3. THEORETICAL CHEMISTRY By J. G. Stamper (The Chemical Laboratory University of Sussex Brighton BN1 9Q4 1967 has not seen any single outstanding achievement in the field of theoretical chemistry. Rather there has been steady progress on a number of fronts some of which are reported here. I have made no attempt to report on progress in scattering theory although this is a very active field of theoretical chemistry, I have commented in some detail on electron correlation Hartree-Fock calcula-tions semiempirical methods and intermolecular forces and there is also a miscellaneous section devoted to several topics which could not be grouped under any single head. At present it is probably fair to say that almost all calculations on the elec-tronic structure of atoms and molecules follow a common basic approach.The problem is thought of as being in two parts the calculation of a molecular-orbital wave-function and a subsequent calculation or estimation of correlation effects. A calculation which leads to a wave-function expressed in terms of molecular orbitals takes account of the electrostatic repulsion between the electrons only by considering that each electron moves in a potential which is the result of averaging the effects of the remaining electrons over their motion. Mathematically this leads to the well-known Hartree-Fock equation. The wave-functions which result from a solution of this equation are suffici-ently accurate for many purposes. They give a very accurate account of the charge distribution in atoms and molecules and a number of other experi-mentally observable qualities (dipole moments diamagnetic susceptibilities etc.).They also give good results for the energy of the system expressed in terms of percentage accuracy. However the total energies of most atoms and molecules are so large compared with chemical energies that the few percent inaccuracy of the Hartree-Fock energy is comparable for example with the bond energy of the molecule. Thus Hartree-Fock calculations of bond energies are usually only accurate to about 50 % and in some cases e.g. F, a known stable molecule is actually predicted to have a negative bond energy. Some other atomic and molecular properties are also badly predicted from the Hartree-Fock wave-functions the nuclear hyperfine coupling constant fQr phosphorous being an example.The remaining energy beyond the Hartree-Fock result is due to the fact that an electron in a molecule actually moves in the instantaneous field of the remaining electrons and the motion of the electrons is thus correlated. Electron Correlation.-A number of methods have been used to introduce electron correlation into atomic and molecular wave-functions. The two most used methods dating back to the very early days of quantum mechanics ar 24 J. G. Stamper the introduction of inter-electronic distances (rij) as explicit variables in the wave-function and the method of configuration interaction (C.I.). In the latter, the wave-function is expressed as a linear combination of determinantal wave-functions in which the leading term is the ordinary molecular orbital wave-function.The application of both methods to simple systems such as the hydro-gen molecule and the helium atom though laborious presents no fundamental difficulties and has been carried out repeatedly with increasing accuracy. When attempts are made to deal with larger systems however both methods quickly become intractable in their basic form. In the case of the rij method the number of rij co-ordinates increases as i N ( N - 1) where N is the number of electrons, while the number of degrees of freedom of the system increases only as 3N so that for quite a small value of N the rij’s cease to be independent and the method fails. In the case of configuration interaction the number of configurations needed for a given accuracy increases rapidly with N so that the complete problem soon becomes unmanageable.One way through this impasse is to deal with the correlation in small sub-groups of electrons independently. The idea originated with theoretical physicists’. working on problems of nuclear structure and its application to chemical systems was probably first suggested by Sinanoglu3 and S Z ~ S Z . ~ Sinanoglu pointed out that pair correlations were likely to be the most impor-tant and that each electron correlated mainly with one other in atomic and molecular systems (the one in the same molecular orbital). He also calcu-lated the correlation energy in the beryllium atom’ using these ideas but esti-mating certain terms. 