Dec., 1962
QUAXTUM THEORY OF ATOMS, MOLECULES, AND
ber (80) of quadruple substitution configurations, such configurations do not represent inter-shell effects. Configurations of the ls2sxy type also could be included in A, and therefore they do not necessarily contribute inter-shell correlation. The particular configurations of this type in Q (10, 11, 14, and 21) were excluded from A to obtain orthogonal lS geminals. Inter-shell correlation energy might be most suitably defined as the difference between the energy of the optimum geminal product function (where the geininals are not subject to any orthogonality or symmetry conditions) and the exact non-relativistii: energy. In the transition from the optimum geminal product to the exact function the energy improves, not because any new configum tions are introduced, but because the constraints on the coefficients are removed. There does not seem to be any reason to revise
THEIR
INTERACTIOKS 2283
the earlier conclusion that the inter-shell correlation energy in the beryllium atom is negligible. Thus the energy of the optimum geminal product function should be considerably lower than the energies obtained thus far by configuration interaction. NOTEADDEDIN PRoor.-The total energy of the beryl-
lium atom may be decomposed into the separate contributions from the two electron pairs.4 In the SCF approximation, the energies of the I( and L electron airs are found to be 13.571 and 1.002 a. u., respectiveIy. $he former result may be compared with the corresponding energy in Be+S, 13.61130 a. u. [C.C. J. Roothaan, L. M. Sacbs, and A. W. Weka, Ren. Mod. Phya., 32, 186 (1960)l.
Acknowledgments.-T. L. A. wishes to express his appreciation to Professor Harry G. Day and the members of the Chemistry Department a t Indiana University for the generous hospitality extended to him and his family during their stay in Bloomington.
SOME ASI’ECTS OF THE QUANTUM THEORY OF ATOMS, MOLECULES, AKD THEIR INTERACTIONS1 BY OKTAYSINANOGLU Sterling Chemistry Laboratory, Yale Unibersity, New Hauen, Connecticut Received May 88, 196s
A quantitative and systematic basis for semi- and non-empirical theories of (a) atomic and r-electron spectra, (b) heats of formation, reltttive energies of the different isomers or conformations of large molecules, and (0) intermolecular forces a t all R is developed. Various effects in the exact wave function and energy of a many-electron system are examined. The major effects are the properties including electron correlation of separate shells or molecular orbital pairs. Effects of correlation on Hartree-Fock SCF orbitals are found to be negligible. The simple transformation of MO’s into localized ones transforms the correlation energy of a saturated molecule exactly into the sum of bond correlation energies and non-bonded attractions. This allows the energy of, e.g., a C-C bond or a lone pair to be obtained separately in the same way a8 in Hs. Also the effects of molecular environment on localized correlations are studied. They are within the constancy of experimental bond energies.
Among the properties that quantum chemistry must deal with are: (a) electronic spectra, (b) thermodynamic quantities such as heats of formation, relative energies of different isomers or conformations of a molecule, and (c) intermolecular forces. Semi-empirical theories of both atomic and Telectron spectra are based on orbital pictures (Hartree-Fock or approximations to it such as simple MO). I n khetje, cores and their instantaneous polarizations by valence electrons are left out. In the r-electron case, an empirical “T-electron Hamiltonian”, E, is used.2 The expressions for the energies obtained on these orbital theories are parametrized, in atomic spectra F and G integrals; in the ?r-electron case a,/3 and one or two center coulombic integrals are left to be determined empirically. The empirical vaIues give quite good agreement with experiment, but values calculated directly from orbitals lead to large errors which are known to be due to electron correlation. 2-4 (1) This work was supported by a grant from the National Science Foundation. (2) (a) M. G. Mayer and A. L. Sklar, J. Chsm. Phyr., 6 , 645 (1938): (b) R. Pariser and R. G. Parr, ibid., 41, 466 (1953). (3) W.Kolos, ibid., 87, 591, 592 (1957). (4) M. J. S. Dewar and C. E. Wulfman. ibid., 19, 158 (1958).
