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Hellmann—Feynman Theorem in Curvilinear Coordinate Systems
1.See, for example, M. Benston and B. Kirtman, J. Chem. Phys. 44, 119 (1966).
2.H. Hellmann, Quantenchemie (Deuticke and Co., Leipzig, Germany, 1937), p. 285;
2.R. P. Feynman, Phys. Rev. 56, 340 (1939).
3.There is no need to give special consideration to the possibility that the parameter is involved in the limits of integration. Such dependence can always be adsorbed into J by the use of step functions.
4.To show that the middle term vanishes one must invoke the Hermiticity of
5.That the statement of the generalized Hellmann‐Feynman theorem is unaltered by the presence of J was emphasized to us by R. Sack.
6.A. C. Hurley, Proc. Roy. Soc. (London) A226, 179 (1954).
7.An exception occurs when is independent of q. Then the third term vanishes from (1). This is the case considered in Ref. 1.
8.Note that if H is Hermitian in the sense of Footnote 4, then H is Hermitian in the sense that
9.It is this theorem which is explicitly under discussion in S. T. Epstein, J. Chem. Phys. 42, 3813 (1965)
9.since, as discussed by S. T. Epstein and J. O. Hirschfelder, Phys. Rev. 123, 1495 (1964), especially around their Eq. (17), wavefunctions derived by unitary transformation automatically contain the factor.
9.Also it is implicitly the theorem under discussion in for example J. O. Hirschfelder, W. Byers Brown, and S. T. Epstein, Advan. Quantum Chem. 1, 255 (1964), especially pp. 305–307, since no reference is made there to J.
10.More generally (4) implies (3) if ψ satisfies the hypervirial theorem . This theorem is of course satisfied by an exact wavefunction and may or may not be satisfied by a variational one
10.[S. T. Epstein and J. O. Hirschfelder, Phys. Rev. 123, 1495 (1964)].
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