with S representing the atomic-orbital overlap matrix. For restricted closed-shell references with α (spin up) and β (spin down) spin components being degenerate, spin adaptation by unitary transformation enables to define singlet excitation amplitudes and energies, X N sing and Ω N sing , with total spin S = 0 and spin quantum number m S = 0, which are decoupled from the corresponding triplet excitation amplitudes and energies, X N trip and Ω N trip , with total spin S = 1 and m S = 0. In the following, spin-adapted matrices will be indicated with lower indices M Sm S , labeling total spin S and spin angular momentum m S . The eigenvalue equations for singlet and triplet spin are given after spin summation within the Sternheimer formalism as