Generalized Brillouin theorem ,

Because the variation principle is involved, certain matrix elements disappear as in ordinary SCF theory, and such relations have recently been referred to as generalized Brillouin theorems.38 A major review of the MCSCF method has been given by Wahl 39 the practical limit on the number of configurations seems to be around 50 at present, but energy results compare extremely favourably with those of traditional Cl calculations invoking many more configurations, and other calculated properties are encouraging. 

F. Grein and T. C. Chang, Chem. Phys. Lett., 12,44 (1971), Multiconfiguration Wavefunctions Obtained by Application of the Generalized Brillouin Theorem. 

The Eq. (4-4) actually indicates the generalized-Brillouin theorem. This theorem implies that a function. S ) AC) is the basis function for describing the excited states. Let us consider an excited function,... 

Finally, again for Li2, Table V also illustrates that at the end of the iterative process, at sweep 7, the sum of En and Eo 1 must be null, a consequence of the Generalized Brillouin theorem. 

It seems in this way that throughout EJR formalism one is able to find some sort of generalized Brillouin theorem for any arbitrary expression involving Quantum Mechanical objects. 

Eq. (155) is referred to as the generalized Brillouin theorem and as the Brillouin-Levy-Berthier theorem . Eq. (156) implies that the converged CSF expansion coefficients are an eigenvector of the Hamiltonian matrix. 

The second condition is the generalized Brillouin theorem, first derived by Levy and Berthier. ... 

The XifS are the matrix dements of X with respect to the f s i, j being creation and annihilation operators for spinorbitals (pi and (pj. The condition for optimum orbitals is then the generalized Brillouin theorem ... 

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. 

From the energy expression for the HPHF function (23), the pseudoeigenvalue equations for the HPHF orbitals may be easily deduced resorting to the Generalized Brillouin Theorem expression [10], This theorem has been shown to be valid for any MCSCF function [17] and, in particular, for a two determinant SCF function [7,8] ... 

W. H. E. Schwarz and T. C. Chang, Multiconfiguration wave functions for highly excited states by the generalized Brillouin theorem method, Int. J. Quant. Chem. Symp. 10 91 (1976). 

Roos BO. Multiconfigurational (MC) Self-Consistent-Field (SCF) Theory. In Widmark P-O, editor. European Summer School in Quantum Chemistry, Book II. Lund Lund University 2009. Grein F, Chang TC. Multiconfiguration wavefunctions obtained by application of the generalized Brillouin theorem. Chem Phys Lett. 1971 12 44. 

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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