently been comprehensively reviewed by Duncan,15 with references to calculations on many molecules and many numerical examples. 1.2 ANHARMONIC FORCE FIELD CALCULATIONS The cubic and quartic anharmonic force constants in eqn (1) become important as the displacements from equilibrium become larger. Their effects appear in higher- order vibration-rotation interactions and in anharmonic spacings of the vibrational energy levels; they are usually calculated from a perturbation treatment of the vibration-rotation hamiltonian. Most anharmonic force constant calculations treat the cubic and quartic terms in the p.e.f.together, because they affect the vibrational energy levels in the same order of magnitude (the cubic terms are larger, but contribute to the energy levels in the 2nd order of perturbation theory; the quartic terms are smaller, but contribute in 1st order). Very few calculations have been attempted in which higher than quartic force constants have been considered. The customary calculation relating cubic and quartic force constants to the spectrum is quite involved. There are two main steps. The first is a non-linear co- ordinate transformation from the geometrically defined internal coordinates Ri to the normal coordinates Qr in terms of which the vibration-rotation hamiltonian is formulated. This transformation has been discussed by Hoy, Mills and Strey.It may be conveniently written in terms of an L tensor, for which eqn (3) is an approxi- mation involving the linear terms only. The exact transformation may be written:IAN M. MILLS 11 The elements of the L tensor must be determined from a preliminary harmonic force field calculation and then used to transform the p.e.f. in eqn (1) to a similar expansion in terms of the normal coordinates Q,, involving the normal coordinate force con- stants, co, (harmonic terms), P)rst (cubic), and qrstu (quartic) etc. : V/hc = 2 3urqr2 + (1/3!) 2 qrstqrqsqt + (7) r rst Hereg, = ( 2 n ~ ~ ~ / h ) ~ Q , . The effect of the non-linear nature of (6) is that quadratic terms in (1) contribute to quadratic, cubic, quartic, and all higher terms in (7) ; cubic terms in (1) contribute to cubic and all higher terms in (7); and so on.The second step is a perturbation calculation, or contact transformation, in which the vibration-rotation hamiltonian is transformed to an effective hamiltonian in which the coefficients are the spectroscopic constants (such as the arB constants describing the vibrational dependence of rotational constants, and the xr, constants describing the anharmonic contributions to the vibrational energy levels). Although the perturbation calculation is quite complicated, the results are generally applied in the form of analytical formulae for the spectroscopic constants in terms of coefficients in the original hamiltonian, particularly the force constants wry qrsr, yrstu etc.Most of the important formulae are available in the literature.12 Often, further complica- tions arise due to resonances, which require parts of the vibration-rotation hamiltonian to be related to the observed spectrum by matrix diagonalization, or by a mixture of matrix diagonalization and perturbation theory. When both steps of this calculation are made, the spectroscopic observables may be predicted from a trial anharmonic force field. It is finally necessary to reverse the calculation and calculate the force field from observed data. The only general method of achieving this is by a least-squares refinement of the force field to fit the data. All of the difficulties of harmonic force field calculations arise once more in an exaggerated form: too many parameters in the force field, highly correlated results, and unknown errors incurred through the neglect of higher order effects- these are the limiting features of all attempts to determine a general p.e.f. from vibra- tion-rotation data.Our own calculations in this field have been confined to quite simple molecules on which there is an abundance of spectroscopic data (H20, H2S, H2Se, SO3, PF3, AsF,, SbF,, HCN, HCP, HCCH, NCCN and HCCCN).14*16-18 We have also made use of trial calculations as an aid to the analysis and assignment of microwave spectra, in situations where we would hardly claim to have determined an anharmonic force field (e.g., CF3H16 and SiF3H19). Other research groups have made similar calcula- tions with somewhat similar results ; particularly Overend and coworkers 20*21 and Kuchitsu, Morino and coworkers.22 My own feeling is that the greatest success of calculations of this kind lies (i) in the successful interpretation of high-resolution spectra, and (ii) in giving more reliable information on the equilibrium structure and the harmonic force field.I would say that there are relatively few cases where the cubic and quartic anharmonic force constants have been experimentally determined with precision, except for those associated with bond stretching where they can be reliably estimated from assumptions of Morse-like potential curves. Indeed, one general result of anharmonic force field calculations is simply to demonstrate the expected results that bond stretching coordinates give rise to Morse-like sections of the potential surface, and that the " valley-bottom " for an angle bending coordinate tends to follow ;I curvilinear path representing bending at constant bond length.More profound generalizations are hard to make, and I think not very profitable. As a single example to illustrate the quality of calculations of this kind, one may12 POTENTIAL SURFACES FROM VIBRATION-ROTATION DATA consider recent attempts to determine the p.e.f. for the bending vibrations of acetyl- ene,21p23 and their interactions with bond stretching. I choose this example because there has been a lot of precise spectroscopic work on many isotopic species of acetyl- ene; it is hard to imagine that such an abundance of experimental data will become available for any other four-atomic molecule.Also the high symmetry greatly TABLE I.-TERMS IN THE P.E.F. OF ACETYLENE INVOLVING THE ANGLE BENDING COORDINATES. a = HCC bend, z = phase angle between HlClC2 and ClC2H2 bending, Y = CH stretch and R = CC stretch + faat a,cc,cos z V = 3fua (a12 + .2’) simplifies the calculation and reduces the number of force constants involved. In fact, all the signs are favourable for a successful anharmonic force field calculation. The appropriate part of the expansion of the p.e.f. is shown in table 1. The relationship between the force constants and the observed data is shown diagram- matically in table 2 and fig. 1. The best estimates of the force constants to fit the observed data are shown in table 3. The fit to the observed data is good [for this, and other details, see ref.(23). However, this fit has been obtained with a quartic force field in which two of the four TABLE 2.-DIAGRAMMATIC REPRESENTATION OF THE OBSERVED SPECTROSCOPIC CONSTANTS AND THEIR MLATION TO THE FORCE CONSTANTS IN TABLE 1 a 4 - F 4 4 = fa, - faat 05 - F55 = faa + faanIAN M. MILLS 13 oo 2* oo 2O pl'l l"llF1 22 oo 2O oo oo l1 l1 oo oo oo Cr) 0 rn l- ? e4 I4 (0 FIG. 1 .-The low-lying bending vibrations of acetylene, to illustrate the constants x1414, x1415, x.1515 and Y ~ ~ . Some of the observed transitions are marked with approximate vibrational origins for C2Hz in cm-l; those around 612 cm-l have been observed in the Raman spectrum, and those around 730 cm-l in the infrared spectrum. TABLE 3.-BEST ESTIMATES OF THE FORCE CONSTANTS IN TABLE 1, WITH STANDARD ERRORS FROM THE LEAST SQUARES CALCULATION IN PARENTHESES, OBTAINED BY STREY AND MILLS^^ AND BY SUZUKI AND OVEREND.*= force force constant units Sand M S and 0 constant units Sand M Sand 0 f a a aJ 0.2510(5) 0.258 8 frua aJA-l -0.67(4) -0.39 frah -0.52(4) 0.01 faaaa aJ 1.418(43) 0.515 2 fRucc -0.34(3) -0.76 fad 0.0925(5) 0.100 1 h a a s -0.02(1) -0.09 Luau' (a) (b) fRaa 0.27(1) 0.36 faaci~a' + 0.1 60( 17) &a,*,, - ) (a) (b) (a) constrained; (6) S and 0 ' s results ar0 not comparable14 POTENTIAL SURFACES FROM VIBRATION-ROTATION DATA angle-bending force constants are constrained to zero.Moreover, the uncertainties in the cubic force field, and the comparison between the two independent calculations, suggest that the cubic and quartic force constants have not been well determined.The correlation matrix shows that the force constants are highly correlated, and the standard errors are probably an optimistic estimate of uncertainty by a factor of 5. However, in this case (and in all our experience) one can say that the harmonic force field is much more precisely determined as a result of including the cubic and quartic terms in the analysis. The difficulties which these results illustrate are partly inherent in attempts to determine potential surfaces from vibration-rotation spectroscopy, arising from the fact that higher order force constants relate to the shape of the surface further and further from equilibrium, which is the part of the surface to which the data are least sensitive. They also arise from the fact that the convergence of the power-series expansion of Vin normal coordinates, in terms of which the hamiltonian is formu- lated, is not sufficiently rapid.The difficulty is that the zero’th-order model of vibration-rotation theory, used to establish the basis functions, is based on harmonic vibrations in rectilinear normal coordinates ; but even for one-quantum excitation many bending vibrations follow a significantly curvilinear path, to such an extent that the deviations cannot be comfortably handled by perturbation theory. The procedure described above involves using a hamiltonian formulated in recti- linear normal coordinates, truncated at quartic terms in the potential. If it were possible to formulate the hamiltonian directly in curvilinear normal coordinates, the equivalent truncation would give a much better representation of the p.e.f.for large displacement in the angle bending coordinates, and recent work has shown that the use of (drlr) as a bond stretching coordinate, rather than the displacement 6r itself, also gives improved convergence of the potential energy expansion.24 The use of curvilinear coordinates, however, greatly complicates the kinetic energy terms in the h a m i l t ~ n i a n , ~ ~ . ~ ~ and there is as yet no widely accepted development of the yerturba- tion treatment in this format. However Handy and Kern and co- workers,28 and Whiffen and have all developed variational methods of handling the vibration-rotation hamiltonian which allow one to use a potential energy expansion directly in curvilinear coordinates, as discussed by Handy in a later paper presented at this Discussion.2. LOW FREQUENCY VIBRATIONS When a molecule has one or more low-frequency mode of vibration, the presence of hot bands often make it possible to determine the vibrational energy levels over many quanta in the low-frequency mode. Such vibrations are also often observed in combination bands in near-infrared and Raman spectra; again in microwave spectra hot bands may be observed over many quanta, so that in general more detailed information is available on that section of the potential surface associated with the low-frequency mode. Interpreting such data, however, poses its own problems. By their nature low frequencies imply large amplitudes of vibration, for which curvilinear paths tend to play a more important role; as discussed above, the customary theory of molecular vibrations is based on rectilinear normal coordinates and is thus ill-adapted to large- amplitude vibrations.Large amplitudes also tend to involve highly anharmonic potential surfaces, for which a power series expansion as in eqn (1) may not give a satisfactory representation; yet it may not be easy to find an analytical function that gives a suitable parameterization of the surface. Finally it may be difficult to deter-IAN M. MILLS 15 mine both the precise form of the coordinate involved in a large amplitude vibration, and the zero-point averaging effect of the remaining vibrations. 2.1 RING PUCKERING VIBRATIONS Four membered ring molecules show a low frequency out-of-plane puckering vibration which illustrates many of these problems.Ring puckering vibrations are usually very anharmonic, and in many cases vibrate in a double minimum potential with two equivalent non-planar equilibrium configurations. The subject has beel\ recently reviewed by L a f f e r t ~ . ~ ~ Cyclobutane (C4H8) is the parent molecule of the series, and provides a good ex- ample. The puckering vibrational energy levels have been determined over the first 10 quanta from the Raman spectrum3' and from near-infrared combination band^.^^*^^ By treating this vibration separately from the rest, and solving the one- dimensional anharmonic oscillator problem exactly (usually by expanding the hamiltonian in a harmonic basis set), it is found that the observed puckering energy levels can be fitted to an anharmonic double-minimum potential of the form V/hic = -Ax2 + BX4. (8) Here x represents the puckering coordinate (usually taken as half the separation of the ring diagonals).Since this two-parameter function gives a good fit to the first 10 vibrational intervals, one might think that the potential surface in this coordinate is well determined. In the case of cyclobutane the barrier to inversion is found in this way to be 515 5 5 cm-I (see Lafferty3' for other examples). However, although the barrier height may be quite reliably determined, the exact form of the puckering normal coordinate (e.g., the amount of CH2 rocking vibration mixed with ring puckering) is not known, so that the correct reduced mass associated with ring puckering is uncertain.This in turn introduces uncertainty into the horizontal scale of the puckering potential function. Furthermore, the path of the coordinate is almost certainly curvilinear in the sense that the C-C bond lengths remain constant as the ring puckers : this implies a coordinate-dependent effective mass, which is a complication that is omitted from most ring-puckering potential function calculations. Mal10y~~ has discussed these problems at length. In prin- ciple, the vibrational energy levels of an isotopically substituted species (C4D8 in this case) give information on the form of the coordinate, but even this is complicated by the fact that the form of the puckering coordinate may change on isotopic substitution. Malloy concludes that barrier heights are generally more reliable than are the other features of the puckering potential function.Four membered ring molecules of lower symmetry, such as oxetane (trimethylene oxide, C3H60), thietane (C3H6S), and silacyclobutane (C3H6SiH2), exhibit far infrared puckering spectra, and microwave spectra which may be analysed to give precise rotational constants in many excited vibrational states. These give further informa- tion on the form of the puckering potential, because it is found that the effective rotational constants show an anomalous vibrational dependence on the puckering quantum number.30 This may be interpreted by equations of the form where q is a scaled (usually dimensionless) puckering coordinate, and the averages are calculated using the exact anharmonic wavefunctions in the puckering coordinate.The observed vibrational dependence of the rotational constants A,, B, and C, proves to be a sensitive measure of the anharmonic wavefunction, and hence of the16 POTENTIAL SURFACES FROM VIBRATION-ROTATION DATA puckering potential function, as shown by the example of oxetane which has been most comprehensively In this case, the microwave spectrum has also been analysed in sufficient detail to give the vibrational dependence of the distortion constant^.^^^^^ These have also been successfully interpreted by using the exact anharmonic wavefunctions in the puckering coordinate. The analysis also gives information on the degree of mixing of CH2 rocking into the puckering coordinate, as shown by Mallinson and Mills.40 However, despite the fact that five different partially deuterated isotopic species of oxetane have been studied in detail, there still remain small inconsistencies in the observed isotopic dependence of the various data.2.2 LOW-FREQUENCY BENDING VIBRATIONS OF A LINEAR CHAIN In my own laboratory we have recently been studying three examples of quasi-linear molecules, involving a low wavenumber, large amplitude anharmonic bending vibra- tion. These are fulminic acid (HCN0),41 silylisocyanate (SiH3NC0),42 and carbon- suboxide These examples involve a two-dimensional anharmonic oscillator (in contrast to the one-dimensional example of the four-membered ring molecules). Once again, information on the potential function can be obtained from both the vibrational energy levels and the vibrational dependence of rotational constants, and the experimental data available on these molecules are summarized in table 4.The TABLE 4.-sOTJRCES OF INFORMATION ON THE LOW FREQUENCY BENDING POTENTIAL FUNCTION OF HCNO, SiH,NCO, AND C302. THE BODY OF THE TABLE GIVES REFERENCE NUMBERS TO PAPERS ON THE APPROPRIATE SPECTRA (ABSENCE OF A REFERENCE INDICATES ABSENCE OF THAT KIND OF DATA) HCNO, SiH,NCO C302 v5 v 10 V l molecule and vibration: DCNO vibrational energy levels from: 1. near i.r. combination and hot bands 47 2. far i.r. spectra 44 46 3. Raman spectra 46 1. microwave spectra 45 42 rotational constants from: 2. high resolution i.r. 44 47 diffraction 48 49 vibrationally averaged distances from gas electron analysis of the data has been made in a similar manner to that described above for ring puckering vibrations, and the combination of vibrational and rotational data leads to the potential surfaces illustrated in fig.2. With SiH3NC0 there is no direct observation of the vibrational energy levels, and the potential function and energy levels shown in the figure have been inferred from rotational structure. However for both C302 and SiH3NC0 a gas electron diffrac- tion study has been made (as indicated in the table); this essentially confirms the other evidence on the potential surface. Details of the analysis are given in ref. (41), (42) and (43). Although we regard the main features of these surfaces as well-determined (partic- ularly the height of the central potential hump in GO2 and SiH,NCO), the exact form of the bending coordinate is less well determined; this reflects uncertainty on theIAN M.MILLS 80. 60. 40. 17 33 0 - A 1 1 2' c - A t 22 A - T 2od 01 -20 201 ' I *--[ i -20 O W , > , I , 1 I 0' 10' 20° 30° 40. 0 10. 20' 3O0 40° 0. 10' 20' 30. 40. 's i 1": O=C&crC=O 3- -c=o .-.. ,--.I . . H-!-C&N=O FIG. 2.-The potential energy surface for the large amplitude bending vibrations of HCNO, GO2, and SiH3NC0. In each case the coordinate involved is bending of the linear chain, the major component of the bending being as indicated below each figure. The bending coordinate is doubly degenerate, and the potential surface should be visualized in three dimensions, with a circular valley and a central hump.The figure shows a radial section of this surface. appropriate reduced mass and hence on the horizontal scale in the figures. Also, the curvilinear nature of the bending coordinate has not been properly accounted for in relating the potential surface to the observed spectrum, as in the example of ring molecules. Another feature of these potential surfaces is illustrated by fig. 3 for C,O,. For this molecule fairly high resolution infrared spectra of the fundamentals v4 and v2 are available, and these show hot bands and combination bands involving the large amplitude bend v7. From the resulting vibrational and rotational constants we have been able to determine43 the effective bending potential function in v7, in both the 820 810 780 775 760 J.. -r ground stole 30 40{ 2 0 4 ---I # Go loo ?CO JC" 40' FIG.3.-The bending potential surface of c302 in the doubly degenerate large amplitude vibration v7, and the effect on this surface of exciting one quantum of v2 and one quantum of v4. The form of v2 and v4 is illustrated in the diagram.18 POTENTIAL SURFACES FROM VIBRATION-ROTATION DATA ground vibrational state of all other modes, and in the excited states v4 = 1 and v2 = 1. The resulting surfaces are shown in fig. 3. The interesting feature is that in both cases there is a significant change in the shape of the v7 potential surface on exciting another vibration; the hump at the linear configuration is doubled in the v4 = 1 state but almost removed in the v2 = 1 state. This draws attention to the point that the potential surface in any low-frequency large-amplitude vibration is really an efective potential obtained by averaging over all the remaining high frequency vibrations, in a manner analogous to the adiabatic separation of electronic from nuclear degrees of freedom.A similar comment applies to all of the previous examples: in each case the section of the potential surface which is determined should be thought of as vibration- all) averaged over the remaining normal modes. 3. DISCUSSION The limitation of vibration-rotation spectroscopy to a small region around the equilibrium configuration in the potential surface suggests that the most fruitful comparison with other methods of study will be with ab initio calculations. This, indeed, is proving to be true, since there are now many papers appearing which report the ab initio calculation of both harmonic and anharmonic force constants for small molecules, and it is clear that sufficiently careful calculations give good results. There have been fewer attempts to calculate potential surfaces for large amplitude vibra- tions, such as those described in section 2 of this paper, perhaps because the energy changes involved are smaller, implying the need for an even greater degree of precision in the calculation.In this paper I have tried to emphasize two features of the difficulties in determining potential surfaces from vibration-rotation data. The first is the difficulty of knowing the exact form of the coordinate in terms of which some section of the potential surface may have been determined.I began by describing this as the fundamental difficulty of harmonic force field calculations, and I finished by emphasizing that this is the least certain feature of large-amplitude potential surfaces such as those in fig. 2 and 3. The second is the importance of vibrational averaging effects. Kuchitsu, in a following paper, will emphasize the importance of these effects on molecular structure determinations; fig. 3 shows that they can make an important contribution to the effective potential surface. If we were to compare an ab initio calculation of the bending potential in C302 with the experimental surface, we would have to face these two problems in making a comparison. From the ab initio end, it would be necessary not only to recalculate the energy for each value of the large amplitude coordinate, but also to recalculate the local potential surface with respect to all other vibrational coordinates.It would then be necessary to determine the vibrational wavefunctions (including anharmonic effects) in the remaining coordinates, holding the large amplitude coordinate frozen, and use them to average the potential energy function. In this way an effective poten- tial in the large amplitude mode might be determined. From the experimental end, it is clearly desirable to determine the effective potential as a function of as many other parameters as possible (e.g., isotopic substitution, vibrational excitation) in order to provide the information necessary to extrapolate back to the true potential surface. At present there are no examples where an analysis of this kind has been made from either the experimental or the ab initio approach.I am grateful to my colleagues for numerous discussions of these problems, andIAN M . MILLS 19 particularly to Dr. A. G. Kobiette and Mr. J. A. Duckett for permission to quote results from our forthcoming paper on C301. I also thank Dr. Robiette for a critical reading of this manuscript. A. G. Maki, J. Mol. Spectr., 1973, 47, 2 17. D. Van Lerberghe, J. J. Wright and J. L. Duncan, J. Mol. Spectr., 1972, 42, 251. C. Camy-Peyret, J. M. Flaud, G. Guelachvili and C. Aniiot, hfol. Phys., 1973, 26, 825. G. Guelachvili, Nouv. Rev. Opt. Appl., 1972, 3, 317. S. M. Freund, G. Duxbury, M. Romheld, J. T. Tiedje and T. Oka, J. Mol. Spectr., 1974,52, 38.H. E. Radford, K. M. Evenson and C. 3. Howard, J. Chem. Phys., 1974, 60, 3178. T. H. Edwards and S. Brodersen, J . Mol. Spectr., 1975, 54, 121. ’ A. Cabana, L. Lambert and C . Pepin, J. Mol. Spectr., 1972, 43,429. ’ A. S. Pine, J . Opt. SOC. Amer., 1976, 66, 97. lo R. J. Butcher, D. V. Willets and W. J. Jones, Proc. Roy. SOC. A , 1971, 324, 231. l1 A. F. Krupnov and A. V. Burenin, in Molecular Spectroscopy: Modern Research, ed. K. N. l2 I. M. Mills, in Molecular Spectroscopy: Modern Research, ed. K. N. Rao (Academic Press, l3 A. R. Hoy, I. M. Mills and G. Strey, Mol. Phys., 1972, 24, 1265. l4 I. M. Mills, in Theoretical Chemistry Vol. I , ed. R. N. Dixon (Specialist Periodical Reports l5 J. L. Duncan, in Molecular Spectroscopy ed. R. F. Barrow, D. A.Long and D. J. Millen, Vol. l6 A. R. Hoy, PkD. Thesis (University of Reading, September 1972). l7 M. Bertram, Ph.D. Thesis (University of Reading, September 1973). l9 A. R. Hoy, M. Bertram and I. M. Mills, J. Mol. Spectr., 1973, 46, 429. 2o M. A. Pariseau, I. Suzuki and J. Overend, J . Chem. Phys., 1965, 42, 2335. 21 I. Suzuki and J. Overend, Spectr. Acta, 1969, 25A, 977. 22 Y. Morino, K. Kuchitsu and S. Yamamoto, Spectr. Acta, 1968, 24A, 335. 23 G. Strey and I. M. Mills, J. Mol. Spectr., 1976, 59, 103. 24 G. D. Carney, L. A. Curtiss and S. R. Langhoff, to be published. 25 J. T. Hougen, P. R. Bunker and J. W. C. Johns, J . Mol. Spectr., 1970, 34, 136. 26 C. R. Quade, J. Chem. Phys., 1976, 64, in press. 27 R. J. Whitehead and N. C. Handy, J. Mol. Spectr., 1975, 55. 356. 28 G. D. Carney and C. W. Kern, Int. J. Quant. Chem., 1975, Symp. No. 9, 317. 2 9 A. Foord, J. G. Smith and D. H. Wiffen, Mol. Phys., 1975, 29, 1685. 30 W. J. Lafferty, in Critical Evaluation of Chemical and Physical Structural Itformation, ed. D. R. Lide and M. A. Paul (National Academy of Sciences, U.S.A., 1974). 31 F. A. Miller and R. J. Capwell, Spectr. Acta, 1971, 27A, 947. 32 J. M. R. Stone and I. M. Mills, Mol. PIzys., 1970, 18, 631. 33 T. Ueda and T. Shimanonchi, J. Chem. Phys., 1968, 49, 470. 3J T. B. Malloy, J. Mol. Spectr., 1972, 44, 504. 35 S. I. Chan, J. Zinn and W. D. Gwinn, J. Chem. Phys., 1961, 34, 1319. 36 H. Wieser, M. Danyluck and R. A. Kydd, J. Mol. Spectr., 1972, 43, 382. 3’ R. A. Kydd, H. Wieser and M. Danyluck, J . Mol. Spectr., 1972, 44, 14. 38 R. A. Creswell and I. M. Mills, J. Mol. Spectr., 1974, 52, 392. 39 P. D. Mallinson and A. G. Robiette, J. Mol. Spectr., 1974, 52, 413. 40 P. D. Mallinson and I. M. Mills, Mol. Phys., 1975, 30, 209. 41 J. A. Duckett, A. G. Robiette and I. M. Mills, J . Mol. Spectr., 1976, in press. 42 J. A. Duckett, A. G. Robiette and 1. M. Mills, J . Mol. Spectr., 1976, in press. 43 J. A. Duckett, A. G. Robiette and I. M. Mills, J. Mol. Spectr., 1976, in press. 44 B. P. Winnewisser, M. Winnewisser and F. Winther, J . Mol. Spectr., 1974, 51, 65. 45 B. P. Winnewisser and M. Winnewisser, J . Mol. Spectr., 1975, 56, 471. 46 L. A. Carriera, R. 0. Carter, J. R. Durig, R. C. Lord and C. C. Milionis, J. Chem. Phys., 1973, 47 A. W. Mantz, P. Connes, G. Guelachvili and C. Amiot, J. Mol. Spectr., 1975, 54, 43. 48 C. Glidewell, A. G. Robiette and G. M. Sheldrick, Chetn. Phys. Letters, 1972, 16, 526. 4 9 M. Tanimoto, K. Kuchitsu and Y . Morino, Bull. Chem. SOC. Japan, 1970, 43, 2776. Rao (Academic Press, N.Y., 1976, Vol. 2). N.Y., 1972, Vol. 1). of Chemical Society, London, 1974). 3 (Specialist Periodical Reports of Chemical Society, 1975). P. D. Mallinson, to be published. 59, 1028.

