Hartree-Fock approach Brillouin theorem

There are some additional reasons which make the contribution of monotransferred terms uniquely important. As assumed before, the MO s used are the Hartree-Fock or other SCF ones so that the values of Ho,p of monoex. terms are small, since the Brillouin theorem 55> requires that the matrix element between the ground state and a monoexcited state in the Hartree-Fock approach should vanish in an isolated molecule. In addition to this, the denominator of the second-order term... 

Apart from practical and computational sides, the SCF model has both intuitive appeal and formal mathematical advantages the Hartree-Fock solution is a well-defined one, with known mathematical properties (Brillouin theorem, Hellmann-Feynman theorem etc.). As an example one may mention the invariance of the coupled perturbed Hartree-Fock solution under a translation of the gauge of an external perturbing magnetic field 78). Such properties give the SCF approach a certain special position. 

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. 

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