(ii) A convenient expression (Eq. 27) of the S-W kernel in terms of the spin coherent state projection operator. Even though it is straightforward to prove its equivalence to the original expression in Eq. 26, this new expression in Eq. 27 can drastically simplify a lot of derivations, such as deriving the identity property of the kernel (Eq. 28), the S-W kernel in the diabatic representation (Eq. 52), the S-W transform of projection operator (Eq. 53), and can be potentially useful for future theoretical derivations. It also drastically simplified the proof of the equivalence (see Sec. VI C) between the EOMs used in GDTWA approach (Eq. 98c) and the EOMs in the MMST variables (Eq. 117).