Matrix elements of the third-order spin-adapted reduced Hamiltonian

A formalism recently developed to obtain a matrix representation of the second-order spin-adapted reduced Hamiltonian [J. Karwowski, W. Duch, and C. Valdemoro, Phys. Rev. A 33, 2254 (1986)] and derived from the symmetric-group approach to configuration-interaction methods has been generalized to the case of the third-order spin-adapted reduced Hamiltonian. As in the second-order case, the matrix elements are linear combinations of generalized two-electron integrals with coefficients explicitly related to traces of products of the orbital occupation-number operators. The third-order spin-adapted reduced Hamiltonian matrix has been block diagonalized according to the irreducible representations of the symmetric group $S_3$.