Brillouin’s theorem in the Hartree–Fock method: Eliminating the limitation of the theorem for excitations in the open shell

It is well known that Brillouin’s theorem (BT) holds in the restricted open-shell Hartree–Fock (ROHF) method for three kinds of single excitations, c → o, c → v, and o → v, where c, o, and v are the orbitals of the closed, open, and virtual shells, respectively. For these excitations, the conditions imposed by BT on the orbitals of a system under study are physically equivalent to the conditions imposed by the variational principle, and this provides a fundamental meaning of BT. Together with this, BT is not satisfied for some excitations of the kind o → o, in which both orbitals participating in excitation belong to the open shell. This limitation of BT is known, for example, for the helium atom, where BT is satisfied for excitation from the ground state S01 (1s2) to the state S11 of the configuration 1s12s1 and is not satisfied for excitations S11 → S01 and S11 → S21 (2s2). In this work, we prove that Brillouin’s conditions for two latter excitations cannot be related to the fundamental conditions imposed by the variational principle due to specific symmetry restrictions. Based on this finding, we give a rigorous proof of fulfillment of BT for the alternative o → o excitation, which takes in the helium atom the form S11 → S31, where both the initial and excited states are treated as arising from the same open-shell configuration 1s12s1, and the state S31 is described by the symmetry-adapted ROHF wave function Ψ(S31) = [Ψ(S21) − Ψ(S01)]/2. The new formulation of BT obeys all the necessary variational and symmetrical conditions, and its validity is illustrated by the results of computations of atom He and molecule LiH in their singlet states arising from different closed-shell and open-shell configurations performed using both ROHF and limited configuration interaction methods.