1967 has seen the first application of this approach to a larger atom by Nesbet.6 In principle the method consists of selecting a group of electrons (say the pair in a 1s orbital) and solving a modified Schrodinger equation (the Bethe-Goldstone equation) for the detailed motion of these electrons in the average potential of the remaining electrons (the so-called Fermi sea or Hartree-Fock sea).Any method may be used to solve the Bethe-Goldstone equation but in practice Nesbet uses a variational method. An important constraint on the solution is that it must be orthogonal to the Hartree-Fock solution. This is very difficult to achieve if a variation function containing r j j is used though this would otherwise be the obvious choice. In consequence Nesbet uses a configura-tion interaction calculation which also allows formulation of the problem in a very simple way.Let Q be the solution of the Hartree-Fock equation and let @$ be a con-figuration (Slater determinant) in which electrons have been excited from spin-orbitals a b . to spin-orbitals i,j . . Then the complete C.I. wave-function can K. Brueckner Phys. Rev. 1954,% 508; 1955,97 1353. H. A. Bethe and J. Goldstone Proc. Roy. SOC. 1957 A 238,551. 0. Sinanaglu J . Chem. Phys. 1962,36,706. L. Szasz Z. Naturforsch. 1960,15a 909; Phys. Rev. 1962,126 169. D. F. Tuan and 0. Sinanogly J. Chem. Phyi. 1964,41,2677. R. K. Nesbet Phys. Rev. 1967,155,51,56 Theoretical Chemistry 25 be written where the summations run over all possible values of abc . . . ijk . . . . The solution of the Bethe-Goldstone equation for the pair of electrons in spin-orbitals a and b is i j where the coefficients C:; are found by the variation principle using the full Hamiltonian operator for the system i.e.by a straightforward matrix diagonali-zation procedure. If the energy obtained as a result of this calculation is Eab, and Eo is the energy obtained from the Hartree-Fock wave-function Nesbet defines the pair correlation energy for the pair ab as The process can then be repeated for all other pairs of electrons and the total pair correlation energy e12 found. e 1 2 = z e a b a b (4) This approach can be extended formally to three electron and larger groups in an obvious way and the total energy of the system then appears as The advantage of this is that although calculation of all the energy increments e requires a solution of the complete C.I.problem for the system it seems certain that each set of increments will be much smaller than the previous set so that a well-ordered set of approximations results. Expectation values of other operators can be treated in the same way as the energy. Nesbet has applied the method to the beryllium and neon atoms with very satisfactory results some of which are given in Tables 1 and 2. Nesbet did not calculate many three-electron terms for neon. The one expected to make the largest contribution gave an approximate e of +O*OOO261 which must be too large since a variational principle is being used and which is fairly small rela-tive to the accuracy of three places of decimals for which he was aiming. These results present a number of interesting features which cannot be commented upon here except to point out that they support the use for calcu-lations of moderate ‘chemical’ accuracy of the idea that each electron correlates principally with one other.Essentially the same approach has been followed by S z a s ~ ~ who has carried out calculations on the beryllium atom using explicit rij functions instead of C.I. He has treated 1s 1s and 2s 23 correlation only and obtains values of eab for these of -0-0423 a.u. and -00445 a.u. which agree satisfactorily with ’ L. Szasz and J. Byme Phys. Rev. 1967,158 34 26 J. G. Stamper TABLE 1 Correlation energies in Be Electron Group eab or eabe (a.u.) 1s 1s 1s 2s 1s 2s 2s 2s Total pairs l s l i 2s ls2s 2s Total triples Total correlation energy Experimental correlation energy -0.041827 -0*000813 - 0.0021 19 0.0453 5 1 + O-ooOo30 + 0.000428 + 0.O009 1 6 - 0.093042 - 0.092 126 - OW3897 TABLE 2 Correlation energies in Ne Total e for all Type of Pair No.of Pairs pairs of type (a.u.) K-shell K-shell 1 - 0.039932 K-shell Lshell 16 - 0.025026 Lshell L-shell 28 - 0.3 17269 - 0.382227 Total pair energy -Experiment a1 correlation energy - - 0.393 the values of Nesbet in Table 1. An interesting finding of Szasz is that the pair energies appear not to be quite additive. The correlation energy obtained by treating the 1s 1s and the 2s 2s pairs simultaneously is less by about 0.003 a.u. than the sum of the energies obtained by treating the two pairs separately. Hartree-Fock Calculations.