Relative energies of saturated molecules also are calculated well empirically. Pitzer6 assumed constant bond energies and considered zero point vibrational energies and van der Waals attractions between non-bonded regions. In this way, the heats of formation and isomerization of saturated hydrocarbons were obtained to within 0.01 e.v, (0.2 kcal./mole). Intermolecular forces are given quite well around the equilibrium configuration by Hartree-Fock MO as shown, for example, by R a n d 6 on He-He interaction. At large separations van der Waals forces are due mainly to correlations in the motions of electrons.5 In the usual London dispersion theory the complete basis set for the composite system of interacting molecules which occurs in the usual infinite sum of the second-order perturbation energy is taken as all products ILak+bl of the eigenfunctions of separate atoms. This, of course, assumes not only that there is no overlap between the ground state wave functions of the atoms but also none for all the virtual excited states. Consider, for example, two hydrogen atoms 3 A. apart both with the same spin. There is no (5) K.8.Pitzer, Aduan. Cham. Phya.. 8, 59 (1959): K.9. Pitzer and E. Catalano, J. Am. Cham. Soe., ’18, 4844 (1956). (6) B. J. Ransil. J . Cham. Phyr., 84, 2109 (1961).
2284
Vol. 66
OKTAY SISASOGLU
overlap between t,he ground state orbitals. However, since Bohr radii are proportional to n2, even the next virtually excited state (2~,),(2p,)~t,hat would contribute to London dispersion energy shows very large overlap. Thus, the usual theories of long range van der Waals forces even at considerable distances are not on a sound basis. The situation is certainly worse in non-bonded intramolecular attraction. I n all these semi-empirical theories use is made of “the chemical picture.” Separate groups such as inner and outer shells or localized bonds are assumed; only interactions within or between some groups are evaluated, often semi-empirically. Such quantum chemical problems can be separated into an orbital part and the remaining electron correlation part.’ The framework of the above semi-empirical theories is based on orbitals. But both large errors in non-empirical calculations and some important effects such as the non-bonded att’ractions are due mainly to electron correlation. Therefore, the semi-empirical theory should be based not on orbital pictures but on a theory t,hat considers electron correlation in a detailed way. Non-empirical Theories.-The configuration interaction (CI) method is meant for the removal of actual or near degeneracies. However, iionempirical calculations on more than two-electron systems usually are done by CI too, even though many configurations may have to be mixed in as for He atom.9 I n ext,ending the Hylleraas’ “r12-coordinate met,hod’’s that deals successfully with such correlations in He to n-electron systems difficult integrals’o arise if all ~ j ’ sare introduced into trialfunctions in the usual manner.ll In addition to the computational difficulties, these methods treat each atom or molecule as a different numerical problem without making use of the physical distinctness of various correlation effects and the existence of separate groups of electrons. In the orbital picture, concentric shells such as 6 , ?r already are separated. For saturated molecules, Lennard-Jones12 showed that an &IO determinant already contains localization of electrons of the same spin with respect to one another. One goes t,o such a localized description by a simple unitary transformation that leaves the original det’erminantunchanged. There have been at.tempts to put elect.ron correlation into these localized groups directly.l3 (7) The best orbitals that give good electron densities and simplify the remaining electron-electron effects are those of Hartree-Fock method. This will be taken here as the standard starting point. Following Lowdin (ref. 8) the remaining non-relativistic error will be defined as “electron correlation.” (8) P. 0. Lowdin, Advan. Chem. Phys., 2 , 207 (1959). (9) See, e.