Generalized Hellmann-Feynman theory

This is just the expectation or average value of the interaction operator, 6h, taken over the electron density function of the isolated molecule, as would have been anticipated from perturbation theory. This is a simple, powerful and extremely general result, which provides a basis for discussion of all atomic and molecular properties that involve a first-order response, and is applicable even when we do not possess exact wavefunc-tions it is usually referred to as a generalized Hellmann-Feynman... 

In this section, we want to derive expressions for the second derivatives of the energy with respect to two components Pq,... and P. .. of the general electromagnetic field without relying on perturbation theory. According to the Hellmann-Feynman theorem, Eq. (3.7), the second derivative of the energy is equal to the first derivative of the expectation value of the derivative of the Hamiltonian for a non-zero value of the field, P 0, i.e. 

Equation (13.5.8), which represents the generalization of the Hellmann-Feynman theorem to coupled-cluster wave functions, is shown in Exercise 13.1 to give size-extensive first-order properties. For variational wave functions, the Hellmann-Feynman theorem contains the real average value of the operator V (4.2.51) rather than a transition expectation value as in (13.5.8). Likewise, to ensure teal properties, we may in coupled-cluster theory work in terms of the manifestly real expression... 

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