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What is the best expression of the second‐order sum‐over‐state perturbation energy based on the Hartree‐Fock wavefunction?
1.J. O. Hirschfelder, W. B. Brown, and S. T. Epstein, Adv. Quantum Chem. 1, 255 (1964).
2.J. H. Epstein and S. T. Epstein, J. Chem. Phys. 42, 3630 (1965);
2.T. J. Dougherty, T. Vladimiroff, and S. T. Epstein, J. Chem. Phys. 45, 1803 (1966).
3.Y. Kato and A. Saika, J. Chem. Phys. 46, 1975 (1967).
4.For example, T. H. Dunning and V. McKoy, J. Chem. Phys. 47, 1735 (1967);
4.J. C. Ho, G. A. Segal, and H. S. Taylor, J. Chem. Phys. 56, 1520 (1972); , J. Chem. Phys.
4.T. Shibuya and V. McKoy, J. Chem. Phys. 58, 500 (1973)., J. Chem. Phys.
5.(a) H. J. Silverstone and M. L. Yin, J. Chem. Phys. 49, 2026 (1968);
5.(b) S. Huzinaga and C. Arnau, Phys. Rev. A 1, 1285 (1970);
5.S. Huzinaga and C. Arnau, J. Chem. Phys. 54, 1948 (1971).
6.K. Morokuma and S. Iwata, Chem. Phys. Lett. 16, 192 (1972).
7.(a) A. Dalgarno, Adv. Phys. 11, 281 (1962);
7.(b) R. M. Stevens, R. M. Pitzer, and W. N. Lipscomb, J. Chem. Phys. 38, 550 (1963);
7.(c) J. I. Musher, Ann. Phys. 32, 416 (1965);
7.(d) D. F. Tuan, S. T. Epstein, and J. O. Hirschfelder, J. Chem. Phys. 44, 431 (1966);
7.(e) P. W. Langhoff, M. Karplus, and R. P. Hurst, J. Chem. Phys. 44, 505 (1966); , J. Chem. Phys.
7.(f) G. Diercksen and R. McWeeny, J. Chem. Phys. 44, 3554 (1966); , J. Chem. Phys.
7.(g) J. I. Musher, J. Chem. Phys. 46, 369 (1967); , J. Chem. Phys.
7.(h) A. T. Amos and J. I. Musher, Mol. Phys. 13, 509 (1967);
7.A. T. Amos and J. I. Musher, J. Chem. Phys. 49, 2158 (1968);
7.(i) T. C. Caves and M. Karplus, J. Chem. Phys. 50, 3649 (1969)., J. Chem. Phys.
8.The method of summation is similar to that given in (a) H. P. Kelly, Phys. Rev. 136, B896 (1964);
8.(b) See also Eq. (8) of J. I. Musher, Chem. Phys. Lett. 6, 33 (1970).
9.See, for example, B. Kirtman and M. L. Benston, J. Chem. Phys. 46, 472 (1967).
10.K. Hirao and H. Kato, Chem. Phys. Lett, (to be published).
11.The unitary transformations within occupied and/ or virtual orbitals can be written as a special case of the unitary transformations among singly excited configurations.
12.A. T. Amos, J. I. Musher, and H. G. F. Roberts, Chem. Phys. Lett. 4, 93 (1969).
13.J. N. Murrell, M. A. Turpin, and R. Ditchfield, Mol. Phys. 18, 271 (1970).
14.In this last case, the orbitals and are not necessarily the canonical orbitals chosen at the beginning. They are actually the and modified orbitals obtained after the self‐consistent procedure mentioned in the text.
15.K. Hirao, “How to resolve the orbital ambiguity to obtain the orbital set which is stable to excitation” J. Chem. Phys. (to be published). The author acknowledges Dr. Hirao for sending the preprint prior to publication.
16.For open shells, see, K. Hirao and H. Nakatsuji, J. Chem. Phys. 59, 1457 (1973).
17.J. M. Schulman and J. I. Musher, J. Chem. Phys. 49, 4845 (1968).
18.When the solution of the modified HF equation (35) is inserted into the “uncoupled HF” equation of Dalgarno,7a Eq. (11), it becomes identical, in the one‐HF‐orbital case, to Eq. (25).
19.In the case of hydrogen atom, this unitary transformation of the HF virtual orbitals leads to the ordinary exact excited orbitals. Namely, in this case, the modified HF orbitals are equivalent to the exact excited orbitals (see Ref. 24).
20.In the ordinary coupled HF formalism,7 one obtains first the perturbed HF equation by applying the variational principle to the wavefunction (36) in the perturbed field Then, one expands and with respect to the order of perturbation and obtain the coupled HF equation from the first‐order relation. This leads to the second‐order energy of the coupled HF theory.
21.The second‐order orbital correction contributes to the second‐order energy in the form which vanishes identically due to the Brillouin theorem.
22.(a) H. D. Cohen and C. C. J. Roothaan, J. Chem. Phys. 43, S34 (1965);
22.H. D. Cohen, J. Chem. Phys. 43, 3558 (1965); , J. Chem. Phys.
22.(b) J. A. Pople, J. W. MczIver, and N. S. Ostlund, J. Chem. Phys. 49, 2960 (1968); , J. Chem. Phys.
22.(c) See also, H. Nakatsuji, K. Hirao, H. Kato, and T. Yonezawa, Chem. Phys. Lett. 6, 541 (1970).
23.In comparison with the D matrix in Sec. III, Eq. (20a), the matrices and are not diagonal, but the sum, is a diagonal matrix as defined by Eq. (42).
24.H. Nakatsuji and J. I. Musher, J. Chem. Phys. 61, 3737 (1974), following paper.
25.R. Ditchfield, N. S. Ostlund, J. N. Murrell, and M. A. Turpin, Mol. Phys. 18, 433 (1970).
26.H. Nakatsuji, J. Chem. Phys. 59, 2586 (1973).
26.See also, H. Nakatsuji, H. Kato, and T. Yonezawa, J. Chem. Phys. 51, 3175 (1969)., J. Chem. Phys.
27.See Eq. (V.1)–(V.4) of Ref. 1.
28.If we include, e.g., doubly excited configurations into of Eq. (58), a peculiar result occurs. Let us denote these doubly excited terms as The variation of these terms depends only on the last two terms of Eq. (58), and the minimization of with respect to leads to , where is the HF However, the left‐hand‐side integrals, which reduce to electron repulsion integrals, are not generally zero. This contradiction arises from the restriction of the zeroth‐order function in Eq. (58) to the HF In other words, doubly excited terms as well as other higher terms are allowed to appear in the first‐order correction only when the zeroth‐order wavefunction is an exact wavefunction or some over‐HF wavefunction which satisfies the generalized Brillouin relation like Eq. (A). However, this is not the present case. Therefore, the first‐order wavefunction is written uniquely by the sum of the singly excited configurations, so long as we choose the HF as the zeroth order wavefunction in Eq. (58).
29.If the form of the doubly excited terms of is free from the singly excited terms of it may be written as where is an independent parameter from the coefficients of the singly excited terms of The variation of given by Eq. (59) with respect to leads, as in Ref. 28, to the equation , which can never be satisfied actually. As discussed in Ref. 28, this contradiction means that the doubly excited second‐order terms free from do not occur in the of Eq. (59), so long as we restrict the in Eq. (59) to the HF
30.This corresponds to the Hylleraas variational principle1 of the second‐order energy, which is applicable only when the zeroth order wavefunction is an eigenfunction of the zeroth‐order Hamiltonian.
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