Ilec., 1962
ELECTROXIC STRUCTT-KIC OF SIMPLE MOLECULES
2329
no data from any member of the methyl-substituted in their excited states; ( 5 )has become an established benzenes were iised. The results of the spin density tool for the chemist. Acknowledgments.--1he implementation for the work are summarized in Table I11 and the good Univac 1105 of the ?r-electron program was in the agreement is gratifying indeed. main due to Dr. H. N. Schmeising, who, in the Conclusions process, produced a comprehensive matrix interIt seems quite clear that a-electron theory: pretive for the Univac 1105. Dr. 0. W. Adams and (1) can be coiiveniently used in a form which is Mr. R. L. Miller played a significant role in the properly based on quantum mechanics; (2) has been evolution of the program. We also were aided by given considerable impetus by the advent of the Dr. R. L. Flurry, Jr., and Mr. R. Blomquist was relarge fast electronic digital computer; ( 3 ) is suffi- sponsible for the “molecular diagram” computer ciently profound that a variety of molecular proper- output program. We received material support ties can be understood within a single theoretical from the following: generous grants of Univac 1105 framework while a t the same time sufficiently computer time from IIT; a grant from the National simple that it js tractable; (4) already has brought Science Foundation; and a grant from the National into new prominence and provided the theoretical Institutes of Health, for all of which we are grateha4s for the chemistry of conjugated molecules f ul .
ELECTRONIC STRTJCTURE OF SIMPLE MOLECULES’ BY LELAKDC. ALLEN I h p a y t m n t of Chemistry, Princeton University, Princ~ton,N. ,I. AND
ARSOLDAI. KARO
Department of ChPniiatr71, Lawrence Radaatzon Laboratory, LivermorP, Cnlif. Recezved June 26, 1062
Recent theoretical and computational developments in the direct solution of Schradinger’s equation are reviewed. The relationship between direct solutions of Schrodinger’s equation and semi-empirical calculations also is discussed.
I. Introduction Direct solutions of Schrodinger’s equation are defined as those attempts to solve the non-relativistic many-electron problem without approximation. Although most of the theoretical framework was worked out in the decade from 1930 to 1940, intensive efforts to explore, extend, and test the theory date from about 1951. The present article is concerned primarily with development’sin the last two years and with ;systems of more than two electrons.* A tabulation of molecular calculations reported during the last two years was to have been included but is omitted due to circumstances beyond our control. It will be published in J . Phys. Chem. in the near future and will bring a previous tabulation up to date. 11. General Formulations.-During the past two years there have been numerous papers devoted purely t.o the theory of many-electron systems. Perturbation Theory.-Many-particle pert’urbation t8heoryfor electronic problems has been act,ively pursued by L O ~ v d i nand ~ ~ S i n a n ~ g l u . This ~ ~ work follows the field theoretic approach developed by Rrueckner, Pines, Gell-Mann, Hubbard, Bogolnhov, Goldstone, Hugenholtz, and others.? (1) 9.M. K. wishes t o acknowledge t h e support of t h e U. S. Atomic Energy Commission. L. C. A. wishe8 t o acknowledge the financial assistance of the Petroleum Research F u n d of the I m e r i c a n Chemicnl Society. ( 2 ) L. C. Allen a n d 9. M. Karo, Rev. M o d . Phys., 3 2 , 275 (1960). (3) (a) P.-O. Lowdin, Technical Notes 47 a n d 48, “Quantiim Chemistry Group,” Uppsnlrt Univerdity, Uppsala, Sweden: (b) 0. Sinanoglu, Proc. Roy. SOC.(London), A260, 379 (1961). (4) D. Pines, “The Nany-Body Problem,” W. A. Benjamin, Inc., Yew York, X. Y., 19fil.
