GENERAL DISCUSSION Prof. Dr. W. C. Nieuwpoort (Rijksuniversiteit te Groningen) said With regard to Pople’s paper even an optimized minimum basis set offers a rather poor description of atomic charge densities. This is true for the inner parts as well as for the outer parts of the atomic wave-functions. Extension of the basis with diffuse functions to describe the outer parts better will generally lead to changes in the exponents of the functions already present. Precisely such an extension takes place in a molecular calculation where in a sense every atomic minimum basis set is extended by the functions on other centres. The question then is how much of the exponent changes observed must be attributed to these purely atomic effects and how much of it has to do with molecule formation.This can be investigated by repeating one or some of the calculations with just the functions but no charges present at the various centres but one. Dr. G. Doggett (Uniuersity of Glasgow) said I would question the reliability of using Mulliken gross atom charges for discussing bond ionicities. The partitioning of the overlap density in a population analysis is not unique. In general the Mulliken partitioning does not even maintain the centroid of the overlap density. In calcula- tions on HCN McKendrick and myself found that the variation in the effective atomic charge on H with changing CH internuclear separation is sensitive to the method used for partitioning the overlap densities for some molecular configurations different methods of partitioning can yield effective atomic charges of opposite sign.Prof. M. Randid (Zagreb) said In Pople’s paper a comparison is made of atomic electron populations obtained by the semi-empirical INDO molecular orbital theory with some values from full SCF calculations using a minimal basis set of exponential- type functions. A total assignment of all electrons in a molecule to atomic orbitals is made by finding gross populations as suggested by Mu1liken.l We in Zagreb became interested in the question of the definition of the population analysis and would like first to stress that although Mulliken definition is universally accepted it contains an artibrary element in assigning the overlap density 2ciacibSab equally between atoms a and b. Other choices are possible and we proposed recently that the overlap density on the bond a-b to be so divided that the centre of the charge does not change.2 Such a definition will lead to different results when the atomic orbitals 4a and 4b are considerably different in their exponential (screening) part.With wider use of double zeta and other limited bases sets the situations with con- siderably different exponents are more common and Mulliken population analysis may lead to a biased distribution of electrons in molecules. The modified definition might provide an essential and important improvement. It would be of interest to know if the moderately good agreement of the comparison of atomic electron populations reported by Pople between the semi-empirical and the full SCF calculations is affected by the limitations of the Mulliken population analysis. R. S. Mulliken J .Chem. Physics 1955 23 1833. T. Zivkovii and N. Trinajstib Chem. Physics Letters 1968 2 369. 54 GENERAL DISCUSSION 55 Dr. D. B. Cook (University of ShefJieZd) said Since the question of the convergence of Gaussian expansions for STO’s has been raised I would like to describe my own experience in this field. If one looks at an SCF calculation using this Gaussian expansion method then in order to reproduce the total energy one-body density matrix etc. of the full STO calculation a Gauss STO ratio of at least 6 1 is needed as Pople points out. Thus a calculation on a fairly large molecular system is soon swamped by the large number of two electron integrals which arise. However if one examines the convergence of the integrals to the STO values some important conclusions emerge.It is found that the one electron integrals converge very slowly but the two electron repulsion integrals converge rapidly to the correct values an expansion of two Gaussians per STO yielding results in good agreement with the true STO values. These conclusions are easily understood in terms of the functional form of the Gaussians-in particular their behaviour near the origin. Thus the rather long expansions necessary are due to the poor convergence of the one electron integrals and the large numbers of repulsion integrals which arise are largely redundant since these integrals have converged after two or three terms. We have used these conclusions to develop a “ Mixed Basis ” method in which we use one-electron integrals calculated over STO and a short Gaussian expansion repulsion integrals thus combining the advantages of STO and Gaussian bases.Prof. R. McWeeny (University of ShefJieZd) said I agree with Pople’s remark that a mixed-basis approach cannot be viewed as a satisfactory ab initio procedure. It is put forward as method of closely reproducing the results of or “ simulating ” an ab initio calculation by evaluating all one-electron quantities with high accuracy and estimating all electron interaction effects with somewhat lower accuracy. Such “ simulated ” ab initio calculations 2* retain all the features