where H (Eq. 118) is equivalent to H s (R, P ) (Eq. 86) through the transform defined in Eq. 49 (or equivalently in Eq. 46). The Hamiltonian H in Eq. 118 can be viewed as Eq. 50 with the Wigner transform over only the nuclear DOFs. Further, using the transformation defined in Eq. 49, the nuclear EOMs in Eq. 98 can be expressed as