with the Hamiltonian defined in Eq. 1, and β = 1/k B T . The density matrix under the canonical equilibrium condition is ρeq = e -β Ĥ /Z and the partition function is defined as Z = Tr e Tr n [e -β Ĥ ]. In order to compute the nuclear trace Tr n , we insert the identity 1R = dR |R R |. To compute the electronic trace we use the property of the S-W transform in Eq. 34. This leads to