Here, the effective variance matrix ΣR is non-diagonal, unlike the initial Σ ξ , which indicates the correlation between atomic displacements due to the collective nature of the vibrational modes. The replacement of the variables {p, ξ} → {P, R} is unitary, since dpdξ = L -1 dRLdP = dRdP. Using this distribution, it is possible to simultaneously sample both the linear displacements of the nuclei from the equilibrium geometry and nuclear velocities in polyatomic molecules. This and similar sampling procedures for initial conditions are called Wigner sampling (WS). If we replace l 0 for each vibrational mode with the temperature-dependent l quant from Equation , we can also sample temperature-averaged vibrational states.