Phonon Softenings and the Mott-spin-Peierls Transition in VO$_{2}$

To explore the driving mechanisms of the metal-insulator transition (MIT) and the structural transition in VO$_{2}$, we have investigated phonon dispersions of rutile VO$_{2}$ (\textit{R}-VO$_{2}$) in the DFT and the DFT+$U$ ($U$: Coulomb correlation) band calculations. We have found that the phonon softening instabilities occur in both cases, but the softened phonon mode only in the DFT+$U$ describes properly both the MIT and the structural transition from \textit{R}-VO$_{2}$ to monoclinic VO$_{2}$ (\textit{M$_{1}$}-VO$_{2}$). The present {\it ab-initio} phonon dispersion calculations clearly demonstrate that the Coulomb correlation effect plays an essential role of assisting the Peierls transition in \textit{R}-VO$_{2}$ and producing the spin-Peierls ground state in \textit{M$_{1}$}-VO$_{2}$.