Minimum Basis Self'-consistent Group Function Calculations* BY MARTIN KLESSINGER Organisch-Chemisches Institut 34 Gottingen Germany Received 20th September 1968 The interpretation of complicated wave functions in a way which is illuminating and suggestive to a chemist requires concepts such as localized pairs of electrons. This is realized by writing the wave function as an antisymmetrized product of localized two-electron functions. Thus the separated-pair approximation appears to be of great value in discussing chemical problems parti- cularly in its most simple form which is characterized by the use of a minimum set of basis orbitals. Following a brief review of the self-consistent group function method (SCGF-method) and of the use of contracted gaussians within this approach results of SCGF calculations are reported for methane methyl fluoride ethane ethylene acetylene and cyclopropane.INTRODUCTION The idea of casting the total electronic wave function of a molecule in a form that exhibits localization of pairs of electrons goes back to Hurley Lennard-Jones and Pople who proposed to write the N-electron function "(1 . . . N ) as an anti- symmetrized product of pair functions At the same time they introduced the strong orthogonality condition Y(1 ... N) = d[QA(1,2)@,(3,4) ...I. (1) which leads to the important simplification that only interactions within one pair function iDR or between any two pair functions OR and as enter the energy expecta- tion value. developed a theory of generalized group functions of which the separated pair approximation is a special case.On the basis of this theory a practical method of computing wave functions and energies of polyatomic molecules was developed which takes into account groups of two and more electrons each group (localized or non-localized) being made self-consistent in the field of all others. This self-consistent group function method (SCGF-method) represents a generally applic- able approach strictly defined within the approximation determined by the strong orthogonality condition and leading to localized orbitals which show up a direct dependence on molecular structure. Thus it is hoped that this approach will make possible a genuine understanding and a proper explanation of molecular structure and of chemical regularities and their implications for the chemical behaviour of a molecule. The aim in developing this method was not to calculate absolute energies as accurately as possible but rather to obtain a scheme which is simple enough to be McWeeny * The work reported forms part of the Hubi/itutions.schr$t submitted by the author at the Georg- August-Universitat Gottingen 1968.Financial support by the Deutsche Forschungsgemeinschaft Bad Godesberg is gratefully acknowledged. 73 74 SCGF CALCULATIONS applicable to sufficiently large molecules and which gives results that may be helpful in discussing problems of a chemical nature. At the same time the semi-empirical development of the method was aimed at as it was thought that in spite of the encouraging progress which has been made in computing ab initio wave functions it would for some time be unavoidable to rely on semi-empirical methods in order to achieve what may be called chemical accuracy i.e.to calculate energy differences for large systems accurately to +_ 1 kcal/mole. For semi-empirical methods to be reliable enough to make chemical predictions they have to be put on a firm basis which allows the inherent approximations to be well understood. With this objective in mind non-empirical minimum basis set SCGF calculations were carried out on a sufficiently large number of molecules in order to find out to what extent this method is suited for solving chemical problems and in order to have suitable data to test the necessary approximations for a semi- empirical development. Results of some of these calculations are reported and discussed in the present paper. MINIMUM BASIS SET SCGF CALCULATIONS As a consequence of the strong orthogonality condition eqn.