Testing the Jacob's ladder of density functionals for electronic structure and magnetism of rutileVO2

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 $2p\text{\ensuremath{-}}3d$ 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 ${F}_{x}(s,\ensuremath{\alpha})$ with \ensuremath{\alpha} for small $s$, as in the MGGA_MS functionals, leads to large (probably too large) local magnetic moments in spin-polarized states.