Being a reduced matrix element, u pq does not depend on the spin projections of the states or the excitation operator. To emphasize this property, we drop the corresponding index in excitation operator and replace it with a dot: T 1,• pq . If the transition between the target states (or between the reference and target states) is a spin-flip transition, the spin-tensor form of the one-particle transition density matrix is the transition density matrix γ up to a sign, see Eq. ( ) and . If the transition is spin-conserving, the transition density matrix always can be decomposed into the singlet and triplet components in any spin-adapted basis (i.e., atomic orbitals):