Hartree-Fock theory Brillouin theorem

The second possibility is to use a gradient code, if this is available for the chosen method. The third method is the simplest, namely to evaluate E2 as an expectation value. This method is equivalent to the other two, if Eq satisfies a stationarity condition, like the Brillouin condition of Hartree-Fock theory. For non-stationary approaches, like MP2 or CC, the methods based on differentiations of the expectation value are more reliable. This is related to the validity or non-validity of the Hellmann-Feynman theorem. 

Section 3.3 continues with formal aspects of Hartree-Fock theory. We derive and discuss two important theorems associated with the Hartree-Fock equations Koopmans theorem and Brillouin s theorem. The first... 

In words, singly excited states do not interact with the HF ground state the corresponding matrix element of the Hamiltonian is zero. This famous theorem (Brillouin 1933) plays an important role in the Hartree-Fock theory as well as in more sophisticated methods based on a Hartree-Fock reference state. It can be shown that Eq. (11.1) is a necessary and sufficient condition for to be the exact... 

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