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It is shown that a sufficient condition for an optimal energy variational wave function ψ₀ to satisfy the hypervirial relation (ψ₀, [H, W]ψ₀)=0 is for the trial function ψ to admit variations of the form ∂ψ∂a=(i)Wψ. Here H is the Hamiltonian, W is a Hermitian operator, and a is a variational parameter. Explicit forms of such trial functions are exhibited for several W's. The case in which W generates a point transformation of the coordinates is discussed in detail. Conditions are given for the existence of simultaneous hypervirial theorems.
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Epstein et al. (1961) studied this question.
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