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Variational Methods and Degenerate Perturbation Theory
1.For a recent review see J. O. Hirschfelder, W. Byers Brown, and S. T. Epstein, Advan. Quantum Chem. 1, 255 (1964).
2.Note that with this sort of trial function where η is a small constant, is a possible variation whence implies Similar remarks will apply to other variations which we will be using later, so that we will never need to use explicitly.
3.See, for example, L. Pauling and E. B. Wilson, Jr., Introduction to Quantum Mechanics (McGraw‐Hill Book Co., Inc., New York, 1935), especially Sec. 24.
4.With a fixed basis set one is doing perturbation theory exactly for the operator where and and where P is the projection operator onto the basis set. If then one chooses the basis, as one may always do, so as to diagonalize then all formulas look like the exact sum over states formulas except the sums are finite. [See also H. J. Kolker and H. H. Michels, J. Chem. Phys. 43, 1027 (1965).].
5.If is not exact then all we can prove in general concerning the relation between the and the exact values is that for the ground state is an upper bound to the lowest associated with the same symmetry (see Footnote 11, p. 300 of Ref. 1). If one uses linear Variational parameters then, of course, one can say more [J. K. L. MacDonald, Phys. Rev. 43, 830 (1933); see also Footnote 6].
6.This orthogonality assumption is not essential for the analysis which follows. However, if is a possible variation of and conversely then it is easy to prove from (18) and (19) that and will be orthogonal. Thus given a set which did not yield we can always get orthogonality by entering the Variational principle again with the new Variational function for : , where now the are fixed and the are linear Variational parameters which we are to determine. Incidentally, from MacDonald’s theorem5 this will also ensure, at least for the ground state, that the (new) arranged in order of increasing value, will be upper bounds to the similarly ordered exact
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