One can thus propagate quantum dynamics based on the EOMs outlined in Eq. 98. The initial conditions for the electronic DOFs can be sampled based on the phase space volume element in dΩ (Eqs. 21-22), which provides initial values of {θ n } and {ϕ n }, and then provide the values of {Ω k } through Eq. B2-B4. The nuclear DOFs will be sampled through the initial Wigner density (for example, e -β Ĥg w in Eq. 73). These variables will be numerically propagate according to EOMs in Eq. 98.