#., J. C. Slater, “Quantum Theory of Atomic Structure,” hlcGraw-Hill Book Co., Ino., New York, N. Y., 1961, Vol. 11, p. 48. (10) L. Szitsz, J . Chem. Phgs., 36, 1072 (1961). (11) For “correlation factor” methodssee Lowdin (ref. 8), also R. E. Peirls, “Lectures in Theoretical Physics, Boulder, 1958,” T’ol. I, p. 265 (discussion of trial function of Len2 (1929)), and R. Jaatrow, Phys. Res., 28, 1479 (19551, for the type of trial function discussed by Szasz (ref. 10). (12) J. E. Lennard-Jones, Proc. Roy. Sac. (London), A198, 1, 14 (1949); Ann. Rev. Phys. Chem.. 4, 167 (1953). (13) (a) A. C . Hurley, J. E. Lennard-Jones, and J. A. Pople, Proe. .Roy. SOC.(London). A4a0,446 (1953); (b) P. G. Lykosand R. G. Parr,
These assume a pair function with correlation for each bond and antisymmetrize their product to get the total wave function. The pair functions AK are assumed to satisfy the arbitrary and strong orthogonality condition (AA(i,j), AB(i,l))xl = 0
(1)
where < >xi means integration over electron xi only. This condition is equivalent to assuming the separability of bonds to start with. In addition the method does not allow for non-bonded correlations to be put in in a natural way. Limitations of this method have been discussed previ0us1y.l~ The “Chemical Picture” and Quantitative Theory.-The “chemical” facts have been with us for a long time. Early quantum chemistry started with intuitive pictures based on these facts, but because it introduced many unexamined assumptions to force the theory into these pictures, it almost fell into disrepute. It is now desirable to formulate the quantitative theory such that one starts with an exact but detailed form of a wave function and energy, from this systematically isolates the important correlation effects, but allows adequate means for estimating everything that is left over. Below, such a theory is outlined which provides a common basis for both semi- and nonempirical theory. The semi-empirical theory differs from the non-empirical mainly in the evaluation of tthemajor quantities empirically. Theory is not based on infinite basis set expansions so that other methods such as “t-lZ-coordinate,” “open-shell,” etc., can be used. A basic point of the theory, however, is to separate first the various correlation effects, then apply a different suitable method to each part of an atom or molecule. For example in Be atom one would use ‘‘T~~-COordinate” for 1s2, (29 - 2p2)2 X 2 CI for 2s2, and the core polarization method (see reference 28 below) for ls2s. There is no one method best for all the electrons. Parts of this theory have been developed in a series of articles. 15 We summarize the over-all picture here as well as giving some previously unreported results. On Hartree-Fock Methods.-The HartreeFock method is a very good starting point, It gives the charge distributions well, since it takes out the long-range part of the coulomb repulsions between electrons.l5 Also for this reason the remaining electron correlation effects are simplest if one starts with the Hartree-Fock method. Even though it is difficult to obtain HartreeFock orbitals for large systems, it is still desirable to start a “framework” theory with the HartreeFock method and then to put on the remaining corrections. Any further approximations such as J . Chem. Phya., 24, 1166 (1956); 26, 1301 (1956), have based e, w separation on this work. For later references discussing thls type of trial function, see, e g., T. L. Allen and H. Shull. J . Chem. Phys., 35, 1644 (1961). (14) P. 0. Lowdin, ibid., 35, 78 Ll961); P. G. Lykosand R. G. Parr, abad., 26, 1301 (1956); 0. Sinanoglu, zbid., 35, 1212 (1960); R. MoWeeny, Preprint No. 59, Quant. Chem. Group, Uppsala Univ., Uppsals. Sweden, 1961; L. Szaaz, Phys. Rev., 186, 169 (1962). (15) 0. Sinsno&, J . Chem. Phys., Sl?, 706,3198 (1962); Proc. Nail. Acad. Sci. U.S.,47, 1217 (1961), and earlier papers cited there.
