THE JOURNAL OF
PHYSICAL CHEMISTRY (Registered in U. S . Patent Office)
VOLUME66
(9 Copyright, 1962, by the American Chemical Society)
KUUBER12
DECEMBER 28, 1962
ELECTROS PAIRS IS THE BERYLLIUM ATOW BY T ~ o l r . 4L.~ ALL EX^ Department of Chemistry, University of California, Davis, California -4ND
HARRISON SHULL
Chemistry Department, Indiana University, Bloomington, Indiana Received M a y 19, 1062
The electron-pair approximation in the beryllium atom is investigated using a properly antisymmetrized product function over geminals (electron-pair wave functions). A wave function of this relatively simple type is obtained having an overlap of 0.9998886 with one of the best published functions, a superposition of 37 configurations. As a consequence of the separation into electron pairs, some of the coefficients in the configuration interaction treatment are found to be interrelated.
A major problem connected with wave functions of high accuracy is the difficulty of comprehending their significance in physical terms, and in fact the Hartree-Fock approximation has been described recently as “the last handhold for elementary physical int~iti.on.”~A very promising approach to this problem is the method of electron-pair wave functions or germinal^.^ It should be much more accurate than the Hartree-Fock approximation, and yet a rather high degree of i4suality is retained. Thus ,far specific applications include the water5 and formaldehydes molecules. We felt that it would be useful to test the accuracy of the method on a much simpler electronic system, where .the results could be compared with the highly accurate “superposition of configurations” or “configuration interaction” method. In the beryllium atom the electrons are thought to form two fairly distinct pairs, and therefore it should be an especially favorable case for this approach. A number of configuration interaction studies of this system have been published. The most recent arid accurate are those of Watson’ (1) Supported in p;%rtby grants from the National Science Foundation to each of the authors and in part by a contract with Indiana University b y the Air Force OSR. (2) On leave from the University of California, Davis, for 1959-1960. (3) R. K.Nesbet, Rev. Modern Phye., 88, 28 (1961). (4) T. L. Allen and H. Shull, J . Chem. Phys., 86, 1644 (1961), and references therein. (5) R. McWeeny and K. -1,Ohno, Proc. Roy. Soc. (London), 367 (1960). ( 6 ) J. AI. Parks and R. G. Parr. J . Chem. Phus.. 82, 1657 (1960). (7j R.E. Watson, Phys. Rea., 119, 170 (1960).
and TT’eiss.8 Watson calculated a 37-configuration function using an orthogonal basis set, and obtained a total energy which included 89.4% of the correlation energy. Weiss calculated a 55-configuration function using a non-orthogonal basis, and obtained a total energy containing 93.1% of the correlation energy. Because of its orthogonal basis, the former function 37
@ =
n=l
K,\k,
(1)
provides a convenient starting point for calculating a good approximation to an accurate geminal product function. The geminal product will be most easily handled if the geminals are one-electron orthogonal (2 1 where AK and AL designate the K-shell geminal and the L-shell geminal, respectively. This condition is fulfilled when each geminal is constructed from different orthogonal orbitals. An examination of 9 shows that most of the orbitals can be classified as strictly K-shell or L-shell orbitals. Only three (SI, PI, and p11) are used in both shells. The SI and p11 orbitals are mainly used in the K shell (except for configurations 10, 11, 36, and 37), while the PI orbital appears in the L shell except for configurations 8, 21, and 35. Therefore SI and p11 are assigned to the K shell, PI is assigned to
2281
Jh~*(1,2)A~(1,4) d71 = 0
(8) A. W. Weiss, ibid., 122, 1826 (1961).
THOMAS L. ALLENAND HARRISON SHULL
2282
the L shell, and the configurations mentioned are omitted from the geminal product function. The resulting classification of orbitals is shown in Table
TABLE I1 COMPARISON OF COEFFICIENTS IN Configuration
I.
L shell
K Bhell S I , 811, S I I I , SIV
2s PIJ Pv
a
AND
@
~I'('S)p~r'('S) Ka = 0.0071 K2K3/Ki Ks .0057 KZKI/Ki ,0016 KzKs/K1 PI*('S)drI*('S) Ki9 PII'(('s) ~I'('S) Kop = .00046 KaKia/Ki Rounded off to two significant figures.