ISSN:0301-7249    

DOI:10.1039/DC9776200007

出版商:RSC    

年代:1977

数据来源: RSC

摘要:

Experimental Determination of Average and Equilibrium Structures of Polyatomic Gas Molecules by Diffraction and Spectroscopic Methods BY KOZOKUCHITSU AND KAZUKO OYANAGI Department of Chemistry, Faculty of Science, The University of Tokyo, Bunkyo-ku, Tokyo 113, Japan Received 29th April, 1976 Various definitions and interrelations of the " geometric structures " of polyatomic molecules are reviewed. Analytical procedures for determining well-defined " structures " are discussed. The variations in average nuclear positions caused by isotopic substitution and their influence on the structure are estimated for OCS, HCN, SO2 and H20 as examples. 1. GENUINE AND APPROXIMATE EQUILIBRIUM STRUCTURES FROM SPECTROSCOPY The geometric structure of a semirigid polyatomic molecule can best be defined and compared with the corresponding theoretical structure in terms of the nuclear positions at a potential minimum.A set of these " equilibrium " nuclear coordinates, or the internuclear distances and angles derived therefrom, are called the re-structure. When the rotational constants for the ground vibrational state and those for the excited states of all the fundamental modes are determined, they can be extrapolated to give the equilibrium rotational constants, A,, Be and Ce, from which the re structure can be determined. This method has been applied to a number of linear and bent XY, molecules.1$2 For many other molecules, where the number of independent geometric parameters exceeds that of independent rotational constants, a sufficient number of isotopic rotational constants is also necessary.The re-structures of several linear XYZ, pyramidal XY3 and CH3X molecules have been determined in this way.2 In principle, a similar procedure may be applied to a nonrigid molecule in order to define the equilibrium structure of a semirigid frame as a function of the large-amplitude coordinates. However, no complete experimental analysis of this sort seems to have been made. In most other cases, only the ground-state rotational constants, A,, Bo and C,, are used to derive geometric parameters. Because of complicated vibration-rotation interactions3 however, these rotational constants do not correspond exactly to the ground-state average of nuclear positions. Hence, the so-called r,-structure derived from A,, Bo and Co is not necessarily a good approximation of the re-structure.The experimental precision of ro parameters is often significantly lower than that of the rotational constants. This systematic error can be even larger when some of the geometric parameters have to be assumed. When one of the component nuclei is substituted by its isotope, the isotopic differences between the rotational constants of this species and the parent species can be used to determine the principal-axis substitution (r,) coordinates of this nucleus, whereby a substantial part of the vibration-rotation interaction is compensated.KOZO KUCHITSU A N D KAZUKO OYANAGI 21 Kraitchman’s equations are used for this purpo~e.~*~ When the rotational constants of all the singly substituted species are determined, a complete r,-structure is obtained.Even when complete isotopic substitutions are not practicable, one can use a first- moment equation, or the condition that the cross-products of inertia vanish, to esti- mate the remaining nuclear coordinates. In a less favourable case the rotational constants of the parent species have to be used to obtain a hybrid r,,r,-structure at the expense of a systematic error caused by vibration-rotation interaction. The precision of the Y, coordinates depends on the nuclear positions. The co- ordinates of a nucleus located close to a principal inertial plane carry large uncertain- ties irrespective of the nuclear mass; an empirical equation for this estimation has been proposed by Costains In addition, the signs of such coordinates may be ambiguous, since Kraitchman’s equations provide only their absolute values.Doubly substituted species have sometimes been used to improve the precision of such small coordinates .6 The difference between the r, and re-structures of polyatomic molecules has not been understood as clearly as that for diatomic molecules, though physical significance of the r, coordinates has been discussed by Sorensen et aL7 and Watson.* There still are only a limited number of molecules for which both the r, and re-structures have been determined precisely from experiment and compared with each other ~ritically.~.~ The rs parameters, such as the C-N bond lengths in various cyanides, determined with an empirical correction for vibration-rotation interaction appear to be nearly independent of the isotopic species used to determine the coordinates, and are chemically pla~sible.~ Nevertheless, systematic errors in the Y, parameters should be treated cautiously when one intends to make a critical comparison of r, parameters in different geometric environments for chemical purposes.For example, recent electron diffraction studies suggest that the observed differences in the r, or ro bond lengths in some simple organic molecules are not necessarily close approximations of those in the corresponding re bond lengths.l0-l2 When the Y, coordinates of all the atoms are determined, one can calculate a moment of inertia, I,, using these rs coordinates. Watson8 has pointed out that the equilibrium moment of inertia, I,, is approximately equal to 21,-10, where I, is h/8n2 divided by the corresponding rotational constant for the ground vibrational state.He called the structure derived from 2Zs-10 the mass-dependence (rm) structure. In a few examples he has shown that the r,-structure is a very close approximation of the re-structure except for some parameters involving hydrogen. Since, however, one needs rotational constants for more isotopic species than are necessary for determining the r,-structure, the application of his method has so far been restricted to very simple molecules. 2. ZERO-POINT AVERAGE STRUCTURE FROM SPECTROSCOPY The difference between the ground-state and equilibrium rotational constants is composed of the harmonic part, which depends on the quadratic potential constants, and the anharmonic part, which is a linear function of the cubic constant~.~J~ When the harmonic part is subtracted from the ground-state rotational constants, one obtains the zero-point average rotational constants, A,, B, and C,, which correspond to the zero-point average (r,) nuclear positions.14-16 The r,(X-Y) distance between the nuclear positions X and Y in a polyatomic molecule should not be confused with the zero-point vibrational average of an instantaneous X - Y distance.A simple calculation 1 2 9 1 7 shows that for a molecule with small vibrational amplitudes rz = r e + < W * , (2.1122 EXPERIMENTAL DETERMINATION OF STRUCTURES OF GAS MOLECULES where Az denotes an instantaneous displacement, Ar, of r(X-Y) projected on the equilibrium X-Y axis (taken as a temporary z axis), and 0 denotes an average over the ground vibrational state.Because nuclear vibrations perpendicular to this z-axis exist in a polyatomic molecule, r, does not agree with the real average X-Y distance, re+<Ar),, which is denoted as r,O. The difference between r, and re, i.e., (Az),, can be estimated if cubic anharmoni- city18 is known or assumed. On the other hand, the difference between (Ar), and (Az), can be estimated with sufficient accuracy if the quadratic force field is known (see section 3). The average bond angle can be defined unambiguously in terms of the average nuclear positions. The r, position depends on nuclear masses. Therefore, the isotopic effect must be known precisely when the A,, Bz and C, of other isotopic species are needed for a complete determination of the r,-structure. The r,-structure derived from such isotopic substitutions is sensitive to the estimated (or assumed) isotopic effect on r,- positions.This presents a serious problem in the experimental determination of the r,-stru~ture.~~ In most previous studies the isotopic variation of an r, parameter, 6r, NN ~(Az),, was confused with Thus the dr, of a bond by substitution of one of the atoms by its heavier isotope was erroneously called " bond shrinkage " or " bond shortening ". This confusion is by 110 means trivial, as discussed more quantitatively in section 4. 3. AVERAGE STRUCTURES FROM ELECTRON DIFFRACTION Gas electron diffraction provides information on the thermal average of all the internuclear distances in a molecule.The rg distance is defined aslo where T denotes the vibrational average in thermal equilibrium. It has been shown that17 where KT = ((Ax2> T + (Ay2> T)/2re (3.3) Ax and Ay denote the displacements perpendicular to the equilibrium internuclear axis z, drcent denotes a small displacement due to centrifugal force. One advantage of rg over rz is that the re bond distance can be estimated if bond-stretching anharmoni- city is assumed.21*22 For example, for a group of similar bonds (e.g., for the C-C bonds in hydrocarbons), thermal average displacements, <Ar)=, can be estimated to be nearly equal to one another, so that the observed differences in the rg distances may well be approximated as those in the re distances. On the other hand, since the KT terms do not satisfy geometric conditions, a set of the rg bonded and nonbonded distances cannot define a physically meaningful bond angle without corrections for KT, so-called linear or nonlinear " shrinkage effects ".23 For example, the rg (Y-Y) distance in a linear XY2 molecule is smaller than twice the rg (X-Y) distance.However, such a shrinkage effect can be calculated if the quadratic force field is The thermal average of nuclear positions, ra, derived from the thermal average of internuclear distances, rg, is defined for this purpose;1°J7 rg = rg--KT-drCent. (3.4)KOZO KUCHITSU AND KAZUKO OYANAGI 23 The ra distance corresponds to the distance between thermal average nuclear positions, and when it is extrapolated to zero kelvin, rao = lim rCr = r,O-Ko z r,+(Az),, T-tO the rao structure is practically identical with the r, structure.This formalism provides a basis for an analysis of experimental data derived from electron diffraction in combination with rotational constants in order to determine a consistent average s t r ~ c t u r e . ~ ~ * ~ ~ For a relatively complicated molecule, where nearly equal and inequivalent internuclear distances exist, it is often difficult for electron diffraction alone to determine all the independent geometric parameters accurately without assumptions. In such a case, rotational constants supply precise independent information on geometric parameters. On the other hand, electron diffraction gives important information for the determination of a unique and accurate average structure, which may not be derived from spectroscopy alone even with numerous isotopic substitutions.So far, only the rotational constants of the parent species or those of deuterated species have been used for such a combined analysis, since for non-hydrogen substi- tutions the isotopic variations, Sr,, mentioned in section 2 are so uncertain that their rotational constants cannot improve the accuracy of the derived parameters. For example, in a structure analysis of a~rolein,~' H2C=CH-CH=0, only the rotational constants of the parent species and a part of the D species were used, whereas those of the 13C and l 8 0 species had to be left out. Under these circumstances, it is pertinent to make a reasonable estimate of Sr, in order to make full use of the rotational constants of all the available isotopic species.4. ISOTOPIC DISPLACEMENTS OF NUCLEAR POSITIONS Average nuclear positions in the principal inertial axes, (ai),,, (bi)o and ( c ~ ) ~ , For example, it is known (4.1) are linear functions of the cubic potential constants,13 kSSrSIt. that (ai>o = a4+(Aa,)o = a:+mi't Clci.,' <Qs>o S where a: is the equilibrium position of the nucleus i and Zc$ is an element of the I matrix. The ground-state vibrational average of the normal coordinate of the s'th mode, (Qs)o, is given by28 where us is the normal frequency and g,. is the degeneracy. When the nucleus j is substituted by its isotope, the displacements of the average position of the nucleus i, 6ai = (ai*>-(ai>, etc., (4.3) can be calculated if one assumes that the potential surface remains unchanged.The nucleus i may or may not be equal to j . It follows that (a,*> = a;+ml-* 2 Z(i",'* <Qs*), S (4.4) where the starred quantities can be calculated by use of the quadratic and cubic potential constants obtained with appropriate mass corrections. The isotopic variations in the r, distances and angles, 6r, and SO,, can then be calculated. Numerical examples of four triatomic molecules, for which precise rotational24 EXPERIMENTAL DETERMINATION OF STRUCTURES OF GAS MOLECULES constants, equilibrium structures, and quadratic and cubic potential constants are known (OCS,29 HCN?O S031 andH2032*33), are listed in fig. 1-4. The following remarks can be made on the isotopic variations: S -3.3 -1.3 ~ o* -11.0 c* 4.4 S* o* -126 :* -7.1 -9.5 c* -2.9 s* -7.0 :*-O.l x x -1.6 -28 s* FIG.1.-Isotopic differences in the I;. distances from those in 16012C32S, in units of A. The asterisks represent lSO, 13C and 34S. The arrows indicate that additivity rules hold in multiple isotopic substitutions. - N* H - ' O C - L $ - t HAC* -7 ,: ;14 :* -3 >c, 435 c* -108, N* N +309 -101 "x ;4 , +296 :*-lo3 FIG. 2.-Isotopic differences in the rz distances from those in H12C14N in units of lW5 A. The asterisks represent 13C and lSN. The arrows indicate that additivity rules hold in multiple isotopic substitutions. FIG. 3.-Isotopic differences in the rz distances (in lW5 A) and the BZ angle (in lW3 degrees) from those in 32S1602. The asterisks represent ''0 and "S. The arrows indicate that additivity rules hold in multiple substitutions.0 - 5 ? 4 . 6 k 7 \ -137&~22 O* D D D FIG. 4.-Isotopic differences in the rz distances (in lW5 A) and the 8, angle (in low3 degrees) from those in H2160. The asterisks represents "0. The arrows indicate that additivity rules hold approxi- mately in multiple substitutions. (i) The orders of magnitude of the isotopic variations of r, bond distances are a few thousandths of 1 A when H/D substitution is made, and a few ten thousandths of 1 A or less in other cases. (ii) The dr, caused by multiple isotopic substitutions are additive. For example, the variations of the r, distances in 18013C34S, from 16012C32S is approximately equal to the sum of those in 18012C32S, 16013C32S and 16012C34S from 16012C32S. (iii) When a nucleus is substituted by its heavier isotope, 6rz may not necessarilyKOZO KUCHITSU AND KAZUKO OYANAGI 25 be negative.A typical example is an apparent increase in the r, distance of C-D in DCN in comparison with that of C-H in HCN.* (iv) When a nucleus X is substituted, the 6r, of a bond which is not directly associated with X, say Y-Z, can be comparable with, or even larger than, that directly associated with X, say X-Y. A typical example of this " secondary isotopic varia- tion " is shown in the large dr, (C-S) in OCS when l6O is substituted by "0. The trends (iii) and (iv) are explicable by use of eqn (2.1) and (3.5), Since the first and the second terms have the same sign (positive or negative, respec- tively, when one of the nuclei is substituted by its lighter or heavier isotope), they tend to cancel each other.The secondary variation is mainly ascribable to 6Ko, which accounts for the isotopic difference in perpendicular displacements (see section 2). The second term can outweigh the first term and, in case of substitution with a heavier isotope, make dr, positive. In other words, an apparent " bond stretch " instead of a " bond shortening " can take place. 5. THE r z STRUCTURE DERIVED FROM ISOTOPIC SUBSTITUTION The rotational constants, A,, B, and C,, for various isotopic species can be used to determine the substitution r, coordinates. A schematic diagram is shown in fig. 5. The isotopic displacements in the nuclear coordinates, 6ai, 6bi and 6ci, are I I I 9 structure re Is h., 1 r.7 FIG. 5.-A schematic diagram showing determination of structures from isotopic rotational constants.introduced into the substitution scheme. The Kraitchman coordinates derived therefrom has been formulated by Nygaard and S ~ r e n s e n . ~ ~ The zero-point average coordinate, a,,, derived from the Kraitchman equation with inclusion of the isotopic displacements in all the hai caused by the substitution of a nucleusj is given as a linear combination of 6ai by \ i J where a, jeff denotes the corresponding effective coordinate calculated by a direct application of the Kraitchman equation4 to A,, B, and C, with neglect of 6ai, and m and Am denote the atomic mass and the isotopic difference. The sum is taken over all the atoms including j . The r,-structure can then be calculated from the a,, co- ordinates.The isotopic displacement is multiplied by a large factor particularly when a, is small and/or miaI is large. * The difference, rz(C-D)-r,(C-H), was estimated by Laurie and Herschbach16 to be -0.003A. The discrepancy between their estimate and the present estimate, +0.003A, seems to originate from their neglect of the l2C-I3C isotope differences in the v, parameters when they estimated the r,(C-H) and r,(C-D) distances independently by use of the B, constants of the 12C and "C species.26 EXPERIMENTAL DETERMINATION OF STRUCTURES OF GAS MOLECULES This method has been applied to OCS, HCN, SO2 and H20, as summarized in table 1. The r,-structures of SOz and H20 obtained by the present substitution method are consistent with those derived directly from the A, and B, (or C,) * of the parent molecules, and the r,-structures of OCS and HCN are consistent with any of those derived from the B, constants of any two isotopic species.For comparison, TABLE 1 .-STRUCTURES OF TRIATOMIC MOLECULES * ocs 0-c c-s HCN H-C C-N SO2 s-0 L 0-s-0 D20 t D-0 L D-0-D 1.1543 1.5628 1.0655 1.1532 1.4308 0.9575 119.33 104.51 0.0063 - 0.003 1 - 0.0023 0.0019 0.0015 0.07 0.0039 -0.51 0.003 1 0.001 7 0.0046 0.0038 0.10 0.0107 - 0.02 - 0.0043 0.0065 -0.0031 - 0.0028 0.0026 0.0022 0.10 0.0067 -0.25 * Distances in 8, and angles in degrees. The Y, and rS structures are taken from the references cited in the text. The Y, and rZeff-structures are calculated with and without the isotopic variations, 6r,, given in fig. 1-4, respectively. 7 The r,-structure of D20 has been calculated by use of the rotational constants of D2160, H2l60 and D2'*0.the r,eff-structures derived from azjeff coordinates are also listed in the table. They are close to the r,-structures derived from A. and Bo by use of the ordinary Kraitchman method with neglect of isotopic displacements. The differences between the r, and rZeff distances are a few thousandths of 1 A. Though none of the molecules has small nuclear coordinates (0.1 or less), the differences are one order of magnitude < dr, and are comparable with experimental precision. 6. EXTENSION TO LARGER MOLECULES The rigorous method stated above for estimation of isotopic variations, 6r,, requires precise knowledge of the quadratic and cubic potential constants. Even if the cubic constants are not available, one can use an approximate anharmonic model for this estimation.The simplest model seems to be to use the quadratic force field and the Morse anharmoiiicity parameters, a, of the bonds estimated from the cor- responding diatomic m01ecules.~~*~~ The dr, values for the four molecules listed in fig. 1-4 have been estimated by use of this model. The orders of magnitude of the dr, distances have been estimated correctly, but the agreement is not always quantitative. Furthermore, the isotopic variations in the angles have to be estimated somehow. The potential constants obtained from ab initio calculations, if available, may be used for this purpose. In this respect, the isotopic variations of the r,-structures of H2C0 and C2H4 calculated by Duncan35 from the observed isotopic rotational constants are suggested.The average structure determined by electron diffraction can also supply inde- pendent and accurate information on the dr, parameters. In such favourable cases as CH3CN36 and V0C13,37 dr, parameters can be determined as a part of the variable parameters in a least-squares analysis. The subject of this section will be reported in detail elsewhere. * In principle, there is no inertial defect among A,, B, and C, for a planar molecule.KOZO KUCHITSU AND R A Z U K O OYANAGJ 27 7. SUMMARY A spectroscopic experiment primarily determines nuclear positions, either equili- brium or vibrational average, whereas electron diffraction primarily determines thermal-average internuclear distances. These " position " and " distance " averages can be converted from one to the other if the quadratic force field of the molecule is known at least approximately. A conjoint analysis of electron diffraction data and rotational constants can improve the accuracy of the average structure.When rotational constants of different isotopic species are used to determine the average structure, it is necessary to consider the isotopic dependence of average nuclear positions. Our main future problems are the following: (i) Estimation of the equilibrium structure from average structures. For this purpose, our potential models have to be refined. Particularly, accurate data from high-resolution spectroscopy and from ab initio calculations should be most helpful.(ii) A complementary use of accurate electron diffraction data. There is much to be investigated in this respect before one understands spectroscopic structures of medium-sized molecules, either Y, or ro. (iii) Structure analyses of nonrigid molecules. In order to apply the present method to nonrigid molecules, the theory needs to be modified and the potential function in regard to the large-amplitude motion has to be known. A combined analysis of electron diffraction and microwave spectroscopic data for symmetric internal rotors has been formulated by Iijima.38 Efforts are being made to work out a more general formulation. The authors are grateful to Drs. Lise Nygaard and G. Ole Slarensen, Chemical Laboratory V, University of Copenhagen, for helpful discussions.' Y . Morino, Pure Appl. Chem., 1969,18, 323. Y . Morino and E. Hirota, Ann. Rev. Phys. Chem., 1970, 20, 139. I. M. Mills, Molecuiar Spectroscopy, Modern Research, ed. K. N. Rao and C. W. Mathews (Academic Press, New York, 1972), p. 115. J. Kraitchman, Amer. J. Phys., 1953, 21, 17. C. C. Costain, J. Chem. Phys., 1958, 29, 864; Trans. Amer. Cryst. Assoc., 1966, 2, 157. L. Pierce, J. Mol. Spectr., 1959, 3, 575. J. K. G . Watson, J, Mol. Spectr., 1973, 48, 479. R. H. Schwendeman, Critical Evaluation of Chemical and Physical Structural Information, ed. D. R. Lide, Jr. and M. A. Paul (National Academy of Sciences, Washington, D.C., 1974), chap. 2, p. 94. lo K. Kuchitsu and S. J. Cyvin, Molecular Structures and Vibrations, ed. S. J. Cyvin (Elsevier, Amsterdam, 1972), chap.12, p. 183. l1 R. L. Hilderbrandt and J. D. Wieser, J. Chem. Phys., 1971,55,4648; 1972,56, 1143. l2 K. Kuchitsu, ref. (9), chap. 2, p. 132. l3 H. H. Nielsen, Rev. Mod. Phys., 1951, 23, 90. l4 T. Oka, J. Phys. SOC. Japan, 1960,15, 2274. ' e.g., J. Casado, L. Nygaard and G. 0. Sarensen, J. MuZ. Struct., 1971, 8, 211. D. R. Herschbach and V. W. Laurie, J. Chem. Phys., 1962, 37, 1668. V. W. Laurie and D. R. Herschbach, J. Chem. Pliys., 1962, 37, 1687. l7 Y. Morino, K. Kuchitsu and T. Oka, J. Chem. Phys., 1962,36, 1108. l8 J. Pliva, ref. (9), chap. 5, p. 289. l9 W. LafTerty, D. R. Lide, Jr. and R. A. Toth, J. Chem. Phys., 1965, 43, 2063. 'O K. Takagi and T. Oka, J. Phys. SOC. Japan, 1963, 18, 1174. 21 L. S. Bartell, J. Chem. Phys., 1955, 23, 1219. '' K.Kuchitsu, Birll. Chern. Soc. Japan, 1967, 40, 498, 505.28 EXPERIMENTAL DETERMINATION OF STRUCTURES OF GAS MOLECULES 23 S. J. Cyvin, Molecular Vibrations and Mean Square Amplitudes (Universitetsforiaget, Oslo and Elsevier, Amsterdam, 1968). Y . Morino, J. Nakamura and P. W. Moore, J. Chem. Phys., 1962, 36, 1050; Y. Morino and T. Iijima, Bull. Chem. SOC. Japan, 1962,35, 1661 ; 1963,36,412. 25 K. Kuchitsu, Molecular Structure and Properties, MTP International Review of Science, ed. G . Allen (Buttenvorth, London, 1972), vol. 2, chap. 6, p. 203. 26 A. G. Robiette, Molecular Structures by Difractiun Methods, Specialist Periodical Report, ed. G. A. Sim and L. E. Sutton (The Chemical Society, London, 1973), vol. 1, part 1, chap. 4, p. 160. 27 K. Kuchitsu, T. Fukuyama and Y. Morino, J. Mul. Strzrct., 1969,4,41. 28 M. Toyama, T. Oka and Y . Morino, J. Mul. Spectr., 1964, 13, 193. 29 Y . Morino and T. Nakagawa, J. Mul. Spectr., 1968,24496; A. G. Maki and D. R. Johnson, J. Mul. Spectr., 1973, 47, 226. 30 T. Nakagawa and Y. Morino, Bull. Chern. SOC. Japan, 1969,42,2212; G. Winnewisser, A. G . Maki and D. R. Johnson, J. Mol. Spectr., 1971,39, 149; G. Strey and I. M. Mills, Mul. Phys., 1973, 26, 129. 31 Y. Morino, Y. Kikuchi, S. Saito and E. Hirota, J. Mol. Spectr., 1963, 63, 95; S. Saito, J. Mol. Spectr., 1969,30, 1. 32 K. Kuchitsu and L. S. Bartell, J. Chem. Phys., 1962, 36, 2460. 33 K. Kuchitsu and Y. Morino, Bull. Chem. Sue. Japan, 1965,38, 814. 34 L. Nygaard and G. 0. S~rensen, to be published. 35 J. L. Duncan, Mul. Phys., 1974,28, 1177. 36 K. Karakida, T. Fukuyama and K. Kuchitsu, Bull. Chem. Suc. Japan, 1974, 47, 299. 37 K. Karakida and K. Kuchitsu, Inurg. Chim. Acta, 1975, 13, 113. 38 T. Iijima and S. Tsuchiya, J. Mol. Spectr., 1972,44,88; T. Iijima, Bull. Chem. SOC. Japan, 1972, 45, 1291.