QUANTUMTHEORY OF ATOMS, MOLECULES, A I ~ DTHEIR INTERACTIONS
Dec., 1962
the use of LCAO MO instead of Hartree-Fock orbitals then may be examined in regard to their effects on the more general theory. For a closed-shell system there is a unique Hartree-Fock method. With non-closed shells, however, about 9 variants of Hartree-Fock methods have been discussed. There are many possibilities because one can vary the energy of different combinations af degenerate determinants that arise from a given non-closed configuration or make different approximations to get around complications such as the off-diagonal energy parameters that do not come up for a closed-shell system. All these methods give energies within often less than 0.01 e.v. of one another. The simplest theory both for getting the orbitals and for putting on correlations later is one based on the average energy of a configuration,ls but with each HartreeFock potential modified so as to make it the same acting on all electrons and as symmetric as possible.*’ (For instance, in carbon ls22s22p2configuration, the orbitals obtained this way are the same for all t’he terms 3P, lD, lS; there are no off -diagonal energy parameters Xi, and the HartreeFock potentials are the same for all the electrons.) Even in large saturated molecules the HartreeFock method is a valid starting point. The Hartree-Fock wave function for such a molecule is expected to show regional properties quite unaffected in going from one molecule to another. This is because electrons move in the HartreeFock potentials of all the electrons and also in the field of the nucliei. The net potential of one end of a molecule is a, shielded one and therefore of short range.’* For example, in ethane the average potential of a hydrogen atom on one carbon dies off before it reaches a hydrogen atom on the other carbon. Attempts to build molecular HartreeFock orbitals firom localized and invariant orbitals have been and are being made.19 Clusters of Electrons in the Hartree-Fock “Sea.”-In the Hartree-Fock method each electron moves in the average field of all the other electrons. Consider now all the remaining effects that the residual instantaneous repulsions between electrons will introduce. For simplicity, let us take a closed-shell system. Theory for a nonclosed shell is quite similar. It was shown previously that the exact wave function of a many-electron system’j is $’ = $o =
x
a(123
-v
=
c
1=1
+
40 X ; ($0, X) = 0 (2) k ... S); 1 = l(xl) = ls,, etc.
N
(fl)
+ c {d’l,I + 1>J
(16) See, e.g., J. C. Slater, “Quantum Theory of Atomic Structure,” Vol. I, McGraw-Hill Book Co., Inc., New York, N . Y.,1960. (17) A. J. Freeman, Rev. Mod. Phys., 33, 273 (1960). This combines the advantages of methods discussed by R. K. Nesbet (Proc. Roy. Soc. (London), AISO, 31:! (1955)) and by Slater (ref. 9). See also K. Ruedenberg (ref. 18) lfor a similar starting point for a-electron aystems. (18) K. Ruedenberg, J . Chem. Phys., 34, 1861 (1961). (19) For example, W.H. Adams, ibid., 34, 89 (1961); T. L. Gilbert
(fi,
k)
=
0,
(d’ijj
2285
k ) = 0, . . ., (Oijk-X, k ) = 0 (4)
for k = 1,2,3, i, . . . j , . . . N 4 0 is the Hartree-Fock determinant. Kote that this exact expression contains only a finite number of terms for a finite many-electron system. The functions fi, djj, etc., are not expanded in any basis set; they are in closed forms and are rigorously orthogonal (eq. 4) to all the individual Hartree-Fock orbitals occupied in $o. The successive terms in X correct for the effects of progressively larger numbers of interacting electrons. For convenience, one may speak of “collisions” of successive numbers of electrons in analogy with imperfect gas theory. The “collisions” are caused by the instantaneous repulsions that remain after the average parts have been taken out in the Hartree-Fock orbitals. Equation 3 can be written in a much more detailed fashion. It is easily shown that each term 0’containing a certain number of electrons at, a time actually consists of all possible ant’isymmetrized products of previous terms and an additional new term 0 representing the “collisions” of that many electrons all a t once. For example
ii’ij
= B(fifi)
+
D’ijk = a ( f i f j f k Oijkl
=
a{fifjfkfjl fi
Dijk
aij
+fi
+
djk fifj
+ dij
a
.
.
