Brillouin theorem generalized form

This is the closed form of the generalized Brillouin theorem usually given in C.I. form. > > It holds only for the exact C if of Eq. (20) and with H.F. orbitals. Note that the w s have been reduced to = l/r /s. 

The local Brillouin theorem of the usual self-consistent-field (SCF) theory was also generalized to the following form... 

This condition leads to a generalized form of the local Brillouin theorem of the usual SCF method... 

There are two basic differences in [his] approach which permit all orders to be treated at once. First, the starting point is the Brillouin-Wigner BW) perturbation theory, whose formal structure is much simpler than that of the RS expansion. Secondly, we use a factorization theorem , which expresses the required energy-denominator identities in a simple and general form. ... 

In other words, the stationary condition on V H ) is equivalent to a condition on the matrix elements connecting V with excited Vs formed by substituting an occupied spin-orbital by an unoccupied spin-orbital in all Slater determinants where it appears (destroying determinants in which it does not occur). This is reminiscent of the Brillouin theorem for a 1-determinant wavefunction, but is clearly a generalization it is not, however, in the form given by Levy and Berthier (1968). 

The Hartree-Fock state is thus characterized by a perfect balance between excitations and deexcitations for any pair of orbitals p and q, the interaction with the state generated by the excitation of a single electron from p to g is exactly matched by the interaction with the state generated by the opposite excitation. This result is known as the generalized Brillouin theorem (GBT) [1]. For closed-shell states, all interactions are trivially equal to zero (due to the structure of the Hartree-Fock state) except those with the singly excited states i a) and (10.2.19) then reduces to the special condition (10.2.17). For all other states, we may write the GBT (10.2.19) in the more explicit form... 

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