Efficient treatment of relativistic effects with periodic density functional methods: Energies, gradients, and stress tensors

The implementation of an efficient self-consistent field (SCF) method including both scalar-relativistic effects and spin-orbit interaction in density functional theory (DFT) is presented. We make use of Gaussian-type orbitals and all integrals are evaluated in real space. Our implementation supports density functional approximations up to the level of meta-generalized gradient approximations for SCF energies and gradients. The latter can be used to compute the stress tensor and consequently allow us to optimize the cell structure. Considering spin-orbit interaction requires the extension of the standard procedures to a two-component formalism and a noncollinear approach for open-shell systems. Here, we implemented both the canonical and the Scalmani-Frisch noncollinear DFT formalisms, with hybrid and range-separated hybrid functionals being presently restricted to SCF energies. We demonstrate both efficiency and relevance of spin-orbit effects for the electronic structure of discrete systems and systems periodic in one to three dimensions.

Figure 1

Simulated band structure for (a) an fcc bulk gold crystal (lattice constant $a=4.0800\AA$ [163]) and (b) an fcc bulk lead crystal ($a=4.9508\AA$ [163]). Results for the RIPER module with (2c) and without (1c) spin-orbit coupling are shown together with relativistic 2c calculations using the quantum espresso (QE) code [52]. All computations employ the PBE exchange-correlation functional [151]. We use the dhf-TZVP-2c GTO basis set for RIPER and a plane-wave basis for quantum espresso. Vertical dashed lines mark high-symmetry points of the FBZ.

Figure 2

(a) Top and side views of the two-dimensional InTe honeycomb system with indicated unit cell. The unit cell consists of two In and two Te atoms. (b), (c) Electronic band structure of the FBZ. The orange dashed lines are calculated without spin-orbit coupling, while the solid purple lines include spin-orbit interaction. The black vertical dashed lines mark the $\mathrm{\Gamma }$ and $K$ points of the Brillouin zone. Calculations are performed with the PBE functional [151], the D3-BJ dispersion correction [115, 116], and the dhf-TZVP-2c basis set [152]. See Supplemental Material [199] for results with HSE06 [203, 204, 205].

Figure 3

(a)–(c) Electronic band structure in the FBZ for a linear platinum chain at $d=5.56$ Å. The unit cell consists of two platinum atoms and is exemplarily shown above (b). The black vertical dashed lines in each panel mark the $\mathrm{\Gamma }$ point of the FBZ. Expectation values of the respective spin components are 1.16 [1c UKS, (b) and (c)], 0.82 [2c KU $S_x$, (b)], and 0.71 [2c KU $S_z$, (c)], and the spin vector is fully aligned along the $x$ and $z$ direction, respectively. Note that we assume periodicity along the $x$ direction for one-dimensional systems [150]. The open-shell solutions are energetically favored compared to the respective closed-shell solutions (1c RKS or 2c KR) by about $4\times {10}^{-3}$ hartree (1c UKS), $2\times {10}^{-3}$ hartree (2c KU $S_x$ alignment), and ${10}^{-3}$ hartree (2c KU $S_z$ alignment). The magnetization parallel to the nanowire is thus energetically preferred to the perpendicular one by about ${10}^{-3}$ hartree. Calculations are performed with the PBE functional [151] and the dhf-SVP-2c basis set [152]. (d) Dependence of the energy on the cell parameter in units of hartree per atom for 1c RKS and 1c UKS. (e) Dependence of the energy on the cell parameter in units of hartree per atom for 2c KR, 2c KU $S_x$, and 2c KU $S_z$. (f) Magnetic moment in units of Bohr's magneton ${\mu }_{\text{B}}$ per atom for the spin contribution of 1c UKS, 2c KU $S_x$, and 2c KU $S_z$.