(2) each pair function (D in eqn. (1) may be written in the form @R(1,2) = ~c~’($~(1,2) (3) P as a linear combination of Slater determinants (or suitably coupled sets of Slater determinants) 4; built up from an orthonormal set of orbitals (OAO) rl,r2 . . . sl,sz . . . in such a way that the orbitals ri appear only in the functions 4; of group R the si only in the 4; of group S etc. If the basis orbitals form a minimum set there are just two OAO rl and r2 for each bond pair R of electrons which for a singlet state is therefore conveniently = (ri - r J I J2. ( 5 ) which only one suitably hybridized @dW = 4SCL2) = I s(l)ms(2)P(2) I. A self-consistent ground state is obtained by using an effective Hamiltonian for a two-electron group R which is defined as 2 2 t where the matrix elements of hfff(i) which take into account the interactions with all other groups of the molecule in a self-consistent way are given by (hk)rirj = (hR)rir + C [xRs(rirj) + X,IRxs(rirj) + ~SxFTrirj)] S(#R) M .KLESSINGER 75 with Here X and Ys are a " bond order " and a " polarity " (measuring the amount of charge transferred from orbitals s2 in bond S to orbital sl) respectively and are defined in terms of the first order density matrix by xs = Pi 2 = P; 1 = J2c',s'(c',s' + CS"') (9) Ys = $(PS 1 -E2) = -(cy))2. (All details together with formulae for the calculation of excited states and ionization potentials as well as for the inclusion of many-electron groups are given by Klessinger and McWeeny. 3 The set of OAO r1,r2 .. . s1,s2 . . . is obtained from a set of atomic orbitals (AO) by a transformation with a matrix V which may be written as v= ws,*u. (11) W makes the inner shell orbitals orthogonal to all valence orbitals. Preliminary calculations showed that the polarity of the bonds in a molecule increases if valence orbitals are allowed to mix into the inner shell orbitals. This polarity increase leads to a gain in bond energy which is not sufficient to compensate for the accompanying loss of inner shell energy. Thus the most suitable procedure to construct inner shell orbitals is a Schmidt orthogonalization with respect to all valence orbitals which keeps the inner shell orbitals pure and leads thus to the best value of the total energy.4 of the valence orbitals (which are already orthogonal to the inner shells) i.e.Sb is the overlap matrix of the A 0 after applying the Schmidt orthogonalization. By the Lowdin procedure the orbitals are disturbed as little as possible in a least-squares sense.6 Finally U transforms the OAO into suitable hybrids. U is an unitary matrix so that S 3 represents a symmetrical Lowdin-orthogonalization v = WS,fU = wus,+ (12) (where S is the overlap matrix of the nonorthogonal hybrids) i.e. the hydridized OAO of eqn. (11) are equivalent to orthogonalized hybrids obtained by the same transformation U. W and S;* are entirely determined by the initial choice of A 0 and by the geometry of the molecule whereas U is arbitrary within certain limits and determines how the original basis set is divided up into subsets one for each group of electrons.As a consequence of writing the bond pair functions as linear combinations of Slater determinants i.e. of taking into account configuration interaction within each bond pair the total energy is not invariant with respect to the way in which the original basis is divided up. Therefore in the SCGF method the total energy of a given state of a molecule is a function of hybridization. This leads apart from the deter- mination of the self-consistent group functions to an additional optimization process for the hybridization. At the same time it enables one to determine optimum valence hybrids by applying the variational principle i.e. by an energy criterion. This is of great importance for chemical applications of the method as hybridization is a 76 SCGF CALCULATIONS concept well suited for relating results of quantum mechanical calculations and chemical ideas.Calculations according to the above described method have been performed on a number of small molecules using a minimum basis TABLE COMPARISON OF RESULTS OF SCF AND SCGF BASIS OF THE STO molecule geometry ground-state energy Eo(a.u.1 SCF SCGF HF R = 1.733122 a.u. - 99.4786 - 99.4972 HzO R = 1.8103 - 75.6807 -75.7139 CH R = 2.06742 a.u. - 40.0606 - 40.0980 Nz R = 2.06928 - 108.5736 - 108.6498 CO R = 2.13202 - 1 12.3437 - 1 12.401 9 e = 1050 of Slater A 0 (STO) (table l).? CALCULATIONS USING A MINIMUM hybridization A.Eo(eV) -0.51 A = 0.40" -0.90 a = 102.7'** - 1.02 sp3 hybrid. -2.07 3 = 0.74* - 1 *58 Rc = 0.59* 20 = 0.86* * Bond hybrids bx = N[A(2s)x+(2pz)x] ** cc = angle subtended by bond hybrids -40[ R(;.u.)