Accuracy of ab initio electron correlation and electron densities in vanadium dioxide

Diffusion quantum Monte Carlo results are used as a reference to analyze properties related to phase stability and magnetism in vanadium dioxide computed with various formulations of density functional theory. We introduce metrics related to energetics, electron densities and spin densities that give us insight on both local and global variations in the antiferromagnetic M1 and R phases. Importantly, these metrics can address contributions arising from the challenging description of the $3d$ orbital physics in this material. We observe that the best description of energetics between the structural phases does not correspond to the best accuracy in the charge density, which is consistent with observations made recently by Medvedev et al. [Science 355, 371 (2017)] in the context of isolated atoms. However, we do find evidence that an accurate spin density connects to correct energetic ordering of different magnetic states in ${\mathrm{VO}}_2$, although local, semilocal, and meta-GGA functionals tend to erroneously favor demagnetization of the vanadium sites. The recently developed SCAN functional stands out as remaining nearly balanced in terms of magnetization across the M1-R transition and correctly predicting the ground state crystal structure. In addition to ranking current density functionals, our reference energies and densities serve as important benchmarks for future functional development. With our reference data, the accuracy of both the energy and the electron density can be monitored simultaneously, which is useful for functional development. So far, this kind of detailed high accuracy reference data for correlated materials has been absent from the literature.

Figure 1

Radial distribution function of absolute total density difference from extrapolated DMC [Eq. (8)] around V atom for various theoretical methods using RRKJ pseudopotentials: (a) M1 phase and (b) R phase. As a function of increasing $U$ value the $\mathrm{LDA}+U$ density tends to an improved accuracy, i.e., as $U$ is increased $\mathrm{LDA}+U$ curves get closer to zero.

Figure 2

Magnetization $M=\int |{\rho }^{\uparrow }-{\rho }^{\downarrow }|dV$ in units of $N_e$/formula unit from different methods for the two considered phases: M1 (blue) and R (red). The asterisk symbol refers to PAW results with VASP, where the $U$ values are 3.5 eV (left) and 4.0 eV (right). The other results are obtained with hard norm conserving pseudopotential (OPT) using Quantum Espresso. For the DMC values we have used the spin density from the extrapolated estimator.

Figure 3

$\mathrm{LDA}+U$ spin gap as a function of $U$ for $\text{M}1{\mathrm{VO}}_2$. The energy of the ferromagnetic (FM) and fully spin unpolarized (or nonmagnetic, NM) states are shown relative to the antiferromagnetic (AFM) state. Cubic fits to the data—performed prior to referencing against AFM—are shown as solid lines. Dashed lines correspond to the DMC results from Ref. [18]. Ferromagnetic calculations in the range $U=2–6$ eV were unstable.