This argument leads to the ZPE parameter γ = 1, and without the constraint provided in Eq. 57, even though N n=1 1 2 (q 2 n + p 2 n -γ) is a constant of motion. Most of the existing mapping approaches are based upon this mapping procedure, or a modified one that constraint the operators within the SEO subspace. One can further down-grade these mapping operators into classical variables, for example, using the Wigner transform through the mixed quantum-classical approximation, 112 linearization approximation, or the Husimi representation (coherent state basis) for these mapping variables through the semi-classical approximation or the partial linearization approximation. The MMST mapping formalism should be viewed as a fundamentally different mapping procedure compared to the SU (N ) formalism. This is because that the MMST mapping operators â † n and ân (or qn , pn ) live in a larger Hilbert space than the electronic Hilbert space of the original Hamiltonian. Truncating the larger Hilbert space to include only SEO subspace ruins the simple commutation relation between pn and qn , such that [q n , pn ] = i in the truncated Hilbert space. A detailed expression of [q n , pn ] in the truncated SEO mapping space can be found in Eq. 17 of Ref. 109. Thus, when applying truncation of the mapping Hilbert space (such as done for the mapping formalism in Ref. 24), this additional commutator [q n , pn ] need to be explicitly included to replace γ = 1 in ĤMMST Hamiltonian (Eq. F3), and is required to be evaluated through additional approximations. As opposed to the Stock-Thoss mapping procedure, the starting point of the SU (N ) mapping formalism is completely different. The SU (N ) mapping formalism uses the generators of the su(N ) Lie algebra (more specifically, the GGM basis in Eq. 2-Eq. 4), which exactly preserves the commutation relations among operators as well as the original electronic Hilbert space. As a result, there is no need for additional Hilbert space truncation that ruins commutation relations, nor necessity of projecting back to the subspace as required by MMST formalism. The exact quantum Liouvillian from the current SU (N ) mapping formalism (see Sec. V) is also different than the exact Liouvillian of the MMST formalism. It is interesting that despite a completely different mapping procedure, the same mapping Hamiltonian (Eq. 50) can be obtained upon a variable transformation (Eq. 49) of the mapping Hamiltonian Ĥ( R) s (Ω) in the SU (N ) formalism. In the context of the SU (N ) mapping formalism, these conjugated mapping variables {q n , p n } do not have the meaning of position and momentum of mapping oscillators as suggested in the Stock-Thoss mapping procedure; 8,10 they are simply the real and imaginary components of the expansion coefficients of the generalized spin coherent states in the diabatic basis (up to a global phase) as indicated in Eq. 46.