Two-photon absorption with spin–orbit coupling: Methodology and application to <i>fac</i>-Ir(ppy)3

This paper presents an approximate yet practical methodology for calculating two-photon absorption (TPA) cross sections with time-dependent density functional theory (TDDFT), including spin–orbit coupling (SOC). The approach is based on the second-order perturbative sum-over-states (SOS) expression for the TPA cross section and is referred to as the SOS-SOC-TDDFT method. To investigate the influence of SOC on TPA, we apply our approach to the calculation of TPA cross sections that include singlet and triplet states in fac-Ir(ppy)3, which is a phosphorescent complex known for its strong SOC effects. Due to the high angular momentum orbitals in Ir, SOC induces substantial mixing between many singlet and triplet states, although the ground state is predominantly of singlet character. Our evaluation of the TPA cross section involves constructing transition dipole matrix elements between SOC excited states by transforming the dipole matrix elements of singlet and triplet excited states obtained from scalar-relativistic TDDFT calculations, using the excited-state wavefunctions from SOC-included calculations. Our TPA results reveal that SOC leads to spectral broadening, redistribution of TPA intensities, and activation of spin-forbidden transitions. In particular, the presence of intermediate states with mixed spin character allows new excitation pathways to triplet-dominated two-photon excited states, which are inaccessible in spin-pure calculations. This highlights the importance of including SOC for accurate modeling of TPA for heavy-element systems, with results that are more sensitive to relativity than for one-photon excitation, providing insights for using the present approach in future studies of relativistic effects in nonlinear optical properties.