In the MMST mapping formalism, this parameter is viewed as the zero-point energy (ZPE) parameter of the mapping oscillators. In the SU (N ) mapping formalism, it is the parameter related to the choice of r s . Nevertheless, Eq. 51, which was first derived in Ref. 43, helps to establish the connection between the boundaries on the S-W radius and the boundaries on the γ parameter in the MMST mapping. The constraint of the radius r s ∈ (0, ∞) leads to a corresponding constraint for the ZPE parameter, γ ∈ (-2 N , ∞). The negative values of the ZPE parameter has been proposed in the MMST framework, and simply correspond to r s ≤ r Q = 1. In our own opinion, it might be more intuitive to understand the choice of radius of Bloch sphere 43 (that should be larger than 0) rather than the negative ZPE of quantum mapping oscillators. Using the conjugate variables {q n , p n } defined in Eq. 46, the S-W kernel in Eq. 27 can also be equivalently expressed in the diabatic electronic basis as follows