General framework for calculating spin-orbit couplings using spinless one-particle density matrices: Theory and application to the equation-of-motion coupled-cluster wave functions

Standard implementations of nonrelativistic excited-state calculations compute only one component of spin multiplets (i.e., M-s = 0 triplets); however, matrix elements for all components are necessary for deriving spin-dependent experimental observables. Wigner-Eckart's theorem allows one to circumvent explicit calculations of all multiplet components. We generate all other spin-orbit matrix elements by applying Wigner-Eckart's theorem to a reduced one-particle transition density matrix computed for a single multiplet component. In addition to computational efficiency, this approach also resolves the phase issue arising within Born-Oppenheimer's separation of nuclear and electronic degrees of freedom. A general formalism and its application to the calculation of spin-orbit couplings using equation-of-motion coupled-cluster wave functions are presented. The two-electron contributions are included via the mean-field spin-orbit treatment. Intrinsic issues of constructing spin-orbit mean-field operators for open-shell references are discussed, and a resolution is proposed. The method is benchmarked by using several radicals and diradicals. The merits of the approach are illustrated by a calculation of the barrier for spin inversion in a high-spin tris(pyrrolylmethyl)amine Fe(II) complex.