Pair Correlations.-The high percent’age of the total energy achieved by a Hartree-Fock solution (approximately 99.5%) and the stability of this solution as expressed by Brillouin’s theorem plus the exclusion effects of the Pauli principle and n wide variety of numerical calculations over a period of years has led us to believe that correlation corrections t o Hartree-Fock solutions, adequate for chemical problems, can be obtained with correlations between rather localized, independent pairs of electrons only. Recent treatments by N e ~ b e t , ~ Sinanoglu,E and Lowdin’ have formalized these ideas. Although arising basically from a valence bond description, recent many-electron formulations using two-electron basis functions are closely related to the pair correlation problem. Much of this work derives from the classic paper of Hurley, Lennard-Jones, and Pople.s Parks and Parr,g Allen and Shull,loand McWeeny and Ohno” have made contributions to this area in the last two years. ?;one of this work nor that on pair correlnt’ionsand the many-particle perturbation theory has led to improved computational methods, but expres( 5 ) R. K. Nesbet, Rev. M o d . Phys.,3 2 , 275 (1960). (6) 0. Sinanoglu, J . Chem. Phys.,36, 706 (1962). ( 7 ) P.-O. Lowdin, Phys. Rev., 97, 1509 (1955): Aduan. Clieni. Plrys., 2, 207 (1959); preprint No. 1, July, 1960, Quantum Theory Project, University of Florida, Gainesville, Florida. (8) .4. C. Hurley, J. E. Lennard-Jones. and .J. A . Pople. Proc. Roy. Soc. (London).
AB20, 416 (1953).
(9) J. M. Parks a n d R. G . Parr, J . Chem. Phvs., 26, 3 3 5 (1988):
32, 1657 (1960). (10) T. L. .Illen a n d H. Shull, ibid., 36, 1644 (1961). (11) R. McWeeny and K.A. Ohno, Proe. Hog. S o r . (London), A266, 367 (1900).
2330
LELAND C. ALLENAND ARNOLD M. KARO
sing the results in a variety of ways has led us to a clearer understanding of the many-electron problem. A good deal of our thinking about the correlation problem comes from the unrestricted and extended Hartree-Fock methods and the alternant molecular orbital scheme is one practical result of these concepts. Originating with Coulson and Fischer,I2 the scheme has been generalized by Lowdin's and applied to several molecular systems by Itoh and Yoshizumi," Lefebvre, Dearman, and McConnell,16 and Pauncz, de Heer, and Lowdin.16 Although the recent calculations have been restricted to r-electrons, the results are of general significance. The calculations present fairly good evidence that the alternant molecular orbital scheme for introducing different orbitals for different spins accounts for perhaps half the correlation energy and leads to improvement in such quantities as the unpaired spin density in certain free radicals. Hypervirial Theorem.-Hirschfelder and Coulson15have generalized the virial theorem and have shown that the resultant hypervirial theorem is a consequence of the Heisenberg equation of motion. As pointed out by the authors, this generalization may help in the determination of average values of dynamical variables and in providing criteria for improving wave functions. Density Matrices.-The subject of density matrices continues to receive a considerable amount of attention. McWeenyls and LOwdinlg have been prominent in this endeavor. However, the well known and great problem of ascertaining the necessary and s d c i e n t conditions that must be satisfied to make density matrices correspond to solutions of Gchrijdinger's equation still is unsolved. Colemanz0 has made some progress on conditions for first-order density matrices but the realization of a, complete set of requirements for the first and second-order density matrices seems very remote. Nevertheless, it is quite likely that density matrices and perhaps Lowdin's natural spin orbitals will provide the best cannonical form for comparing wave functions calculated by different methods. Shull has demonstrated how this may be done for the two-electron problem.21 Even a cursory survey of the existing ab initio wave functions reveals that except in a very few cases, we have not abstracted the iiiformation and knowledge available nor do we have a systematic method (other than the total energy) for comparing spacial distributions (12) C. A. Coulson and I. Fischer, Phtl. Moo., 40, 486 (1949). (13) P.-0. L h d m , Nikko Sump. Mol. Phya., 18 (hlaruzen. Tokyo, 1954), Technical Note 67, Quantum Chemistry Group, Uppaala University. Oct., 1961. (14) T. Itoh and H. Yoshizumi, J . Phya. SOC.Japan. 10,201 (1955); J . Cham. Phua., 28, 412 (1955). (15) R. Lefebvre H.H. Dearman. and H. M. McConnell, %bid.,32, 176 (1960). (16) R. Pauncz, -1. de Heer. and P.-0. LBwdin, Technical Notes 55 and 56, Quantum Chemistry Group, Uppsale Unlversity, 1960; J . Chcm. Phya., to be published. (17) J. 0.Hirachfelder and C. A. Coulson, J . Cham. Phvr., 88, 941 (1962). (18) R. Mcweeny, Proc. Rou. SOC.(London), A182, 114 (1955). (19) P . 4 . L6wdln. Phua. Rev., 97, 1474 (1955): Ann. Rev. Phur. Cham., 11, 107 (1960). (20) A. J. Coleman, Can. Math. Bull.. 4, 209 (1961). (21) H. Shull, J . A m . Chsm. Soc., 81, 1287 (1960).