ISSN:0301-7249    

DOI:10.1039/DC9776200020

出版商:RSC    

年代:1977

数据来源: RSC

摘要:

Semi-Classical Methods for Vibrational Energy Levels of Triatomic Molecules BY N. C. HANDY, S. M. COLWELL AND W. H. MILLER* University Chemical Laboratory, Lensfield Road, Cambridge CB2 1 EW Received 29th April, 1976 The method recently proposed by Chapman, Garrett and Miller for semi-classical eigenvalues of non-separable systems has been applied to the triatomics SOz and H20. The Hamiltonian is expressed in normal coordinates, using potentials V = V(Arl, Ar2, A@. The energy levels are compared with corresponding quantum mechanical energy levels. For SO2, the fundamental fre- quencies differ by at most 0.1 cm-l, and for H20 they differ by at most 1.6 crn-l. 1. INTRODUCTION Semi-classical methods have been used extensively in recent years to treat atomic and molecular scattering problems,1*2 but only most recently has attention turned to the problem of determining semi-classical eigenvalues for molecular systems.The goal is to generalise the Bohr-Sommerfeld, or WKB, quantum condition,? to multidimensional systems, which are not separable and thus cannot be quantised as individual one-dimensional systems. Eqn (1) has been of immense utility in analys- ing vibrational/rotational spectra of diatomic molecules and determining their inter- atomic potential functions, and it is anticipated that its multidimensional extension would be of similar use for small polyatomic molecules. The multidimensional eigenvalue problem is actually the oldest problem in “ semi-classical mechanics ”, being the central problem of the old quantum theory.The only truly new approach to it in recent years has been Gutzwiller’s periodic orbit theory (with the important modification of Miller), but although interesting and illuminating in a number of ways it is now clear that the quantum condition obtained by periodic orbit theory has dynamical approximations, beyond the semi-classical approximation itself, inherent in it. The quantization procedure of the old quantum theory3 is first to construct the total (classical) Hamiltonian as a function of the complete set of “good” action variables (or adiabatic invariants) which are the constants of the motion of the system, and then to require them to be integers (or perhaps half-integers). To carry out this construction requires the solution of a Hamilton-Jacobi equation for the non-separable system, and can thus be carried out analytically only within the frame- work of perturbation t h e ~ r y .~ Recently, however, Chapman, Garrett and Miller6 (CGM) have shown how the * J. S. Guggenheim Memorial Fellow on sabbatical leave from Department of Chemistry, t Atomic units are used, unless otherwise stated. University of California, Berkeley.30 SEMI-CLASSICAL METHODS FOR VIBRATIONAL ENERGY LEVELS Hamilton-Jacobi equation in action-angle variables, which appears in Born’s3 formulation, can be cast in a form that permits efficient numerical (i.e., non-pertuba- tive) solution. Application6 of the CGM procedure to a two-dimensional anharmonic oscillator model problem has shown the semi-classical eigenvalues so obtained to be in good agreement with the exact quantum mechanical ones, even when the non- separable coupling is quite strong.Other ways of constructing a non-separable Hamiltonian as a function of the “good” action variables have been given by Marcus and Percival* and their co-workers; this work, based on Keller’sg forniula- tion, deals with the Hamilton-Jacobi equation in Cartesian variables. This paper reports application of the CGM approach to determination of the vibrational eigenvalues of SO2 and H20 ; this is the first non-perturbative calculation of semi-classical eigenvalues for real polyatomic molecules. Since a triatomic molecule in its centre-of-mass coordinate system with zero total angular momentum has three degrees of freedom, these present applications involve one degree of freedom more than the model problem treated originally by CGM6 and Marcus7 and co- workers.Section 2 first summarises the CGM approach and the modifications of the original procedure that have been introduced in the present work. The form of the classical Hamiltonian for a non-rotatory, non-linear triatomic molecule is discussed in Section 3, and Sections 4 and 5 present the results of calculations for SO2 and H20. Comparison with available quantum mechanical eigenvalues is in general quite good, the same level of accuracy typically observed for one-dimensional semi-classical eigenvalues. Section 6 summarises and discusses prospects for future developments and outstanding problems. 2. THE CGM METHOD Because CGM has been recently published, we shall here only summarise their The total (classical) Hamiltonian is written in the form theory, and not give the philosophical reasoning behind the approach.2=Xo+X1 (2) where the reference Hamiltonian So is separable. Formulae are simplest if X0 is taken to be harmonic (but this is not necessary), which we now do, where t is the number of degrees of freedom, and where unit mass has been assumed (i.e., mass-weighted coordinates are used). Action-angle variables 3 ~ 1 0 n, q are ob- tained for this Hamiltonian using an Fl type generatori1 Fl(q, x), where Fl(q, x) = - ~ $ w i x r Z tan qr. i The transformation relations are (4) These relations give = JZcosq, p i = - V 2 i n i ~ sin qi.HANDY, S . M. COLWELL AND W. H. MILLER 31 N. C . On replacing n by n and the relations (6) + 3, in accordance with eqn (l), Xo takes the form t z o = 2 + w, i= 1 become xi = J- cos qi In semi-classical theory, eqn (3, for integer ni, gives the energy levels for this separable system. On the introduction of Z1, the action angle variables It, q are no longer good action angle variables i.e., 3 is no longer a function of n only.CGM, following propose using an F2 type generator F2(N7 q) for a canonical transformation from the n, q variables to the good action-angle variables N, Q for Z. For an F2 type generator, the transformation relations are CGM and Born argue, for independent reasons, that the appropriate form for F2(N, q) is t where G(N, q) is a periodic function in q of period 2n. These authors, therefore, propose the form G(N, q) = i 2' B k exp(i k.q) (1 1) k where the prime on the summation implies that k = 0 is excluded.Here we find it preferable to use the equivalent form In eqn (1 l), kl, k2, . . . take both positive and negative values, but in eqn (12), they are all positive (or zero). The notation in eqn (12) implies that all possible terms such as sin klql cos k2q2 cos k3q3 . . . sin k,q, occur in the summation with different coefficients A. The Hamilton-Jacobi equation to determine G(N, q) is Xl will usually be given as a function of p and x, but use of the relations (8) and (9) will express it as a function of Nand q. From relations (9) and (10) we have and thus the Hamilton-Jacobi equation is32 SEMI-CLASSICAL METHODS FOR VIBRATIONAL ENERGY LEVELS In CGM, the coefficients &, which are implicitly a function of N, were determined by equating to zero all the Fourier components (except k = 0) of eqn (14).In practice this means multiplying by e'ik.q, and then integrating 1' .. . [' dq. The procedure is to impose the quantisation condition by setting N to be a given set of integers corresponding to the energy level required. The Fourier components of (14) then give a set of non-hear equations for &, which can be written Bk =fk(B)* (15) The & will have different values for each set of N. Eqn (15) have to be solved by an iterative procedure, and because of the difficult form of S1, the integrations Jdq referred to above have to be performed numerically. Once the eqn (15) are solved the energy E(N) is obtained by integrating eqn (14) Jdq: Clearly the ultimate test of the success of the theory is that the energy E(N) should have " converged " using only a small number of terms in the expansion, eqn (1 1).The use of the alternative expansion eqn (12) leads to similar non-linear simul- taneous equations for the A k A k =fk(A)- (17) These are obtained by multiplying the Hamilton-Jacobi eqn (13) by one of the functions sin k,q, sin k2q2 sin k3q3 {cos k,q,){cos k,q,)(cos k3qJ * {:: ::> I' and then integrating dq, using numerical methods. The algebra required to obtain eqn (17) is a little more complex than that to obtain eqn (15) but it is quite straightforward. The energy is again obtained from eqn (1 1). CGM propose two methods for solving the non-linear simultaneous eqn (17): (i) If eqn (17) are written in the form * = f(x) (18) and x, denotes the Z'th estimate for x in an iteration procedure, the ( I + 1)th estimate is obtained by the direct substitution method XI -k 1 =f(XJ.(19) In practice it was, in certain cases, found more efficient to use some variants of eqn (19) such as Xl + 1= 3x1 + f(x1)l (i) The Newton-Raphson approach. Let % + 1 = XI + A* Determine A from the solution of the linear simultaneous equationsN . C . HANDY, S . M . COLWELL AND W. H . MILLER 33 In CGM, formula (20) is given in more detail, and in particular it is shown that to evaluate a f s ( n , q) is required. Because in our calculations Zl had a complicated form when expressed in terms of n, q, these derivatives were evaluated numerically, and because eqn (20) is only approximate, it does not matter if a%',/an is only approxi- mate.Thus we write ax, ' an Y (21) a s 1 -- - z l ( n + E ) - Zl(n) an E for a suitable E, which we found to be 0.01. that the straightforward estimate is x = 0. of the action-angle variables It, q has the symmetry In both approaches above an initial estimate for x is required, and it is clear In all the problems discussed here, the perturbation Z1 when expressed in terms Zl(n,q,, . . . qi . . . qi) = Xl(n, 2n - qr, 2n - q2, . . ., 2n - qr) (22) i.e., the perturbation is even with respect to inversion in 4 = (n, n, . . ., n). In the general three dimensional case, the 8 expansion functions 2; klql 2; k2q2 Ei"," k3q3 split into two groups of four: (i) Those even under inversion, viz. cos klq, cos k2q2 cos k3q3 cos klq, sin k2q2 sin k3q3 sin klql cos k2q2 sin k3q3 sin klq, sin k2q2 cos k3q3 and (ii) Those odd under inversion, viz.(23) cos klq, cos k2q2 sin k3q3 cos klql sin k2q2 cos k3q3 sin klql cos k2q2 cos k3q3 sin klql sin k2q2 sin k3q3. It can be shown that the odd terms in G are determined only by the even terms in the perturbation, and similarly the even terms G are determined only by the odd terms in the perturbation. As the perturbations considered here have no odd terms, G will contain no even terms, i.e., only the teriiis in group (ii) above will occur in the expansion for G. This symmetry has therefore halved the number of expansion functions for G, and is the main reason why the expansion (12) is preferred to the expansion (11). The other reason is that in all the calculations reported here, all the coefficients A k obtained from the solution of eqn (17) are real.To summarise, the approach is clear, the main question being whether the method will converge for real systems. In the next section we introduce suitable Hamiltonians for triatomic systems for applications of the method. 3. THE CLASSICAL HAMILTONIAN The ideal form of Hamiltoiiian for this theory is one for which its main part Z,, is separable, and in particular when Z0 has the form (3). Such a Hamiltonian is immediately available for vibrating systems when expressed in terms of normal co- ordinates Ql, 02, & and the corresponding momenta pl, p*, p3 (from here on we34 SEMI-CLASSICAL METHODS FOR VIBRATIONAL ENERGY LEVELS shall refer explicitly to triatomic systems, and further limit our discussion to non- linear triatomics).The form of this classical Hamiltonian is given by Wilson, Decius and Cross,12 and for systems with zero angular momentum, it has the form 3 %P 2 = +(FIZ + 4 2 + Ps2) + *z n,,p@cp -t v (24) where n is the so-called vibrational angular momentum The lbi,k are transformation coefficients from mass-weighted Cartesian displacements to the normal coordinates ok: Qk = 2 Li,kmi’(rai - rli) 2 ri = (rli, rzi, r3i) is the position of the i’th nucleus in molecule fixed coordinates, ri0 is the corresponding equilibrium position and mi is the mass of the i’th nucleus; p is closely related to the inverse of the moment of inertia matrix. Detailed informa- tion may be found in ref. (12). The form of the potential V used in these calculations was In H,O for example, Ar12 is the change in the OH1 bond length from equilibrium, Ar13 is the change in the OH, bond length and A0 is the change in the HOH angle from its equilibrium value.The normal coordinates Qk are found by expanding (27) in Cartesian displacement coordinates up to quadratic powers. If this truncated Yis denoted V,, then in normal coordinates it has the form v, = +(w,2Q12 + w2’Q2’ + w32Q32). (28) To connect with the theory of the previous section, we write and 2 0 = *(Pi2 + PZ2 + P3,) + ~ ( u : Q ~ ~ + a?&’ + w ~ ~ Q ~ ~ ) (29) 2 1 = 2 +Z&3ng + v - Yo. (30) G P Zo has the exact form of eqn (3) and, from the mechanics of the system, it is considered that Xl represents a small perturbation.The explicit forms of the potential V, eqn (27)’ can be found in ref. (13). Calcula- tions were performed on SO2 and H,O. The potentials are (i) SO,: The potential of Kutchitsu and Morino14 given in table 5 of ref. (13). The fundamental frequencies wl, co,, co3 for this potential are 1171, 525 and 1378 cm-l. (ii) H,O: The potential of Hoy, Mills and Strey” given in table 1 of ref. (13). The fundamental frequencies are 3832, 1648 and 3942 cm-l. These two triatomic systems have a further symmetry, which when expressed in terms of normal coordinates is (3 1) mQll 02’ QJ = WQl, Q2l 4 3 ) .N. C. HANDY, S. M. COLWELL A N D W. H. MILLER 35 When this symmetry is described in the action-angle variables n, q, it can be written Jfm q1, q 2 , 43) == q1, q 2 , - q 3 ) (32) where q3 is the angle variable corresponding to e3.This symmetry means that solutions of eqn (17) will give A k as zero for all terms with k3 odd, provided accurate integration methods are employed. Such terms were therefore omitted from the expansion (12). In the next section, results of this semi-classical approach will be compared with the quantum mechanical results. The latter were obtained by Whitehead and Handy l3 when solving the quantum mechanical secular equations where !Ds are appropriate vibrational expansions, written as products of Hermite polynomials of the normal coordinates. It is recalled that 3 a=l (34) ZQM = &Q - ‘ 2 8 Pa.* as first derived by Watson.16 On the right-hand side of eqn (34),% is given by eqn (24), with pi replaced by -ia/a&,.Further details on the form and the number of expansion functions required for convergence are given in ref. (13). 4. SEMI-CLASSICAL ENERGY LEVELS FOR SO2 Before giving details of the calculations and results on SOz, it is necessary to specify the selection of the integration points used to evaluate the integrals occurring in eqns (16) and (17). It appears ideal to use integration points which force the sym- metries arising from eqns (22) and (32). This means that the number of integration points must be even. Furthermore if they are equally spaced then Fourier functions are integrated exactly. Thus M points were used in each of the dimensions ql, q2, q3, with 111 even, and distributed as 3 2M- --)2n. 1 2M’ 2M” ’ 2M (35) It will be recalled that M equally spaced points will exactly integrate sin kq and cos kq fork = 0, I, 2 .. . M - 1 from 0 to 2n. The expansion functions used in G, eqn (12) were all functions occurring in (23), compatible with k3 even and 0 < ki < K, = 1, 2, 3 for some K. Initial tests were performed on the lowest (000) state of SOz. (This notation implies that the good action variables N,, N2, N3 were each set equal to zero.) The results are given in table 1 , where various numbers of integration points and expansion TABLE 1.-sEMI-CLASSICAL VIBRATIONAL ENERGIES E FOR THE (000) STATE OF s02, USING VARIOUS NUMBERS OF INTEGRATION POINTS M AND EXPANSION FUNCTIONS (23), WITH ki < K. E 1crn-l M K 1533.52 4 1 1529.33 6 2 1529.30 8 2 1529.12 8 3 1529.13 10 3 1529.12 10 436 SEMI-CLASSICAL METHODS FOR VIBRATIONAL ENERGY LEVELS functions were used.These results show that effective convergence on this ground state is obtained using the expansion functions for G given in eqn (12) with ki 6 3, and with 8 integration points in each q-dimension. All results on SO2 were obtainable using the direct substitution method for solving eqn (17), each converging in at most 20 iterations. In table 2, results for the vibrational energies for various low-lying states of SO2 TABLE 2.-sEMI-CLASSICAL VIBRATIONAL ENERGIES E, rN Cm-l, FOR VARIOUS LOW-LYING STATES WITH kr < K. OF SO2, USING VARIOUS NUMBERS OF INTEGRATION POINTS M AND EXPANSION FUNCTIONS (23), M = 6 , K = 2 M = 8 , K = 1529.33 1529.12 2045.71 2045.36 2556.56 2555.86 268 6.74 2685.1 1 2889.39 2889.07 * 3060.30 * 3 558.49 4050.08 * 3 M = 1 0 , K = 4 1529.12 2045.35 2555.77 2685.06 2889.06 * * * * Not computed.are presented, also with various numbers of integration points and expansion func- tions. No difficulties were encountered in any of the calculations, but it is doubtful whether sufficient expansion functions and points have been included for the higher bending states 030, 040 and 050. In table 3, the semi-classical energies are compared with the quantum mechanical energies taken from ref. (5), table 7 (for brevity we shall let SC and QM refer to semi- classical and quantum-mechanical results respectively). It will be noted that in the absolute energies the difference between the QM and SC results is about 0.5 cm-l. TABLE 3 .-A COMPARISON OF " SEMI-CLASSICAL " (sc) ENERGIES AND QUANTUM-MECHANICAL (QM) ENERGIES FOR LOW-LYING VIBRATIONAL STATES OF SO2 absolute energylcm-' frequency/cm-l state sc QMb 6" sc QMb (000) 1529.12 1529.60 7.7 - - ( 1 0 2685.1 1 2685.63 22.8 1 155.99 1 156.03 (010) 2045.36 2045.81 16.3 516.24 516.21 (020) 2555.86 2556.21 30.5 1026.74 1026.61 (001) 2889.07 2889.53 25.6 1359.95 1359.93 a 6 is the difference between the zero'th order energy, eqn (3), and the exact SC energy.b ref. (13). A similarity between these results and those obtained by other workers on more idealized systems6g7 can be noticed if we consider the quantity where E(QM) and E(SC) are the QM and SC energies in table 3, and Eo is the zero'th- order energy given by eqn (7). It is seen that p decreases from 7.5 for the (000) state down to 2.0 for the (020) state.This order of magnitude, and its decrease with increasing energy are similar to the results on other systems. In coiiclusion, then, for SO2, the semi-classical approach appears to give resultsN. C. HANDY, S . M. COLWELL AND W. H. MILLER 37 which are virtually indistinguishable from the quantum mechanical results and, which, when one has become accustomed to performing such calculations, are as easy to obtain. 5. SEMI-CLASSICAL ENERGY LEVELS FOR H20 Water is a very much more testing system for the applicability of semi-classical methods because of the increased effect of the perturbation arising from the lightness of the protons. Similar tables to those presented for SO2 are presented for H20. In table 4 low- TABLE d.-SEMI-CLASSICAL VIBRATIONAL ENERGIES E, IN Cm-', FOR VARIOUS LOW-LYING STATES OF H20, USING VARIOUS NUMBERS OF INTEGRATION POINTS M AND EXPANSION FUNCTIONS (23) WITH ki < K.state (000) 4 648.16" 4 645.1 1 a 4 645.46" (010) 6 248.35" 6 242.55" 6 242.67" (020) 7 815.94" 7 804.80" * (loo) 8 378.46ab 8 360.90ab 8 358.13 (001) 8 466.5gb 8 466.Bb * M = 6, K = 2 M = 8, K = 3 M = 10, K = 4 a Non-linear simultaneous eqn (17) solved by successive substitution. b Eqn (17) solved by Newton-Raphson method. * Not computed. lying semi-classical energies are given for various numbers of points and expansion functions. However, this time it was not found possible to solve eqn (17) by the direct substitution method for the (001) energy, and the Newton-Raphson method had to be employed.No difficulties were then encountered (this also applies to other states of H20 not published here). The results are much more sensitive to the number of expansion functions- indeed the results indicate that K = 4 is probably insufficient for complete converg- ence (but the same was true for the QM calculation, because a bigger secular matrix had to be diagonalised to obtain full convergence). In table 5, the results for M = 8, K = 3 are compared with the quantum mechan- ical results. Here the QM energies lie above the SC energies by 6 to 10 cm-l. The frequencies differ by up to 2 cm-'. The quantity ,u defined in eqn (36) varies from TABLE 5.-A COMPARISON OF "SEMI-CLASSICAL" (Sc) ENERGIES AND QUANTUM-MECHANICAL (QM) ENERGIES FOR LOW-LYING VIBRATIONAL STATES OF H20 absolute energy/cm- frequency/cm- state sc Q M b 6" sc Q M b (010) 6 242.55 6 249.33 117 1 597.44 1 597.35 - - ( O W 4 645.1 1 4 651.98 65 (020) 7 804.80 7 811.53 198 3 159.69 3 159.55 8 360.90 8 369.29 178 3 715.79 3 717.36 (001) 8 466.58 8 472.75 185 3 821.47 3 820.77 a 6 is the difference between the zero'th order energy, eqn (3) and the exact SC energy.b Taken essentially from ref. (13). However the results in ref. (13) contain a minor error because the 004 Term in U, in our programs, had an incorrect sign. The results have been recalculcated, with insignificant effect on any of the conclusions reached in ref. (13). They now completely agree with results published by G. D. Carney, L. A. Curtiss, and S. R. Langhoff (to be published in J. Mol. Spectr.).38 SEMI-CLASSICAL METHODS FOR VIBRATIONAL ENERGY LEVELS 10 for the (000) energy down to 3.5 for the (020) energy, therefore being much in agreement with the results for SO2. The overall results for this difficult system remain satisfactory, although computa- tionally considerable difficulty was encountered solving eqn (17), until the Newton- Raphson method was employed, use of which makes the method time consuming, 6.CONCLUDING REMARKS The purpose of this paper has been to see if the method proposed by Chapman, Garrett and Miller for obtaining semi-classical eigenvalues of multidimensional systems is applicable to determining the vibrational energy levels of triatomic mole- cules with realistic potential functions. The results indicate that this is the case, with the accuracy of the semi-classical eigenvalues (as compared to the quantum mechanical ones) being about the same as for the one-dimensional WKB quantum condition.The practicability of the method, i.e., the ease of calculation compared to straight quantum mechanics, depends primarily on whether the direct substitution iteration procedure converges or not-as for all the states of SO, investigated, but not for all those of H,O-then the calculation is as easy as for these low vibrational states as a quantum mechanical one, and would be expected to be easier for more highly excited states. If it is necessary to use the Newton-Raphson iteration proced- ure to solve eqn (17), then one must deal with matrices of order of the number of Fourier coefficients.Since the number of Fourier coefficients roughly corresponds to the number of basis functions in a quantum calculation, one does not expect the semi-classical calculation in this case to be easier than a quantum mechanical one. Since the semi-classical procedure obtains individual eigenvalues directly, it might be especially useful in situations where particular excited states, rather than them all, are of interest. Particularly difficult to deal with semi-classically is the case that the reference system has low order resonances or degeneracy, i.e., the case that for some small integers (kl, k2, k3). It is clear from the form of the CGM equations,6 for example, that the direct substitution iteration procedure cannot be used in this situation. The Newton-Raphson iteration is still possible, however, and has been successfully applied l7 in some cases to obtain semi-classical eigenvalues for the degenerate version (q = co2) of the two dimensional model problem considered in ref.(6) and (7); the semi-classical eigenvalues so obtained, though, are not in as close agreement with the quantum mechanical ones as for the non-degenerate cases. Born’s3 way of dealing with degeneracy is to transform the zero’th order problem into one for which there are no such resonances; in general, however, it appears that this is difficult to do in practice. Miller18 has recently suggested an expansion for the generating function F(q, N ) , different from Born’s, that formally alleviates the difficulty with the degenerate case, but the numerical procedure it suggests does not appear promising.A completely satisfactory way of dealing with the degenerate (or nearly degenerate) case thus does not seem to be at hand; research on these problems is continuing in this 1ab0ratory.l~ There is, however, one type of degeneracy which can be successfully overcome using semi-classical methods : namely transverse degenerate vibrations of a linear molecule. A transformation from appropriate Cartesian to polar coordinates over- comes the difficulty, and use of the normal coordinate Hamiltonian for linear mole- cules makes the problem then ideally suited to methods used in the present paper. k@l+ k2% + k3w = 0 (37)N. C. HANDY, S . M . COLWELL AND W. H. MILLER 39 Therefore, the case of degeneracy which is most often met in practice can be overcome and semi-classical methods may be u w l . S.C.M. thanks the S.R.C. for financial support. M. V. Berry, and K. E. Mount, Rept. Prog. Phys., 1972, 35, 315. W. H. Miller, Adv. Chem. Phys., 1974, 25, 69; 1975,30, 77. M. Born, The Mechanics of the Atonz (1924), (Ungar, New York, 1960). M. C. Gutzwiller, J. Math. Phys., 1971, 12, 343. W. H. Miller, J. Chem. Phys., 1975, 63, 996. S. Chapman, B. C. Garrett and W. H. Miller, J. Clwm. Phys., 1976, 64, 502. W. Eastes and R. A. Marcus, J. Chenz. Phys., 1974, 61, 4301; D. W. Noid and R. A. Marcus, J. Chem. Phys., 1975,62,2119. I. C. Percival, J. Phys., 1974, A7, 794; 1. C. Percival and N. Pomphrey, Mol. Phys., 1976, 31, 917. J. B. Keller, Ann. Phys. (N.Y.), 1958, 4, 180; J. B. Keller and S. I. Rubinow, Ann. Phys. (N.Y.), 1960, 9, 24. lo H. Goldstein, Classical Mechanics (Addison- Wesley, Reading, Mass. , 1950), pp. 288-307. l1 Ref. (lo), pp. 237-247. l2 E. B. Wilson, J. C. Decius and P. C. Cross, Molecular Vibrations (McGraw-Hill, New York, l3 R. J. Whitehead and N. C. Handy, J. Mol. Spectr., 1975, 55, 356. l4 K. Kuchitsu and Y . Morino, J. Chem. SOC. Japan, 1965, 38, 814. l5 A. R. Hoy, I. M. Mills and G. Strey, Mol. Phys., 1972, 24, 1265. l6 J. K. G. Watson, Mol. Phys., 1968, 15, 479. l7 N. C. Handy and W. H. Miller, unpublished results. l 8 W. H. Miller, J. Chem. Phys., 1976, 64, 2880 l9 K. S. Sorbie, to be published. 1955).

