Hellmann-Feynman and virial theorems

The derivation here follows Hurley [179], Given //(f) for some real parameter f, and a variational approximation such that SS(T // T) = 0 and 5(T T) = 0, (he following theorem can be proved if variations /ST include all variations induced by 8%, then 

The notation ) here denotes a mean value, (T T)/(T T). To prove the theorem, let each trial function depend on f, and require /ST = 5f to lie in the Hilbert space of variational trial functions. For variational wave functions and energy values, ( 5T // — E T) + (T /7 — 5T) = 0. For variations driven by 8%, 

If the Hellmann-Feynman theorem is to be valid for forces on nuclei, the Coulomb cusp condition must be satisfied. However, if the nuclei are displaced, the orbital Hilbert space is modified. Hurley [179] noted this condition for finite basis sets, and introduced the idea of floating basis functions, with cusps that can shift away from the nuclei, in order to validate the theorem for such forces. 

As an extension of Noether s theorem to quantum mechanics, the hypervirial theorem [101] derives conservation laws from invariant transformations of the theory. Consider a unitary transformation of the Schrodinger equation, U(H — F)T = U(H — = 0, and assume the variational Hilbert space closed under a 

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Equations (3.20) and (3.21) represent an identity in Hartree-Fock theory. (The Hellmann-Feynman and virial theorems are satisfied by Hartree-Fock wavefunc-tions.) The particular interest offered by (3.21) lies in the fact that 7 = 1 appears to be the characteristic homogeneity of both Thomas-Fermi [62,75,76] and local density functional theory [77], in which case (3.20) gives the Ruedenberg approximation [78], E = v,e,-, while (3.21) gives the Politzer formula [79], E = Vne-... 

Hellmann-Feynman and virial theorems 4.2.1 Generalized Hellmann-Feynman theorem... 

The wavefunction variational principle implies the Hellmann-Feynman and virial theorems below and also implies the Hohenberg-Kohn [25] density functional variational principle to be presented later. 

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