-The Hartree-Fock wave-function that forms the starting point for the electron-correlation calculations is itself the typical goal of most calculations on small molecules.1967 in common with the past 5 or 6 years has seen a number of such calculations of varying degrees of accuracy. Those reported below illustrate various ways in which this kind of calculation can be of interest. The most striking development in this field has been the attempt to obtai Theoretical Chem is try 27 Hartree-Fock wave-functions for quite large molecules which has been made simultaneously by two sets of workers. Preuss and Diercksen’ have carried out calculations for benzene the cyclopentadienyl anion and cyclopropane together with some smaller systems while Clementi et aL9* have treated tkie NH,-HCI reaction system pyridine pyrrole and pyrazine.One of the biggest problems in trying to calculate Hartree-Fock wave-functions for molecules as large as these is the very large number of multicentre integrals which have to be evaluated. If as is usually the case the molecular orbitals are constructed from Slater-type atomic orbitals (STO’s) Ylmrn exp( - cr) the evaluation of these integrals can be done only by summation over rather slowly-convergent series which requires a prohibitive amount of com-puter time. Both these groups of workers have evaded this difficulty by using Gaussian orbitals (GTO’s) (x’J+”z“ exp( -c?)) as a basis in the calculation for all the integrals needed for this type of calculation can be obtained in closed form when a Gaussian basis is used.The use of a Gaussian basis is not new but it has not been popular in the past because a very much larger number of GTO’s than STO’s is needed to obtain a wave-function of comparable accuracy. For example one STO gives an absolutely accurate result for the hydrogen atom while five or six GTO’s are needed for results of ‘chemical’ accuracy. Thus these calculations must employ very large bases which are made tractable by contraction. This process consists of con-structing certain combinations of GTO’s on each centre which are then treated as a unit in the calculation the coefficients of the GTO’s being kept in constant ratio. The combinations are chosen on the basis of calculations on smaller systems and it has been shown that this process does not significantly decrease the accuracy of what are in any case approximate calculations.A number of interesting results follow from these calculations. One is that in all the aromatic compounds the lowest-lying n-orbital is more strongly bound than the highest o-orbitals. For example the second ionization potential of benzene is predicted to be of a o rather than a n electron. The second is that the nitrogen atom in pyrrole carries a net negative charge which is the result of two opposing effects a transfer of electrons from carbon to nitrogen in theo system and a transfer from nitrogen to carbon in the n-system. This suggests that care is required in the use of Ir-electron only cakulations on such a system. A second kind of application of Hartree-Fock calculations is to smaller molecules in cases where they can throw light on experimental problems.Carlson Claydon and Moser,’ ’ for example have carried out approximate self-consistent field calculations on TiN to assist in the elucidation of its complex electronic spectrum by predicting the kinds of transition to be expected. For many diatomic molecules this kind of prediction can be made using simple H. Preuss and G. Dierksen fnt. J . Quant. Chem. 1967,1,349,357,361,365,369 373. E. Clementi J . Chem. Phys. 1967,46,3851,4731,4737. K . D. Carlson C. R. Claydon and C. Mom J . Chem. Phys. 1967,46,4963. l o E. Clementi H. Clementi and D. R. Davis J . Chem. Phys. 1967,46,4725. * 28 J. G. Stamper qualitative considerations. When transition elements are present however, this is not possible and this kind of calculation can be of great utility.A related use of SCF calculations is that by RichardsI2 on CS. He has carried out an SCF calculation comparable in accuracy to that of Nesbet13 on CO and concludes that thiocarbonyl complexes should be more stable than the corres-ponding carbonyls. The lone pair electrons on carbon are more weakly bound in CS than CO suggesting that CS will form stronger CT bonds to metals while the lowest unoccupied n-orbital in CS lies lower than in CO suggesting that back donation is likely to be more effective. Cade and Huoi4 have made exhaustive Hartree-Fock calculations on the diatomic hydrides of all the first and second row elements. These calculations are for almost all the molecules considered of considerably greater accuracy than any previous calculation the energies being accurate to about 0-02 ev.