While orthonormality of excitation amplitudes X N Sm S of same spin as well as orthonormality of both amplitudes with the ground-state MO coefficients C is implied, an orthonormality for amplitudes of different spin, i.e. between X N sing and X N trip , is not given due to the fact that both eigenvalue equations and corresponding constraints are decoupled (Eq. ( )). For mixed matrix elements, as given in Eq. ( ), the Slater-Condon rules therefore do not apply and Eq. ( ) has to be solved relying on the more general Löwdin rules defining the required matrix elements as outlined in Refs. . The one-electron SOC-Hamiltonian ĥSO within the zeroth-order regular approximation (ZORA) is defined as