of a complete non- empirical calculation and attempt to include all integrals except those that are negligible in a fairly strict numerical sense they give results in better accord with those of ab initio work than do the more empirical approximations-unless of course the latter are actually Jitted to the results by purely empirical adjustment of parameter values.The work referred to by Cook4 appears to us t o be a good compromise between theoretical rigour and computational simplicity ; the use of only two Gaussians per Slater orbital appears to be satisfactory and will probably be essential in dealing with larger molecules. Prof. R. McWeeny (University of ShefJieZd) said One major disadvantage of localization criteria based on orbital overlaps or populations is their disregard of the energy which implies that localization may lead to a wave function which is energeti- cally quite poor. In particular one finds that contamination of the high energy inner shell orbitals by admixture with valence orbitals can lead to very inferior wave f ~ n c t i o n s .~ I suggest that such criteria should be applied with caution and then only to valence electrons the valence orbitals being Schmidt-orthogonal to those of the inner shells. A similar difficulty arises with lone-pair orbitals when the corresponding orbital energies are not taken into account and it is for this reason that in recent work we have used an energy criterion. In some cases ( e g when D. B. Cook and P. Palmieri to’be published. * D. B. Cook P. Hollis and R. McWeeny Mol. Physics 1967 13 553. D. B. Cook and R. McWeeny Chern. Physics Letters 1968 1 588. D. B. Cook and P. Palmieri to be published. R. McWeeny and K. Ohno Proc. Roy. SOC. A 1960,255 367. R. McWeeny and G. Del Re Theor. chim. Acta 1968,10 13. 56 GENERAL DISCUSSION there are no lone pairs) the results are not very different from those obtained by overlap or population criteria but in general there are significant differences and the energy criterion appears to be more reliable.The main advantage of the method of Weinstein and Pauncz seems to be that it involves an intrinsic criterion and can therefore be applied even when an approximate wave function is not available. Prof. G. G. Hall (University of Nottingham) said There are two rather different motives for using localized orbitals and it seems to me that they may sometimes lead to different results. The wish for a convenient physical interpretation leads naturally to a minimization of the exchange energy and similar criteria for localized orbitals. On the other hand the desire for a compact treatment of correlation also leads to localization since its main effects are short range.I would like to ask in particular whether any specific criterion for localization arises from the alternant molecular orbital theory. Mr. M. P. Briggs (University of Sussex) said In connection with the point raised by Hall concerning the choice of localized orbitals such that they are a suitable starting point for the improvement of the wavefunction by the introduction of correlation effects I would mention some calculations as yet unpublished by J. G. Stamper and myself. For the two four-electron systems Be and LiH we have taken a limited configuration interaction wavefunction and examined the effect on the total energy of the function of a unitary transformation among the occupied orbitals E (a.u) - 14.62 5 i C F.-14640 0.1 8 (radians) FIG. l.-Plot of E against 8 for Be based on Clementi double-zeta S.C.F. function. of the ground-state configuration. The C.I. function consists of pair-correlations chosen to account for the main features of the correlation error. For example for Be the function is - I,!/ = N( I ls22s2 I +Al I lS22p2 I + A 2 I 2d22F1 +A2 I 2~’~2?1 } where the correlating orbitals 2p 2s’ and 2p’ are single Slater-type orbitals ortho- gonalized to all previous orbitals. Such a function with optimized correlating GENERAL DISCUSSION 57 orbitals gives about 70 % of the difference between the energy of the exact wave- function and that of the first determinant in the C.I. expansion. For a four-electron system the unitary transformation is a single 2 x 2 rotation ; hence the transformed starting orbitals are given by Is = cos e ls+sin e 2s; 2s = -sin 8 1s+cos o 2s.The correlating orbitals are optimized for each value of 8 taken and the energy of the C.I. function evaluated. A typical plot of energy as a function of 8 is shown in the figure for Be based on the double-zeta S.C.F. function of Clementi;' EsCF = 12.56868 (double-zeta). Although such a method does in itself provide a criterion for localization it was our intention to see how far existing criteria give good starting orbitals for this particular sort of improvement to the wavefunction. Results to date show that as expected the best transformations are rather small i.e. that the initial orbitals are well localized. They also show that the orbitals obtained by the method of Edmiston and Ruedenberg are not very good starting functions and in particular that they form a worse starting point than the S.C.F.orbitals (see fig. 1). Dr. R. A. Sack (University of Salford) (partly communicated) The product $A$B of two Slater orbitals based mainly on centres A and B lies between the nuclei