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Generalized Sternheimer Potential
1.For a recent review see J. O. Hirschfelder, W. Byers Brown, and S. T. Epstein, Recent Advances in Quantum Chemistry, P.‐O. Löwdin, Ed. (Academic Press Inc., New York, 1964), Vol. 1, p. 255.
2.For a discussion of possible criteria see the paper “What is ?” by S. T. Epstein to be published in the proceedings of the seminar on “Perturbation Theory and its Application in Quantum Mechanics” held in October 1965.
3.R. Sternheimer, Phys. Rev. 96, 951 (1954).
3.See also R. Makinson and J. Turner, Proc. Phys. Soc. (London) A66, 857 (1953).
4.If, for example, were not symmetric then, though infinite‐order perturbation theory might produce an antisymmetric ψ starting from an antisymmetric it is doubtful that any finite‐order approximation would be antisymmetric.
5.The formula is often quoted (see for example Ref. 1). However, it is clearly meaningless unless the spin dependence of is factorable so that one can “cancel” the spin functions in the last term, i.e., unless the sum in (3) contains but a single term.
6.A simpler way of getting a symmetric potential is to write , where the are simply products of one‐electron spin functions. Then is symmetric. However, in general, it does not commute with
7.If in addition we sum over m then will commute with all components of S.
8. should, of course, also be Hermitian. This condition is discussed in an Appendix.
9.In this case, of course, all the complications of (5) are unnecessary since is also presumably symmetric whence we may use
10.Some of this spin dependence is spurious in the sense that, as defined, unnecessarily annihilates any spin function not contained in Thus if all the are identical, then instead of the formula of Footnote 9, (5) yields , where I is the unit operator as far as spatial coordinates and any spin function in are concerned, but is the zero operator with respect to spin functions not in
11.Note that this many‐body character is in general present even for a simple two‐electron like the single determinant . On the other hand we may recall that by the use of nonlocal potentials, an for a single determinant may always be found in the form of a sum of one‐electron operators
11.[see S. T. Epstein, J. Chem. Phys. 41, 1045 (1964)].
12.A. Dalgarno and J. T. Lewis, Proc. Roy. Soc. (London) A233, 70 (1955). See also Ref. 1.
13.Since the projection operators are in general integral operators, is nonlocal. However, no integral operators occur in the analogs of (7) and (9). Further, some of this nonlocality is spurious in the sense that, as defined, unnecessarily annihilates functions belonging to irreducible representations not contained in
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