Vol. 66
and other theoretically interesting quantities. This seems particularly unfortunate since the principal utility of direct solutions for simple systems has been and remains the testing of theories rather than the prediction of chemically interesting properties. Ill. Parametric Methods and Models.-There is another aspect to the general formulations mentioned above which is quite encouraging and this is our greatly increased understanding of why the highly successful Huckel model and the Pariser and Parr method actually work. In addition to the papers noted above, Parr and LykosZ2have comprehensively analyzed these schemes and Ruedenbergz3has written a definitive set of articles on the problem. The increasing interest of organic chemists in simple electronic structure models and our more extensive knowledge concerning them leads one to believe that there will be a greater interchange between groups of people working at different levels of approximation and a greater impact of a priori methods on chemical problems. IV. Wave Functions for Simple Molecules.By 1960 there were approximatelv 80 molecular wave functions that could be classed as direct solutions to Schrodinger's equation, During the past two years 80 more have been carried out. In spite of their inability to aid in the interpretation of current chemical problems, these calculations provide by far the best guide to a detailed understanding of the electronic structure of molecules. Almost all existing calculations have constructed a manyelectron wave function from one-electron spin-orbitals and almost all have adopted a molecular orbital approach carried out via the Roothaan procedure. Within this framework the wave functions have been expressed either as LCAO or one-center expansions. One-center calculations for methane and hydrogen fluoride have been quite successful but the potential singularities for atoms other than hydrogen are so severe that the LCAO expansion is almost certainly the more efficient. All but about 25 of the existing wave functions use the LCAO expansion. Two Four-Electron Calculations.-A Be atom wave function calculated by Watsonz4and an LiH wave function determined by Ebbingz6 have considerably enhanced our knowledge of many-electron effects and deserve special mention. The important feature of these calculations is the extensive configuration interaction that was carried out and the detailed analysis and testing of particular configurations. They form the best confirmation of our qualitative ideas about pair-correlations. A Molecular Hartree-Fock Solution.-Probably the most exciting development in the past two years has been the achievement of the first true molecular Hartree-Fock solution. The calculations were performed independently by ClementiZ6 (22) R. G. Parr and P. G . Lykos, Tozaa J . Sci., 8 , 135 (1956); R. G. Parr, J . Chsm. Phys.. 38, 1184 (1960). See this latter article for discussion and references to previoue work on the justification of seiniempirical models. (23) K. Ruedenberg. ibid.. 84, 1861, 1878, 1884, 1892, 1897, 1907, (1961). (24) R. E. Watson, Phus. Rea., 119, 170 (1960). (25) D. D. Ebbing, J . Cham. Phus., 86, 1361 (1962). (26) E. Clementi, ibid., 86, 33 (1962).