* The authors present an extended discussion of the results which cannot con-veniently be summarised here but one conclusion which they emphasise is that the difference between their result and the experimental energy is nearly constant over a considerable region near the minimum in the potential energy curve.Thus the correlation energy is almost constant over this region and Hartree-Fock calculations of spectroscopic constants are fairly reliable for these molecules at least. Semi-Empirical Methods.-The authors of the calculations on large mole-cules described in the last section stress that they are only obtaining approxi-mate solutions of the Hartree-Fock equations and it is clear that it will be a long time before exact solutions are available for any considerable number of large molecules.More approximate approaches to calculating the properties of larger molecules are therefore still necessary. The most familiar and successful of such methods is that of Pariser and Parr,15 and Pople16 (the P method) for n-electron systems sometimes referred to as the zero-differential overlap approximation. The principal difficulty in the application of exact non-empirical molecular-orbital calculations to large systems is the very large number of integrals over atomic orbitals which must be calculated stored and handled. In the P method, the number of integrals is drastically reduced by : (i) neglecting all the electron-repulsion integrals in which the same electron appears on two different atoms : (ii) obtaining values for the remaining electron-repulsion integrals by a mixed procedure partly from non-empirical calculation and partly from atomic spectroscopic measurements ; l2 W.G. Richards Trans. Faraday SOC. 1967,63,257. l 3 R. K. Nesbet J . Chem. Phys. 1964,40 3619. l4 P. E. Cade and W. M. Huo J . Chem. Phys. 1967,47,614,649. Is R. Pariser and R. G. Parr J . G e m . Phys. 1953,21,466,767. l6 J. A. Pople Trans. Faraday SOC. 1953,49,1375. * This is the accuracy in the Hartree-Fock energy. There is a much larger difference from experi-mental results due to correlation effects Theoretical Chemistry 29 (iii) treating the nuclei and cr-electrons of the molecule as a fixed core in whose field the a-electrons move.Some of the integrals involving this cort are neg-lected others are obtained from atomic spectral data while one p which corresponds quite closely to the p of Huckel theory is chosen so as to obtain the best fit with experiment. This method has proved very fruitful over the past ten years but it is only comparatively recently that much serious effort has been devoted to extending this kind of approach to other kinds of molecule. One particular exte-sion which appears rather promising is the so called CNDO (complete neglect of differential overlap) method of Pople Santry and Segal.”? 18* l9 In this method all the valence electrons of the molecule are considered ex-plicitly and approximations which are essentially those of the P-method are introduced.There are two features which are new to the theory as compared with the P-method. Firstly p instead of being taken as a constant is set pro-portional to the corresponding overlap integral, p = ptBs where ptB is a constant for a particular pair of atoms chosen to give agreement with more accurate calculations on simple systems.* Secondly in order that the results should be invariant under either a change of axis system or a change from atomic orbitals to hybrid orbitals as a basis (the solution to the full Hartree-Fock equations has this property) it is necessary to ignore differential overlap between different kinds of orbital on the same atom and to set all electron-repulsion integrals involving the same pairs of atoms equal.Although the method was not intially described in the year under review a number of applications and extensions have appeared which suggest that it, or something close to it is likely to be of wide utility in the future. For example it has been applied in its original form to the electronic spectra of small poly-atomic molecules by Kroto and Santry.2’ In their first paper they treat HNO, HCF H2C0 C2H2 and NH, using the virtual orbitals that result from an SCF calculation on the ground state as an approximation to the orbitals occupied in the excited state. As would be expected the excitation energies calculated are rather high (e.g. 6.80 ev for NH3 as against the experimental value of 5-72 ev) but they obtain good agreement in most cases with the observed change in geometry on excitation.The exception is the case of formaldehyde where they predict a planar configuration for both the first singlet and the first triplet excited states. In all cases the geometry depends on a balance between the change in orbital energies on excitation and the change in electron repulsion ” J. A. Pople D. P. Santry and G. A. Segal J . Chem. Phys. 1965,43 S129. J. A. Pople and G. A. Segal J. Chem. Phys. 1965,43 S136. l9 J. A. Pople and G. A. Segal J . Chem. Phys. 1966,44,3289. 