where we have explicitly used the property in Eq. 42 for the S-W transform and the property in Eq. 67 for the Wigner transform. One can in principle derive the Liouvillian expressed in terms of the MMST variables (defined in Eq. 46), using the kernel expressed in the MMST variables and diabatic basis (Eq. 52). However, this leads to a complicated expression of [ Â B] s , as opposed to the simple expression in Eq. 42. We thus decide to use the su(N ) generators to derive the Liouvillian expression. One can then perform the transform (see Sec.VC-D) to obtain an expression with MMST variables. Based on Eq. 83, we identify a state-independent part and a state-dependent part of the Liouvillian, acting respectively on the state-independent and state-dependent components of the operator B. By rewriting the total time-derivative of operator [ B] ws we have