--r ~ -""I I R (a.u.)+ I 1.0 20 3.0 1.0 2.0 30 FIG.1.-Potential energy curves for HF (a) SCF and SCGF results (b) SCF and SCF-CI results of Karo and Allen. A consequence of taking into account some electron correlation within each bond pair a ground state energy is obtained which is lower than the SCF energy calculated with the same set of basis orbitals by an amount of A& = 0.25-0.50eV per two- electron bond. Potential energy curves for the ground state of the HF molecule as calculated by the SCF and the SCGF r n e t h ~ d ~ are shown in fig. la and may be compared with results of an SCF calculation with and without CI by Karo and Allen in fig. l b (the SCF curves are not identical as Karo and Allen used a basis of atomic Hartree-Fock orbitals). That graph shows that by including CI within the bond pair function aR(1,2) (eqn.(3)) about the same effect is obtained as by extensive CI within the basis of SCF-MO. As the SCGF calculations involve at 1- Atomic units are used unless otherwise stated i.e. 1 a.u. = 0.529 x lO-'O m for the unit of length and 1 a.u. = 27.210 eV for the unit of energy. M. KLESSINGER 77 most 3 x 3 secular problems and convergence is rapid they are considerably faster than conventional SCF calculations even without CI. This demonstrates clearly the great value of the SCGF approach in calculating ground state properties. R(CI u.)- USE OF CONTRACTED GAUSSIANS I N THE SCGF APPROACH R (a u.) -+ In principle SCGF calculations of the type described could be carried out for any kind of molecule. But as with all ab initio calculations the determination of all electron interaction integrals makes the use of Slater A 0 for larger molecules prohibitive.During the last few years gaussian type orbitals (GTO) originally introduced into quantum chemical calculations by Boys and by McWeeny,lO as well as pure gaussian orbitals l 1 (GO) have proved successful in calculations on large molecules as the need to use larger basis sets is far more than compensated by the ease with which molecular integrals over the GTO and the GO can be calculated. I I A way of preserving the simplicity of the minimum basis SCGF method in spite of the slow convergence of the GTO is the use of a minimum basis set of contracted gaussians i.e. linear combinations of GTO (LCGTO). In order to be able to extend the SCGF calculations to larger molecules it was thus decided to expand the minimum STO set in terms of the GTO in the manner described explicitly by Huzinaga.In order to find an appropriate compromise between sufficient accuracy and reasonable computing time extensive calculations with different expansions of the basis set were performed for the methane molecule. It turned out that an expansion of the Is 2s and 2p A 0 in terms of five two and three GTO respectively will be adequate for first row atoms whereas for the H-atom an expansion in terms of three GTO seems to be sufficient. This basis set is well suited to reproduce the results of SCGF calculations with a minimum basis set of STO. This may be seen from fig. 2 which shows for the HF molecules the ground state potential energy curves and the optimum hybridization of the bond orbital as a function of the internuclear distance as determined by the SCGF method.A detailed comparison of the results 4 TABLE 2.-cOMPARISON OF THE RESULTS OF SCGF CALCULATIONS ON HF WITH A MINIMUM BASIS OF THE STO (UPPER PART) AND LCGTO (LOWER PART).