Excitation Energies from Real-Time Propagation of the Four-Component Dirac–Kohn–Sham Equation

We report the first implementation of real-time time-dependent density functional theory (RT-TDDFT) at the relativistic four-component level of theory. In contrast to the perturbative linear-response TDDFT approach (LR-TDDFT), the RT-TDDFT approach performs an explicit time propagation of the Dirac–Kohn–Sham density matrix, offering the possibility to simulate molecular spectroscopies involving strong electromagnetic fields while, at the same time, treating relativistic scalar and spin–orbit corrections variationally. The implementation is based on the matrix representation of the Dirac–Coulomb Hamiltonian in the basis of restricted kinetically balanced Gaussian-type functions, exploiting the noncollinear Kramers unrestricted formalism implemented in the program ReSpect. We also present an analytic form for the delta-type impulse commonly used in RT-TDDFT calculations, as well as a dipole-weighted transition matrix analysis, facilitating the interpretation of spectral transitions in terms of ground-state molecular orbitals. The possibilities offered by the methodology are illustrated by investigating vertical excitation energies and oscillator strengths for ground-state to excited-state transitions in the Group 12 atoms and in heavy-element hydrides. The accuracy of the method is assessed by comparing the excitation energies obtained with earlier relativistic linear response TDDFT results and available experimental data.