In essence, the approach is grounded in Wigner-Eckart's theorem that outlines the relationships between the matrix elements involving multiplet components. In the context of SOC, Wigner-Eckart's theorem has been exploited within configuration interaction formalism: for example, an algorithm for generating the entire SOC matrix from only three reduced matrix elements has been described by Fedorov . However, as explained below, Fedorov's approach requires access to several multiplet components (M s =0,±1), a feature which is not commonly available in excited-state codes, and is also affected by the phase issue.