ISSN:0301-7249    

DOI:10.1039/DC9776200029

出版商:RSC    

年代:1977

数据来源: RSC

摘要:

Semi-Classical Theory for Non-separable Systems : Construction of “Good” Action-Angle Variables for Reaction Rate Constants* BY WILLIAM H. MILLER-^ Department of Chemistry, and Materials and Molecular Research Division of the Lawrence Berkeley Laboratory, University of California, Berkeley, Ca 94720 and University Chemical Laboratory, Lensfield Road, Cambridge, England Received 3rd May, 1976 A semiclassical expression for bimolecular rate constants for reactions which have a single activa- tion barrier is obtained in terms of the “ good ” action variables of the (classical) Hamiltonian that are associated with the saddle point region of the potential energy surface. The formulae apply to non-separable, as well as separable saddle points. 1. INTRODUCTION A semiclassical expression for reaction rate constants has recently been derived which is of interest because it has the form of a generalization of transition state theory which correctly takes into account non-separability of the Hamiltonian about the saddle point of the potential energy surface.lg2 Application3 of this theory to the collinear H + H2 reaction gives reasonably good agreement with (numerically) exact results of quantum scattering calculations4 over a wide range of energies near and below the effective threshold for reaction, much better agreement than the usual (i.e., separable) quantum mechanical version of transition state theory.This is of some practical importance, since for chemical reactions with significant activation energy it is the threshold region that primarily determines the thermal rate constant.This semiclassical reaction rate theory 193 involves certain periodic classical trajec- tories of the molecular system and is very similar in its mathematical structure to the periodic orbit theory of semi-classical eigenvalues derived by Gutmiller (see also the important modification by Miller). As has been discussed,6 however, the quantum condition produced by periodic orbit theory, although qualitatively correct, contains dynamical approximations in addition to the semiclassical approximation itself, and semiclassical quantization of non-separable systems is in general more accurately achieved by the Hamilton-Jacobi approach of the old quantum theory.’ In this approach it is necessary to construct the total non-separable Hamiltonian as a function of the complete set of “ good ” action variables (or adiabatic invariants) of the system, and recently a great deal of progress has been made in finding ways to do this non-perturbatively (Le., numerically).The preceding paper in this volume, * Supported in part by the U.S. Energy Research and Development Administration, and by the t Camille and Henry Dreyfus Teacher-Scholar ; J. S. Guggenheim Memorial Fellow on sabbatical National Science Foundation under grant GP-41509X. leave from the Department of Chemistry, University of California, Berkeley, California 97420.WILLIAM H. MILLER 41 for example, describes an application of the Hamilton-Jacobi approach to semi- classical eigenvalues and also gives more references to this topic.Just as the periodic orbit theory for eigenvalues involves implicit dynamical approximations, so does the periodic orbit of the semiclassical reaction rate theory. To overcome these dynamical limitations of periodic orbit theory one would like to construct a Hamilton-Jacobi theory for reaction rate constants analogous to that for the eigenvalue problem. Such is the object of this paper. To motivate the form of the general result section 2 first discusses the separable case. Section 3 then shows that the expression suggested there is also the correct semiclassical result for the general, nsn-separable case. The relation between this new semiclassical rate theory and the earlier one1*3 based on periodic orbits is dis- cussed in section 4. 2. SEPARABLE CASE For notational simplicity, formulae below are all written without taking explicit account of total angular momentum conservation; the modifications necessary to account for this are minor and have been given bef0re.l~~ The rate constant for a bimolecular reaction, such as A + BC --+ AB + C, can be written in the k(T) = (2nh QJ-l dE e-P"N(E); (1) Irn where Qr is the partition function (per unit volume) for reactants, is related to tem- perature in the usual way, and E is the total energy. The exact expression for N(E) is 1,3 N E ) = 2 ISnt,n(E>12, (2) n'.n where the reactive S-matrix elements must in general be determined by carrying out a quantum mechanical scattering calculation.n( nl,n2, . . ., nF-1) and n' are the quantum numbers for the F - 1 internal degrees of freedom of reactants and products, respectively, and F is the total number of degrees of freedom of the complete system in its overall centre of mass.If motion along a reaction coordinate (the F'th degree of freedom) is separable from the P - 1 internal degrees of freedom, then the S-matrix is diagonal, ISn*,n(E>l2 = Jn,,nP(E - &If), (3) where P(EJ is the one-dimensional tunnelling probability for a translational energy E, along the reaction coordinate; En is the energy of internal state n. Since the semiclassical approximation for P(EJ is P(EJ = [l + e2Q(Et)]-1, (4) where 8(Et) is the usual barrier penetration integral, eqn (2) becomes where N(E) = 2 [l + e2e(E-E,) I- ? n=O 2 ° C 2 ... 2 . n=o n,=O n,=O np-,=O42 SEMI-CLASSICAL THEORY FOR NON-SEPARABLE SYSTEMS The action variable for the F'th degree of freedom, n F , is defined in the usual one- dimensional fashion (setting R = I), = iO(E,), (7) so that the expression for N(E) finally takes the form N(E) = 1 11 + e2n Im n~(E,n)]-l, n=O To see the meaning of eqn (8) in a simple example, consider a particle first in a two-dimensional harmonic potential well.The total (classical) Hamiltonian is given in terms of the two action variables (Le., semi-classical quantum numbers) in this case by E(n,,n2) = hw(n1 + 4) + hu2@2 + $1. (9) Making the replacement m2 + ico2 corresponds to inverting the harmonic potential of this degree of freedom, so that the two dimensional well becomes a saddle point region, a harmonic potential well in one direction and a parabolic barrier in the other.The hamiltonian in eqn (9) thus becomes E(n,,n2) = kw,(n + 3) + ifio,(n, + $1, (10) and one notes the important point that there are '' good " action variables associated with a saddle point region of a potential surface just as there are those associated with a minimum. Rather than requiring both action variables to be integers, as is done for the semiclassical eigenvalue problem, one solves the following equation, E = E(n,,n2), (1 1) n2(E,nl) = -3 + iCfim1(nl + - El/(fiwJ, (12) to express n2 as a function of E and n,; for eqn (10) this is and eqn (8) in this case gives J For this harmonic-parabolic saddle point eqn (13) is actually the exact quantum mechanical result, the " bonus " one often obtains with harmonic potential functions in semiclassical theory; i.e., if eqn (13) is substituted into eqn (l), the result is where Q * is the partition function of the " activated complex " and I' is the tunnelling correction : Q* = +hm1/3/sinh (ihml/3) I? = +hco2p/sin ( ~ h u 2 p ) .(1 5 4 (1 5b) The next section shows that eqn (8), derived here for the separable case, is also the correct semiclassical expression for N(E) for the general, non-separable case. In the non-separable situation, however, the total (classical) Hamiltonian is of course not a separable function of the P " good " action variables as in eqn (lo), and theWILLTAM H. MILLER 43 partition functioii for the activated complex and the tunnelling correction thus do not enter as separate factors as they do in eqn (14), and to construct the function E(rz,, n?, .. ., nF) is tlic primary task of the theory. It can be done, for example, using the procedure devised by Chapman, Garrett, and Miller9 for solving the Hamil- ton-Jacobi equation in action-angle variables. (Interestingly, it does not appear that the approach of Eastes and Marcuslo would be possible because the classical trajectories about a saddle point are not quasi-periodic.) One action variable, nF say, is identified as the one associated with the " reactive direction ", and one then solves the equation E = E(n,,n2, . . . , nF> (16) to obtain nF(E,nl,n2, . . . , n,-,). Eqn (8) then gives N(E) and eqn (1) k(T). 3. NON-SEPARABLE CASE Another formally exact expression for the rate constant, completely equivalent to eqn (1) and (2), is', k(T) = Qr-' tr[e-D"s(f)fP], (17) wherefis an operator that defines a surface which divides reactants and products, f is the Heisenberg time derivative off, and 9 is a projection operator that projects onto all states which have evolved in the infinite past from reactants.ll It is easy to see that eqn (17) can be written as eqn (1) with N(E) given by N(E) = 2nh tr[G(E - H>S(f)f?Y], (19) an expression formally equivalent to eqn (2) but more useful for present purposes.The plan is to evaluate eqn (19) semiclassically. For notational simplicity consider the case of two degrees of freedom, i.e., a collinear A + BC + AB +- C reaction. If n, and n2 are the two " good " action variables, then the quantum mechanical states In,n,) are eigenstates of the total hamiltonian, Hln,n,) = E(n1,nJ l W 2 ) .(20) Evaluating the trace in eqn (19) in this representation gives where J , q, and q2 are the angle variables conjugate to n1 and n2, and E(n1,n2) is the total (classical) hamiltonian, in general non-separable, expressed in terms of the " good " action variables. [Superposition of amplitudes from the different 271 intervals of q1 leads to n, being integral, and this is why the domain of q1 is limited to (--n,n) rather44 SEMI-CLASSICAL THEORY FOR NON-SEPARABLE SYSTEMS than (- 00, wavefunctions are l2 Since (nl,n2) and (ql,q2) are conjugate variables, the semiclassical (23) tynln1(q1,q2) = d(27c1-2 e i(nlql + n2q2), so that eqn (21) becomes where the fact has been used that nl is real (an integer) and that n2 may not be, and wherefhas been taken to involve only the (n2,q2) degree of freedom.(The final result cannot depend on the specific choice for$)'' The first factor in eqn (24) is clearly unity, and the second factor is essentially a one-dimensional flux for the (n2,q,) degree of freedom that depends parametrically on nl. The flux through a one- dimensional barrier has the standard form of the frequency v of the (n2,q2) motion, multiplied by the tunnelling, or transmission probability P, p = (1 + e2n Im (1 + CZrr Irn 2)-1 . - - e-2n Im n2 Eqn (24) thus becomes Before proceeding to the final result, it is interesting to note the simple illuminating interpretation eqn (27) has if one expands the tunnelling probability in a geometric series : where Eqn (27) then reads (284 (28@ (29) P = e-20 - e-4Q + e-60 .. . 0 = ~t Im n2. ( n l n 2 [ 6 ~ f P l n l n 2 ) = ve-,O - V ~ - ~ O + ve-@ . . ., and each term can be thought of as the flux contribution of one of the classical trajectories that comes from reactants and crosses the dividing surface f = 0. The trajectories (classical trajectories in complex time 12) are pictured in fig. 1 ; the reactive ones contribute positive flux and the non-reactive ones negative flux. The Appendix shows how eqn (29) results directly from eqn (24) if one makes a particular choice for f. The final expression is now obtained by substituting eqn (27) into (21), where n2(E,n,) is determined by E = E(n1,n2). This is eqn (8) for the case F = 2.WILLIAM H. MILLER 45 FIG. 1.- Sketch of the potential energy along a reaction coordinate through a saddle point region and the different classical trajectories [in complex time-see ref.(12)] that come from reactants and pass through the saddle point region at least once. P is the probability associated with the trajectory in each case. 4. RELATION TO PERIODIC ORBIT THEORY It is easy to see how the periodic orbit expression for the rate constant is recovered if one uses eqn (8) with the periodic orbit quantum condition6 for determining E(nl,n2, . . . , nF). Periodic orbit theory determines E(nl,n2, . . . , nF) implicitly by the equation6 where @(E) is the action integral for one pass around the periodic orbit and (coi(E)) are the F - 1 stability frequencies. For the case of a saddle point 0 is pure imagin- ary 9 @(E)/ti 2ie(E); (33) so that eqn (32) gives F- 1 27C(nF + *) = i[2e(E) - 28'(E) 2 hui(E)(ni +)I, i= 1 or F- 1 which, with eqn (8), is the exprsssion obtained' via periodic orbit theory for N(E). 5.CONCLUDING REMARKS This paper has shown how a semiclassical approximation for the rate constant for a bimolecular reaction with a single activation barrier can be constructed in terms of the complete set of " good " action variables associated with the saddle point of the potential energy surface. This new expression for the rate constant bears the same relation to the earlier semiclassical rate expression in terms of periodic orbits that the old quantum theory for semiclassical eigenvalues bears to the periodic orbit quantization rule. Since the expressions based on the " good " action variables46 SEMI-CLASSICAL THEORY FOR NON-SEPARABLE SYSTEMS of Hamilton-Jacobi theory should in general be more accurate than their periodic orbit approximations, this new expression for the rate constant is expected to be more accurate than the one based on periodic orbit theory. APPENDIX If in eqn (24) f is chosen as f= - sin q2, then the effect of 9 is simply to require q, > 0. The operator f is 1 8 . where (n& =- i aq2' This then gives (setting h= 1) which is eqn (27). W. H. Miller, J. Chem Phys., 1975, 62, 1899. Mechanical Transition State Theory, 1976. S . Chapman, B. C. Garrett and W. H. Miller, J. Chem. Phys., 1975, 63, 2710. D. G. Truhlar and A. Kuppermann, J. Chem. Phys., 1972, 35, 2232. M, C. Gutzwiller, J. Math. Phys., 1971, 12, 343. W. H. Miller, J. Chem. Phys., 1975, 63, 996. N. C. Handy, S. M. Colwell, and W. H. Miller, Faraday Disc. Chem. SOC., 1976, 62, pre- S. Chapman, B. C. Garrett, and W. H. Miller, J. Chem. Phys., 1976, 64, 502. 'See also, W. H. Miller, Accounts. Chenz. Res., Importance of Non-Separability in Quantum ' M. Born, The Mechanics of the Atom (Ungar, New York, 1960). ceding paper. l o W. Eastes and R. A. Marcus, J. Chem. Phys., 1974, 61,4301. l1 W. H. Miller, J. Chem Phys., 1974, 61, 1823. l2 W. H. Miller, Adv. Chem. Phys., 1974, 25, 69; 1975, 30, 77.

ISSN:0301-7249    

DOI:10.1039/DC9776200040

出版商:RSC    

年代:1977

数据来源: RSC

摘要:

