Structural energetics of VO$_2$ under strain*

Attaining a proper first-principles description of structural energetics in VO$_2$ is a necessary condition for fully characterizing the metal-to-insulator transition. Of the existing methods which have been employed for total energies, only DFT+$U$ applied under unorthodox conditions (ie. non-spin-polarized (NSP), with an unreasonably small $U$) has shown promise. In particular, this DFT+$U$ properly captures the qualitative and quantitative energy difference between the monoclinic (M$_1$) and rutile (R) phases. We explain why this unorthodox procedure works by studying a minimal model of the structural energetics based on the Peierls-Hubbard model; where the exact solution may be compared to Hartree-Fock, single-site dynamical mean-field theory (DMFT), and cluster DMFT. Furthermore, we use this DFT+U approach to calculate the strain phase diagram at T=0, including the R, M$_1$, and M$_2$ phases.