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.