and hence any expansion in functions centred on one nucleus at a time requires a relatively large number of terms to approximate the higher moments of the distribution. It is known 3 9 that can be expressed as an integral of charge distributions with centres P lying along AB resembling extended Gaussians when P lies between the nuclei and becoming weaker and at the same time more concentrated as P approaches A or B. Hence as a logical first approximation $A$B can be replaced by one or two Gaussians with centres along AB supplemented by point charges at A and B (for orbitals with Z>O the Gaussians have to be multiplied by appropriate angular factors and the point charges replaced by multipoles).Exchange integrals evaluated by such an approximation will be the more accurate the further the two charge products are separated. When one of the nuclei is common to both orbital products the Coulomb interaction of the point charge in the approximation to the one product with the whole of the other product can be evaluated in one operation as a l-electron 2-centre integral. So far I have only been able to derive explicit formulas and results for products of 1s orbitals with the same exponents. Taking the abscissas of A and B as * a the approximations are (i) point charges AS each at A and B a Gaussian integral approx.(i) approx. (ii) accurate 5 (hl,hZ ; h3,h4) 0.03023 0.03054 0.030682 (hlh2 ; hlh3) 0.03534 0603517 0-03 5 694 of total charge 4 S a t the midpoint ; (ii) point charges & S each at A and B Gaussians of total charge -& S each at x = &(a/ J7). Here S is the overlap integral and the width of the Gaussians is chosen so as to yield the same second moments as for $A$n. for two of the exchange integrals occurring in the methane molecule I obtained the results given Using the same data and notation as Shavitt and Karplus E. Clementi Z.B.M. J. Res. Dev. 1965 9 2. C. Edmiston and K. Ruedenberg Rev. Mod. Physics 1963 35 457. P. J. Roberts Proc. Physic. SOC. 1966 89 63. I. Shavitt and M. Karplus J. Chem. Physics 1965 43 398.4R. A. Sack Int. J. Quantum Chem. 1967 IS 369. 58 GENERAL DISCUSSION below. The values calculated with the aid of a Curta calculating machine and 4-figure tables are accurate to within 1-2 %; the relatively poorer result for the approximation (ii) in the 3-centre integral is probably due to the greater interpenetra- tion of the charge clouds. Further work along these lines is in progress. Prof. J. A. Pople (Carnegie-Mellon University Penn.) said I would emphasize the importance of rotational invariance in LCAOSCF schemes involving integral approximations. To be useful in chemical applications such methods must be appropriate for general three-dimensional molecules with no particular spatial symmetry where no unique choice of Cartesian axes is possible. Methods leading to results which depend on the choice of axes should in my opinion be avoided.I would also mention some work related to that of Doggett and McKendrick which has been developed at Carnegie-Mellon University principally by Dr. M. D. Newton. This is a general method of integral approximation in which two-centre differential overlap products are projected onto the space defined by a set of one- centre functions AA centred on the atoms A. Thus if +; is an atomic orbital on A and 4; is another on B the product +f+ is approximated as a sum over the sets of functions AA and AB. The coefficients dpvu and dLvu are chosen so that the integral of the square of the difference between the two sides of this equation is minimized. This method is thus described as projection of diatomic diferential overlap.It is rotationally invariant and all four-centre integrals are reduced to two-centre coulomb integrals involving the A’ functions. The set AA is taken to include the one-centre products +?#:. If the basis set q!$ consists of a minimal set of Slater-type orbitals (1s for H and 2s 2s 2p for Li to F) the products +?+; are spanned by a larger set of Slater-type functions (1s for H and Is 2s 2p 3s 3p 3d for Li to F). If AA is limited to this then each diatomic differential overlap product in (1) is written as a sum of up to 28 terms. Calculations at this level of approximation lead to total energies within 0.1 hartree of accurate values. In a more refined version of the theory the set A* is supplemented by additional 2s’ 2p’ functions for H and additional 2s‘ 2p’ 2d’ functions for heavier atoms.The expression (1) may then contain up to 46 terms. This reduces the error of calculated total energies to about 0.01 hartree. Prof. G. G. Hall (University of Nottingham) said The nub of the discussion is that the variation expression for the energy is accurate to second order in the wave- function. When the Hamiltonian and the trial functions depend on a perturbation parameter A then = wo+Aw1+A’W,+ ... being an identity in A has each term accurate to second order. This gives e.g. Wl = IJ$o*W,dz + j$T(H- Wo)$od.c + j3x(H- ~o)$ldt)/p:$od~ which is accurate to second order whereas the first term generally is accurate only to first order. belongs to the domain of the trial functions used in determining $o then the remaining terms vanish and the first term which is often assumed to be the only term becomes accurate to second order. On the other hand if