where [ Â( R)] ws is a Wigner transform of the nuclear DOFs (defined in Eq. 65) and a S-W transform of the electronic DOFs in the SU (N ) representation (defined in Eq. 25 or Eq. 34). The time evolved expectation value [ B(t)] ws is written using the quantum Liouvillian L to update [ B(0)] ws . The exact expression of L is derived in Sec. V. From now on, we will simply denote Â(0) as Â and B(0) as B.