* minimum basis of the STO S-C** R = 1.25 R = 1-50 Eo - 99'24802 - 99.43002 HZ -3.51874 - 2.95060 X 0.9514 0.9725 Y 0.2900 0.1763 P 1.4801 1.1317 jZopt 0.7207 0.5165 %s 34.1 8 21.06 R = 1.73312 - 99.4971 7 - 2.54864 0.9741 0-0908 0.8076 0.3962 13.57 R = 2.00 - 99.5 1889 - 2.2092 1 0.9588 0.0184 0.4719 0.3006 8-29 R = 2-50 - 99.50128 - 1.79913 0.8831 0.001 8 0,1814 3.19 - 0.0520 R = 3-99 - 99.47089 - 1.56785 0.7385 - 0.0563 - 0.1763 0.1044 1 *08 R = 3.50 (a.u.) - 99.44982 - 1 ~44075 0.5473 - 0,0341 -0.1396 0.0541 0.3 1 minimum basis of the LCGTO S-"* minimum basis of LCGT0,C.R.-<*** R = 3-50 (a.u.) EO - 99.05520 - 99.30212 - 99030304 - 99.24467 - 99,12027 - 99.34837 - 99.30542 - 99.23 1 10 - 1,43525 R = 1.25 R = 1.73311 R = 2.50 R = 1 -25 R = 1.73312 R = 2.50 R = 3.50 - 3.57046 - 2.59238 - 1.81334 - 1.42471 - 3.64547 - 2.67497 - 1.83335 0,4786 X 0,9566 0.9776 0.8903 0,5605 0.9658 0.9735 0.8799 Y 0.2755 0,0746 - 0.0724 - 0.0468 0.2422 0.1272 0.0172 - 0.0066 0.1 156 I.1 1 5426 0.8523 -0.0129 -0.1886 1 -5290 1.1952 0.6046 Jopt 0.7691 0.4239 0.2006 0.061 9 0-7507 0.4335 0.1936 0.0425 %s 37.17 15-23 3.86 0.38 36-04 15.82 3.61 0.18 *Ground-state energy Eo and effective bond energy HzT in a.u.dipole moment p in D bond order X and polarity parameter Y optimum hybridiza- **S.-C = orbital exponents according to Slater rules CF(~S) = 8.7 C~(2s) = CF(2p) = 2.6 C~(ls) = 1.0.***C.R.-C = orbital exponents according to Clementi and Rairnondi,l3 C~(1s) = 8.6501 C~(2s) = 2.5638 C~(2p) = 2.5500 c ~ ( 1 ~ ) ~1 1-20. tion parameter hopt (cf footnote to table 1) and percentage s-character of the bond orbital %s. M . KLESSINGER 79 obtained with these two kinds of basis functions is made possible by the data in table 2. These data also show that calculated binding energy and optimum hybridiza- tion are little affected by the choice of the basis orbitals whereas calculated charge distributions and dipole moments are sensitive to the choice of the orbital exponents as is well known for SCF calculations Similar results were obtained for the water molecule where binding energy and optimum hybridization appear to be insensitive to the choice of basis orbitals and unexpectedly to the magnitude of the valence angle.The minimum of the total energy is obtained when the valence angle coincides with the hybrid angle (the angle between the bond hybrids) i.e. there is no indication for the existence of bent bonds in the equilibrium ground state of the water molecule. SOME RESULTS OF SCGF CALCULATIONS A fully automatic computer programme MOLCAL has been developed for the IBM 7040 which requires only the following input data the molecular co-ordinates a specification of the basis functions and some data indicating the symmetry of the system. By means of this MOLCAL programme SCF and SCGF calculations were carried out for the following molecules methane methyl fluoride ethane eclipsed and staggered ethylene acetylene and cyclopropane.The results of the ground state total energy Eo for an expansion of the STO with exponents given by Clementi and Raimondi l3 together with 5s = 1.20 for the hydrogen atom in terms of the LCGTO as described in the previous section are collected in table 3 and compared with the best available calculations of other authors. TABLE 3. SCGF GROUND-STATE ENERGIES Eo(a.U.) ENERGY LOWERING A&,(eV) AS COMPARED TO SCF CALCULATIONS WITH SAME BASIS SET AND RESULTS OF OTHER AUTHORS. molecule EoWGF) AEO (eV) Eo (other methods) - 40.0606 (SCF ST0)Q - 40.3 12 (ASPG NO)b methane CH4 -40.1022 - 1.22 methyl ff uoride CH3F - 138.2500 - - ethane stagg. C2H6 - 79.0107 - 1 *48 - 79.1478 (SCF Hartree-Fock-AO)C ethane ecl.C2H6 - 79.01 19 - 1.65 - 79.1438 (SCF Hartree-Fock-AO)C ethylene CzH4 - 77.7814 - 1.47 - 78.0063 (SCF ext. basis of LCGTO)d acetylene C2H2 - 76.5733 - 1-75 - 76.7418 (SCF ext. basis of LCGTO)e cyclopropane C3H6 - 1 16.6797 - 1.05 - 1 16-02 (SCF G0)f * M. Klessinger and R. McWeeny J. Chem. Physics 1965,42,3343. b R. Ahlrichs and W. Kutzelnigg Chem. Physics Letters 1968 1 651. c W. H. Fink and L. C. Allen J. Chem. Physics 1967 46,2261. d J. M. Schulman J. W. Moskowitz