GENERAL DISCUSSION Dr. D. S. Marynick and Dr. D. A. Dixon (Harvard University) said: We have per- formed calculations on inversion barriers for the AH3 molecules PH3, SH3 +, ASH, and SeH3+ with a minimum basis set (MBS) of Slater orbitals. The valence shell expo- nents and geometry were fully optimized and the results are given in table 1. TABLE 1 A PH3 calculations property MBS PDZ-SCF PDZ-CI Alkcal mo1-I 39.8 36.77 36.55 1.444 1.403 1.41 3 93.9" 95.3 1 O 93.72" angle R'IA" 1.405 1.370 1.377 RIA B AH3 MBS calculations property PH3 AsH3 SH3 -I- SeH3+ 1.444 1.528 1.374 1.465 Alkcal rno1-I 39.8 43.8 22.5 35.4 RIA angle R'/& 1.405 1 A67 1.343 1.444 93.9 93.6 100.13 95.06 (I Bond length for planar AH3 species with D3* symmetries. All bond angles are 120". In order to calibrate these results, we have also performed calculations on the in- version barrier for PH3 using a polarized double zeta (PDZ) basis set plus configura- tion interaction (CI).Our initial basis was the double zeta (DZ) basis of Roetti and C1ementi.l We then optimized the exponents for the hydrogen 1s and 2s orbitals at the DZ level. A 2p function (3 components) on H was added to this basis as was a 3d orbital (5 components) on P and the latter exponent was optimized for this basis. Geometry optimization at the PDZ-CI level was done for the planar and pyramidal geometries. Our CI included all single and double excitations from the valence shell to all virtual orbitals. The calculated geometry of PH3 is in good agreement with expximent for which R is 1.42 A and the bond angle is ~93.2O.~ The barrier at the SCF level is 36.8 kcal mol-l, which is very close to the value of 36.7 kcal mol-1 obtained by Lehn and M ~ n s c h , ~ who usedalargegaussian basis without CI.The barrier obtained by Ahlrichs et aL4 is slightly higher (38 kcal mol-I). We note that our SCF energy of -342.4869 au is 0.03 au lower than the lowest previously reported energy. The CI correction is ---0.157 au for PH3 in both forms and we find essentially no correction to the barrier height (a lowering of 0.2 kcal mol-I). Ahlrichs et aL4 find a significant 3 kcal mole' decrease in the barrier when using their CEPA C. Roetti and E. Clementi, J. Chem. Phys., 1974, 60, 4725. M. H. Sirvetz and R. E. Weston, Jr., J. Chem. Phys., 1953, 21, 898. J. M. Lehn and B.Mimsch, Mol. Pliys., 1972, 23, 91. K. Ahlrichs, F. Keil, H. Lischka, W. Kutzelnigg and V. Staeinmler, J. Chnii. Plzys., 1975,63,455.48 GENERAL DISCUSSION correction, which we feel is too large. We note here that our final SCF-CI energy is -342.6437 au for PH,. We find a significantly lower barrier than Pettke and Whit- ten1 found (40.1 kcal mol-l) using a smaller basis and a partial CI. Comparison of the barrier for PH3 obtained from the best calculation with the MBS barrier shows that the MBS barrier is -10% too large. However, we feel that our relative barrier heights for the AH3 molecules we have studied are qualitatively correct and we note that they are significantly larger than observed for first row atoms such as NH,, where the barrier is -6 kcal mol-1.2 We are now doing large basis cal- culations with some CI on the remaining three AH3 molecules to accurately determine their inversion barriers.This is especially useful for the AH, molecules with large barriers as these barriers can not now be determined by experiment. Prof. W. Kutzelnigg (Boclzum) said: I would like to stress the complementarity of theory and experiment. There are properties which are directly (which does not necessarily mean : easily) accessible from quantum chemical calculations while their extraction from experimental data poses problems of principle. An example is the equilibrium (re) geometry of a molecule, i.e., the geometry for which the electronic energy has its minimum. (The equilibrium geometry is a well-defined quantity only if the Born-Oppenheimer or at least the adiabatic approximation is applicable.) The theoretical calculation of the r,-geometry requires an additional step, namely the averaging over zero-point vibrations.Froin experiment, rz (or some other average geometry) is directly accessible and the extrapolation to re is a rather complicated pro- cedure. Provided that theory and experiment are of comparable accuracy (which at present is possible for some small molecules) one has to conclude that theory is the method of choice for the determination of re and experiment the one for rz. Refined quantum chemical ab-initio methods including correlation effects, like the coupled electron pair approximation (CEPA), yield re values for XH bonds that are accurate3 to within 0.003 A whereas the differences between re and rz are of the order of 0.02 A.Something similar holds for harmonic force constants. There is in principle no difficulty whatsoever to get them from quantum chemical calculations as the second derivatives of the energy at the equilibrium geometry. (In an analogous way higher order force constants can be calculated.) The harmonic vibration frequencies are easily derived from the harmonic force field. A theoretical calculation of the observ- able, " anharmonic " vibration frequencies, or rather of the vibronic energy levels, requires an additional computational step either using the higher order force constants or via a direct quantum mechanical treatment of the vibrational problem with the full calculated surface. On the other hand, experiment furnishes the anharmonic vibra- tion frequencies directly whereas their '' harmonization " as well as the extraction of the force field are nontrivial problems, especially when the number of equations is smaller than the number of unknowns.We conclude that quantum chemistry is the method of choice for the determination of harmonic force constants. In fact CEPA type calculations yield CH force with an accuracy of -l%, but so far overestimate force constants of double bonds by -10%. Significant progress is expected within the next few years. It should also be mentioned that, in view of the accuracy possible at present with both theory and experiment, one has to realize that, e.g., experimental rotational barriers (if they are really static barriers and not just activation energies) refer to an J.D. Petke and J. L. Whitten, J. C/zenz. Phys., 1973, 59, 4855. R. M. Stevens, J. Chern. Phys., 1974, 61,2086. W. Meyer and P. Rosmus, J . Cliem. Pliys., 1975, 63, 2356.GENERAL DISCUSSION 49 average over fast vibrations, but calculated ones to a lowest-energy path. The same is true for many other properties. Prof. D. H. Whiffen (Newcastle upon Tyne) said: Kuchitsu has referred to r, quan- tities which are essentially those discussed by Lucas as vibrational expectation values. These structures require only harmonic force fields, in addition to the primary ob- servations, and are not difficult to evaluate with observations from most of the regular techniques, that is rotational spectroscopy, diffraction methods or n.m.r.in liquid crystals. Such structures are slightly dependent on molecular isotopic species, but nevertheless form a useful basis for intercomparison of different techniques which should give the same answer to high accuracy as discussed by Lucas. Prof. I<. Kuchitsu (Tokyo) said: I agree essentially with Whiffen: (a) The r, nuclear positions constitute a well-defined structure at least for a semi-rigid molec~le.~*~ (I discussed previously4 the problem remarked upon by LucasS that " vibrationally averaged distances and angles '' do not obey the rules of geometry.) (b) In principle, the same structure can be derived from different lines of experiment. (c) Since each experimental method measures a different function of parameters related to molecular geometry, the effective vibrational averages in general differ from one another, and a correction has to be made to derive the r, structure.(d) The corrections are essentially " harmonic ", however, and only an approximate estimate of anharmonicity is suffi- cient to make the correction (except in a nonrigid case). On the other hand, as I pointed out in the text, the systematic uncertainty in the r, structure derived from any experimental method can be much larger than the random experimental error. One has to be very cautious, particularly when the r, structure is derived from a combination of various isotopic rotational constants. If neglected, the very small isotopic dependence of the r, nuclear positions can be magnified several orders and cause a large systematic uncertainty in the r, structure.This systematic uncertainty may be overlooked unless one of the following steps is taken: (a) deter- mination or estimation of the re or rm structure by use of rotational constants of vibrationally excited states and various isotopic species, (b) use of other experimental techniques such as gas electron diffraction, and (c) use of a reasonable model of the anharmonic potential function to estimate isotopic displacements of the r, nuclear positions. Dr. N. C. Handy (with DP. K. Sorbie) (Cambridge) said: I wish briefly to introduce an alternative approach to the method presented in our paper. It is based on the work of Marcus and co-workers, but has some simplifications which make it much more practical. In these problems, for most energies and initial conditions, the motion in phase- space is quasi periodic. The exception occurs near dissociation when the trajectories in phase-space become ergodic.In this brief note, we shall not discuss the latter, although such trajectories do occur with our particular potential at high energies. A quasi-periodic trajectory will close to any accuracy after a sufficient time. We denote such a trajectory I'. From the equation following eqn (13), we have N. J. D. Lucizs, Mol. Phys., 1972,23,825. T. Oka, J. Pliys. SOC. Japan, 1960,15,2274. e.g., K. Kuchitsu, J. Chem. Phys., 1968, 49, 4456. K. Kuchitsu, Bull. Chent. SOC. Japan, 1971, 44, 96. N. J. D. Lucas, Mol. Phys., 1971,22,147; 1972,23, 825.50 GENERAL DISCUSSION From eqn (1 l), G is a periodic function, and hence the last term in eqn (36) is zero.If, during I‘, qt increases by 2nMi, where Mi is an integer, we have dq, = Ni2nM,. (37) The alternative method is therefore to adjust the energy and initial conditions until eqn (37) are obeyed, with Ni = integer + 3. Hamilton’s equations are integrated, step by step, in the variables n, q, accumulating at each step the 1.h.s. of eqn (37), until the trajectory closes to a specified accuracy. We have applied the method to the two dimensional system = *(Pi2 + PZ2) + $(dx12 3- m22x22) - &Xi2& (39) with mi2 = 0.9, 0122 = 1.6 and E = 0.08. The semiclassical eigenvalues Esc determined by this method for certain values (N1,N2) are given in table 6. Also in the table are the quantum eigenvalues E,, TABLE 6.-A COMPARISON OF SEMICLASSICAL ENERGIES, Esc (DETERMINED BY THE “ TRAJEC- TORY ” METHOD), AND QUANTUM MECHANICAL ENERGIES, EQM, FOR THE MODEL HAMILTONIAN EQN (39) eigenvalue no.1 2 53 54 55 56 59 100 115 133 Esc 1.1051 2.3673 15.1710 15.2149 15.28 18 15.4952 15.8819 20.6208 22.2606 23.8654 EQM 1.1058 2.3679 15.1692 15.21 75 15.2834 15.4959 15.8854 20.6382 22.2738 23.9963 Eo 1.1068 2.3717 15.653 15.653 15.653 15.653 16.825 21.345 23.242 25.140 obtained by diagonalising a 290 x 290 secular matrix, using harmonic oscillator functions as a basis. The escape energy for this system is 25.3125; it has approx- imately 150 bound states. Because the higher quantum values have not fully converged, it is probable that the corresponding semiclassical eigenvalues E,, are more accurate.Also in the table are given the zero’th order eigenvalues E,. We are now applying this method to the potential surfaces of triatomic molecules. Prof. J. N. Murrell (Sussex) said: Although the semiclassical approach to poly- atomic vibrational energy levels has introduced some interesting physical concepts it has not yet led to a direct method of determining polyatomic potentials analogous to the RKR method for diatomics. Until such a procedure is found one must solve the “ inverse problem ” of determining a potential from spectroscopic frequencies by the indirect method of adopting a functional form of the potential with parameters ai, calculating eigenvalues for different parameter sets, and selecting values of ai which give the best agreement between calculated and observed frequencies. This is the same procedure that must be used for a quantum mechanical approach to the prob- lem.In view of the possible limitations in the type of potential amenable to semi- classical calculations it is likely that the quantum mechanical solution of the “ inverse problem ”, using perturbation or variation methods, will be the computationally simpler approach unless a direct method becomes available.GENERAL DISCUSSION 51 Dr. M. Shapiro ( Weizmann Institute of Science) and Dr. G. G. Baht-Kurti (Bristol University) said: In view of the interest expressed at this conference, and, in particular, in the papers of Mills and of Kandy, Colwell and Miller, in the calculation of vibrational-rotational energy levels for triatomic and polyatomic molecules, we would like to present the basic ideas of a new quantum mechanical method for performing such calculations.The method was first proposed in an article, by one of us (M.S.), published in 1972.l We are presently implementing the method in a form suitable for the calculation of vibrational-rotational energy levels of triatomic molecules. We expect that the method will prove competitive in both accuracy and computational speed with other available methods. As a simple example consider the calculation of the vibration levels of a one-dimensional oscillator whose potential is of the qualitative form illustrated in fig. 1. The Schro- We will attempt here only to illustrate the basic ideas behind the method. R FIG. 1 dinger equation, whose solution yields the bound state energies and wavefunctions for negative energies is : At positive energies, above the dissociation limit the wavefunctions are continuum wavefunctions and the energies (cb) span a continuous spectrum.In order to find the bound state energies in which we are interested we construct a fictitious scattering problem, involving two additional fictitious " electronic " states or channels of the system. The qualitative form of the potential curves of these addi- tional states is illustrated in fig. 2. I n the fictitious problem the three " electronic " states are coupled to each other through a non-physical potential energy matrix of the form : M. Shapilo, J. Clieni. Pitys., 1972, 56, 2582; see espccially section 111.52 GENERAL DISCUSSION I >r m W c W L \ .\ \\ / scattering \ '\\I enerqy R FIG. 2 We will be interested in calculating the probability of a transition or the T matrix element, between states 3 and 2. We can see from the form of the potential energy matrix that these states are not directly coupled to each other. The only mechanism which will permit a transition between them involves excitation from state 3 to 1, fol- lowed by de-excitation from 1 to 2. This is illustrated below 1 The coupled Schrodinger equations corresponding to our fictitious scattering problem are : J. The expression for the T matrix element for a transition from state 3 to 2 can be shown to be1: M. Shapiro, J. Chem. Phy~., 1972, 56,2582; see especially section 111.GENERAL DISCUSSION 53 where x2- and x3+ are the solutions to the homogeneous equations obtained from eqn (3) by setting V13 = VZ1 = 0.The first term in the above corresponds to a sum- mation over the bound states of V,, and the second term to an integration over its continuum states. The important thing to note about the expression for T2t3 is that the first term " blows up " whenever E = eb, i.e., whenever the energy becomes equal to a bound state energy. At the bound state energies T263 has aJirst order pole. Because we know the analytic behaviour of T2+3 in the vicinity of the pole the exact position of the bound state is very easy to 1ocate.l The method is readily generalised to the case of a triatomic molecule. In the case of water, for example, we treat the system as an oxygen atom scattering off H2.The T matrix element we need is easily calculated using any of the several available scatter- ing programs. We are presently applying the method to the calculation of the vibrational-rotational states of water. Some preliminary calculations have been com- pleted and we hope to publish a detailed description of the theory together with some results in the near future. Prof. R. N. Dixon (Bristol University) said: In their paper Handy, Colwell and Miller have exploited the power of semi-classical methods for obtaining the density of states for a system with an assumed potential function, as have a number of other recent authors. In many instances in spectroscopy the essential problem is the reverse, that is, to derive a potential function from a known spectrum of transitions.It is well known2 that in this inverse problem a knowledge of the energy levels is not sufficient, and that additional data such as vibration-rotation interaction constants are necessary for the derivation of a unique potential function. In electronic transitions band intensities provide such additional data through the use of the Franck-Condon principle. The intensity distribution has long been used with a very simple semi-classical deconvolution for deriving repulsive potential func- tions for diatomic molecule^.^ I should like to encourage Handy, Miller and their co- workers to extend their theories to include the semi-classical formulation of expecta- tion values for such properties as rotational coupling constants and transition probabilities. Such a development would prove invaluable in the interpretation of the spectra of molecules such as NO, or SO, for which the Born-Oppenheimer separation is a poor first approximation.Thus in NOz most recent work has attempted to give a detailed classification of individual energy levels, but has not been able to give any detailed interpretation of the derived energy level manif~ld.~ In this case the visible absorption spectrum involves at least three strongly coupled electronic states. The dimensionality of these problems is such that I am sure that only semi-classical methods are likely to make progress towards the goal of deriving satisfactory potential functions for the various interceding states. Prof. R. J. Le Roy (Waterloo) said: The paper of Handy et aZ.,5 and much of the discussion thereof, was concerned with the problem of determining a reliable multi- dimensional analogue of the first-order WKB quantum condition which has been so M.Shapiro, J. Chern. Phys., 1972, 56,2582; see especially section 111. I. M. Mills, in Molecular Spectroscopy: Modern Research, Vol. I, ed. K. N. Rao (Academic Press, N.Y. 1972) and in Theoretical Chemistry, Vol. 1, ed. R. N. Dixon (Specialist Periodical Reports of the Chemical Society, 1974). G. Herzberg, Spectra of Diatomic Molecules (D. Van Nostrand, N.Y., 1950), p. 392. C. G. Stevens, M. W. Swagel, R. Wallace and R. N. Zare, Chem. Phys. Letters, 1973, 18,465; T. Tanaka, R. W. Field and D. 0. Harris, J. Mol. Spectr., 1975,56, 188. N. C. Handy, S. M. Colwell and W. H. Miller, this Discussion.54 GENERAL DISCUSSION successfully applied to one-dimensional problems. It, therefore, seems appropriate to bring attention to two problems associated with the use of semiclassical quantization conditions in one dimension, as they may also affect multi-dimensional problems.The first is simply the fact that the accuracy of results obtained using the first-order WKB approximation may sometimes be inadequate, particularly for a species with a small reduced mass. For example, we found that the exact (numerical) quantum mechanical eigenvalues of a first-order RKR potential curve (the result of an inversion procedure based on the first-order WKB quantum condition) for the ground state of HF differed by up to several cm-l from the experimental data on which the RKR curve was based.Moreover, Kirschner and Watson showed that second-order WKB procedures may be required for calculating spectroscopically accurate potential curves and centrifugal distortion constants for large reduced mass species like CO. While my preceding remark shows that a “ better than first-order WKB ” treat- ment is often required in one-dimensional eigenvalue problems, my second point is the warning that the second- and fourth-order WKB quantum conditions almost always have non-physical singularities at potential asymptotes. The general WKB quantum condition may be written as v + + = %\R2 LE - Y(r)]* dR + A,(E) + A,(E) + . . . 7di R1 where the i-th order contribution A,(E) is an integral of the potential V(R) and its derivatives on the interval between the two classical turning points at energy E, R1 and R2.To any given order i, the allowed WKB eigenvalues are the values of E for which the sum of terms on the right hand side of eqn (l), truncated after A,(E), is precisely half integer. However, we have recently found2 that for a potential which varies as an inverse power of distance as it approaches an asymptote at energy D, V(R) N D - C,,/Rn, if the power n > 2, then As(E) cc -(D - E)-‘n-2’’2n A,(E) cc (D - E)-3(n--2)’a. Thus, at an asymptote of this type, the third-order quantum condition predicts that (v + 5) -+ - while the fifth-order criterion predicts that (v + 5) --f + a. Since the former is absurd and the latter is incorrect, it is clear that these higher-order quantum conditions cannot be used in this region. In the one-dimensional case, this instability of the higher-order quantization condition only manifests itself at energies extremely close to the potential asymptote, but its importance in multi-dimensional problems is not known.Prof. D. H. Whiffen (Newcastle upon T’ne) said: The work of Handy and others has a long term goal which is to use spectroscopic observations to delineate the poten- tial surface employing some technique such as a least squares adjustment to obtain optimal values of the parameters which describe the surface. This step will require a weighting scheme for the observations used as input. Beaton and Tukey3 have raised the question as to whether the traditional least squares scheme with weights chosen, S. M. Kirschner and J.K. G. Watson, J. Mol. Spectr., 1974, 51, 321. A. E. Beaton and J. W. Tukey, Critical Evaluation of ChemicaZ and Physicnl Structural Informa- tion, ed. D. R. Lide and M. A. Paul. Nat. Acad. Sci. 1974, p. 15. * S. M. Kirschner and R. J. Le Roy, 1976, to be published.GENERAL DISCUSSION 55 somewhat subjectively, after inspection of experimental errors, is always appropriate. The appropriate choice of weights is especially difficult when the observations are of different natures or even with different dimensions and units as wave-numbers for vibrational levels, expressed in cm-l, and frequencies for rotational &values, expressed in MHz, from microwave observations. Indeed for the latter the errors of computa- tion from a given potential surface still exceed the experimental uncertainties which make observational errors unsuitable for suggesting a weight matrix.Recently I have experimented with a scheme, which differs from that of Beaton and Tukey, and has so far proved valuable, although it has not yet been tested for problems outside the area of fitting potential surfaces. Traditionally the diagonal weight matrix element, Wii, for the ith observation is put equal to l/ai2 where a, is the typical error expected for the ith observation. The new scheme replaces this by Wii = 1/[(AcJ2 + (Aei)2]. Here is the difference between the ith observation and its calculated value at the current values of the parameters to be determined. Aei is the expected error in the calculation of the ith observation determined in the usual way from the Jacobian matrix and the variance-covariance matrix for the parameters.This latter involves a preliminary weight matrix based on chosen ci, which should be appropriate in absolute value as well as relative values. (Aei)2 is thus the diagonal element of the variance- covariance matrix for the observations. In principle the influence of the initially chosen ci could be removed by repeated iteration at this stage using the revised Wit to deduce improved (AeJ2 but it is doubtful if this is profitable although such a process would not be very expensive in time; it would have the merit of making the scheme truly objective but has not yet been undertaken. Apart from the choice of W, standard non-linear least squares procedures are used. In miscellaneous use, especially in the fitting of potential surfaces, the results from such a scheme are found to agree with those of a sensible weighting scheme within their mutual standard deviations.One merit, its objectivity has been referred to above, but in practice an equal merit at least comes from a treatment of rogue results. These are incorrect features arising from any cause including copying errors, mis- assignment or truncation errors of any nature which affect special individual calculated values in a specially sensitive way. The existence of (Aei)2 in the denominator of W,, downweights such rogues whereas the standard least squares tries very hard indeed to fit these quantities at the expense of worsening the fit elsewhere. Trials with de- liberate errors in input observations with the new scheme always gave values of parameters close to those most acceptable and suggested that the poor fit to the deliberately erroneous information was exceptional.The same input with standard least squares gives a poor overall fit which does not indicate why this should be. It is interesting that in current work on the potential surface of carbonyl sulphide, the lower vibrational levels are more accurately calculated than the higher levels, even though this bias was not introduced via the a,; such a bias was deliberately introduced in a subjective manner in previous work,l since the truncation errors of the calculation made this pattern appropriate. Prof. I. C. Percival (London) said: It is nice to see Handy, Colwell and Miller applying semiclassical methods to real molecules, going beyond the earlier work on models and bringing the theory into contact with experiment.I will discuss the application of semiclassical methods to the forward problem of determining energy levels from a potential surface and also the inverse problem of determining a potential surface from an energy spectrum. * A. Foord, J. G. Smith and D. H. Whiffen, Mol. Phys., 1975,29,1685.56 GENERAL DISCUSSION In either case it is important to distinguish the principles and the methods. For one degree of freedom, the Bohr-Sommerfeld quantization rule is the semiclassical principle upon which the RKR method is based. For a semiclassical theory of systems with many degrees of freedom, where the hamiltonian may not be separable, the first requirement is to generalize the principle, that is Bohr-Sommerfeld quantization.Two distinct generalizations are that of Einstein, Brillouin and Keller (EBK) and that of Gutzwiller, the latter being based on closed classical trajectories. The first is right and the second is wrong, as was implied by Marcus1 at a Discussion of the Faraday Society, Three methods of obtaining energy levels from potential energy surfaces are based on EBK quantization. The method of Marcus and his collaborators uses stepwise integration of trajectories, which are not necessarily closed. The method of Miller and his collaborators is based on the solution of the Hamilton-Jacobi equation for the action function. The methods I have used with Pomphrey at Queen Mary College are based on a variational principle.Since all the methods are based on EBK quantiza- tion the results obtained from them should in principle be the same apart from the inevitable numerical errors. By contrast the Gutzwiller method generally gives different results. Semiclassical quantization of bound states is discussed in a forthcoming review.2 So much for the forward problem. Now for the inverse problem, the question of the extension of RKR to polyatomic molecules. The inverse problem is tough, so it is convenient to take it in two stages. 1. A method for the solution of model non-separable problems. 2. The generalization of the method to the vibration-rotation spectrum of polyatomic molecules. Consider the first. In the case of a diatomic molecule, the vibrational spectrum alone does not provide a unique potential, and the same applies to polyatomic molecules, and to simple models.This problem of uniqueness is overcome by putting an additional constraint on the potential; that it should be a polynomial in th, p s q uares of the co-ordinates. The example chosen was a particle of unit mass moving in the potential V(x, y ) = a1dc2 + a20x4 + aoly2 + a11x2y2 + a12x2y4 + cxoZy4 + a21x4~2 + a22x4y4, with a10 = 0.5, a01 = 0.25, a20 = 0.0015, all = -0.003, aO2 = 0.001, and the remaining coefficients zero. 21 energy levels were calculated (semiclassically) and then the semiclassical inversion method was used on these levels to derive all eight " unknown " coefficients of the above potential form. All values were obtained correctly with an error of less than 2 x The time of computation was less than 16 s on a CDC 6600. The method used resembles a semiclassical version of the quanta1 method of Foord et al.and Wl~iffen.~ It differs in three respects 1. It is semiclassical. 2. It uses a classical analogue of the Hellmann-Feynmann theorem to accelerate the least-squares procedure. R. A. Marcus, Faraday Disc. Chem. SOC., 1973,55,71. I. C. Percival, Adv. Chent. Phys., 1977. A. Foord, J. G. Smith and D. H. Whiffen, Mol. Phys., 1975,29,1685; D. H. Whiffen, Mol. Phys., 1976,31,989.GENERAL DISCUSSION 57 3. It is in a much more primitive form. Work is in progress on stage 2, an extension of the method to real triatomic molecules. Dr. J. N. L. Connor and Dr. W. Jakubetz (University of Manchester) said: We would like to mention an extension of the simple WKB quantization formula that has recently been used to calculate complex angular momentum eigen~alues.l-~ These eigenvalues are required in Regge Pole theories of elastic scattering phenomena.*-’ Consider the radial Schrodinger equation with a potential of the Lennard-Jones type.Imposing an “ outgoing wave only ” boundary condition leads to the following semiclassical quantization formula for the Regge Poles I,. where n = 0, 1,2, . . . and a(2,) and e(lJ are the complex turning points. Results using this formula are compared with exact quantum ones in the table below.3 A Lennard-Jones (12,6) potential was used with collision parameters of exact quantum semi-classical I2 Re I,, Im l,, Re I,, Im I,, 0 1 80.01 2 21.219 1 80.01 5 21.21s 1 179.239 24.035 179.242 24.034 10 175.074 50.561 175.076 50.560 20 176.187 79.644 176.189 79.643 75 210.626 194.336 2 10.628 194.335 E = 4.0 x 10-21J, Y, = 4.0 x 10-lOm, rn = 4.377 x 10-23g and E = 2.0 x 10-20J.These parameters correspond approximately to the elastic scattering of K by HBr. Agreement between the semiclassical and exact results is seen to be very good. Dr. M. Tabor (University of Bristol) said: Miller has stated that “ there are ‘ good ’ action variables associated with a saddle point region of a potential surface just as there are those associated with a minimum ”. This result has been obtained by making one (in the case of a two dimensional system) of the classical frequencies of motion complex. This is another example of analytic continuation of a classical equation, in this case the Hamilton-Jacobi equation, still yielding quantum mechanic- ally useful results.Solutions of the Hamilton-Jacobi equation in real space imply the existence of so called “ invariant tori ” [see e.g., Appendix 26 of Arnol’d & Avez (1968)].8 These are n-dimensional regions of phase space imbedded in the 2n - 1 dimensional energy shell and contain an infinite family of classical orbits, i.e., the orbits are not isolated. It is not yet known if this holds for complex frequencies. If it does, then a delicate point arises concerning the connection between the current theory and the earlier one N. Dombey and R. H. Jones, J. Math. Phys., 1968, 9, 986. J. B. Delos and C. E. Carlson, Phys.Rev. A., 1975,11, 210. J. N. L. Connor, W. Jakubetz and C . V. Sukumar, J. Phys. B. Atom. Mol. Phys., 1976,9, 1783. J. N. L. Connor, Chenz. Soc. Rev., 1976, 5, 125. C. V. Sukumar and J. N. Bardsley, J. Phys. B. Atom. Mol. Phys., 1975, 8, 568. C. V. Sukumar, S. L. Lin and J. N. Bardsley, J. Phys. B. Atom Mol. Phys., 1975, 8, 577. J. N. L. Connor and W. Jakubetz, Chern. Phys. Letters, 1975, 36, 29. V. I. Arnol’d and A. Avez, Ergodic Problems of Classicnl Mechanics (W. A. Benjamin, N.Y., 1968).58 GENERAL DISCUSSION based on Gutmiller periodic orbit theory. The latter is now known to be incorrect (see remarks at this session by Percival) and we have shown that this is because in the case of a torus of orbits, i.e., when action variables exist, the method of stationary phase as employed by Gutzwiller (and Miller) cannot be applied since the classical orbits are not It thus remains to be seen whether periodic orbit theory and the action variable theory of reaction rates are in fact mutually exclusive or that the former, as claimed, merely represents an approximate version of the latter. The semiclassical eigenvalue method used by Handy, Colwell and Miller is also based on the existence of invariant tori. For systems of more than two degrees of freedom (the minimum dimensionality for nonseparability) there will not only be the obvious problem of increased computational difficulty but also problems related to the more complicated structure of phase space. For two degrees of freedom the two dimensional tori divide the three dimensional energy shell. For higher dimensionality we no longer have this simple topological result and gaps between tori can connect up widely differing regions of phase space. This suggests the possibility of increased con- vergence difficulties in the above method. It should also be mentioned that if the gaps between tori are of the order tid (d degrees of freedom) further problems arise due to the existence of the ‘‘ Irregular Spectrum ” predicted by Per~ival.~ M. V. Berry and kf. Tabor, Proc. Roy. Soc., A, 1976,349,101. M. V. Berry and M. Tabor, J. Phys. A., 1976, in press. I. C. Percival, J. Phys. B., 1973, 6, 1229.

ISSN:0301-7249    

DOI:10.1039/DC9776200047

出版商:RSC    

年代:1977

数据来源: RSC

摘要:

