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Perturbation Theory of the Constrained Variational Method in Molecular Quantum Mechanics
1.Unless the operator commutes with the Hamiltonian, in which case the error is also of order
2.A. Mukherji and M. J. Karplus, J. Chem. Phys. 38, 44 (1963).
3.Y. Rasiel and D. R. Whitman, J. Chem. Phys. 42, 2124 (1965).
4.D. R. Whitman and R. Carpenter, Bull Am. Phys. Soc. 9, 231 (1964).
5.It is assumed that μ is a possible value of M; that is, μ does not exceed a bound of M. In the case of linear variation functions of the type (12) this is equivalent to assuming that μ lies between the smallest and largest eigenvalues of the matrix representative M of M in the basis set.
6.A. C. Hurley, Proc. Roy. Soc. (London) A226, 179 (1954).
7.This slightly confusing situation has arisen because the true energy of the system is not E(λ) in general (except for ) but , , . The energy difference is therefore of order and the “first‐order” term is essentially spurious.
8.J. O. Hirschfelder, W. Byers Brown, and S. T. Epstein, Advan. Quantum Chem. 1, 255 (1965).
9.In this section the Born‐Oppenheimer fixed‐nucleus approximation is assumed and all formulas are in conventional atomic units.
10.Since μ has three components there are in general three constraints, and strictly speaking the multiple‐constraint theory developed in a later section is required.
11.W. Byers Brown, Proc. Cambridge Phil. Soc, 54, 251 (1958).
12.E. Steiner, Ph.D. thesis, University of Manchester, England, 1961.
13.L. Salem, J. Chem. Phys. 38, 1227 (1963).
14.This is not strictly the original virial theorem, but a closely related theorem derived by homogeneous scaling. The original virial theorem leads to and is only equivalent to the scaled form quoted in the text if the Hellmann‐Feynman theorem is valid.
15.J. O. Hirschfelder, J. Chem. Phys. 33, 1762 (1960).
16.S. T. Epstein and J. O. Hirschfelder, Phys. Rev. 123, 1495 (1960).
17.For the virial theorem , and for the Hellmann‐Feynman theorem for the total force .
18.For the virial theorem the mode is homogeneous scaling of the electronic coordinates. For the Hellmann‐Feynman theorem for the total force it is uniform translation of the electronic position coordinates in an arbitrary direction.
19.Note that the coefficient of in (97) does not vanish if when C and D are nonzero constants.
20.See A. Dalgarno, “Stationary Perturbation Theory”, in Quantum Theory, edited by D. R. Bates (Academic Press Inc., New York, 1961), Vol. 1 Chap. 5, p. 171.
21.T. Sasakawa, J. Math. Phys. 4, 970 (1963).
22.P.‐O. Löwdin, J. Mol. Spectry. 10, 12 (1963);
22.P.‐O. Löwdin, 14, 112 (1964).
23.J. M. Robinson, Jr., Ph.D. thesis, University of Texas, Austin, 1957.
24.Robinson’s work has been surpassed in accuracy by the recent 28‐term calculation, using a mixed orbital basis set, of J. C. Browne and F. A. Matsen, Phys. Rev. 135, A1227 (1964). It was chosen by Rasiel and Whitman because it gave a good energy but a poor dipole moment, and was therefore ripe for improvement.
25.D. P. Chong has pointed out an error in the values quoted by Rasiel and Whitman3 for the forces, and has recalculated them. The figures given here are the recalculated values.
26.Y. Rasiel, Ph.D. thesis, Case Institute of Technology, 1964, and private communication.
27.This corresponds to a value of the polarizability parallel to the LiH axis of which is over twice as large as the approximate value calculated by H. J. Kolker and M. Karplus, J. Chem. Phys. 39, 2011 (1963). These authors think their method already overestimates α, so that the replacement of by its experimental value would be in error by over a factor of two.
28.The exact value of the coefficient given by Eq. (51), is 4 for LiH. This is a warning that the use of theoretically or experimentally known values of the exact coefficients in estimating the effects of constraints may be very wide of the mark.
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