+
C-ijk)
(5)
+ ... +
+ . . + dij + . . . fiki
cijkl]
The product terms represent correlations taking place at the same t.ime but in different, regions of space, whereas a non-separable term such as Cijkl accounts for the correlations of all four electrons a t once (a four-body “collision”). Again, by analogy t’o imperfect gas theory the latter are “linked clusters,” whereas products are referred tmo as “unlinked clusters.” Corrections to the Hartree-Fock energy which represent various physical correlat’ion effects can be obtained explicitly by substituting X, eq. 3, in the variat’ional expression. l 5 The explicit, evaluation of the energy in this manner is greatly facilitated by the use of diagrams as in imperfect gas theory. Many different theories can now be derived from this detailed variational energy expression by minimizing different portions of it.20 If only the first two terms of eq. 3a are retained and the energy varied so as to obtain optimum f i and d i j , one get’s essentially a Brueckner theory for finite systems. This is also similar t o a method discussed by Fock, et ~ 1 . 2 ~ (to be published: this work was pointed out to us by P. G . Lykosl. See also 9. F. Boys, Rev. Mod. Phys., S I , 296 (1960). (20) -4 most systematic way of using the variation method is t o get trial functions by minimizing the physically significant o r large portions of /, then using these in the complete expression. Various perturbation theories come out as specid cases of this approach. 0. Sinano& J . Chem. P h y s . , 84, 1237 (1961). (21) V. Fock,M.Veselov, and M. Petrashen, J . E z p t l . Theor. P h y s . . USSR, 10, 723 (1940).
2286
OKTAY
SINANOGLU
If all the terms are retained, coupled equations are obtained which make self-consistent with the remaining effects. This is similar to Slater’s generalized SCF method.22 These theories are unnecessarily complicated and stress the part of X which is not physically the most important. We examine below all the different effects in X and in the energy and then take out the most significant parts, but estimate the remaining effects. Effect of Correlation on Orbitals.-Since the average coulombic field of all the electrons on a given one are implicit in the Hartree-Fock orbitals of 40, the fi in eq. 3 represents corrections to the Hartree-Fock orbital, i, due to the residual, instantaneous repulsions, Le., due to electron correlation. If X is approximated by perturbation theory, the first-order X1 rigorously does not contain23any fl. Since XI gives both the second and third-order energies, any effect of fi on energy would show up in the fourth and higher orders. Without being limited to perturbation theory nor to an infinite summation of single excitations, fi for a closed shell system can be obtained easily. A strong test of how much correlation would modify a Hartree-Fock orbital would be on He atom where the two electrons are quite tightly packed. Any trial function such as one of Hylleraas’ with rl2 for He atom can be rewritten as in eq. 2 and then a further Schmidt orthogonalization (see ref. 23) splits the correlation part into and the two-electron parts as in eq. 3. Thus, just by simple integrations from the known wave functions of helium, one obtains the fi completely. This was donez4using the simple trial function of Lowdin and Redei25for He and the fi was found to be about 4% of the Hartree-Fock orbital where it is largest,. This5 however is due to the inadequacy of the trial function. If it is subtracted out, a better trial function is obtained as evidenced by an energy lowering of 0.019 e.v. Similarly when a more accurate trial function, a Hylleraas’ sixterm one, is used, one obtains an fi everywhere considerably smaller than in the previous case. I n other types 0%correlation effects which give large energy errors,f, may beunimportant, too. For example, the main part of the correlation error in the outer shell of the Be atom is due to the new degeneracy of 2s2 with 2p2. But this is a double excitation and does not give any fi (for a study of f, in H2 near dissociation see ref. 24). Hartree-Fock orbitals give good electron densities. This supplemented by the above results leads one to drop fl from eq. 3 for closed shells. This eliminates all the products in eq. 5 that involve fi. The resulting energy expressions are much simpler than they would have been otherwise. Many-Electron Correlations.-Any three electrons picked out in a many-electron system will have a t least two spins alike. These will stay away from each other’s Fermi hole. Now, the repulsion, 1/rii, that an electron i exerts on another one, say (22) J. C. Slater, Phys. Rev., 91, 528 (1953). (23) 0. Sinanoglu, PPOC. Roy. SOC.(London), 8160, 379 (1961). In CI language, this means there are no single excitations which mix with +o (Brillouin theorem). (24) 0. Sinanoglu and D. Tuan, J . Chem. Phya., to be published. (25) P. 0. LGwdin and L. Redei, Phya. Rev., 114,’752 (1959).