2o R. S. Mulliken J. Chim. phys. 1949,46 497. ” H. W. Kroto and D. P. Santry J. Chem. Phys. 1%7,47,792,2736. Such an approximation is not of course a new idea. It goes back at least to 194920 and probably much earlier 30 J. G. Stamper terms and for formaldehyde these two are very similar in magnitude so that the lack of success in prediction is not very surprising.The experimentally estimated barrier to inversion in the singlet state is only 0.081 ev. In their second paper they use a simple procedure to optimise the molecular orbitals for the excited state and obtain somewhat improved values for the excitation energies. They are also able to obtain a non-planar equilibrium con-figuration for the singlet excited state of formaldehyde with an out-of-plane angle of 15" (experimental value 3 1 "). The CNDO approximations have also been used to attack this problem by Dixon22 who using a slightly different set of parameters (CNDO/l instead of CND0/2) and including some one-centre exchange terms obtained comparable results.The one-centre exchange terms are important in accounting for singlet-triplet separation and also in planar n-free radicals in allowing some spin density to appear ino-orbitals thus accounting for isotropic hyperfine coupling. Pople et aL2 have carried out calculations similar to Dixon (but using CND0/2) and have also looked at the question of spin densities in methyl and ethyl radicals. They do not attempt to predict the absolute values of the coupling constants but obtain quite satisfactory results for the ratios of these constants in the two radicals. The original CND0/2 set of approximations have been used in a study of d-orbital participation in bonding in some compounds of elements of the second row of the periodic table.24 The CNDO method does not give any definite method for treating d-orbitals so the authors have made three series of calcu-lations two using different and extreme ways of representing the d-orbitals and the third omitting them altogether.A number of interesting though perhaps rather tentative conclusions are drawn from these results the most striking of which is that the geometries of the molecules they consider which include the celebrated case of the T-shaped ClF, are unchanged by the inclusion of d-orbitals in the basis-set of atomic orbitals. On the other hand charge distributions and dipole-moments are considerably affected by the inclusion of d-orbitals. The most noticeable effect is in compounds where the second row element is in a higher than usual valence state (SO:- SFs PF5 ClF,).When d-orbitals were omitted a large positive charge accumulated on the central atom ; d-orbitals besides strengthening theo bonds allowed a good deal of this charge be back donated through rbonding. The CND0/2 approximations are also able to give a satisfactory account2' of bond lengths in first row diatomics (to 0-05 A) though not of the force constants or dissociation energies in these molecules. A similar set of approximations has been used by Dewar and Klopman26 '* R. N. Dixon Mol. Phys. 1967 12 83. " J. A. Pople D. L. Beveridge and P. A. Dobosh J . Chem. Phys. 1967,47 2026. 24 D. P. Santry and G. A. Segal J . Chem. Phys. 1967,47 158. 25 G. A. Segal J . Chem. Phys. 1967,47,1876. 26 M. J. S. Dewar and G. Klopman J . Amer. Chem. SOC.1967,89,3089 Theoretical Chemistry 31 in a discussion of a large number of hydrocarbons the main difference being the inclusion of certain one-centre and two-centre electron-repulsion integrals neglected in CNDO. Although this procedure destroys the formal invariwce under a change of axis system the authors have found that their results are invariant to the accuracy of their calculations. The results are very encouraging. Not only are the heats of formation of saturated and unsaturated hydro-carbons predicted to an accuracy of “chemical” size (usually better than 4 kcal.) but the correct conformations are predicted for ethane cyclohexane, 2-butene and 1,3-butadiene. On the other hand the calculations predict that n-alkanes should be more stable than branched-chain alkanes which is in contradiction with experimental results.Unfortunately the extended approximations of both Dewar and Klopman,26 and Pople et ~ 2 1 . ~ ~ lack the theoretical justification of the plain CNDO method. The success of the P-method for n-systems has been explained27 by showing that its approximations are equivalent to starting from a symmetrically orthogonalised set of atomic orbitals as basis. Dah12* has shown that the CNDO method can also be rationalised on this basis-but that the various partial extensions cannot. However the proof of the pudding is in the eating and the partial successes of various authors lead one to say with Dewar and Klopman that “this kind of approach has exciting possibilities”. Interatomic Forces.