and L representing the angular momentum matrix and ∆V SO l (r, r ) representing the non-local SOC correction pseudopotential contribution to the remaining scalar relativistic local and non-local terms. Explicit formula are given in Ref. . The reduction to one-electron matrix elements over molecular orbitals relying on the Löwdin formula and integration of spin can be performed as for the all-electron Hamiltonian. Diagonalizing the resulting matrix of spin-orbit matrix elements (SOME) H NM Sm S ,S m S of dimension Ñ × Ñ with Ñ = (1 + N + 3N) gives the SOC-corrected excitation energies Ω Ñ as well as the perturbed eigenfunctions Ψ Ñ . Resulting SOC-corrected eigenvectors can be used to compute the corresponding oscillator strengths f len Ñ within the length or velocity representation, with rζ indicating the dipole operator and pζ the momentum operator. Both representations are equal for exact wave functions not including a non-local potential at the complete basis-set limit or for approximate wave functions if fulfilling the Thomas-Reiche-Kuhn sum rule at the complete basis-set limit. Explicit working equations for the AMEW ansatz are given in the supplementary information.