PI'('s) SI'('S)
TABLE I CLASSIFICATION OF ORBITALS Is,
Yol. 66 Aa
A
=
= = =
0 0078 ,0069 ,0018 .00050
This method predicts that five of the new configurations will have coefficients of magnitude dx -0.0031; larger than 0.001 : PI~(~S)SISII('S), f I I , fIII ( l S ) ~ ~ ~ ~ ~ 0.0017; ~ v ( l S P ) I , ~ ( ~ S ) P I ~ I ~0.0015; ( ' S ) , PI'g11, g111 ('S)PIIPIII('s),0.0015; PI~('S)~ISIV(~S), 0.0015. It also is convenient to omit configuration 14 The overlap integra! of the geminal product from the geminal product because of the vector function with the 37-configuration function, f A*. coupling problem. (It could be included at a cPd.r, is easily calculated because of the orthogonal later stage in the calculations by applying a IS basis. Its value is 0.9998886. (Variation of the projection operator to the resulting geminal prod- coefficients to maximize the overlap integral did not give any significant improvement.) Thereuct function.) We can now write first approximations to AH fore the two functions are almost identical, and they should have closely similar energies. This result and AL is to be contrasted with the overlap integral of the Ar((1,2) = AqrK(KllS2f K$rr2 &SI' SCF function with 9,which has a value of KI = 0.9575824. K&12 K ~ P I I I ~KIZPIIPIII K I & I I ~ The energy contribution of each configuration KI~SIII' Ki&r12 Ki8PIv2 KBOSISII when added to the configuration interaction function may be obtained from Table I11 of ref. 7. For KzzgII' KBPIIIPIV KB~SIIISIV the configurationsomitted from the geminal product K ~ ~ S I S I VKZSPIIPIV Kzsd11d11.r K ~ O ~ I I I ~function (8, 10, 11, 14, 21, 35, 36, and 37) the total energy increments are 0.00347 a.u., or 3.7%. of K3idrv2 K33SIV2 K u ~ I I I ~ ) / K (3) I the total correlation energy. Since a configuration usually contributes less t o the energy of the final A~(3,4)= N L ( K ~ ~ sK2p1~ ~ Kiad12 function than it does when first included, in the 37-configuration function these configurations probK27pv2 KZSPIPV) (4) ably account for somewhat less than 3.70/, of the I n each configuration the functions with different correlation energy. The slight differences in the ml and mo values are vector coupled to form a coefficients of configurations 6, 9, 19, and 32 IS function. The K,'s are taken from Table I11 undoubtedly have a negligible effect on the energy. of reference 7, and the normalization constants The new configurations introduced by the geminal N g and NL are 1.0008179 and 0.9991075, respec- product function also must have nearly optimum tively. coefficients, as indicated by the coefficient test in The geminal product functionQA = [AK(~,~)AL-Table 11, and therefore they should improve the (3,4)] consists of 105 configurations. These in- energy significantly, Thus we estimate that clude 29 of the 37 configurations in 9,of which 25 89.4 - 3.7 = 85.7y0 is a lower limit to the correla(1-5,7,12, 13,15-18,20,22-31,33, and 34) appear tion energy of this geminal product function. with rewonable coefEicients because the correspondIn a geminal product n-ave function, electrons in ing Kn's have been used in writing AK and AL. different geminals affect one another by their The other 4 configurations (6,9,19, and 32) appear average distribution only. Therefore inter-shell automatically, and a comparison of their coefficients correlation cannot be included in a geminal product with the corresponding Kn's provides a test of the function, and in some systems this limitation may validity of this approach. For example, the coef- be of importance. For the beryllium atom, the ficient of configuration 6 is K2K3/KI (disregarding inter-shell correlation energy has been variously is almost unity). In this way the found to be negligible," and amounting to about N R N ~ which , following approximate relationships are derived 5% of the correlation energy incorporated into @ (plus an unknown amount from the remaining Ka &KdKi, Kg N K2K4/Ki1 energy).' The latter estimate is based (5) correlation on the correlation energy introduced by ls2sxy Kie KzKdKi, Kat 1 :KaKidKi and quadruple substitution configurations (that The numerical values (shown in Table 11) are in is, configurations using neither 1s nor 2s orbitals), reasonably good agreement. Thus the electron although it was pointed out that this interpretapair method is able to predict correctly the sign tion is not strictly realistic. Since A, which cannot and approximate magnitude of four coeEcients in have any inter-shell correlation, has a large num@ from six other coefficients.'O (IO) Similar results have been obtained independently by 0. Sinanop11, PIII, prv dIr, dm, dIv
+ + + +
+ + + +
+ + + + + + + + + + + + + +
(9) Brackets denote the partial antisymmetrization operator. It should be emphasized that the geminal product function is a completely antisymmetric wave function satisfying the Pauli exclusion principle.
i l u , Proc. Natl. Acad. Sci. U. S..47, 1217 (1961)and, J . Chem. Phya., 86, 706 (1962).
(11) J. Linderberg and H. Shull, J . Mol. Spectry., 6, 1 (1960).
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).
