The Hellmann-Feynman Theorem for Approximate Wavefunctions

The approximate wavefunctions o) described in the previous sections of this chapter depend on molecular orbital coefficients c p and possibly also on some kind of configuration or determinant coefficients C o that together are here denoted as Ci. The energy of a molecule in all these approximate methods can be expressed as the following asymmetric expectation value 

In the case of the variational methods, SCF, MCSCF and CI, o0 = l o) and we have the normal expectation value. For the non-variational methods such as Mpller-Plesset perturbation or coupled cluster theory, the energy is calculated as a transition expectation value, where I FoO = 

Let us now consider again the case of a Hamiltonian H ), which depends on a perturbation symbolized by the real parameter A. Both sets of wavefunction coefficients will depend on A and the wavefunction thus indirectly also 

In addition to the wavefunction parameters, c p and Cofe, also the basis functions Xfi can depend on the perturbation. This will be the case when the perturbation 

The derivative of the electronic energy L o(A, Cj(A) ) with respect to the real parameter A is then 

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