Vol. 66
j, deviates any instant from the average value (the i, j part of the total Hartree-Fock potential). This “fluctuation potential” is what causes “electron correlation,”28 and is of short range in going from orbital to orbital.16 Thus, only at the most two electrons can “see” each other at once. Others remain outside the Fermi hole which is larger than the range. This causes more than two electron correlations, Oijk, etc., in eq. 5 to be small. The angular potential shown in ref. 15 for boron is similar to those that act between the p-electrons also in C, N, 0, F, and Ne. I n neon there are two separate tetrahedra of like spins which in +o move uncorrelated with one another. But only two electrons of opposite spins can come together a t one point, so even here pair correlations are the significant ones. These considerations also are supported by the recent values of correlation energies of first row atoms estimated e m p i r i ~ a l l y . ~ ~ In large systems when electrons get delocalized near-degeneracies arise just as the energy levels of a particle-in-a-box get closer together as the box gets larger. Even in such delocalized cases the molecular orbital pair correlations are the significant ones due to the orthogonality (&j,k) = 0, eq. 5. Electrons i, j while correlating cannot go into the already occupied orbital k. For example, the (ylg)2 correlation in Hz is about 7.7 e.v. near dissociation. This reduces to 0.5 to 1 e.v. in He-He, because now (crlu) is occupied and the dlZfor ( u , ) ~no longer contains (crlu)z as its main part. This also has the effect of making the correlation between any three (in Hez+) or four electrons at a time (in He-He) small. These arguments apply also to the many-electron terms that arise in energy even when just ~ j ’ s and their products are taken into account in X (see ref. 15). Main Correlation Effects.-Then the energy of a closed shell system reduces to the sum of the variational expressions of independent two-electron systems in the Hartree-Fock medium.I5 This allows calculations to be made on all the first row atoms and their ions with essentially 11 distinct pair functions of which only 5 or 6 are significant. These now provide the new building blocks to be used iTith slight changes in parameters in many atoms and molecules. Such calculations, also with estimates of the many-electron terms, etc. , that are neglected, are in progress in this writer’s L a b o m tory. Once the coupling terms are shown to be small, these results also provide a basis for semiempirical methods. Inner, Outer Shells.-The above theory naturally separates inner and outer shells in atoms and molecules without concealing what the neglected terms are. In a n-electron system if just that portion of the total energy corresponding to 7r-electrons is minimized according to the theory,’5 one derives (26) Note the definition in ref. 7. This should not be confused with, J. W. Linnett’s use of the word. His usage does not distinguish between all the “orbital average polarization’’ and Fermi hole effects, already contained in the Hartree-Fock do, and the residual charge fluctuations. (27) E. Clementi, J . Chem. Phys.. in press. The 2p.a2pYa correlation in nitrogen atom is only 0.1 e.v. e.@.,
QUANTUM THEORY OF ATOMS,MOLECULES, AXD
Dec., 1962
rigorously the semi-empirical “n-electron Hamiltonian.” The form of the energv -” including correlation is now just the same1sp28as in the- Hartree-Fock N
approximation
(Jij
-
Kij
+
q) instead of
2>3
N
(JI, - K1J. These q’s are independent
just c-3
of one another. They do not depend on o i j k , t?,, etc., of X, but only on t&j. This justifies the parametrization of spectra mentioned above. Note also that the above theory not only introduces correlations into each shell, but also contains the correlations (instantaneous core polarizations) between different shells. These are analogous to van der Waals’ attractions.28 Localized Groups.-The determinantal function #J, was so far expressed in terms of molecular orbital pairs. How are these related to the localized electron-pair bonds or lone pairs of a saturated molecule? It is not necessary now to start anew with arbitrary assumptions to discuss localized groups. The transformation of Lennard-Jonesl2 which took do from the molecular to the localized orbital description also translates the correlation energy and X from one picture to the other. It has been shown rigorously that the exact correlation energy of a saturated molecule i s given by the sum of bond or lone pair correlation energies and non-bonded van der Waals’ attraction^.