Potential Energy Surfaces for Ion-Molecule Reactions. Intersection of the 3A2 and 2B1 Surfaces of NH; BY CHARLES F. BENDER Laser Division, Lawrence Livermore Laboratory, University of California, Livermore, California 94550 AND JAMES H. MEADOWS AND HENRY F. SCHAEFER 111 Department of Chemistry, Lawrence Berkeley Laboratory University of California, Berkeley, California 94720 Received 10th May, 1976 The N+ + H2 system is one of the few ion-molecule reactions for which detailed molecular beam studies have been carried out. To complement this experimental research, we have performed a theoretical study of two of the low-lying NH; potential energy surfaces. The intersection and avoided intersection (for C, geometries) of the lowest 3Az and 3B1 surfaces allows a pathway by which the ground state of NH t may be accessed without a potential barrier.The electronic structure calculations employed a double zeta plus polarization basis set, and correlation effects were taken into account using the newly developed Vector Method (VM). To test the validity of this basis, additional self-consistent-field studies were performed using a very large contracted gaussian basis N( 13s 8p 3d/9s 6p 3d), H(6s 2p/4s 2p). The 3Az surface, on which N+ and Hz may approach, has a sur- prisingly deep potential minimum, -60 kcal mol-l, occurring at r,(NH) - 1.26 8, and B,(HNH) - 43'. Electron correlation is responsible for about 15 kcal of this well depth, which appears fairly insensitive to extension of the basis set beyond the double zeta plus polarization level. The line of intersection (or seam) of the 3Az and 3B1 surfaces is presented both numerically and pictorially.The minimum energy along this seam occurs at -51 kcal below separated N+ + H2. Thus for sufficiently low energies one expects N+ - H2 collisions to provide considerable " complex formation ". Fur- ther molecular beam experiments at such low energies (<0.5 eV) would be of particular interest. Simple ion-molecule reactions have provided some of the most fascinating examples to date of the interplay between different potential energy surfaces of a single chemical system.l Most noteworthy in this regard are the molecular beam studies of Mahan and co-~orkers,~-~ who have carefully investigated, among other systems, the Cf + H2, Nf + H2 and O+ + H2 reactions.These reactions are particularly appealing as prototypes, since they are sufficiently simple to be studied by both electronic structure theory5 and classical6 or semiclassical7 dynamics. In addition, the use of qualitative electronic correlation has also proved to be very helpful in understanding these simple reactions ; alternatively the experiments may serve as testing grounds for simple molecular orbital theory. A reasonable starting point for our discussion is the N+ + H2 electronic state correlation diagram of Fair and Mahan.3 This diagram is reproduced with their permission in fig. 1. As discussed by Fair and Mahan (and el~ewhere~*'~ in regard to the C+ + H2 reaction) the key feature in the interpretation of low energy (say less than -3 eV) molecular beam results is the intersection of two low-lying potential energy surfaces.For the Nf + H2 case (in C,, symmetry) these are the 3B1 and 3A2 surfaces. The 3B1 state is known11-14 to be the ground state ofNHz, the nitrenium60 POTENTIAL ENERGY SURFACES FOR ION-MOLECULE REACTIONS ion, while the 3A2 state is less understood. However, on the basis of orbital symmetry considerations l v 8 and earlier theoretical work9 on C+ + H2, the 3B1 surface is expected to be quite repulsive as the N+ initially approaches H2. The deep well of the 3B1 surface is “ protected ” from N+ - H2 collisions on the same surface by means of this large barrier. However, the 3A2 surface should be either much less repulsive9 or attractivelo as N+ approaches H2. And since the two surfaces are both of 3A” symmetry as soon as the N+ ion moves off the Hz perpendicular bisector, the C,, crossing of surfaces becomes an avoided intersection.If there are points along this crossing of 3Br and 3Az surfaces which lie at energies near or below the N+ + H2 asymptote, then there exists a barrier-free pathway 3A2 3 3A“ + 3B1 (1) for the formation of ground state NHT from separated N+ and Ha. Such a pathway for the analogous situation with respect to C+ + H2 has been recently demonstrated unequivocally in the important theoretical work of Pearson and Roueff.lo In their communication, Pearson and Roueff lo bring to light a critical ingredient in the proper theoretical treatment of this problem. That is, polarization functions5 (d functions on carbon andp functions on the hydrogen atoms in their case) critically affect the energy at which the seam or line of intersection occurs.Their finding is pertinent to the present discussion since Gittins and Hirst l5 have recently reported single configuration self-consistent-field (SCF) results for N+ + H2 using a basis set which is quite well-chosen and flexible 16*17 but lacks polarization functions. Gittins and Hirst conclude that access to the deep 3B1 potential well may be possible with only a small barrier, in the order of 4 kcal mol-? By comparison of the effects of polarization functions in the C+ + H2 sy~tem,~*~O it would appear likely that this barrier should disappear completely. The present paper, then, builds on the Gittins- Hirst work15 but goes well beyond it for the N+ + H2 system by the use of larger basis sets and the direct inclusion of correlation effects.These two theoretical exten- sions should allow for a meaningful comparison with the molecular beam experiments of Fair and Mahan.3 THEORETICAL APPROACH Two basis sets of contracted gaussian f ~ n c t i o n s ~ ~ ’ ~ were used here. The first was a standard Huzinaga-Dunning double zeta plus polarization (DZ + P) set, designated N(9s 5p ld/4s 2p Id), H(4s lp/2s lp). The polarization function exponents were 0.8 (nitrogen d functions) and 1.0 (hydrogen p functions), and a scale factor of < = 1.2 was used on the hydrogen s functions. This first basis is essentially the same (except for the obvious replacement of the C basis by one appropriate to N) as that used by Pearson and Roueff,lo and was used for both SCF and configuration interaction (CI) calculations.Since we were initially quite surprised by Pearson and Roueff’s demonstration lo of the critical importance of polarization functions, it was decided to test whether further extensions of their basis would be of qualitative importance to the shape of the N+ + H2 potential surfaces. Therefore, following the recent work of Meadows1* on CH2, a very large basis was adopted: N(13s 8p 349s 6p 3 4 and H(6s 2pj4s 2p). The polarization functions had gaussian orbital exponents a = 1.6, 0.8 and 0.3 for the nitrogen d functions and a = 1.4 and 0.25 for the hydrogen p functions based on past experien~e.~*~”~~ The nitrogen sp functions and hydrogen s functions were the appropriate primitive gaussian basis sets of van Duijneveldt,22 contracted to provide maximum flexibility in the valence region.That is, the five s functions with largestCHARLES F . BENDER, ET A L . 61 orbital exponents ai were grouped together according to the nitrogen atomic 1s orbital, and an analogous procedure followed for the three nitrogen p functions with largest exponents. Based in part on Clementi and Popkie's of the water molecule with many basis sets, we estimate that the present basis set for NHZ should yield total energies within 0.005 hartrees (-3 kcal) of the Hartree-Fock limits for the 3Az and 3Br potential surfaces. Relative errors, of course, should be much smaller. The electron configuration for the two states of primary interest are la! 2ai lb; 3a1 lbl 3B1 (2) la; 2a: 3a; lbz lbl 3A2 (3) and restricted SCF t h e ~ r y ~ ~ .~ ~ has been applied to both of these states. We also note that the first excited electronic state of NH; is of 'Al symmetry and several two-configuration C, la! 2ai lb; 3ai + C2 la! 2ai lbi lb: (4) SCF studies of this state were also made. Finally, it should be noted that the source of the large barrier in the N+ + H2 3B1 approach is the fact that for large N+ - H2 separations the la! 2a; 3 4 1 bl 4a, 3B1 (5) configuration, rather than (2), dominates the wave function. Here we report two such tests, the first with the N+ and H, species separated by a distance R = 100 bohr radii. R is the distance between the N+ ion and the H2 bond midpoint, while Y will designate the H-H internuclear separation.For R = 100, r = 1.4 (essentially the equilibrium internuclear separation of H2) the 3A2 SCF energies are -55.011 59 and -55.021 23 hartrees, the difference being 0.009 64 hartrees or 6.0 kcal mol-'. Secondly we report a point near the equilibrium 3A2 geometry, namely R = 2.0 and r = 1.8 bohr, where the two basis sets yield SCF energies -55.078 23 and -55.091 57 hartrees. The difference in the latter case is larger, 0.013 34 hartrees or 8.4 kcal mol-I. It is certianly not surprising that the near Hartree- Fock basis yields somewhat lower relative energies as N+ and H2 approach. And if SCF basis set errors are directly transmitted to CI results, one would expect our DZ + P basis to yield CI dissociation energies for N+ - H2 about 2.5 kcal less than the exact values.Of course, in the present case, the uncertainties in our treatment of the correlation problem are roughly of that same order of magnitude. In any case the potential surface differences arising from the two basis sets are small, about an order of magnitude less than those found by Pearson for CHZ in going from the DZ to the DZ + P basis set. Electron correlation was taken into account variationally using the newly developed vector method (VM) of Bender and co-workers.26 The CI calculations were carried out with the early version of the VM code. That is, all Slater determinants differing by one or two spin orbitals from (2) for the 3B1 calculations or (3) for the 3A2 calcula- tions were included. In this way 1810 and 1824 determinants were respectively employed in the 3B1 and 3A2 variational procedures.The above was carried out with the usual restriction that the lal orbital (essentially nitrogen 1s) be doubly occupied in all determinants. It is now well-established5 that such a CI procedure will provide at least 90% of the attainable valence shell correlation energy in cases A number of direct SCF comparisons of the two basis sets were made.62 POTENTIAL ENERGY SURFACES FOR ION-MOLECULE REACTIONS (such as the present) where the wave function is qualitatively described by a single determinant SCF wave function. Use of the near Hartree-Fock basis was restricted to the location of the equilibrium geometries of the 3A2, 3B1, and 'Al electronic states. With the DZ + P basis, a regular grid of points (available from the authors on request) for both the 3B1 and 3A2 states was mapped out.These were all combinations of R = 3.0, 2.5, 2.0, 1.75, 1.5 and 1.25 bohrs with Y = 1.2, 1.4, 1.6, 1.8, 2.0, 2.2, 2.4 and 2.6 bohrs for a total of 6 x 8 = 48 points on the surface. To provide a reference point for the relative energies quote hereafter, we note that for the 3A2 state R = 100, Y = 1.4, the SCF and CI energies with the DZ + P basis were -55.011 58 and -55.123 29 hartrees. Thus for separated N+ plus H2 the calculated correlation energy is 0.11 1 708 hartrees. As we shall see, the correlation energy increases as the N+ and H2 are brought together. SINGLET-TRIPLET SEPARATION I N THE NITRENIUM ION Before going on to the primary purpose of this research, let us make a brief digression.Although the 3B1 - lAl separation in NH; is not known experimentally, there have been at least four theoretical predictions of this quantity. On the ab initio side the groups of Morokuma,l' Hayes12 and Harrison13 have predicted 45, 36 and 45 kcal mol-l, with the 3B1 state the lower lying in each case. More recently Haddon and Dewar14 have used their semi-empirical MIND0/3 method to predict 31 kcal mol-1 for this quantity. For comparison with these results, the near Hartree-Fock basis was used to predict the 3B1 and lAl equilibrium geometries. For the 3B1 state the predicted structure was r,(NH) = 1.018 A, B,(HNH) = 143.3', corresponding to a total SCF energy of -55.229 65 hartrees. The two-configuration SCF description (4) of the 'Al state yields re(NH) = 1.033 .$, B,(HNH) = 108.2' and a total energy of -55.183 29 hartree.Thus the singlet-triplet separation AE is predicted to be 0.046 36 hartrees = 29.1 kcal mol-l. The best means of evaluating the reliability of the above prediction is by compari- son of analogous theoretical procedures with experiment for CH2, for which an accurate - lAl) has recently become a~ailable.'~ The experimental value of 19.5 & 0.7 kcal may be compared with the 10.9 kcal obtainedl8 for CH2 by the method described in the previous paragraph. Thus it is evident that a two-configuration description of the lAl state overcompensates for the fact that the 3B1 state has less correlation energy. For CH2 the use of a single configuration SCF treatment of the 'Al state yields a separation of 24.8 kcal, too large as expected. More precisely the experimental result lies 61.9% of the way from the two-configuration lAl result to the one-configuration lAl result.With the above in mind, we carried out single configuration (la: 2a: lb: 3at) SCF calculations on NHZ, yielding r,(NH) = 1.032 A, O,(HNH) = 109.6", and E = -55.158 38. The singlet-triplet separation obtained in this way (44.7 kcal) is con- siderably greater than the two-configuration result, 29.1 kcal. It seems quite certain that the exact nitrenium separation lies between the two, and if the same 61.9% criterion is used, a semi-empirical prediction of 38.8 kcal is made. Partly because of the semi-empirical nature of our prediction and also because of the use of a Hartree- Fock limit basis, we suggest that the 38.8 kcal value is probably the most reliable prediction made to date.CHARLES F.BENDER, ET AL. 63 REGION OF INTERSECTION OF THE 3 A 2 AND 3B1 SURFACES Certainly the most interesting result found here is the rather deep potential well associated with the 3A2 state of NHf. Such a deep well is not anticipated from the C+ - H2 calculations of Liskow, Bender and Schaeferg or the correlation diagram (fig. 1) of Fair and Mahan. Such a well is implicit in the work of Pearson,l0 but he - 3 - 2 t- - -41- -t -6 4- i I: -6 FIG. 1,-Correlation diagram of Fair and Mahan3 for the N+ + H2 system. does not report the predicted 2B2 (analogous to the NH; 3A2 state) dissociation energy relative to separated C+ + H2. We do know that Pearson's CH; 2B2 state must be bound by at least 15 kcal, since that is the lowest energy at which the 2B2 and 2A1 electronic states are degenerate. Using the near Hartree-Fock basis, the 3A2 state of NHS is predicted by SCF theory to have an equilibrium geometry r,(NH) = 1.207 A, B,(HNH) = 46.4'.This small bond angle is characteristic of the early approach of N+ to H2; and the predicted equilibrium geometry corresponds to a near Hartree-Fock energy 44.6 kcal below separated N+ and H2. Using the DZ + P basis the 3A2 minimum is less precisely located since the grid (see previous section) is relatively sparse in this region (note that the density of grid points is greatest near the intersection of the 3A2 and 3B1 surfaces). With this disclaimer we note that the DZ + P SCF minimum is predicted by a 9-point fit to lie at r,(NH) = 1.25 A, 8, = 42', with energy 44.3 kcal below N+ + HZ.Realistically the true SCF minimum with this basis probably occurs about 2 kcal higher, when one considers the direct comparisons (between DZ + P and near Hartree-Fock basis sets) of the previous section. Similarly the CI equilibrium geometry is r,(NH) = 1.26 A, 0, = 43", and lies 60.4 kcal below N+ + H2. The lowest actual calculated point on the 3A2 surface occurs at R = 2.0 bohrs, Y = 1.8 bohrs (or r(NH) = 1.161 A, 8 = 48.5') for both SCF and CI methods. These points lie 41.8 and 56.8 kcal below the comparable asymptotic calculations and make it quite clear that electron correlation contributes -15 kcal to the well depth. If in turn this 15 kcal is added to the near Hartree-Fock well depth of 44.6 kcal, one obtains 59.6 kcal as the predicted dissociation energy relative to N+ + H,.In any case a value of -60 kcal for the dissociation energy consistently appears on the basis of the present theoretical research. A dissociation energy this large (nearly 3 eV) must be considered surprising as it certainly cannot be justified in terms of a classical electrostatic picture.64 POTENTIAL ENERGY SURFACES FOR ION-MOLECULE REACTIONS The 3A2 and 3B1 surfaces are illustrated in fig. 2 and 3. Note, of course, that since the region of interest here is that near the intersection, the actual position of the 3B1 NH; equilibrium geometry is not included. The fact that the 3B1 surface becomes very attractive in that direction is however quite clear.Also apparent is the large barrier ( -75 kcal) associated with the Woodward-Hoffmann forbidden least motion9 insertion of N+ into H2. To complement the two contour maps and the line of intersection indicated on each, table 1 gives some numerical values for the line of 10 R ( N - CM I/ bohrs FIG. 2.-3Az potential energy surface for NHf. R(N - CM) is the distance from the nitrogen nucleus to the H2 centre of mass. Contours are labelled in kcal mol-l relative to infinitely separated N+ plus HZ. Note that contours energetically below 25 kcal are labelled in 5 kcal intervals, while those above 25 kcal are spaced by 25 kcal. It seem quite clear from previous work9*'' on the related C+ - H2 system that the C,, approach along the 3A2 surface is by far the most likely to lead to the bound NH+, species.In this light one can make a rough picture of one important aspect of the dynamics. First, as fig. 2 implies, high energy C2, collisions will tend to be ~nreactive.~ That is, with r(H-H) fixed at 1.4 bohrs, the 3A2 surface becomes quite repulsive rather quickly. For example, at R = 1.5 bohrs the surface lies 35 kcal above separated N+ + H2. Therefore, a key feature leading to complex formation is the necessity that the collision occurs slowly enough such that the H-H separation can become sufficiently large to reach the area of the line of intersection. Inspection of fig. 1 and 2 or table 1 shows that the line of intersection reaches zero kcal relative energy at about R M 1.52 bohrs, r M 1.67. In other words the H-H separation must increase by nearly 0.3 bohrs - 0.15 A for the line of intersection to become dynamically meaningful in low energy collisions.A final noteworthy point is that the line of intersection for the N+ - H2 system passes through much lower relative energies (50 kcal as compared with 15 kcal) thanCHARLES F. BENDER, ET A L . 65 R ( N - C M ) bohrs / FIG. 3.-3B1 potential energy surface for NH;. R(N - CM) is the distance from the nitrogen nucleus to the H2 centre of mass. Contours are labelled in kcal mo1-I relative to infinitely separated N+ and H2. Note that contours energetically below 25 kcal are labelled in 5 kcal intervals, while those above 25 kcal are spaced by 25 kcal. TABLE LINE OF INTERSECTION OF THE LOWEST 3A2 AND 3B1 POTENTIAL ENERGY SURFACES OF NH; .THESE POINTS ARE GIVEN IN TWO COORDINATE SYSTEMS FOR EASE OF INTERPRETATION. AS NOTED IN THE TEXT, R IS THE DISTANCE BETWEEN THE N+ NUCLEUS AND THE Hz BOND MIDPOINT. ENERGIES ARE CONFIGURATION INTERACTION (CI) ENERGIES RELATIVE TO SEPAR- Rlbohrs 1.30 1.35 1.40 1.45 1.50 1.55 1.60 1.65 1.70 1.75 1.80 1.85 1.90 r / bohrs 1.240 1.330 1.425 1.523 1.623 1.726 1.834 1.946 2.061 2.179 2.301 2.426 2.556 ATED N+ + H2. r (N-H)/ii 0.762 0.796 0.83 1 0.867 0.902 0.939 0.976 1.014 1.052 1.091 1.130 1.171 1.212 B(HNH)/deg 51.0 52.4 53.9 55.4 56.8 58.2 59.6 61.1 62.4 63.8 65.2 66.5 67.9 mergy/kcal mol- 171.0 116.6 72.6 37.6 10.4 - 10.4 -26.1 - 37.3 - 44.7 -49.1 - 51 .O - 50.8 -48.8 the corresponding line of intersection for the C+ - H2 system." A naive interpreta- tion of this comparison would suggest that at low energies one should observe more complex formation for the N+ than the C+ reaction.At this point, however, we believe that detailed dynamical studies are called for. This work on N+ - H2 and Pearson's researchlo for C+ - H2 appear to provide rather accurate predictions of66 POTENTIAL ENERGY SURFACES FOR ION-MOLECULE REACTIONS some of the crucial potential surface features, and the greatest uncertainties are now of a dynamical nature. Of course, more information concerning these surfaces would be welcome, especially concerning the slopes of the two lowest 3Aff surfaces (arising from 3Az and 3B1 in C,, point group) in the region of their avoided inter- section. We thank Prof. Bruce H. Mahan for helpful discussions, encouragement, and patience.We also benefited from many illuminating conversations with Dr. Peter K. Pearson. The computational burden was shared by the Lawrence Livermore Laboratory CDC 7600 and the Harris Corporation Series 100 minicomputer, sup- ported by National Science Foundation Grants GP-39317 and GP-41509X. B. H. Mahan, Accounts Chem. Res., 1975, 8, 55. B. H. Mahan and T. M. Sloane, J. Chem. Phys., 1973, 59, 5661. J. A. Fair and B. H. Mahan, J. Chem. Phys., 1975,62, 515. K. T. Gillen, B. H. Mahan and J. S. Winn, J. Chem. Phys., 1973, 58, 5373. H. F. Schaefer, The Electronic Structure of Atoms and Molecules: A Survey of Rigorous Quantum Mechanical Results (Addison-Wesley, Reading, Massachusetts, 1972). D. L. Bunker, Accounts Chem. Res., 1974,7, 195.W. H. Miller, Adv. Chem. Phys., 1974, 25, 69. R. B. Woodward and R. Hoffmann, The Conservation of Orbital Symmetry (Verlag Chemie, Weinheini/Bergstr., 1970). D. H. Liskow, C. F. Bender and H. F. Schaefer, J. Chem. Phys., 1974, 61,2507. lo P. K. Pearson and E. Roueff, J. Chem. Phys., 1976,64, 1240. l1 S. T. Lee and K. Morokuma, J. Amer. Chem. SOC., 1971, 93, 6863. l2 S. Y . Chu, A. K. Q. Siu and E. F. Hayes, J. Amer. Chem. SOC., 1972,94, 2969. l3 J. F. Harrison and C. W. Eakers, J. Arner. Chem. SOC., 1973,95, 3467. l4 M. J. S. Dewar, R. C. Haddon, W.-K. Li, W. Thiel and P. K. Weiner, J. Amer. Chem. Soc., l5 M. A. Gittins and D. M. Hirst, Chem. Phys. Letters, 1975, 35, 534. l6 S. Huzinaga, J. Chem. Phys., 1965,42, 1293. l7 T. H. Dunning, J. Chem. Phys., 1970,53, 2823. la J. H. Meadows and H. F. Schaefer, J. Amer. Chem. SOC., 1976, 98, in press. l9 B. Roos and P. Siegbahn, Theor. Chim. Acta, 1970, 17, 199. zo S. Rothenberg and H. F. Schaefer, J. Chem. Phys., 1971,54, 2765. 21 T. H. Dunning, J. Chem. Phys., 1971,55, 3598. 22 F. B. van Duijneveldt, IBM Research Report RJ 945, 1971. (Available from: Research Lib- 23 E. Clementi and H. Popkie, J. Chem. Phys., 1972,57, 1088. 24 W. J. Hunt, T. H. Dunning and W. A. Goddard, Chem. Phys. Letters, 1970,6, 147. 25 E. R. Davidson, Chem. Phys. Letters, 1973,21, 565. 26 R. F. Hausman, S. D. Bloom, and C. F. Bender, Chem. Phys. Letters, 1975, 32,483. 27 P. F. Zittel, G. B. Ellison, S. V. O’Neil, E. Herst, W. C. Lineberger and W. P. Reinhardt, J . 2a J. 0. Hirschfelder, C. F. Curtis, and R. B. Bird, Molecular Theory of Gases and Liqaids (Wiley, 1975,97,4540. rary, IBM Research Laboratory, San Jose, California 951 93.) Amer. Chem. SOC., 1976, 98, 3731. New York, 1954).

ISSN:0301-7249    

DOI:10.1039/DC9776200059

出版商:RSC    

年代:1977

数据来源: RSC

摘要:

