The spin-restricted ensemble-referenced Kohn-Sham (REKS) method is based on an ensemble representation of the density and is capable of correctly describing the non-dynamic electron correlation stemming from (near-)degeneracy of several electronic configurations. The existing REKS methodology describes systems with two electrons in two fractionally occupied orbitals. In this work, the REKS methodology is extended to treat systems with four fractionally occupied orbitals accommodating four electrons and self-consistent implementation of the REKS(4,4) method with simultaneous optimization of the orbitals and their fractional occupation numbers is reported. The new method is applied to a number of molecular systems where simultaneous dissociation of several chemical bonds takes place, as well as to the singlet ground states of organic tetraradicals 2,4-didehydrometaxylylene and 1,4,6,9-spiro[4.4]nonatetrayl.
As a number of symmetry relations, e.g., holds for the microstate densities in Eq. (2), only 8 microstates out of 12 need to be calculated explicitly.
In the limit of non-interacting particles, i.e., , Eq. (1) becomes equivalent to the exact ensemble energy as first given by Perdew et al.10 and Lieb.11 As demonstrated in Ref. 18 (see also Refs. 25 and 26), the energy expression obtained for differs from Eq. (1) only in that there occur single determinant (sd) energies of the usual wavefunction theory with the electron-electron interaction scaled down by the factor . As the density matrices , are idempotent, integrating these energies with respect to , while constraining the density by external potential,25,26 results in the single determinant density functional energies occurring in the standard KS theory. In the practical application of Eq. (1), the energies are calculated by suitable density functional approximations.