^^ The same transformation applies also to the variational expression15 involving sums of correlation energies of molecular orbital pairs. The latter then becomes the sum of variational expressions for individual bonds or lone pairs and for just the individual van der Waals attractions. Some simple coupling, terms arise too. These have been estimated30 to be about 0.1 e.v. in CHI by making the transformation29 on the configuration interaction ( N O CI) nave function of N e ~ b e t . ~ l This rneans2Othat each of these energy terms can be minimized independently. For instance, one can obtain the energy including correlation of just a CH bond or a lone pair by taking a trial function, e.g., containing rlzjust as in the Hz molecule. On the other hand, the part of the energy which corresponds to a van der Waals attraction between bonds also can be obtained by itself. Since this theory does not make any of the assumptions about the independence of basis functions of groups or about the range of the internuclear separations as in the us!ial theories discussed at the beginning of this article, ione now also has a theory of intermolecular force; applicable at all R. Such applications are being carried out on He-He and hydrocarbons. 3L
(28) 0. Slnanoilu, J . Chem. Phys., 88, 1212 (1960). (29) 0 Sinanoglu, abad.. S7, 191 (1962). (30) 0. Sinanoglu and V. McKoy, $bad., t o be published. (31) R. K. Nesbet zbrd., 32, 1114 (1960).
THEIR
INTERACTIONS 2287
How sensitive are these transformed localized functions and energies to the rest of the molecular environment? Just the orbital part of this question has been investigated by Lennard-Jones and co-workers. correlated the ionization potentials of the hydrocarbons ethane to decane assuming constant localized orbitals. Ionization potentials are not much influenced by electron correlation. I n heats of formation and conformations of molecules on the other hand, correlation is all important? Detailed t h e ~ r y ~ ~ -shows ~ O that the correlation energy of a bond depends on the rest of the molecule through (a) the Hartree-Fock potential of the whole molecule and (b) by exclusion effects as in eq. 4. The localized correlation functions29 are still orthogonal in the sense of eq. 4 to the entire molecular orbitals. However, in this orthogonality the only significant contribution from an MO is from the localized orbital part for just that bond. The parts of the Hartree-Fock potentials from other regions are shielded by the nuclei. Moreover, “tight correlation energies” as in helium and HI (similar to those in inner-shells, C-H, C-C single bonds, lone pairs, etc.) are remarkably insensitive to the oTer-all potential. The ls2 correlation* changes only by 0.012 e.v. in going from Li+ to Be+2. Thus the effect of Hartree-Fock potential changes on localized correlations are expected to be a good deal less than 0.01 e.v. The exclusion effects from other orbitals also are expected to be small. We had already estimated the exclusion effect of 2s on the Li+ correlation energy to be about 0.004 e.v.28 Thus, the theory enables one to put such well established empirical facts as additivity and constancy of bonds on a quantitative basis with the inclusion of many-electron effects. Another potential use of the transformations of correlation energy described above is in relating correlation shifts of the levels based on HartreeFock MO theory in the electronic spectra of mole cules to the correlation energies of localized groups which can be estimated muchmore readily, Inn-.lr* transitions, for instance, one could estimate the correlation energy in lone pairs, then by the transformation make estimates of correlation energies between delocalized molecular orbitals. Finally, the theory described above also can be formulated in very much the same way relativistically. Once shells are separated, relativistic effects can be left in the inner shells where they are known to be imp0rtant.~~,33One then deals with non-relativistic results for the outer shells obtained upon further approximation. Thus one finds a quantitative and systematic basis for the “simple quantum. chemistry” as well as a “chemical” simplification in “molecular quantum mechanics.” (32) G. G. Hall, Proc. Roy. SOC.(London), A206 541 (1951). (33) A. Frdrnan, Rev. Mod. Phye., 83, 317 (1960).