-The interaction between atoms or molecules at moderately large distances is an obvious situation in which a perturbation-theory approach is likely to be fruitful.Indeed such calculations were carried out in the very early days of quantum mechanics by a number of workers, especially London.29 These early workers almost all used a simple product of the wave-functions of the two atoms (or molecules) as the unperturbed wave-function of the system. Such an approach cannot be correct as by the Pauli principle the wave-function for the system must be antisymmetric under any interchange of electrons and electrons cannot therefore be associated with particular atoms. These two different starting points do not lead to significantly different results provided that the overlap between the wave-functions on the two atoms is negligible so that the original method of calculation is satisfactory at large interatomic distances.The interactions of non-bonded atoms and molecuks at intermediate distances have in recent years become of considerable interest since quite accurate potential energy curves for such systems have been derived from atomic beam and other data. Consequently a number of attempts have recently been made to apply perturbation techniques to the situation in which the overlap is not negligible. When this is attempted in a straightforward way a difficulty arises because 27 I. Fisher-Hjalmars J . Chem. Phys. 1965,42,1962. 28 J. P. Dahl Acta Chem. Scad. 1967,21,1244. 29 F. London 2. Physik 1930,63,245 32 J. G. Stamper the unperturbed wave-function is not an eigenfunction of the unperturbed Hamiltonian for the system.This can easily be seen in the case of two hydrogen atoms A and B. Here the simple product function is $ = hA(l)hB(2) while the correct antisymmetrical function is Y = lsA(l)l~B(Z) - 1sA(2)1sB(1). The unperturbed Hamiltonian is the sum of two hydrogen atom Hamiltonians one for each electron HO = HA(1) + HB(2) (6) Here Jr is an eigenfunction of Ho HoY! = 2E,ls~(l)ls~(2) - ~SA(~)HA(~)~SB(~) - ~SB(~)HB(~)~SA(~) (8) If we now carry out a perturbation calculation in the usual way putting H = H o + h V E = E o + h E + Y = Yo + h Y 1 + HoYo + AHOYl + AVY0 = EoY0 + AEIYo + AEOY, (9) (10) HoYo = EoYo (11) we obtain to first order in h In ordinary perturbation theory we have so that all the terms in (10) are of first or higher order in h.In this case where (11) does not hold (10) contains terms of both zero and first order in A. The different orders of perturbation theory are therefore not well defined and what terms should be included in a calculation “correct to second order” is not clear. An additional difficulty arises when an attempt is made in the usual way to calculate the perturbed wave-function in terms of excited-state wave-functions for the unperturbed system. This is because the set of all antisym-metrised products of excited-state functions is overcomplete* and therefore there is no unique way in which the perturbed wave-function can be expressed in terms of them. During the year a number of authors have attempted to solve this problem and several expressions have been obtained for the first and second order energies in such a system.Amos and Musher3’ and Murrell and Shaw3’ 30 A. T. Amos and J. I. Musher Chem. Phys. Letters 1967 1,149. 31 J. N. Murrell and G. Shaw J . Chem. Phys. 1967,46 1768. A complete set is one such that any function can be expressed in terms of it. An overcomplete set is such a set with additional functions in it. It follows that some functions in such a set are in fact linear combinations of other members of the set Theoretical Chemistry 33 avoid the problem of overcompleteness by using an antisymmetrised product only for the unperturbed function and by using simple products of excited-state functions to express the perturbed wave-function.Amos and Musher3' use a dummy coefficient analogous to h in equations (9) to (11) above to define the order of perturbation theory while Murrell and Shaw31 use a projection-operator technique. The two approaches however have been shown to lead to equivalent results.32 An alternative approach which has also received attention during the year is that due initially to Eisenschitz and London.33 This work has been revived and rewritten in a modern form by van der A ~ o i r d ~ ~ who uses the overcomplete set of antisymmetrised products but gives a specific method for choosing the expansion coefficients. Hir~chfelder,~~ in the course of a rather heavily mathe-matical survey of the whole problem