Coupling coefficients "symmetry dilemma" in the restricted open-shell Hartree-Fock method

Two opposite conclusions are known on the symmetry of the vector coupling coefficients (VCCs), a(mn) and b(mn), in the restricted open-shell Hartree-Fock (ROHF) method. The first one states that the VCCs are symmetric for all spectroscopic terms, i.e., a(mn) = a(nm) and b(mn) = b(nm). An opposing statement is that the "non-Roothaan" terms,arising from the degenerate open-shell electronic configuration gamma(N), can be characterized by non-symmetric VCCs matrices only: \\a(mn)\\ not equal \\a(nm)\\, \\b(mn)\\ not equal \\b(nm)\\. This article presents a detailed analysis of the VCCs symmetry problem. A general approach to the VCCs determination has been developed leading to non-symmetric VCCs for gamma(N) systems with dim gamma greater than or equal to 3. The main purpose of this work is to eliminate the contradiction arising in the ROHF theory when the latter is applied to highly symmetric open-shell molecules and atoms.