  The compound CuTe (vulcanite) undergoes a quasi one dimensional charge
density wave (CDW) at $T< T_{\mathrm{CDW}}=335$ K with a $5\times1\times2$
periodicity. The mechanism at its origin is debated. Several theoretical works
claimed that semilocal functionals are unable to describe its occurrence and
ascribed its formation only to strong electron-electron interaction. Moreover,
the possible role of quantum anharmonicity has not been addressed. Here, by
performing quantum anharmonic calculations, we show that semilocal functionals
correctly describe the occurrence of a CDW in CuTe if ultradense electron
momentum grids allowing for small electronic temperatures are used. The
distortion is driven by the perfect nesting among 1D Fermi surface sheets
extending in the $k_y$ direction. Quantum anharmonic effects are important and
tend to suppress both the distortion and $T_{\mathrm{CDW}}$. The quantum
anharmonic structural minimization of the CDW phase in the generalized gradient
approximation leads, however, to distorted Te-Te bond lengths in the low
temperature phase that are $21\%$ of the experimental ones at $T=20$ K. This
suggests that, even if the electron-electron interaction is not crucial for the
mechanism of CDW formation, it is relevant to accurately describe the
structural data for the low-T phase. We assess the effect of correlation on the
CDW by using the DFT+U+V approximation with parameters calculated from first
principles. We find that correlation enhances the Te-Te distortion, T$_{CDW}$
and the total energy gain by the distortion.