An Ab Initio Potential Surface for the Reaction N++H,-NH++H BY MARTIN A. GITTINS AND DAVID M. HIRST* Department of Molecular Sciences, University of Warwick, Coventry CV4 7AL MARTYN F. GUEST Science Research Council Atlas Computer Laboratory, Chilton, Didcot, Oxon OX11 OQY AND Received 3rd May, 1976 Ab initio coiifiguration interaction calculations using Dunning’s gaussian basis set are reported for the potential energy surface for the reaction N+ + H2 --t NH+ + H. For the collinear approach of N+ to H2 the 3X- surface has a shallow minimum and the 311 surface is repulsive. For CZu geometries the 3Bz surface is strongly repulsive and the 3A2 surface has a shallow minimum. The 3B1 surface has a deep well which is not adiabatically accessible at low relative energies. An avoided crossing in C, symmetry between the 3A” surfaces correlating with the 3Az and 3B1 surfaces gives an adiabatic route to the deep potential well, but trajectory hopping is thought to be significant except at the lowest relative energies.Preliminary work on the fitting of an analytic function to attractive surfaces is discussed. 1. INTRODUCTION Ion-molecule reactions are of particular interest for several reasons. For the primary ion beam one has a greater degree of control over the translational energy than is the case for a neutral beam. This means that it is possible to study ion- molecule reactions over a much larger range of relative translational energies than is possible for neutral reactions. A second feature of interest is that many ion-molecule reactions show a change in mechanism as the relative translational energy is changed. Potential energy surfaces for ground and excited states are often comparatively close so that, even at the lowest energies, non-adiabatic processes may compete effectively with adiabatic reactions.Finally, because most ion-molecule reactions do not have an energy barrier, reactivity and dynamics can be studied in the absence of the domin- ating influence of the activation barrier. The reaction N+ +H2+NH++ H (1) has been studied recently by Fair and Mahanl in an ion-beam collision apparatus in the range of initial relative energies from 0.79 to 2.8 eV. Above 2 eV asymmetric product distributions were obtained, indicating that the reaction proceeds by a direct mechanism. As the relative energy is lowered, the intensity peak moves towards the centre-of-mass velocity, and the intensity contours become more symmetric.At 0.79 eV the distribution is quite symmetric, suggesting that at this relative energy the reaction proceeds through a long lived complex. It is surprising that this behaviour is only observed below 2 eV because the 3B1 ground state of symmetric NH2+ lies68 AN A B ZNZTZO POTENTIAL SURFACE approximately 6 eV below the energies of reactants and products. This is in contrast to the reaction C++H2+CH++H (2) which shows considerable symmetry up to 3.5 eV,2 despite the fact that the potential well of CH2+ is only 4.4 eV deep. Fair and Mahan interpreted their observations in terms of qualitative correlation diagrams constructed with the aid of ab initio calculations for NH,3 NH+ and NH2+ and the ab initio potential surfaces of Liskow et aL6 for reaction (2).For the col- linear approach of N+(3P) to H2(lZ;) they concluded that the potential surface of lowest energy ("-) would have neither a deep well nor a high potential barrier. For configurations of C2" symmetry, surfaces of 3A2, 3B1, and 3B2 symmetry correlate with the reactants. It was considered unlikely that complex formation would result from the 3B2 surface, which gives highly excited configurations. The 3B1 surface does have a deep well, but this is thought to be inaccessible at low relative energies because of tlie existence of a substantial potential barrier arising from an avoided crossing between two 3B1 surfaces, one correlating with the reactants and the other with N(2D) + H2+(2Z3.The 3A2 surface, on the other hand, was thought to be rather flat. However, in C, symmetry the 3A2 and 3B1 states relax to 3A" symmetry, and a conical intersection gives a route to the potential well which does not involve any substantial activation barrier. The main features of the collinear and C2, cases have been confirmed by prelimin- ary calculations ' using the unrestricted Hartree-Fock method. We report here a more thorough investigation of the potential surfaces thought to be important in this reaction. 2. A B INITIO CONFIGURATION INTERACTION CALCULATIONS The calculations reported here were made using the ATMOL 3 integral and SCF packages and the MUNICH configuration interaction programg*lo as implemented at the Atlas Computer Laboratory with Dunning's l1 (9s5p/4s2p) and (4s/3s) Gaussian basis sets for nitrogen and hydrogen respectively.The MUNICH system employs the bonded function formalism for configuration interaction. In the configuration interaction calculations we included all single and double excitations, with the exception that the lowest orbital was always doubly occupied and that the highest virtual orbital was excluded. For the collinear approach of N+(3P) to H2('Z:) to give 3Z-(NHH)+ we consider the 1293 configurations derived from the configurations and la2 2a2 3a2 ln,l lnyl la2 202 3 d 401 lzx1 lnyl, (3) (4) It was necessary to include root function (4) to describe correctly the dissociation of 3Z- (NHH)+to NH+(4C-) + H because the restricted Hartree-Fock function dis- sociates to NH(3C-) + H+.A limited number of points on the 311 surface were cal- culated using the SCF orbitals from the 3Xc- calculations with the root functions la2 202 3a2 In1 401 (5) and la2 202 30' In1 402 (6) to allow for the virtual orbitals for the 3C- state not being optimum for the 311 state.MARTIN A . GITTINS, ET AL. 69 For geometries of C,, symmetry we considered the 3B2, 3B1 and 3Az surfaces. The 3B2 surface was calculated using the 546 configurations derived from the function la12 2a12 3a12 lb,' 4a11. la12 2a12 3a12 lbll 1b21. la12 2a12 1 bZ2 3a11 lbll (7) (8) (9) For the surface of 3A2 symmetry 505 configurations were derived from the function The 3B1 ground state of (HNH)+ has the configuration and correlates, at large N-H2 distances, with N(2D) + H2+(2Z:g+) whereas the reactants N+ (3P) +.H2('Eg+) correlate with the excited configuration la12 2a12 3aI2 1 bll 4a11. (10) Thus the transition from reactants to 3B1 (HNH)+ proceeds through an avoided intersection between surfaces corresponding to configurations (9) and (10). In order to describe properly the lowest 3B, surface we included the 859 terms derived from configurations (9) and (10). The case of C, symmetry is more complicated. We considered the 2556 configura- tions derived from the functions and Configurations (11), (13) and (14) are the A" equivalents in C, symmetry to functions (8), (9) and (10) respectively and configuration (12) is included to allow for correct dissociation to NH+(4C-) + H. Fig. 1, 2 and 3 illustrate the coordinate systems used for linear, C,, and C, geo- metries respectively.FIG. 1 i" R ( N-MI + H*-*H r (HH) FIG. 2 f" s--:'") H*-r(HHl-) ' FIG. 370 A N AR I N I T I O POTENTIAL SURFACE -54.9 We also considered some geometries of C, symmetry with the configuration illustrated in fig. 4. * N \ - c H H* f-) t---+ 0 '1 '2 FIG. 4 u) 0) ? f -55.0- !? - L Q ). C a, 3. POTENTIAL ENERGY SURFACES Fig. 5 shows a cut through the potential surfaces for the collinear case as N+ approaches H2 with the H-H distance fixed at 1.5 a.u. (1 a.u. = 5.291 77 x m). The 3Xc- surface has no activation barrier, and has a shallow well of depth approxi- - 55.2 I I I I t -55.1 \ mately 1.5 eV; it therefore is unlikely to contribute to complex formation. The 311 surface is repulsive and need not be considered further.Cuts through the three surfaces for C,, geometry with H-H distances of 1.5 and 2.0 a.u. are shown in fig. 6 and 7. Motion on the 3B2 surface clearly does not contri- bute to the dynamics as this surface is initially strongly repulsive. There is evidence for the existence of an avoided crossing with a bound surface of high energy. The 3A2 surface is similar to the collinear 3 X - surface in having no activation barrier and a shallow well of depth of approximately 1.7 eV. Thus the entrance channel for collisions of C,, symmetry is slightly more favourable energetically than is the collinear approach. The 3B1 surface clearly exhibits the avoided crossing between surfaces correlating with N+(3P) + H2(%:) and with N(2D) + H,+(2C:).There is a substantial energy barrier of height approximately equal to 3.6 eV in the entrance channel making theMARTIN A . GITTINS, ET A L . 71 -54.60 -5L.70 -5L.80 Ln 4, 2 b ‘-54.90 B c -c x c -55.00 -55 10 - 55.20 10 2 0 3.0 4.0 5.0 6 0 R (N-MI FIG. 6.-Energy (in hartrees) of C2,(HNH)+ as a function of distance R(N - M) between N and mid-point M of HI - H2 for r(HH) = 1.5 a.u. 3B1 ground state of (NHH)+ inaccessible, in CZU symmetry, to collisions with low relative energy. Fig. 8 shows the variation of energy for the 3B1 and 3A2 states as a function of angle for the case where the N-H distance is 2.0 a.u. The energy of the 3B1 state falls rapidly as the bond angle is increased from 60” to give a well depth of approximately 5.66 eV for a bond angle of 150-1 55”.The potential well is flat with res- pect to variation in bond angle as observed by other worker^.^*^^-^^ The intersections of the 3A, and 3B1 curves in fig. 6 and 7 occur on the repulsive part of the 3A2 potential. It is not until the H-H distance has been stretched to about 3.0 a.u. that the 3B1 and 3A2 curves are attractive at the point of intersection. An avoided crossing in C, symmetry giving a low energy path to the potential well would be expected to occur in this region of the surface when the geometry is distorted slightly. Fig. 9 shows sections of the 13A” and 23At’ &Is surfaces for the configuration of fig. 3 with 8 = 90” and r(HH) = 1.5 and 3.0 a.u. The minima on the 13A” surface are shallow, and the avoided intersections between the two surfaces occur in the repulsive region.Thus, reactive collisions with these sorts of geometries will proceed by a direct mechanism. Calculations with 8 = 120” and r(HH) = 3.0 a.u. gave results similar to those with 8 = 90”. It is for geometries having the configuration of fig. 4 that the avoided crossing between the 3A” surfaces correlating with the 3B1 and 3A2 states in Czu symmetry gives a low energy path to the deep potential well of 3B1 (NH,)’. Fig. 10 shows cuts through the 1 3Att and 2 3A” surfaces for r(HH) = 4.0 a.u. with rl = 2.5 and r2 = 1.5. The avoided crossing now occurs on the attractive portions of the surfaces. Similar potential curves were obtained for r(HH) = 3.0 a.u. (rl = 1.0, r2 = 2.0) and r(HH) =72 -54.60 -54.70 -54.80 IA aJ b- c ;i P c .;-54.9 a c -55.00 -55.1C -55.2( AN A B INITIO POTENTIAL SURFACE I I ~~ ~~ ~~ O 2 .o 3.0 L.0 5.0 6'0 R (N-M)/a.u.FIG. 7.-Energy (in hartrees) of CZD (HNH)+ as a function of distance R(N - M) between N and mid-point M of HI - H2 for r(HH) = 2.0 a.u. -5L.S v) -55.c E aJ - L CI Jz -. 2. a C u k-55.1 -55. z -55.: I I I I I 0 63 90 120 150 180 8 /degrees Frc,. &-Variation of energy (in hartrees) of C2,(HNH+) as a function of angle 0 for r(NH) = 2.0 a.u.MARTIN A . GITTINS, ET AL, 73 -54.90 -55 00 In W L c L 0 L % -. ? c -55.10 I_-__- - .A- I 1 5 2 0 2 5 3 0 R (NHl/a u FIG. 9.-Energy (in hartrees) of C,(HNH)+ for 6 = 90" as a function of R(NH) - r(HH) = 1.5 a.u.; - - - - - - - - r(HH) = 3.0 a.u. -54.0 - 54.9 VI -55.0 23 L \ x 9 5 -55.1 - 55.2 -55.3 .I 1 2 3 L 5 R3/a.u FIG.lO.-Energy (in hartrees) of C,(HNH)+ for perpendicular approach for r1 = 2.5, r2 = 1.5 as a function of R3.74 AN A B ZNZTIO POTENTIAL SURFACE 3.5 a.u. (rl = 1.5, r2 = 2.0). Table 1 summarises the separation between the potential surfaces. It can be seen that the splitting is very small for r = 4.0 a.u. (8.4 kJ mol-l) and increases as r(HH) decreases from 4.0 a.u. Thus, for slightly off-centre perpen- dicular collisions, motion on the 1 3A’r surface gives a route, which does not involve an activation barrier, to the deep potential well. TABLE 1 r(HH)/a.u. R3/a.u. 2.0 2.25 2.5 2.75 3.0 E/eV 0.5 1 .o 2.0 (4 (6) .-DETAILS OF AVOIDED INTERSECTION OF 3A” SURFACES 3 .OO 3.50 4.00 E(13A”) - E(2 3A”)/kJ mol-1 11 8.9 127.2 28.5 54.5 127.8 48.3 8.4 107.7 55.8 241 153.6 92.0 surface hopping probability 0.53 0.90 0.64 0.93 0.73 0.95 Our ab initio potential surfaces thus confirm the qualitative conjectures of Fair and Mahan.’ Neither CZc nor linear geometries give a sufficiently deep potential well to account for the symmetric product distributions they obtained at low relative energies.However, our calculations for geometries of C, symmetry clearly demon- strate the existence of a conical intersection which gives an adiabatic route to a deep well. Thus, trajectories which pass through this intersection and into the potential well will contribute to a symmetric product distribution. In order to get a rough idea of the importance of transitions between the 3A‘‘ surfaces at the avoided crossing, we made estimates of the probability of potential surface hopping as a function of relative energy, using the approximations of Bausch- licher et aZ.15 to the Landau-Zener-Stuckelberg forniulation.16-18 Using their approximations, the hopping probability P for a relative energy E is given by P = e~p[-(E,/E)~-l (15) where with AE, being the separation between the surfaces and AE,” the curvature of the separation AE at the avoided intersection.The results are included in table 1. They are very approximate because we did not explore the region of the avoided inter- sections in sufficient detail to obtain accurate values for the curvature AE,”. Even at low energies, there is a very high probability of hopping from the lower to the upper surface for r(HH) = 4.0 a.u. Thus the majority of reactive trajectories passing through this part of the potential surface will not visit the potential well.For the case of r(HH) = 3.5 a.u., the hopping probabilities are lower and fall off more rapidly as the energy decreases. Surface hopping is clearly still important here, but at lower relative energies a significant proportion of trajectories will remain on the lower surface. These rough estimates of the hopping probabilities give us some understanding of why the direct mechanism giving an asymmetric product distribution persists to very low relative energies, despite the fact that the potential well is so deep.MARTIN A . GITTINS, ET A L . 75 4. ANALYTIC FITTING OF ATTRACTIVE POTENTIAL SURFACES Before an ab initio potential energy surface obtained from a limited number of calculations for a series of geometries can be used for dynamical calculations, it has to be expressed in a form such that either the energy or the gradient of the energy can be obtained over a continuum of nuclear geometries.Many workers have considered this problem and some of their efforts have been reviewed briefly.19 The two basic approaches are either interpolation or fitting the set of ab initio points to some analytic function. Sathymurthy and Raff l9 investigated the use of three dimensional spline inter- polation and concluded, from a comparison with an analytic function, that although the method was not sufficiently accurate to produce a point-by-point match in classical trajectory calculations, the total cross sections, energy partitioning and spatial dis- tribution were in good agreement with those obtained with the analytical potential.Thus spline interpolation methods may be useful for dynamical calculations but in our experience their application is not straightforward. Spline interpolation in three dimensions requires a complete rectilinear grid of points. Therefore, a coordinate system is required in which the entire three dimensional surface is represented in a rectilinear form. Most coordinates do not do this. Those that do turn out to involve the calculation of a large number of points well away from any reaction co- ordinate and careful selection of grid intervals. Finding a suitable analytic function can be a very difficult procedure. Yarkony et aL2* were unable to obtain a satisfactory fit to their surface for the (HF)2 system.The extended London-Eyring-Polyani-Sato21 surface has been successfully used by Polyani and Schreiber 22 to fit the ab initio collinear surface of Bender et aZ.23 for FHH and this function may well be suitable for fitting potential surfaces having an activation barrier. However, the LEPS function is somewhat inflexible, and Bowman and K~pperrnann~~ have proposed a method involving the rotation of a Morse function and cubic spline interpolation which they believe overcomes the deficiencies of the LEPS function. Little seems to have been done for surfaces which do not have an activation barrier. However, Sorbie and Murrell 25 have recently proposed, for stable triatomic species, an analytic potential which is capable of reproducing both the equilibrium and asymptotic properties of the molecule.The parameters in their potential were obtained by fitting it to the experimental force constants. Their function may be equally useful for fitting a set of ab initio points for an attractive surface. The poten- tial for the molecule ABC is of the form (17) V(R1R2R3) = ‘C/AB(R1) + VBC(R2) + vAC(R3) + VABdRlRZR3) where VAB etc. are the asymptotic diatomic potentials and VABC(RiR2R3) is a suitable three-body function for which they proposed the form (1 - tanh 3y2S2)(1 - tanh 3y3S3). (18) A is a constant, P is a polynomial containing up to quartic terms and y i are variable parameters. The potential is expressed in terms of Si which are displacements from the triatomic equilibrium configuration.In the case of NH2+, if in addition to the yi and the coefficients in the polynomial P, A is varied, there are 24 parameters. In our experience these can best be optimized by using Fletcher’s modification26 to the algorithm of Marquardt.”76 AN A B INITIO POTENTIAL SURFACE Using this function we were able to fit 302 points obtained in previous unrestricted Hartree-Fock calculations for this system with a least squares deviation of 21.4 kJ mol-l. While this is not particularly good, it does indicate that the potential of Sorbie and M~rre11~~ is a useful starting point for the fitting of a set of ab initio points for attractive potential energy surfaces. It should be possible to refine the function to obtain analytic surfaces sufficiently good for dynamical calculations.We are grateful to the Science Research Council for a Research Studentship to one of us (M. A. G.) and for the provision of computer time at the Atlas Computer Laboratory. J. A. Fair and B. H. Mahan, J. Chem. Phys., 1975,62, 515. B. H. Mahan and T. M. Sloane, J. Chem. Phys., 1973,59, 5661. H. P. D. Liu and G. Verhaegen, Int. J. Quant. Chem., 1971,5, 103. H. P. D. Liu and G. Verhaegen, J. Chem. Phys., 1970,53, 735. S . Y. Chu, A. K. Q. Siu and E. F. Hayes, J. Amer. Chem. Soc., 1972, 94,2969. ti D. H. Liskow, C. F. Bender and H. F. Schaefer 111, J. Chem. Phys., 1974,61, 2507. ’ M. A. Gittins and D. M. Hirst, Chem. Phys. Letters, 1975, 35, 534. M. F. Guest and V. R. Saunders, ATMOL 3, Science Research Council, Atlas Computer Laboratory, Chilton, Didcot, Oxon. G. H. F. Diercksen and B. T. Sutcliffe, Theor. Chim. Acta, 1974,34, 105 lo M. F. Guest and W. R. Rodwell, ATLAS C.I., Science Research Council, Atlas Computer Laboratory, Chilton, Didcot, Oxon. l1 T. M. Dunning, J. Chem. Phys., 1970,53,2823. l2 S . T. Lee and K. Morokuma, J. Amev. Chem. Sac., 1971,93, 6863. l3 J. F. Harrison and C. W. Eakers, J. Amer. Chem. Soc., 1973,95, 3467 l4 M. J. S. Dewar, R. C. Haddon, W. K. Li, W. Thiel and P. K. Weiner, J. Amer. Chem. Soc., l5 C. W. Bauschlicher, S. V. O’Neil, R. K. Preston, H. F. Schaefer I11 and C. F. Bender, J. Chem. l6 L. D. Landau, Phys. 2. Sowjetunion, 1932, 2,46. l7 C. Zener, Proc. Roy. Sac. A, 1932, 137, 696. l8 E. C. G. Stuckelberg, HeIv. Phys. Actu, 1932, 5, 369. l9 N. Sathymurthy and L. M. Raff, J. Chem. Phys., 1975, 63,464. *O D. Yarkony, S. V. O’Neil, H. F. Schaefer 111, C. P. Baskin and C. F. Bender, J. Chem. Phys., 21 P. J. Kuntz, E. M. Nemeth, J. C. Polyani, S. D. Rosner and C. E. Young, J. Chem. Phys., 1966, 22 J. C. Polyani and J. L. Schreiber, Chem. Phys. Letters, 1974, 29, 319. 23 C. F. Bender, S. V. O’Neil, P. K. Pearson and H. F. Schaefer 111, Science, 1972,176, 1412. 24 J. M. Bowman and A. Kuppermann, Chem. Phys. Letters, 1975,34,523. 25 K. S. Sorbie and J. N. Murrell, Mol. Phys., 1975,29, 1387. 26 R. Fletcher, U.K.A.E.A. Research Group Report, No. AERE-R6799 (1971). 27 D. W. Marquardt, J. SOC. Zndust. AppI. Math., 1963, 11,431. 1975,97,4540. Phys., 1973, 59, 1286. 1974, 60, 855. 44, 1168.

ISSN:0301-7249    

DOI:10.1039/DC9776200067

出版商:RSC    

年代:1977

数据来源: RSC

摘要:

