Open-Shell Self-Consistent-Field Molecular-Orbital Theory for Transition-Metal Clusters

Detailed formulations are developed for the theoretical calculation of molecular-orbital (MO) wave functions and electronic excitation energies for transition-metal clusters. The procedures are based upon the spin- and symmetry-restricted Hartree-Fock-Roothaan self-consistent-field (SCF) equations for open-shell systems. All diagonal electrostatic matrix elements have been derived to permit determining MO's for all states of all electronic configurations arising from the $d^n$ states of the free metal ion. In addition, configuration-interaction matrix elements among all states occurring in the $d^2$ and $d^3$ ($d^8$ and $d^7$) systems are also presented. The frozen-core approximations and some aspects of achieving converged SCF MO's are briefly discussed.