Relativistic adiabatic time-dependent density functional theory using hybrid functionals and noncollinear spin magnetization

We report an implementation of adiabatic time‐dependent density functional theory based on the 4‐component relativistic Dirac–Coulomb Hamiltonian and a closed‐shell reference. The implementation includes noncollinear spin magnetization and full derivatives of functionals, including hybrid generalized gradient approximation (GGA) functionals. We avoid reducing the generalized eigenvalue problem to half the dimension involving the square of excitation energies since this may introduce spurious roots and also squares the matrix condition number. Rather we impose structure in terms of hermiticity and time reversal symmetry on trial vectors to obtain even better reductions in terms of memory and run time, and without invoking approximations. Further reductions are obtained by exploiting point group symmetries for D2h and subgroups in a symmetry scheme where symmetry reductions translate into reduction of algebra from quaternion to complex or real. For hybrid GGAs with noncollinear spin magnetization we derive a new computationally advantageous equation for the full second variational derivatives of such exchange‐correlation functionals. We apply our implementation to calculations on the ns2 → ns1np1 excitation energies in the Zn, Cd, and Hg atoms (n = 4–6) and (vertical) excitation energies of UO 22+; and we test the performance of various functionals by comparison with experimental data (group 12 atoms) or higher‐level computational results (UO 22+). The results indicate that the adiabatic local density approximation (ALDA) is a good approximation for some GGA functionals, but not all. Furthermore, the results also indicate that ALDA is an extremely bad approximation for hybrid functionals, unless one only employs ALDA for the pure DFT contribution to the exchange‐correlation kernel and retains the fraction of exact exchange; we denote this approximation ALDAh. © 2009 Wiley Periodicals, Inc. Int J Quantum Chem, 2009