and C. Hollister J. Chem. Physics 1967 46 2759 e J. W. Moskowitz J. Chem. Physics 1965 43 60. f H. Preuss and G. Diercksen Ini. J. Quantum Chemistry 1967 1 361. In all calculations optimum hybrids were determined and found to point along the bond axes ; even in cyclopropane the direction of the CH-bond orbitals coincides with the line joining the C and H atoms.Bent bonds consequently do arise only when the valence angle is smaller than 90° as for the CC-bonds in cyclopropane where the deviation of the hybrid direction from the valence direction is calculated to be co = 21.1" in good agreement with maximum overlap considerations which lead to w = 2 1 ~ 4 O . l ~ Energy localized orbitals (LMO) were determined according to the Edmiston- Ruedenberg procedure for ethylene and acetylene by Kaldor.16 As for the triple 80 SCGF CALCULATIONS bond in N2 and CO the LMO for the CC double and triple bond respectively turn out to be equivalent orbitals corresponding to a description of multiple bonds in terms of bent " banana bonds " whereas the SCGF method gives in these cases a lower energy for a description of a multiple bond in terms of a a-bond and one or two n-bonds.17 Whereas the CH-bond orbitals in ethylene determined by the Edmiston-Ruedenberg method form an angle of 11 1 O the SCGF method leads to an hybrid angle of 117" in good agreement with the experimental valence angle.Similar differences between the LMO and SCGF-orbitals are found for acetylene where the SCGF method leads to 45 % s-character of the CH-bond orbitals (corres- ponding to nearly pure sp hybridization) compared to 33 % s-character (sp2 hybridiza- tion) of the LMO. In all cases the SCGF value of the total energy is lower than the SCF value cal- culated with the same basis orbitals by AEo = 0.2-0.4 eV per two-electron bond. For ethane the total energy lowering is 1.48 eV for the staggered conformation and 1.65 eV for the eclipsed conformation.Thus while the SCF calculation gives a barrier of internal rotation of 3 kcal/mole with the staggered conformation being the more stable one the SCGF calculations leads to a value of 0.6 kcal/mole favouring the eclipsed conformation. This seems to suggest that the SCGF method is not well suited for calculating potential energy barriers. The reason for this may be that the localizability of electron pairs is different for the two conformers. For cyclo-propane the SCGF energy is calculated to be only 0.12eV per bond lower than the SCF energy. Thus in open-chain compounds electron pairs appear to be better localized than in cyclic compounds exhibiting ring strain. TABLE 4.-EFFECTIVE INNER SHELL ENERGIES HJ$ OF c- AND F-ATOMS IN DIFFERENT MOLECULES (ENERGIES IN eV).K shell of the C-atom K shell of the F-atom molecule H$ (eV) molecule H: (ev) methane -712.98 HF R = 1-25 a.u. - 1575.96 ethane stagg. - 71 3.09 R = 1.733 - 1577.57 ethane ecl. -713.13 R = 2.50 - 1579.86 methyl fluoride - 71 5.87 R = 3.50 - 1579.59 cyclopropane - 71253 methyl fluoride - 1578.21 ethylene - 714.07 acetylene -715.45 Table 4 gives the effective inner shell energies which lie for the carbon atom between H$f = -712.5 eV and = -715.5 eV; they depend on hybridization and on the nature of the neighbouring groups and are thus slightly different from molecule to molecule. Within an homologous series however they are approximately constant. The values of H:tf for the fluorine atom show that the internuclear distance is of greater influence on the actual value of this quantity than the replacement of the hydrogen in HF by a methyl group.In table 5 the effective framework integrals a, and Priri (cf. eqn. (7)) the effective bond energies HZF and the bond orders X, are collected for CC 0- and n-bonds; the same quantities for the CH bonds together with the polarity parameter YCH are given in table 6. These data clearly reflect different bonding situations in different M . KLESSINGER 81 molecules pCc is for cyclopropane appreciably smaller than for ethane so that the ring strain in the former compound becomes apparent in alower bond order Xc and a smaller absolute value of the bond energy H$$ compared with ethane. The CC a-bonds in ethylene and acetylene are considerably stronger than the o-bond in ethane due to the shorter internuclear distances.The effective bond energy of the n-bonds is roughly half that of a a-bond. TABLE 5.