Relativistic time-dependent density functional theories
The foundations, formalisms, technicalities, and practicalities of relativistic time-dependent density functional theories (R-TD-DFT) for spinor excited states of molecular systems containing heavy elements are critically reviewed. These include the four-component (4C) and exact two-component (X2C) variants (4C/X2C-TD-DFT) that treat both scalar relativistic effects and spin–orbit couplings (SOC) to infinite order, and a composite two-component variant (sf-X2C-S-TD-DFT-SOC) that treats scalar relativistic effects to infinite order via the spin-free part of the X2C Hamiltonian (sf-X2C) but SOC to first order via the Douglas–Kroll–Hess type of spin–orbit operator resulting also from the spin separation of the X2C Hamiltonian. Except for the common adiabatic approximation, the most essential ingredient for all the three variants of R-TD-DFT is the noncollinear exchange–correlation kernel that is invariant with respect to rotations in spin space. It is unfortunate that 4C- and X2C-TD-DFT cannot be made fully symmetry adapted for open-shell systems except for some special cases. Yet, this is possible for closed-shell systems by working with both double point group and time reversal adapted molecular spinors. In particular, the spinor Hessian can be made real-valued in this case, such that the 4C/X2C-TD-DFT eigenvalue problems can be solved in the same manner as nonrelativistic TD-DFT, a point that is discovered here for the first time. By contrast, sf-X2C-S-TD-DFT-SOC can access spinor excited states of both closed- and open-shell systems because spin symmetry is fully accounted for in the spin-adapted TD-DFT (S-TD-DFT). Possible further developments of R-TD-DFT are also highlighted.