Potential Energy Surfaces for Simple Chemical Reactions : Application of Valence-Bond Techniques to the Li +HF+LiF +H reaction BY GABRIEL G. BALINT-KURTI* AND ROBERT N. YARDLEY? School of Chemistry, Bristol University, Bristol BS8 1TS Received 3rd May, 1976 Ab initio multi-structure valence-bond calculations have been performed to determine the potential energy surface governing the reaction Li + HF --t LiF + H. Results for both linear and non-linear nuclear geometries are presented. The system is a prototype for many heavier alkali metal plus hydrogen halide reactions which have been studied using the crossed molecular beams technique. The ab initio valence-bond results are improved by applying corrections, within the framework of the orthogonalized Moffitt (OM) method, for the atomic errors present.The orbital basis set used was of double zeta quality, and was augmented by some extra orbitals. Preliminary calculations on the neutral and ionic diatomic species were performed to ensure the adequancy of the valence- bond structure basis sets used and care was taken to ensure that the basis sets provided an adequate description of F-, HF- and LiF-. The endoergicity of the reaction, ignoring the zero-point vibra- tional energies, was predicted by the ab initio and OM methods to be 5.8 and 2.5 kcal mol-' respectively, as compared with the experimental value of 2.6 kcal mol-I. Besides the ground state potential energy surface, several surfaces for excited electronic states have been calculated and are presented. The relationship of the ground state potential energy surface to the reactive cross section, and its variation with energy, is discussed.The ground and excited state potential energy surfaces are compared with previously proposed models and a mechanism for the production of alkali metal ions in hyperthermal alkali metal atom-hydrogen halide collisions is proposed. 1. INTRODUCTION The very first successful crossed molecular beam experiment was performed by Taylor and Datzl in 1955 on the reaction, K + HBr -+ KBr + H. In the period since these first experiments many groups of workers have performed progressively more refined experiments on the K + HBr system,2-6 on its isotopic variants K + DBr4 and K + TBr7, and also on the reverse reaction.8 The differential reactive cross sections for these reactions do not appear to fit into an easily rationalis- able pattern>** and there has been some speculation9 as to the type of potential energy surface which would be capable of simultaneously explaining the results of all of the experiments.Besides these experiments there have also been several experi- rnentsl0-l4 on related members of the family of reactions M + HX 3 MX + H. As an aid to the interpretation of these experiments we have calculated a potential energy surface for the Li + HF system. This is the lightest member of the M+ HX family of reactions and, besides being of interest in its own right, the surface might serve as a prototype for the heavier members of the family. The calculations were t Present Address: Department of Chemistry, University of Sheffield, Sheffield S3 7HF.78 POTENTIAL ENERGY SURFACES FOR SIMPLE CHEMICAL REACTIONS performed using two different methods.The first of these was an ab initio multi- structure valence-bond method. While the second method, which is called the orthogonalised Moffitt (OM) method, involved the application of corrections for the known atomic errors present in the ab initio calculation. Before examining the tri- atomic LiHF system, preliminary calculations were performed on all the constituent diatomics 15*16 so as to determine which atomic orbital and valence-bond structure basis sets should be used in the triatomic calculations. In section 2 of this paper we sketch very schematically the basic ideas behind the different methods used and outline the atomic orbital and valence-bond structure basis sets.We also summarise the results of the preliminary diatomic calculations. In section 3 the results for the col- linear ground state surfaces are presented and discussed, while in section 4 we present the ground state surfiaces for non-linear geometries. Section 5 is devoted to a dis- cussion of some of the low-lying electronically excited states of the system. In section 6 the relevance of the present results to reactive scattering experiments on the alkali metal atom-hydrogen halide family of systems is discussed, and comparison is made with previously published work. In this last section we also discuss collisions at hyperthermal energies and propose a mechanism for the production of alkali metal atom ions in such collisions.2. THEORETICAL BACKGROUND AND DESCRIPTION OF BASIS SETS The first step in the calculation is to construct approximate atomic eigenfunctions for all the atomic states which will play a role in the calculation. The approximate atomic eigenfunctions for different atoms are then multiplied together and anti- symmetrised to form composite functions (CF’s). These composite functions form the basis set for the expansion of the total electronic wavefunction of the system. The CF’s are constructed so as to be eigenfunctions of S, with the same eigenvalue. They are, however, not in general eigenfunctions of S2. The computer program has the facility to form linear combinations of CF’s which are eigenfunctions of S2 and to use these functions, which are generally just the traditional valence-bond structures, as the basis set for the expansion of the total electronic wavefunction for the system.The theory of the method has been described in detail elsewhere17 and the computer program which performs the ab initio multi-structure valence-bond calculations is available for general use.” The approximate atomic eigenfunctions are constructed from an atomic orbital basis set. In the present calculations we use atomic orbitals of double zeta quality, augmented by some additional orbitals to add flexibility to the basis set. These are, in fact, the same orbitals as those used in the preliminary diatomic calculations which have been published e1~ewhere.l~ The most notable aspect of the approximate atomic eigenfunction basis set is that both F-(lS) and H-(’S) were represented by functions which included a three-fold intra-atomic configuration interaction.This made it possible to obtain reasonably accurate electron affinities for fluorine and hydrogen. The calculated values for these quantities being respectively 0.1042 a.u. and 0.021 75 a.u., while the experimental values are 0.1273 a.u. and 0.027 75 a.u. Besides the extra s and p orbitals on fluorine and the extra s orbital on hydrogen which were required for the intra-atomic configuration interaction just mentioned, the following “ extra ” orbitals were also added to the atomic orbital basis set: 1. A set of p orbitals on hydrogen to allow for polarisation and a “ contracted ” 1s orbital to allow for distortion.GABRIEL G.BALINT-KURT1 AND ROBERT N. YARDLEY 79 2. 1s Gaussian orbitals at the centres of the LiF and HF bonds. 3. An extra very diffuse orbital on fluorine (exponent 0.0001). The last orbital was included to permit a correct description of HF- which auto-ionises at small H-F separations.16 The diffuse orbital effectively permits an electron to escape. In table 1 we tabulate the calculated and experimental dissociation energies for the diatomics HF, LiH and LiF [see ref. (15) for details]. As can be seen from the TABLE 1 .-DIATOMIC DISSOCIATION ENERGIES a large CF basis set small CF basis set ab initio OM ab initio OM experiment HF 0.202 1 0.208 2 0.190 1 0.198 3 0.224 7 LiH 0.080 97 0.081 02 0.079 93 0.079 96 0.092 46 LiF 0.185 8 0.199 1 0.180 9 0.194 3 0.220 6 (‘) All quantities in atomic units.1 a.u. of energy = 4.359 828 x 10-l8 J = 27.21 1 648 eV = 627.52 kcal mol-l. ( b ) The large CF basis set consisted of 113 functions and the small one of 28. (=) The large CF basis set consisted of 34 functions and the small one of 25. (‘) The large CF basis set con- sisted of 90 functions and the small one of 20. table, the valence-bond calculations yield between 82% and 95% of the experimental dissociation energies. These results compare quite favourably with other types of calculations in which much more extensive atomic orbital basis sets were used. Clearly for a triatomic system it is possible to construct a much greater number of CF’s, from the same set of approximate atomic eigenfunctions, than for a diatomic system. We, therefore, used the smaller diatomic CF basis sets as the starting point for the construction of our CF basis sets for the triatomic system.The CF basis set for LiHF was constructed in such a way that if any one of the atoms is removed to a large distance then the potential energy curves which result from a calculation are just those of the small basis set diatomic calculations for both the ground and excited states.15*16 Also included in the basis set were the CF’s needed to provide a reliable description of Li+ + HF- and H+ + LiF-.16 In the linear nuclear configuration, separate CF basis sets were constructed for I: and ll symmetries. These basis sets consisted of 278 and 175 CF’s respectively. For non-linear geometries two separate basis sets were again constructed, one of A’ symmetry which was even with respect to reflection in the molecular plane, and one of A” symmetry which was odd.The A’ basis set consisted of 447 CF’s and the A” of 267 CF’s. For non-linear geometries linear combinations of the CF’s were taken to form doublet spin states. The final basis sets consisted of 321 functions of 2A’ symmetry and 183 of 2A’’. One criterion for the reliability of a potential energy surface for a chemical reaction is the calculated value of the exo- or endo-ergicity of the reaction ( i e , the difference in energy between reactants and products ignoring the zero-point vibra- tional energy). In the present case the ab initio and OM methods yield 5.8 and 2.5 kcal mol-1 respectively for the endo-ergicity while the experimentalvalue is 2.6 kcal mol-’.The excellent agreement of the OM method with the experimental value is due to a fortuitous cancellation of errors. 3. THE COLLINEAR GROUND STATE SURFACE The surface which is perhaps of the greatest interest in discussing the Li + HF 3 LiF + H80 POTENTIAL ENERGY SURFACES FOR SIMPLE CHEMICAL REACTIONS reaction is that corresponding to the 2C+ ground state of the collinear Li - F - H geometry. Fig. 1 and 2 show contour maps of this surface as calculated by the ab initio multistructure valence-bond and OM methods respectively, while the corres- ponding numbers are presented in tables 2 and 3. The most significant point to 2 3 4 5 6 7 LI -F separation/a.u. FIG. 1 .--Contour map of the ground state, linear Li - F - H 'C+ potential energy surface calculated by the ab initio method.The zero of energy is taken to be the calculated energy of the separated ground state atoms. Energies are in atomic units. 5 ' 5 0 I i '. 1 I- I I I 1 1 2 3 4 5 6 7 Li-F separation / a.u. FIG. 2.-Contour map of the ground state, linear Li - F - H 'T,+ potential energy surface calculated by the OM method. The zero of energy is taken to be the experimental energy of the separated ground state atoms. Energies are in atomic units. notice when comparing the two calculations is that they agree on all qualitative aspects of the surface, and differ only slightly from a quantitative viewpoint. Fig. 3 shows a cut along the entrance valley of the reaction. As a lithium atom approaches an HF molecule from the fluorine end, with the HF internuclear distance fixed at its equilibrium value, the potential energy surface possesses a small but definite well.This well is predicted to have a depth of 2.2 kcal mol-' by the OM method and 1.6 kcal mol-1 by the ab initio method. Examination of the valence-bond wavefunctionGABRIEL G . BALINT-KURT1 AND ROBERT N . YARDLEY 81 1 .o 1.25 1.5 1.76 2.0 2.25 2.5 2.75 3 .O 3.5 4.0 4.5 5 .O Table 2" Ab initio GROUND STATE SURFACE FOR LINEAR Li - F - H 2.0 2.5 3.0 3.2 3.5 3.8 4.0 4.5 5.0 6.0 0.7178 0.4746 0.4169 0.4102 0.4066 0.4064 0.4069 0.4085 0.4096 0.4104 0.2942 0.0504 -0.0085 -0.0158 -0.0202 -0.0211 -0.0210 -0.0201 -0.0194 -0.0191 0.1541 -0.0893 -0.1487 -0.1561 -0.1608 -0.1620 -0.1619 -0.1611 -0.1605 -0.1602 0.1233 -0.1199 -0.1794 -0.1869 -0.1916 -0.1927 -0.1927 -0.1917 -0.1909 -0.1904 0.1336 -0.1105 -0.1697 -0.1769 -0.1813 -0.1821 -0.1818 -0.1805 -0.1795 -0.1785 0.1527 -0.0969 -0.1538 -0.1592 -0.1611 -0.1600 -0.1588 -0.1562 -0.1544 -0.1528 0.1560 -0.0989 -0.1541 -0.1574 -0.1546 -0.1479 -0.1430 -0.1334 -0.1284 -0.1248 0.1531 -0.1050 -0.1606 -0.1634 -0.1594 -0.1505 -0.1436 -0.1263 -0.1123 -0.0999 0.1510 -0.1100 -0.1662 -0.1691 -0.1647 -0.1552 -0.1477 -0.1284 -0.1107 -0.0848 0.1497 -0.1156 -0.1731 -0.1761 -0.1716 -0.1618 -0.1540 -0.1335 -0.1 143 -0.0828 0.1499 -0.1180 -0.1763 -0.1793 -0.1748 -0.1649 -0.1570 -0.1361 -0.1165 -0.0841 0.1503 -0.1190 -0.1776 -0.1806 -0.1761 -0.1661 -0.1581 -0.1371 - 0.1173 -0.0845 0.1506 -0.1195 -0.1781 -0.1811 -0.1765 -0.1664 -0.1584 -0.1373 -0.1 174 -0.0845 a All quantities are in atomic units.1 a.u. of length = 0.529 177 x 10-lo m. The zero of energy is the calculated energy of Li('S) + F('P0) + H('S). in the region of the well confirms our intuitive expectations that the well arises from the polarisation of a lithium atom by the permanent dipole moment of HF. A cut, through the same two surfaces along the exit valley of the reaction, with the LiF separation now fixed at its calculated equilibrium position, is shown in fig. 4. In order for the hydrogen atom to depart and for the reaction to be completed, the HF separation must increase from its equilibrium value, which is at the position of R H F YF 1 .o 1.25 1 .5 1.76 2.0 2.25 2.5 2.75 3 .O 3.5 4.0 4.5 5 .O TABLE 3" OM GROUND STATE SURFACE FOR LINEAR Li - F - H 2.0 2.5 3.0 3.2 3.5 3.8 4.0 4.5 5.0 6.0 0.6999 0.4569 0.3994 0.3927 0.3892 0.3891 0.3897 0.3914 0.3926 0.3934 0.2806 0.0369 -0.0219 -0.0291 -0.0334 -0.0341 -0.0339 -0.0328 -0.0320 -0.0316 0.1433 -0.1004 -0.1596 -0.1670 -0.1715 -0.1725 -0.1723 -0.1713 -0.1706 -0.1701 0.1142 -0.1294 -0.1888 -0.1962 -0.2008 -0.2018 -0.2016 -0.2004 -0.1995 -0.1987 0.1257 -0.1189 -0.1781 -0.1853 -0.1895 -0.1902 -0.1899 -0.1884 -0.1871 -0.1860 0.1455 -0.1047 -0.1616 -0.1670 -0.1687 -0.1675 -0.1662 -0.1634 -0.1615 -0.1596 0.1480 -0.1078 -0.1632 -0.1664 -0.1634 -0.1564 -0.1512 -0.1407 -0.1351 -0.1311 0.1444 -0.1149 -0.1708 -0.1737 -0.1696 -0.1606 -0.1536 -0.1360 -0.1211 -0.1062 0.1417 -0.1204 -0.1772 -0.1801 -0.1758 -0.1663 -0.1589 -0.1395 -0.1217 -0.0945 0.1398 -0.1268 -0.1850 -0.1881 -0.1837 -0.1740 -0.1663 -0.1460 -0.1269 -0.0960 0.1397 -0.1296 -0.1887 -0.1918 -0.1875 -0.1777 -0.1699 -0.1492 -0.1298 -0.0981 0.1399 -0.1308 -0.1903 -0.1935 -0.1891 -0.1793 -0.1714 -0.1505 -0.1310 -0.0989 0.1402 -0.1313 -0.1909 -0.1941 -0.1897 -0.1798 -0.1719 -0.1510 -0.1313 -0.0991 a All quantities are in atomic units.The zero of energy is the experimental energy of Li(2s> + F('P0) + H('S). the inner minimum in fig. 4, to infinity. There is clearly a sizeable barrier to the reaction which must be surmounted as the hydrogen atom departs and the system moves along the exit valley of the surface. The height of this barrier with respect to the reactants Li + HF is 21.3 & 0.5 kcal mo1-1 on both the ab initio and OM surfaces. The quoted uncertainty arises frem the fact that an insufficient number of calculations were performed to determine the barrier height more precisely.Fig. 5 shows the energy profile along the reaction coordinate. The reaction co- ordinate is taken to correspond to a lithium atom approaching to within its equili- brium distance of the fluorine atom while the HF distance remains fixed, and then82 POTENTIAL ENERGY SURFACES FOR SIMPLE CHEMICAL REACTIONS 3 4 5 6 Li -F separation 1 a.u. FIG. 3.Variation of potential energy along a cut in the ab initio and OM ground state, linear Li - F - H 'X+ potential energy surfaces with the H - F separation fixed at 1.76 a.u. The zero of energy is taken to be the calculated or experimental energy of the separated ground state atoms as appropriate. - 0.1 L -0 16 3 0 \ m i 2 -0.18 +- 0 W > 0 W x 0) W c ._ c - L L -0.20 1 I I 1 - 2 3 4 5 6 H - F separation 1a.u.FIG. 4.-Variation of potential energy along a cut in the ab initio and OM ground state, linear Li - F - H 'Z+ potential energy surfaces with the Li - F separation fixed at 3.2 a.u. The zero of energy is taken to be the calculated or experimental energy of the separated ground state atoms as appropriate.GABRIEL G. BALINT-KURT1 AND ROBERT N. YARDLEY 83 -o""rT--- - - 016/- I -2 -1 -_ . .. .. . i I 1 1 reaction coordinate l a u FIG. 5.-Energy profile along reaction coordinates of the ab initio and OM, linear, ground state Li - F - H potential energy surfaces. The zero of the reaction coordinate is taken to be at the coriier of the reaction (RLiF = 3.2 a.u., RHF = 1.76 a.u.). For negative values the reaction coordin- ate corresponds to (3.2 - RLIF) arid for positive values to (RHF - 1.76).the departure of the hydrogen atom while the LiF separation stays fixed at its equili- brium distance. That is, we treat the reaction as if there were no cutting-the-corner. As can be seen from the contour maps of the surface (fig. 1 and 2), this should be a reasonable approach. Fig. 5 demonstrates that by far the greater part of the barrier to reaction must be surmounted in the exit valley, after the system has turned the corner. 4. THE GROUND STATE SURFACE I N NONLINEAR The potential energy surfaces for the reaction were examined at selected nuclear configurations with the Li - F - H angle fixed at 135" and 90". In table 4 we compare the ground state potential energy surface as calculated by the OM method along the entrance channel of the reaction for the three different angles examined.In all three cases the H - F separation is held fixed at its equilibrium value of 1.76 a.u. while the Li - F separation is varied. We see from the table that the form of the surface is similar along the entrance valleys at all three angles. The depth of the shallow well in the entrance channel increases marginally (by about 0.25 kcal mol-') as the system is bent out of the linear configuration but then starts to decrease as the LiFH angle decreases below 120". This is demonstrated in curve A of fig. 6 where we show the variation of the ground state potential energy surface with angle for fixed H - F and Li - F separations. In curve A the internuclear separations correspond to the bottom of the shallow well in the entrance valley.We see that, as stated above, this curve starts to rise as the LiFH angle decreases below 120" and the potential NUCLEAR CONFIGURATIONS84 POTENTIAL ENERGY SURFACES FOR SIMPLE CHEMICAL REACTIONS TABLE 4.vARIATION OF THE GROUND STATE POTENTIAL ENERGY SURFACE ALONG ENTRANCE VALLEY OF THE Li + HF 3 LiF + H REACTION FOR DIFFERENT ANGLES OF APPROACH.- THE CALCULATIONS WERE PERFORMED USING THE ORTHOGONALTZED MOFFITT METHOD. THE H-F SEPARATION IS FIXED AT ITS CALCULATED EQUILIBRIUM VALUE (1.76 a.U.).' LiFH angle RLiF 180" 135" 90" 2.0 2.5 3 .O 3.2 3.5 3.8 4.0 4.5 5.0 7.0 9.0 11.0 co 0.1142 -0.1294 -0.1888 -0.1962 -0.2008 -0.2018 -0.201 6 -0.2004 -0.1995 -0.1986 -0.1985 -0.1984 -0.1983 - -0.1225 -0.1 88 1 -0.1963 -0.2013 -0.2023 - 0.2020 - 0.2006 -0.1994 - -0.1983 - -0.1098 - 0.1 829 -0.1925 - 0.1987 -0.2002 -0.2001 -0.1990 -0.198 1 -0.1982 -0.1984 -0.1983 - ' All quantities are in atomic units.The zero of energy is the experimental energy of the infinitely separated ground state atoms. -017 3 0 \ ul E z - 0 18 0 Q, > 0 c ._ c + -0 19 e !? x c al -0 20 \ - ', I I I t I I L L I I L i - F - H a n g l e / deg. FIG. 6.Variation of ground state Li - F - H potential energy surface with LiFH angle as calculated by OM method. The internuclear separations are &xed at: (A) RLiF = 4.0 a.u., RHF = 1.76 a.u. (B) RLiF = 3.2 a.u., R H ~ = 1.76 a.u.GABRIEL G. BALINT-KURT1 AND ROBERT N. YARDLEY 85 increases sharply for angles below 90".Curve B shows the variation of the potential with angle when the Li-F and H-F separations are fixed at values corresponding to the corner of the reaction path. In this case the potential is at first nearly constant as the angle is decreased, and then starts to rise at angles smaller than 120". In table 5 the ground state potential energy surface is compared along the exit TABLE s.-vARIATION OF THE GROUND STATE POTENTIAL ENERGY SURFACE ALONG EXIT VALLEY OF THE Li + HF -+ LiF + H REACTION FOR DIFFERENT LiFH ANGLES. THE CALCULATIONS WERE PERFORMED USING THE ORTHOGONALIZED MOFFITT METHOD. THE Li - F SEPARATION IS FIXED AT ITS CALCULATED EQUILIBRIUM VALUE (3.2 a.u.) (I LiFH angle RHF 180" 135" 90" 75" 1 .o 1.25 1.5 1.7 1.76 1.9 2.0 2.25 2.4 2.5 2.75 3 .O 3.5 4.0 4.5 5 .O 6.0 8 .o co 0.3927 - 0.0291 -0.1670 - 0.1 949 -0.1962 - 0.1921 -0.1853 - 0.1670 -0.1646 -0.1664 - 0.1737 - 0.1801 -0.1881 -0.191 8 -0.1935 - 0.1941 -0.1944 -0.1943 - 0.1942 - - 0.1659 -0.1948 -0.1963 - -0.1867 -0.1698 - 0.1665 -0.1675 - 0.1737 -0.1798 -0.1878 -0.191 6 - - - 0.1942 - -0.0157 -0.1 577 - 0.1897 -0.1925 -0.1894 -0.1808 -0.1788 -0.1789 -0.181 7 -0.1852 -0.1901 - 0.1926 - - - -0.1945 -0.1942 - 0.1820 0.1850 0.1858 0.1821 0.1820 0.1827 0.1858 - ~~ All quantities are in atomic units.The zero of energy is the experimental energy of the infinitely separated ground state atoms. valley of the reaction for different LiFH angles. At the first three angles considered the potential displays the same qualitative shape as shown in fig. 4. The height of the barrier to reaction, however, decreases quite dramatically with decreasing LiFH angle.Thus the barrier on the perpendicular surface (90") is about 12.2 kcalmol-'-a full 9 kcal mol-I lower than on the linear surface! 5. THE LOW-LYING ELECTRONICALLY EXCITED STATES OF LiHF Fig. 7 and 8 show some of the lower lying electronically excited states of the system, calculated using the OM method, along the entrance valley of the reaction Li + HF -+ LiF + H. Fig. 7 corresponds to a linear geometry with the lithium atom approaching the fluorine along the HF molecular axis, while fig. 8 corresponds to a perpendicular approach with the LiFH angle fixed at 90". The lowest curve in each corresponds to the ground state potential energy surface and correlates at large Li - F separations with a lithium atom and an HF molecule both in their ground states.The next group of states, two of them in fig. 7 and three in fig. 8, correlate86 POTENTIAL ENERGY SURFACES FOR S I M P L E CHEMICAL REACTIONS 3 5 7 9 Li -F separation / a u FIG. 7.-Cut through the five lowest linear Li - F - H potential energy surfaces as calculated by OM method with H - F separation fixed at 1.76 a.u. The zero of energy is taken to be the experi- mental energy of the separated ground state atoms. 1 1 - 1 7 1 1 7 6 L i --F - 1 i 2A’ / FIG. &--Cut through the six lowest perpendicular Li - F - H potential energy surfaces (LiFH angle = 90’) as calculated by OM method.GABRIEL G . BALINT-KURT1 AND ROBERT N. YARDLEY 87 at large distances with an excited lithium atom in its 2Po state plus a ground state HF molecule.At 9.0 a.u. the next state in both fig. 7 and 8 is found to correspond mainly to " Lif + HF(XIC+) + electron ". This comes about because HF- auto-ionises at small internuclear separations. As mentioned in section 2 our basis set included an extremely diffuse orbital which was intended effectively to permit the electron to escape and thus describe this auto-ionisation process. The highest curve in both figures corresponds, at an Li-F separation of 9.0 a.u., mainly to Li+ + HF- with the electron localised on the HF- molecule. Our preliminary calculations16 on the HF- ion confirm that the two states just discussed do in fact occupy their correct asymptotic positions relative to the other curves, although at an Li - F separation of 9 a.u.both curves are still rising and have not yet reached their corrected asymptotic values. In figs. 9 and 10 we show some of the low-lying electronically excited states of the system, calculated using the OM method along the exit valley of the reaction. In H-F separation h u . FIG. 9.-Cut through the nine lowest linear Li - F - H potential energy surfaces as calculated by OM method with Li - F separation fixed at 3.2 a.u. fig. 9 the system has a linear geometry while in fig. 10 the LiFH angle is fixed at 90". We have already commented on the fact that the barrier to reaction is much smaller for the perpendicular than for the linear nuclear arrangement. This barrier arises from an avoided curve crossing between states which can, to a first approximation, be described as Li(2S) + HF(X% +) for small H - F separations and LiF(XIX +) + H(2S) for larger separations. In the case of any avoided crossing of curves or surfaces the possibility of surface-hopping, or of a non-adiabatic transition, between the two surfaces arises. The probability of such a transition is increased by a close approach of the two surfaces.In the present case the two lowest surfaces approach closest to each other at an H - F separation of around 2.25 a.u. in the linear configuration.88 POTENTIAL ENERGY SURFACES FOR SIMPLE CHEMICAL REACTIONS 2 3 L 5 H -F separation / a u. I FIG. 10.-Cut through the seven lowest perpendicular Li - F - H potential energy surfaces, cor- responding to doublet spin states, as calculcated by OM method with Li - F separation fixed at 3.2 a.u.At this H - F separation the energy gap between the surface is 0.77 eV (17.8 kcal mol-') in the linear configuration and 1.73 eV (39.8 kcal mol-') in the perpendicular con- figuration. We see, therefore, that non-adiabatic transitions between these two surfaces will take place preferentially in the linear nuclear configuration. At large HF internuclear distances the nine curves shown in fig. 9 correspond in order of increasing energy to the states: 1. 'II -+ LiF(lII) + H(2S) 2. 2rI,411 -+ LiFCII) 3. H(2S) 3. 2Z+ -+ LiF(XIX+) + H(2,S') 6. 2X+ -+ LiF(2lC+) + H(2S) 7. 2C.+,4X+ + LiF(23X+) + H(2S) The curves shown in fig. 10 clearly have the same asymptotes, but only doublet states were examined for non-linear nuclear configurations.In comparing the curves along the entrance and exit valleys (figs. 7 and 9, or 8 and 10) it is very noticeable that there are no available low-energy excited states of the products. Indeed the threshold energy for producing excited products is 6.1 eV (calculated from experimental quanti- ties) and below this energy the only observable processes will be: 1. Reaction to produce products in their ground electronic state. 2. Non-reactive collisions to produce Li(2Po) + HF(XIX+). 3. Non-reactive collisions to produce " Li+ + HF(X'Z+) + electron ". 5.2c+:c+ -+ ~ i ~ ( 1 3 c + ) + H(~s)GABRIEL G . BALINT-KURT1 A N D ROBERT N. YARDLEY 89 6. DISCUSSION The main result of the present calculations is that they predict a ground state potential energy surface for the Li + HF LiF + H reaction which has a barrier to reaction in the exit valley.The barrier is predicted to decrease substantially as the LiFH angle decreases and the system becomes progressively more bent. Polanyi and co-workers 19s20 have performed trajectory calculations on systems with barriers to reaction in the exit valley. They find that the probability of reaction is increased to a much greater extent by the presence of vibrational energy in the reactant diatomic than by the presence of an equivalent amount of relative transla- tional energy. The fascinating experiments of Brooks and co-workers 10~11~13 have shown that, for the reaction K + HCl 4 KC1 + H, increasing the relative trans- lational energy available to the system has little effect, whereas excitation of the HCl with one quantum of vibrational energy results in an increase of the reactive cross section by two orders of magnitude.These experiments, together with Polanyi's calculations, lead to the conclusion that the potential energy surface governing the K + HCl reaction most probably has a barrier along its exit valley, and is, therefore, at least qualitatively similar to the surface reported here for its lighter analogue. In an attempt to reconcile the seemingly inconsistent, experimentally 4 9 determined, differential reactive cross sections for K + HBr and its isotopic analogues K + DBr and K + TBr, Roachg has proposed a model potential energy surface, together with an approximate discussion of the dynamics for the system. Roach's model potential for the K + HBr system is strikingly similar, on a qualitative level, to the one reported here for Li + HF.The main difference in the two potentials lies in their variation as the system is bent out of the linear geometry. Roach identifies two important nuclear geometries whose corresponding energies determine the ease with which reaction will occur. These geometries are for the LiHF system RHF = 1.76 a.u., RLiF = 3.2 a.u. which corresponds to the corner of the reaction, and the position of the maximum of the barrier along the exit valley. Using a very crude description of the dynamics of the system, it is the relative trans- lational motion which provides the energy to reach the corner of the reaction, while the vibrational energy in the reactant diatomic (i.e., HF) provides the energy difference between the corner and the top of the barrier. The variation of the energy at the corner of the reaction with LiFH angle is shown in fig.6(B). We see that this is essentially flat at first and starts rising steeply for angles smaller than 90". The variation of the energy at the top of the barrier can, to some extent, be deduced from table 5. From this table we see that the energy at the top of the barrier at first hardly varies as the system is bent out of a linear configuration and then decreases sharply between 135" and 90". Calculations of this quantity have only been performed at the four angles shown in the table. It seems reasonable to hypothesise that the energy at the top of the barrier will eventually rise as the angle is decreased below 75", and that the size of the hump (ie., the difference in energy between the corner and the top of the barrier) will continue to decrease and also that the hump itself might eventually vanish.The form of these changes of the potential with angle differ from those assumed by Roach. It seems likely, however, that the experimental results on K + HBr and its isotopic analogues could have been satisfactorily recon- ciled using a potential qualitatively similar to the LiHF potential presented here. There have been two previous calculations on the Li + HF system. The first of these was performed by Lester and Krauss2' and was an LCAO-SCF calculation which explored only the entrance valley of the reaction, with the H - F separation fixed at its equilibrium value. Our calculations are in good agreement with theirs90 POTENTIAL ENERGY SURFACES FOR SIMPLE CHEMICAL REACTIONS in that they predict a small well as the lithium approaches the fluorine end of the HF diatomic.Both calculations predict that the depth of the well increases marginally when the LiFH angle changes from 180" to 135" and then decreases sharply on going to 90". Our calculations predict a somewhat shallower well in the linear nuclear geometry but a deeper one for 90". The other previous set of calculations on the system were some performed by Baht-Kurti and Karp1~s.l~ These calculations used a smaller atomic orbital basis set, which was of a poorer quality than that used here. They also did not include any intra-atomic configuration interaction to improve the H- and F- wavefunctions as do the present ones.While several qualitative features of the potential surface reported in the previous work are similar to those reported here, there are important quantitative differences. These differences, which are evident in both the ab initio and OM calculations, are thought to arise mainly from an inadequate description of the F- wavefunction in the previous work. In the present case all features of the ab initio and OM potential surfaces are very similar. This is due to the fact that the approximate atomic eigenfunctions which are used to build up the molecular basis functions are all of a reasonably good and uniform q~a1ity.l~ Besides the experiments at thermal energies, experiments on alkali metal atom- hydrogen halide systems have been performed at hypthermal energies12*14 in the range 1 - 20 eV.In both these sets of experiments alkali metal ions were observed. Lacmann and Herschbach l2 in their experiments also looked for the production of electronically excited potassium atoms from K + HCl collisions, but failed to find them in significant quantities. The production of the alkali metal ions has been discussed in terms of avoided curve crossings which are represented as occurring as the alkali metal atom approaches the hydrogen halide molecule. Comparing Lac- mann and Herschbach's schematic picture of the low-lying potential energy curves of the system [fig. 3 of ref. (12)] with our calculated curves along the entrance valley of the reaction (fig.7 and S), we see that the figures show no qualitative similarity. As a consequence of this incompatibility we would like to suggest a possible alternative mechanism for the production of the alkali metal ions in alkali metal atom-hydrogen halide collisions. This mechanism involves the rapid departure of the hydrogen atom when the alkali metal atom has approached close to the halogen. The departing hydrogen atom takes away only a very small proportion of the initial relative kinetic energy, and therefore leaves the alkali halide molecule in a meta-stable state with enough " vibrational " energy to dissociate. This dissociation takes place to produce ionic fragments with a high p r ~ b a b i l i t y . ~ ~ , ~ ~ From our previous discussion we see that the departure of the hydrogen atom will be favoured (for the Li + HF system) for collisions having small LiFH angles.The viability of this mechanism depends on the continued decrease of the height of the hump in the exit valley with decreasing angle for angles below 90". At present we have had to hypothesise that this is the correct form of the potential, a more definitive discussion must await the outcome of further calculations. It should be noted that this proposed mechanism also rationalises the absence of excited alkali metal atoms in Lacmann and Hersch- bach's experiments, as such atoms are not produced by the dissociation of highly vibrationally excited alkali halide molecules. The authors are indebted to the Science Research Council for a grant of computer time and would like to express their gratitude to the staffs of the Atlas and Rutherford Computer Laboratories for their assistance.In particular, we are grateful for the provision of the ATMOL 2 integral evaluation package which was used in this work. R. N. Y. also wishes to thank the S.R.C. for financial support.G A B R I E L G . BALINT-KUR'I'I A N D ROBERT N. YAKDLEY 91 E. H. Taylor and S . Datz, J. Chem. Phys., 1955, 23, 1711. D. Beck, E. F. Green and J. ROSS, J. Chem. Phys., 1962,37,2895. J. R. Airey, E. F. Green, K. Kodera, G. P. Reck and J. Ross, J. Chem. Phys., 1967,46, 3287. K. T. Gillen, C. Riley and R. B. Bernstein, J . Chem. Phys., 1969, 50, 4019. C. Maltz, N. D. Weinstein and D. R. Herschbach, Mol. Phys., 1972, 24, 133. D. S. Y. Hsu and D. R. Herschbach, Furuduy Disc. Chern. Soc., 1973,55, 116. D. R. Herschbach in Proceedings of the Coilfereizce on Potential Energy Surfaces in Chemistry, ed. W. A. Lester Jr., 1970, IBM Research Laboratories, San Jose, California, Publication RA18. A. C. Roach, Chem. Phys. Letters, 1970, 6, 389. ' L. R. Martin and J. L. Kinsey, J. Chem. Phys., 1967, 46, 4834. lo T. J. Odiorne and R. P. Brooks, J. Chem. Phys., 1969, 51,4676. l1 T. J. Odiorne, P. R. Brooks and J. V. V. Kasper, J . Chem. Phys., 1971, 55, 1980. l2 K. Lacmann and D. R. Herschbach, Clzenz. Phys. Letters, 1970, 6, 106. l3 J. G. Pruett, F. R. Grabiner and P. R. Brooks, J. Chem. Phys., 1974,60,3335; 1975,63, 1173. l4 C. E. Young, R. J. Beuhler and S . Waxler, J. Clietn. Phys., 1974, 61, 174. l5 R. N. Yardley and G. G. Balint-Kurti, Mol. Phys., 1976, 31, 921. l6 R. N. Yardley, Ph.D. Thesis (Bristol University, 1975). l7 G. G. Balint-Kurti and M. Karplus in Orbital Theories of Molecules and Solids, ed. N. H. March (Clarendon Press, Oxford, 1974). MULTIBOND A, Distributed by Quantum Chemistry Program Exchange, Chemistry Depart- ment, Indiana University, Bloomington, Indiana 47401, U.S.A. l9 D. S . Perry, J. C. Polanyi and C. W. Silson Jr., Cheni. Phys. Letters, 1974, 24, 484. 2o D. J. Douglas, J. C. Polanyi and J. J. Sloan, Chem. Phys., 1976, 13, 15. 21 W. A. Lester, Jr. and M. Krauss, J. Cheni. Phys., 1970, 52, 4775. 22 R. S. Berry, T. Cernoch, M. Caplan and J . J. Ewing, J . C'hern. Phys., 1968, 49, 127. 23 Von R. Hartig, H. A. Olschewski, J. Troe and H. G. Wagner, Ber. Bimsenges. Phys. Chem., 1968,72,1016.

ISSN:0301-7249    

DOI:10.1039/DC9776200077

出版商:RSC    

年代:1977

数据来源: RSC