-PROPERTIES OF CC-BONDS AS CALCULATED BY THE SCGF METHOD (ENERGIES IN ev) molecule a-bonds ethane stagg. ethane ecl. cyclopropane ethylene acetylene ethylene acetylene n-bonds 0% - 24.38 - 24.39 - 23.43 - 24.99 - 23.32 - 19.63 - 20.06 Bcc - 11.57 - 11.58 - 10.43 - 16.40 - 21.59 - 3.93 - 4.90 xcc 0.9872 0.9872 0.9822 0.995 1 0.9978 0.891 3 0-93 19 - 54.41 - 54.43 - 50.44 - 64.10 -71.26 - 35.07 - 37.20 TABLE 6.-PROPERTIES OF CH-BONDS AS CALCULATED BY THE SCGF METHOD (ENERGIES IN ev) molecule methane ethane stagg. ethane ecl. methyl fluoride cyclopropane ethylene acetylene 0% - 24.84 - 24.85 - 24.83 - 25.86 - 25.35 - 25.59 - 27.94 MH 'CH xCH =CH H2tY - 23.31 - 11.60 0.9800 0.0980 - 53.58 -23.36 - 11.68 0.9805 0.0918 -53.66 - 23.35 - 11.71 0.9806 0.0914 - 53.68 -23.92 - 11.73 0.9797 0.1045 -55.42 - 23.46 - 11.82 0.9809 0.0950 - 54-52 -23.61 -11.98 0.9812 0.0960 -55.20 - 23.76 - 12.36 0.9768 0.1496 - 58.87 Whereas all CC bonds of the molecules considered in this paper are nonpolar by symmetry the CH bonds show a polarity ranging from a transfer of 0-09 electrons from the hydrogen to the carbon atom in ethane to a transfer of 0.15 electrons also from H to C in acetylene.But as the atomic moment of the bond hybrid by far overcompensates the moment due to charge transfer the calculated bond moment lies for all CH bonds considered between pCH = 1.40 D and pcH = 1.77 D the C atom being the positive end of the dipole in all cases. The change of polarity of the CH bond in going from methane to acetylene as predicted by Cou1son,18 is not observed in these calculations.The influence of hybridization at the carbon atom on the properties of a CH bond is also apparent from table 6 ; in ethylene the calculated polarity parameter is YcH = 0.096 in acetylene Y, = 0.150. The CH-bond orbitals in cyclopropane show the same hybridization as the CH-bond orbitals in ethylene; the calculated bond orders polarity parameters and bond moments of the CH-bonds in these two molecules are nearly identical. The highest value of the polarity parameter was calculated for acetylene in accord with the relative acidity of the CH-bonds in this compound. DISCUSSION From the numerical results of the preceding sections the SCGF approach cannot compete with those methods which were particularly designed for obtaining absolute 82 SCGF CALCULATIONS energies as accurately as possible.But the SCGF approach compares as far as ground-state properties are concerned favourably with the more commonly used approximate SCF method and it offers a number of advantages which may be important. (1) The description emphasizes throughout the structure of the system in terms of chemically significant parts (bonds lone pairs etc.) and the interactions between them. Time-consuming localization procedures are therefore not required. This much facilitates the interpretation of the calculated wavefunctions and makes similarities between different molecules immediately apparent. At the same time it is recognized that a bond (CH for example) is not quite the same in two different molecular situations.The nature of the differences which is related to deviations from chemical " additivity rules " are indicated by the bond order and polarity parameters which take account of the environment in a self-consistent way. (2) As in the SCGF approach the total energy is not invariant with respect to the way in which the basis set is divided up into subsets for individual groups of electrons optimum valence hybrids can be determined using the variation principle i.e. by means of an energy criterion. (3) The individuality of different groups is reflected in the relative weakness of the coupling between them. This results in a rapid convergence of the iterative calculation in