Comparative studies of density-functional approximations for light atoms in strong magnetic fields

For a wide range of magnetic fields, $0\le B\le 2000$ a.u., we present a systematic comparative study of the performance of different types of density-functional approximations in light atoms ($2\le Z\le 6$). Local, generalized-gradient approximation (GGA; semilocal), and meta-GGA ground-state exchange-correlation (xc) functionals are compared on an equal footing with exact-exchange, Hartree-Fock (HF), and current-density-functional-theory (CDFT) approximations. Comparison also is made with published quantum Monte Carlo data. Though all approximations give qualitatively reasonable results, the exchange energies from local and GGA functionals are too negative for large $B$. Results from the Perdew-Burke-Ernzerhof ground-state GGA and Tao-Perdew-Staroverov-Scuseria (TPSS) ground-state meta-GGA functionals are very close. Because of confinement, self-interaction error in such functionals is more severe at large $B$ than at $B=0$, hence self-interaction correction is crucial. Exact exchange combined with the TPSS correlation functional results in a self-interaction-free (xc) functional, from which we obtain atomic energies of comparable accuracy to those from correlated wave-function methods. Specifically for the B and C atoms, we provide beyond-HF energies in a wide range of $B$ fields. Fully self-consistent CDFT calculations were done with the Vignale-Rasolt-Geldart (VRG) functional in conjunction with the PW92 xc functional. Current effects turn out to be small, and the vorticity variable in the VRG functional diverges in some low-density regions. This part of the study suggests that nonlocal, self-interaction-free functionals may be better than local approximations as a starting point for CDFT functional construction and that some basic variable other than the vorticity could be helpful in making CDFT calculations practical.

Figure 1

Atomic total energies for the He atom fully spin-polarized state $1s2p_{-1}$ in magnetic fields from Hartree-Fock and DFT calculations with different functionals.

Figure 2

Atomic total energies for the Li atom fully spin-polarized state $1s2p_{-1}3d_{-2}$ in magnetic fields from Hartree-Fock and DFT calculations with different functionals.

Figure 3

One-electron and two-electron energies for the He atom fully spin-polarized state $1s2p_{-1}$ in magnetic fields from Hartree-Fock and different DFT exchange-correlation functionals.

Figure 4

The ratio of exchange energy $E_x$ over electron self-interaction energy $E_{\text{SI}}$ for the He atom singlet state $1s^2$ in magnetic fields from Hartree-Fock and different DFT exchange functionals. The ratio is exactly $-1$ for Hartree-Fock calculation.

Figure 5

The ratio of exchange energy $E_x$ over electron self-interaction energy $E_{\text{SI}}$ for the He atom fully spin-polarized state $1s2p_{-1}$ in magnetic fields from Hartree-Fock and different DFT exchange functionals.

Figure 6

Various quantities (electron density $n$, paramagnetic current density ${\mathbf{j}}_p$, vorticity $\nu$, and the magnitude of the current correction to the exchange-correlation energy density, $|g{\nu }^2|$, in the VRG functional) for the He atom $1s2p_{-1}$ state in $B=1\text{a.u.}$ All quantities are evaluated from the LDA KS orbitals and plotted along the $z$ and $\rho$ axes (cylindrical coordinates).