The initial conditions for an LSC simulation span 2(F +N ) variables, two phase space variables for every F electronic states and N nuclear modes. Those variables are sampled from A W , where Â = e -β Ĥ /Z is the thermal density operator with β = 1/k B T and Z = Tr[e -β Ĥ ]. We assume that initially the nuclear and electronic variables can be treated independently, which allows us to sample the nuclear variables from [ρ n ] W (R, P) and the electronic ones from [ρ e ] W (x, p), where ρn and ρe are the nuclear and electronic density operators, respectively (i.e., Tr n [ρ n ] = Tr e [ρ e ] = 1), and the notation [...] W indicates a Wigner transform. Making a harmonic approximation for the N nuclear vibrational normal modes yields an analytic expression for the nuclear Wigner distribution,