contrast to the situation usually encountered in Hartree-Fock theory. In addition there is always a good starting approximation available and the dimension of the secular problems which have to be solved is generally low ; in the minimum basis calculations the secular determinants are never larger than 3 x 3.This again leads to short computing times. As with all ab initio calculations by far the most time-consuming step is the determination of the molecular integrals. (4) The wave-function for any single group in the molecule may be refined indefinitely by adding further basis orbitals in order to include correlation effects. Results of calculations on Hz04 N2 and CO suggest that the localized pair functions obtained by the SCGF method are better suited for including intrapair correlation effects than e.g. the Ruedenberg LMO or other localized orbitals. For the methane molecule it is possible to estimate the amount of correlation energy recovered by the SCGF calculation Ahlrichs and Kutzelnigg l9 calculated the intrapair correlation energy of any single CH-bond in methane by a direct determination of the approximate natural orbitals of a localized pair of electrons in the Hartree-Fock field of all other electrons; they obtained a value of 0.6 eV and estimated the correct value to be 0.76eV.To the calculated value of 0.6 eV orbitals of a-symmetry with respect to the bond-axis contribute 0.48 eV the remaining 0.12 eV come from orbitals of n-symmetry. In the minimum basis SCGF calculations where only a,-orbitals are considered the calculated intra-pair correlation for each CH-bond is 0-26 eV i.e. roughly half the value obtained by direct determination of the NO; the computer time necessary for the SCGF calculation is shorter by a factor of approximately 20.From the same calculation by Alhrichs and Kutzelnigg one may conclude that the best possible separated-pair wave function for methane allows only half of the total correlation energy to be recovered i.e. nearly half of the correlation in CH is interpair correlation. Similar results concerning the ratio of intrapair to interpair correlation were obtained by Sinanoglu.20 This seems to set a limit to the SCGF approach as far as the exact determination of absolute energy values is concerned. ( 5 ) By employing throughout orthogonal basis orbitals which are strongly localized the neglect of all many-centre exchange and hybrid intergrals (the zero differential overlap approximation) can be justified.Thus the SCGF approach lends itself admirably to semi-empirical development as suggested by Klessinger M. KLESSINGER 83 and McWeeny3 and further investigated by Cook Hollis and McWeeny.21 It is hoped that such a semi-empirical method based on the SCGF approach could become most useful in discussing chemical problems. A. C . Hurley J. Lennard-Jones and J. A. Pople Proc. Roy. SOC. A 1953 220,436. R. McWeeny Rev. Mod. Physics 1960 32 335. M. Klessinger and R. McWeeny J. Chem. Physics 1965,42 3343. M. Klessinger J. Chem. Physics 1965 43 Sl17 and references therein. P. 0. Lowdin J. Chem. Physics 1950 18 365. B. C. Carlson and J. M. Keller Physics Rev. 1957 105 102. ' M. Klessinger Chem. Physics Letters 1968 2 562. * A. M. Karo and L. C. Allen J. Chem. Physics 1959 31,968. S. F. Boys Proc. Roy. SOC. A 1950 200 542. lo R. McWeeny Acta Cryst. 1953 6 631. l 1 J. L. Whitten J. Chem. Physics 1966 44 359 H. Preuss Mol. Physics 1964 8 157. *3 E. Clementi and D. L. Raimondi J. Chem. Physics 1963 38,2686. l4 C. A. Coulson and T. H. Goodwin J . Chem. SOC. 1962,2851 ; 1963 3161. l 5 C. Edmiston and K. Ruedenberg Rev. Mod. Physics 1963 35 457; J. Chem. Physics 1965 l 6 U. Kaldor J. Chem. Physics 1967 46 1981. S. Huzinaga J. Chem. Physics 1965 42 1293. 43 s97. M. Klessinger J. Chem. Physics 1967 46 3261. C. A. Coulson Valence (Oxford University Press Oxford 2nd ed. 1961) p. 219. l9 R. Ahlrichs and W. Kutzelnigg Chem. Physics Letters 1968 1 651. 2o 0. Sinanoglu and B. Skutnik Chem. Physics Letters 1968 1 699. 21 D. B. Cook P. C. Hollis and R. McWeeny Mol. Physics 1967 13 553.