Exact exchange in relativistic spin-density-functional theory: Exchange splitting versus spin-orbit coupling

Recently, the concept of the exact orbital-dependent exchange was introduced into relativistic spin-density-functional theory (RSDFT) within the collinear limit [D. Ködderitzsch et al., Phys. Rev. B 77, 045101 (2008)]. In this contribution we further expand this exact exchange (EXX) formalism by (i) extending the basic equations to the general noncollinear form of RSDFT and (ii) discussing in detail the solution of the coupled integral equations resulting from orbital-dependent functionals in the framework of RSDFT. The EXX scheme is then applied to open-shell atoms in order to study (i) the relative importance of exchange splitting and spin-orbit coupling, (ii) the consequences of the exact exchange for atomic hyperfine constants, and (iii) the relative stability of the $3d^{n-1}4s^2$ and $3d^{n}4s^1$ configurations in case of the $3d$ transition-metal elements. In particular, it is demonstrated that the exact exchange, when combined with the orbital-dependent random-phase approximation for correlation, yields $s\text{−}d$-transfer energies which are clearly superior to the values obtained with conventional density functionals.

Figure 1

(Color online) KS eigenvalues of $3d$ subshell for $3d$ elements with occupation $3d^{n}4s^2$: exact x-only results (EXX) versus LDA (Ref. 55) results.

Figure 2

(Color online) Same as in Fig. 1 for $4d$ elements with occupation $4d^{n}5s^2$.

Figure 3

(Color online) Same as in Fig. 1 for $5d$ elements with occupation $5d^{n}6s^2$.

Figure 4

(Color online) Same as in Fig. 1 for $6p$ elements with occupation $6p^n$.

Figure 5

(Color online) Hyperfine constants $a$ for lanthanide elements: exact x-only (EXX), LDA,[55] and PBE-GGA (Ref. 57) results versus experimental data. Lines are only drawn to guide the eyes.

Figure 6

(Color online) Deviation of hyperfine constants $a$ from experimental value $a_{\text{expt}}$: exact x-only results (EXX) versus LDA (Ref. 55) and PBE-GGA (Ref. 57) data for $3d$-transition-metal elements.

Figure 7

(Color online) As in Fig. 6 for $4d$-transition-metal elements.

Figure 8

(Color online) As in Fig. 6 for $5d$-transition-metal elements.

Figure 9

(Color online) Low-lying levels of chromium atom: exact x-only results (EXX) versus data obtained from exact exchange plus PBE-GGA (Ref. 57) for correlation $\left(\text{EXX}+\text{PBE}\right)$ as well as complete PBE-GGA and experimental numbers (Ref. 63). The experimental $3d^5\left(6S\right)4s{{}^{5}S}_2$ state is only 0.7 mhartree lower than the lowest state $\left(J=0\right)$ of the $3d^{4}4s^2{}^{5}D$ multiplet and can therefore not be resolved on the present scale.

Figure 10

(Color online) $s\text{−}d$-transfer energies of $3d$ elements: exact x-only results (EXX) versus data obtained from exact exchange plus $\text{RPA}+$ for correlation $\left(\text{EXX}\text{−}\text{RPA}+\right)$, PBE-GGA (Ref. 57) and experimental numbers (Ref. 63) ($\text{RPA}+=\text{RPA}+$ second-order exchange within the LDA).