has extended the work of van der Avoird to form what he designates as the HAV method and has given a variational principle analogous to the Hylleraas principle36 in conventional perturbation theory for finding the first-order perturbed wave-function and the second-order energy without expanding in terms of any set of excited states.All the approaches agree that the first-order energy (in the case of neutral spherically-symmetric species) corresponds to the Heilter-London (valence-bond) chemical-binding energy which is zero in the absence of overlap while the second-order energy is made up of two types of contribution-the familiar van der Waals dispersion energy due to interaction between an instantaneous dipole in one atom with an induced dipole in the other and additional exchange-type terms. The various approaches differ in detail.The decision as to which of these approaches is the most satisfactory will probably have to await the results of numerical applications which have so far been few. Murrell and S h a ~ ~ ~ have applied their method to the interaction of two helium atoms. The potential energy that they obtain has a minimum energy of -3.3 x ev) at an internuclear distance of 5.5 a.u. (= 2.9 A) and appears to agree with the experimentally derived poten-tial-energy curve to within the limits of experimental error. The general importance of the overlap terms is indicated by a calculation of McQuarrie and Hirschfelder3* on H2+. Using a form of perturbation theory published previo~sly,~' differing considerably in approach and to a smaller extent in results from HAV they obtain the energy accurately up to a distance of 4 a.u.(2 A) while traditional van der Waals forces which consist in this case a.u. (= 8-9 x 32 R. E. Johnson and S. T. Epstein University of Wisconsin Theoretical Chemistry Institute Reports 33 R. Eisenschitz and F. London Z . Physik 1930,60,491. 34 A. van der Avoird Chem. Phys. Letters 1967 1 24. 35 J. 0. Hirschfelder Chem. Phys. Letters 1967 1 325 363. 36 See for example J. 0. Hirschfelder W. Byers-Brown and S. T. Epstein in Adv. Quantum Chem., 1964 I 3' J. N. Murrell and G. Shaw Mol. Phys. 1967,12,475. 39 J. 0. Hirschfelder and R. Silky J . Chem. Phys. 1966,45,2188. NO. WIS-TCI-266 1967. D. A. McQuarrie and J. 0. Hirschfelder J . Chem. Phys. 1967,47,1775 34 J. G. Stamper of a charge-induced dipole interaction can only account for the energy of the system up to 15 a.u.Miscellaneous Topics.-Lower bounds and related topics. Essentially all theoretical chemical calculations involve the use of the variation theorem to find an approximate wave-function for a system. Such calculations therefore, give an upper limit to the energy of the system-and it would often be of interest to have additional information about the accuracy of the result of the calculation such as a lower limit to the energy or limits on the accuracy of the expectation values of other quantities calculated from the wave-function. A number of lower bound formulae exist:' but are all awkward to apply because the expectation value of the square of the Hamiltonian has to be evaluated. This is very much easier to do with Gaussian than with Slater orbitals (cf p.27) and Schwarts4' has investigated the kinds of lower bound which can be expected in Gaussian calculations.Taking a Gaussian calcula-tion of the energy of the hydrogen atom as a simple example a calculation which gives an upper bound-i.e. the ordinary 'calculated' energy-correct to 6 x kcal. mole-' has a lower bound 20 kcal. mole-' too low. In passing this illustrates the way in which quite inaccurate wave-functions can lead to very good expectation values of the energy. Wilson and c o - w ~ r k e r s ~ ~ * ~ ~ have also been looking at the lower boundary question and have used an approximate version of the method of Bazley and to investigate the accuracy of some derived values. Their approximation is only justified when the wave-function is a rather accurate one as it involves neglecting terms of second-order in E(E' = 2 - 2s where S is the overlap of the approximate wave-function with the true one) so they have applied it only to accurate H2 and He wave-functions.For example they calculate the magnetic susceptibility of He from the P e k e r i ~ ~ ~ wave-function as -(la8913 & 3 x cm3 mole-' the limits of accuracy not being large enough to bring it into agreement with the experimental value of (- 1.93 & 0.01) x Group theory of non-rigid molecules. Although group theory is a well-estab-lished and commonly used tool of the theoretical and practical chemist its application to non-rigid molecules and especially those in which there are several possible nuclear configurations with minimum energies (ethane, CH,BH etc.) is less well e ~ t a b l i s h e d .