Dec., 1982
ELECTRONIC STRUCTUREOF SIMPLE MOLECULES
and NesbetZ7 with identical results. Hydrogen fluoride was chosen because we have more extensive theoretical knowledge on this molecule than any other. The results obtained were quite unexpected and have far-reaching significance for our understanding of molecular correlation energy and binding energy. This wave function predicts force constant, equilibrium separation, dipole moment, and the derivative of the dipole moment to within a few per cent of the experimental results and reasonable values for excitat,ion energies-all improvements over previous calculations carried out by diverse methods. And, of greatest interest, the predicted binding energy is */3 of the experimental value, in sharp contrast to the figure '/4 which has become almost a cliche for the typical LCAO wave function. This molecular Hartree-Fock wave function is not, of course, an LCAO expansion and the improvement is achieved through polarization of the atomic orbitals (obtained by adding u and T basis functions of one higher I-value to the outer orbitals). Since the molecular wave function is a Hartree-Fock solution and since the separated atoms are Hartree-Fock solutions, the error in the binding energy is the molecular correlation energy and these calcuilations are our first accurate estimate of this important quantity. The value is only 1.8 e.v., although :it frequently has been assumed to be 2.5 times as much, and is close to the value yielded by typical simple configuration interaction calculations, although these have not started with molecular Hartree-Fock solutions. On the other hand, there has been very little exploration of the appropriate excited basis orbitals to use for configuration interaction and it is possible that accurate wave functions may be obtainable by simple, rapidly converging configuration interaction treatments. V. Molecular Integrals.-Mounting one-electron basis orbitals a t the various nuclear sites in a molecule still seems to be the most efficient and readily interpretable procedure for setting up molecular wavie functions, even though this leads to the formidable many-center, two-electron integral problem. It is still true, as it has been for the past decade, that multi-center integrals are by far the greatest problem in the direct solution of Schrodinger's equation. It also is well known that (27) R. K. Nesbet, J Chem. Phys., 36, 1518 (1962).
2331
the difficulty arises through use of the exponential basis function form rne-~~rY1lm,'(&p)where n, I , m are integral, a > 0, and Y is a spherical harmonic. Because exponentials form natural solutions to the atomic central field problem, single or linear combinations of this type of function are the most widely employed basis orbitals (about 95% of all existing wave functions) and four different techniques for their evaluation are under continuing development by workers at four institutions. Massachusetts Institute of Technology.-The Barnett-Coulson expansionz8about a single center is employed by Barnett and co-workers. University of Chicago.-The two-dimensional integration technique developed by McLean and Roothaan28is used by Roothaan and co-workers. Columbia University-IBM Watson Laboratory.-4scheme exploiting the analytic integral transform between exponential and gaussian functions (rne-ar2. Yllml(O,v) has been worked out by Shavitt.ao This is the newest and in some ways the most promising method. Cambridge University.-Boys and co-workers3 use a direct gaussian expansion of the exponentials. The first three groups are now writing the majority of their computer programs for the IBM 7090 while Boys employs the Cambridge University Edsac 11. ,4t present it is possible to obtain for exponential basis functions all integrals to the required accuracy for a set of several first row atoms on a line or a first row atom surrounded by any number of hydrogen atoms. Because all integrals over gaussian basis funct,ions may be expressed analytically, these are the most attractive alternative to exponentials. There has been only a small amount of experimentation with gaussian orbitals but it generally has been thought that the number of gaussians required for an adequate representation of molecular wave functions was prohibitively large. Sew evidence32 indicates that this viewpoint probably is incorrect and a revival of interest in this approach is taking place. (28) M. P. Barnett snd C. 4.Coulson, Phd. Trans. Roy. SOC. (London), A943, 221 (1951). (29) A. D.MoLean, J . Chem. Phya., 39, 1595 (1960). (30) I. Shsvitt and M. Karplus, zbtd., 36,660 (1962). (31) 5. F. Boys and G.B. Cook, Rev. Mod. Phys., 32, 285 (1960). (32) L. C. Allen, J . Chem. Phvs., 37, 200 (1962).
