Testing the Jacob's ladder of density functionals for electronic structure and magnetism of rutile ${\mathrm{VO}}_2$

We employ semilocal density functionals [local spin-density approximation (LSDA), Perdew-Burke-Ernzerhof (PBE) generalized gradient approximation (GGA), and meta-GGAs)], LSDA plus Hubbard $U$ (LSDA+$U$) theory, a nonlocal range-separated Heyd-Scuseria-Ernzerhof hybrid functional (HSE06), and the random-phase approximation (RPA) to assess their performances for the ground-state magnetism and electronic structure of a strongly correlated metal, rutile ${\mathrm{VO}}_2$. Using recent quantum Monte Carlo results as the benchmark, all tested semilocal and hybrid functionals as well as the RPA (with PBE inputs) predict the correct magnetic ground states for rutile ${\mathrm{VO}}_2$. The observed paramagnetism could arise from temperature-disordered local spin moments or from the thermal destruction of these moments. All semilocal functionals also give the correct ground-state metallicity for rutile ${\mathrm{VO}}_2$. However, in the ferromagnetic (FM) and antiferromagnetic (AFM) phases, LSDA+$U$ and HSE06 incorrectly predict rutile ${\mathrm{VO}}_2$ to be a Mott-Hubbard insulator. For the computed electronic structures of FM and AFM phases, we find that the Tao-Perdew-Staroverov-Scuseria (TPSS) and revised TPSS (revTPSS) meta-GGAs give strong $2p\text{−}3d$ hybridizations, resulting in a depopulation of the $2p$ bands of O atoms, in comparison with other tested meta-GGAs. The regularized TPSS (regTPSS) and meta-GGAs made simple, i.e., MGGA_MS0 and MGGA_MS2, which are free of the spurious order-of-limits problem of TPSS and revTPSS, give electronic states close to those of the PBE GGA and LSDA. In comparison to experiment, semilocal functionals predict better equilibrium cell volumes for rutile ${\mathrm{VO}}_2$ in FM and AFM states than in the spin-unpolarized state. For meta-GGAs, a monotonic decrease of the exchange enhancement factor $F_x(s,\alpha )$ with α for small $s$, as in the MGGA_MS functionals, leads to large (probably too large) local magnetic moments in spin-polarized states.

Figure 1

Two magnetic states of rutile-type ${\mathrm{VO}}_2$: (a) the ferromagnetic state and (b) the antiferromagnetic state. Note that the big blue balls are V atoms and the small red ones are O atoms, respectively. The green solid arrows represent the directions of local magnetic moments of V atoms. The octahedra represent ${\mathrm{VO}}_6$ structural units.

Figure 2

(a) Relationship between global crystallographic coordinates $\left(X\text{−}Y\text{−}Z\right)$ and local coordinates $\left(x\text{−}y\text{−}z\right)$ of V atoms and (b) irreducible representations of $3d$ orbitals in $O_h$ and $D_{4h}$ point groups under the local coordinate system $\left(x\text{−}y\text{−}z\right)$.

Figure 3

Energy differences between two magnetic states (FM and AFM) and the spin-compensated nonmagnetic state are calculated using the (a) experimental lattice constants (Expt) (see also Table 1 for the precise values) and (b) fully relaxed geometry (Opt). Also shown is (c) the energy difference between FM and AFM states. Note that for the RPA the values marked as “Opt” are obtained by minimizing the total energy with respect to the PBE cell volume. The FNDMC calculation of Ref. [14] predicts −300 meV/${\mathrm{VO}}_2$ molecule for FM-PM and AFM-PM states.

Figure 4

Computed local magnetic moments of V atoms by different exchange-correlation functionals in the (a) FM and (b) AFM states. The horizontal line represents the exact value of the local magnetic moment carried by a free $\mathrm{V}{}^{4+}$ cation. The fractional coordinates for ${\mathrm{V}}_I$ and ${\mathrm{V}}_{II}$ are (0, 0, 0) and (0.5, 0.5, 0.5), respectively.

Figure 5

Weighted up-spin band dispersions of FM rutile-type ${\mathrm{VO}}_2$ calculated by the semilocal functionals: (a) PBE, (b) TPSS, and (c) MGGA_MS2 and also (d) HSE06. The size of the black dots represents the weights of $3d$ orbitals of V atoms. The solid blue lines are the energy bands of bulk rutile ${\mathrm{VO}}_2$. The fractional coordinates for these special high-symmetry $k$ points are $Z$ (0, 0, 1/2), $A$ (1/2, 1/2, 1/2), $M$ (1/2, 1/2, 0), Γ (0, 0, 0), $R$ (0, 1/2, 1/2), and $X$ (0, 1/2, 0). The Fermi level is set to zero at the top of the valence band.

Figure 6

Relative positions of O $2s$ bands computed by semilocal functionals in FM and AFM states. The $2s$ band positions obtained from the LSDA are set to zero. The value of $F_x(s,\alpha )$ is evaluated with the assumption that $s$→0 and α = 0 for the spherical-like single-orbital region.

Figure 7

Calculated spin-polarized density of states of FM rutile-type ${\mathrm{VO}}_2$ from different exchange-correlation functionals: (a) PBE, (b) TPSS, (c) MGGA_MS2, and (d) HSE06. The angular-momentum-projected densities of states on the $3d$ orbitals are also shown for the up- spin direction. The vertical line refers to the Fermi level. The notation for five $3d$ states of the V atom is defined in crystallographic Cartesian coordinates $\left(X\text{−}Y\text{−}Z\right)$.

Figure 8

Computed percent errors of equilibrium cell volume in three different magnetic states by semilocal and nonlocal functionals. For the RPA, the equilibrium cell volume is obtained by fitting the total energy versus volume curve to the equation of state.

Figure 9

The α dependence of meta-GGA exchange enhancement factors for different $s$ values. The LDA and PBE approximation are represented by horizontal lines because both exchange functionals show no dependence on α.

Figure 10

Position dependence of α near a V nucleus in rutile ${\mathrm{VO}}_2$. The all-electron $3p$ and $3d$ orbitals for a free $\mathrm{V}{}^{+4}$ ion, in arbitrary units, are also shown. These orbitals are obtained by solving the nonrelativistic Schrödinger equations using the PBE approximation for the exchange-correlation energy in the OPIUM code [62].