Brillouins theorem

The Hartree-Eock method is related to a number of theorems, for example, the Brillouin theorem (Equation 2.32). We form anew Slater determinant by substituting one of the occupied spin orbitals, pi, by an unoccupied one, to obtain a new Slater determinant, called hf is orthogonal to all occnpied W . We calculate oMhIo) and find 

The conclusion is that the spin orbitals that satisfy Equation 2.30, and thereby Equation 2.33, are the best possible ones from the point of view of total energy and thus the desired optimal Hartree-Eock spin orbitals. 

It also follows from Brillouin s theorem that the total energy cannot be improved in a configuration interaction (Cl) calculation that involves only the ground state 

Slater determinant and the singly substituted Slater determinants. The off-diagonal matrix elements in the Cl calculation are equal to zero by Equation 2.33. However, it is meaningful to involve all singly substituted Slater determinants in a calculation of the excited states. 

If the number of electrons is even, the spatial functions will be the same for one spin-up and one spin-down spin orbital, since Equation 2.30 is satisfied for both spins. Whether different orbitals for different spins give a better energy is an old problem in quantum chemistry, which we will not discuss here. 

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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... 

The disappearance of matrix elements between the HF reference and singly excited states is known as Brillouins theorem. The HF reference state therefore only has nonzero matrix elements with doubly excited determinants, and the full Cl matrix acquires a block diagonal structure. 

When using HF orbitals for constructing the Slater determinants, the first matrix elements are zero (Brillouins theorem) and the second matrix elements are just... 

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... 

If, in addition, singly excited states with respect to ri 0 are included, it can be shown (Brillouin theorem >) that the electronic ground state will still be described by the single closed-shell determinant wave function zlg of energy q. 

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. 

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. 

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