The Brillouin Theorem

Let be the Hartree-Fock ground-state wave function and a singly excited state which differs from by substituting an occupied spinorbital by a virtual one. Then the following statement holds (Brillouin theorem)  

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 

Hartree-Fock wave function, and, in fact, the most general derivation of the Hartree-Fock equations is possible through the Brillouin theorem which can be proved directly from the variation principle (Mayer 1971,1973,1974). We shall not prove here the complete equivalence of the Hartree-Fock equations and Eq. (11.1), it will be shown only that the Brillouin theorem is fulfilled for the Hartree-Fock wave function. The proof will make use of second quantization which helps us to evaluate the matrix element easily. To this goal, Eq. (11.1) should be rewritten in the second quantized notation. The ground state is simply represented the Fermi vacuum  

Substitution of the singly excited determinant of Eq. (11.2) and the second quantized form of the Hamiltonian into the left-hand side of Eq. (11.1) yields  

We have to evaluate now the matrix elements which is a rather simple task. The matrix element of the one-electron part is trivial  

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These integrals will he non-zero only for double excitations, according to the Brillouin theorem. Third- and fourth-order Moller-Plesset calculations (MP3 and MP4) are also... 

I mentioned the Brillouin theorem in earlier chapters if rpQ is a closed-shell HF wavefunction and represents a singly excited state, then... 

Brillouin, L., ActualiUs sci. et ind. No. 71, La m thode du champ self-consistent." The Brillouin theorem introduced, b. 

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... 

An isospin basis possesses another important physical property - it expands the applicability of the Brillouin theorem to the excited nilNlnilNl configurations [45, 124]. 

The use of isospin basis allows one to widen the domain of applicability of the Brillouin theorem for excited states. In this approach it is also possible to account for part of the correlation effects. There are cases (e.g., configurations of the type niln2lNl,nil4l+1n2lN2 (n2 = m + 1) of multiply charged ions) where the isospin quantum number is fairly exact [31, 32]. 

In dealing with the MO-LCAO wave function no additional assumptions concerning the vibronic matrix elements are necessary. The evaluation of the total molecular energy exactly copies the lower sheet of the adiabatic potential. This is a consequence of the well-known fact that the Hartree-Fock equations are equivalent to the statement of the Brillouin theorem the matrix elements of the electronic Hamiltonian between the ground-state and... 

The Brillouin theorem states that if FHF represents a closed-shell molecule, then singly excited configurations such as Wt do not interact with /HF, i.e. < /HF IPj) = 0 and thus Cj = 0. Hence equation (14) becomes... 

For any variational wavefunction which is not near the Hartree-Fock limit the Brillouin theorem is irrelevant, and even for those of Hartree-Fock accuracy low-lying important excited states may invalidate the conclusions drawn from it. The statement that values of one-electron properties are expected to be good because of the Brillouin theorem should therefore be regarded with caution. 

The SuperCI itself is usually quite stable, but involves solving a non-orthogonal Cl of a considerable dimension, with each Brillouin state containing the same number of determinants as the Valence Bond wavefunction, which is rather time consuming. The SuperCI matrix can be approximated by its first row (the Brillouin theorem elements) and the diagonal at a considerable time saving. Then the Brillouin state coefficients by are estimated following... 

In HF theory, one has the Brillouin theorem (BT) stating that singly excited configurations do not interact with the ground state determinant [130], The proof commonly proceeds by utilizing the properties of the HF wave function. An alternative route was followed by Mayer [131, 132, 133] who derived the BT directly from the variational principle, permitting one to obtain the HF... 

These equations constitute the nonlocal part of the Brillouin theorem for geminals. They can be formulated as the APSG wave function is stationary with... 

Linear terms are absent because of the Brillouin theorem. The coefficients Ap p. and Bap p, can be calculated by equating the nonzero matrix elements of the RPA Hamiltonian [Eq. (122)], in the basis of Eq. (121), to the corresponding matrix elements of the exact Hamiltonian [Eq. (23)] in the same basis. From the translational symmetry of the mean field states it follows that the A and B coefficients do not depend on the complete labels P = n, i, K and P = n, /, K1, but only on the sublattice labels /, AT and /, K. The second ingredient of the RPA formalism is that we assume boson commutation relations for the excitation and de-excitation operators (Raich and Etters, 1968 Dunmore, 1972). 

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