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

Abstract

ABSTRACT: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. BAST, JENSEN, AND SAUE 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 ns 2 → ns 1 np 1 excitation energies in the Zn, Cd, and Hg atoms (n = 4-6) and (vertical) excitation energies of UO 2+ 2 ; and we test the performance of various functionals by comparison with experimental data (group 12 atoms) or higher-level computational results (UO 2+ 2 ). 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.

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

“…In Table 1 31 Wang et al, 62 and Kühn et al, 66 suggesting that the computational levels in these studies are equally sufficient for the considered molecules. We recall that the main differences between our data and those of earlier theoretical studies lies in the choice of the one-electron basis set.…”

Spin contamination analogy, Kramers pairs symmetry and spin density representations at the 2-component unrestricted Hartree–Fock level of theory

“…Actually, the rigor definition of DFT at the general 4c level should account for (spin and electron) density current instead of electron and spin density [30,31,32,33]. Nevertheless, spin magnetization (which is directly represented by the NCOL approach) is nowadays used as an approximation in the 2c/4c spin-DFT and spin time dependent DFT (spin TD-DFT) calculations [28,29,34,35,36,37,38,32,33]. Besides DFT/TD-DFT, spin and spin density have an important role in the polarized neutron (PN) scattering theory, both from the theoretical and from the experimental point of view [39,40].…”

“…Wang and Liu [34] have shown 2c COL, NCOL and unpaired electron spin populations of p-like atoms. Bast et al [38] have presented the spin magnetization of the first excited state of mercury atom, without (COL) and with (NCOL) the inclusion of spin-orbit effects. The representation of the unpaired (open shell) electron density in the GCHF regime (where one is unable to distinguish between up and down spinors) is a further point which has not been discussed yet in the literature.…”

Relativistic four-component calculations of Buckingham birefringence using London atomic orbitals

“…[60]. Extensions to include spin-density contributions have furthermore been described [70], also in conjunction with the use of perturbation-dependent basis sets [61].…”

“…We have employed the full derivatives of the functionals provided by the XCFun library [71,84]. Spin-density contributions [70] to XC matrix elements have been ignored.…”