~ ~ ~ 47 In particular Long~et-Higgins~~ maintained that it was not possible to construct the correct symmetry groups for these molecules in terms of the usual symmetry operations of rotation, reflection etc. but instead developed a method in which the symmetry opera-x cm3 mole-'. *' G. L. Caldow and C. A. Coulson Proc. Cambridge Phil. SOC. 1961,57 341. *' C. M. Roenthal and E. B. Wilson,jun. Phys. Rev. Letters 1967,19 143. 43 P. Jennings and E. B. Wilson jun. J . Chem. Phys. 1967,47 2130. '* N. W. Bazley and D. W. Fox Rev. Mod. Phys. 1963,35,712. 45 C. L. Pekeris Phys. Rev. 1959,115 1216. *' H. C. Longuet-Higgins Mol. Phys. 1963,6,445. *' J. T. Hougen Canad. J .Phys. 1964,42 1920. M. E. Schwartz Proc. Phys. SOC. 1967,90,51 Theoretical Chemistry 35 tions were permutations of identical nuclei together with the inversion. The full set of operations so obtained is then broken down into feasible operations (e.g. the operation corresponding to the inversion in ammonia) and unfeasible operations (e.g. the one corresponding to the inversion in methane). In this way a single group is obtained uniquely for all configurations of the molecule. This year Altmann4* has produced a new approach to the problem in which he works in terms of the common idea of symmetry operations as motions of the moleculethough the word motion has to be rather carefully defined. In his method he distinguishes between two kinds of operation-ordinary operations such as rotation and reflection which carry the molecule into a new position but the same configuration and isodynamical operations which transform one nuclear configuration into a &fleerent equivalent configuration.The appropriate symmetry group for the molecule called by Altmann the Schrodinger supergroup contains both kinds of operations and is different in different nuclear configurations. In some cases (e.g. CH,BH2) all the various supergroups are isomorphous with each other and with the Longuet-Higgins group foi the same molecule. In other cases (e.g. (CH,),B) the different supergroups are not isomorphous and only one of them in this case the one for random orientations of the methyl groups is isomorphous with the Longuet-Higgins group. Altmann does not discuss the physical significance of these differences but it appears that one consequence is a change in the degeneracy of some electronic energy levels of the molecule on change of configuration which might well lead to new kinds of vibronic interaction.Natural Geminals. The concept of natural orbitals was introduced about ten years ago by Lowdin4” 5 0 and they have since proved of considerable use in the interpretation of wave-functions for simple systems. They are defined as those orbitals which diagonalise the first-order density matrix. Their interest arises from two properties; (i) a CI wave-function for the system converges to the accurate wave-function most rapidly if a basis of natural orbitals is used, (ii) it is possible to transform any wave-function certainly any CI wave-function into natural orbital form and this form therefore provides a useful way of comparing and analysing various wave-functions.It is possible to extend the idea to functions capable of containing more than one electron and in particular Barnett and Shullsl have calculated natural geminals for a number of four-electron systems. A geminal is a function capable of containing two electrons and natural geminals are obtained analo-gously to natural orbit& by diagonalising the second-order density matrix, and they have analogous proper tie^.^^ It is not yet clear exactly how these natural geminals are to be interpreted particularly the ones with small occupa-48 S. Altmann Proc. Roy. Soc.1967 A 298,184. 49 P. 0. Lowdin Phys. Rev. 1955,97 1474. 50 P. 0. Lowdin and H. Shull Phys. Rev. 1956,101 1730. G. P. Barnett and H. Shull Phys. Rev. 1967,lB. 61. T. Ando Rev. Mod. Phys. 1963,36,690; A. J. Coleman Canad. Math. Bull. 1961,4 209. B 36 J. G. Stamper tion number (i.e. making a small contribution to the wave-function) but one of their properties seems to stand out. If a Hartree-Fock wave-function is converted to natural geminal form, the geminals are not precisely defined. The first two in the case of Be for example. can be taken as either an electron pair in Is Is2 and an electron pair in 2s 2s2 or as any linear combination of these two. When correlation is introduced the linear combinations are defined and are found rather surprisingly to be (Is’ + 2s’) and (ls2 - 2s2) together with correlation terms in each case. It seems therefore that the best geminal expansion for Be and presumably for other systems does not start from a ls2 2s2 base