The property in Eq. 30b means that the S-W transform preserves the identity in the electronic Hilbert subspace, contrarily to the Wigner transform of the identity operator in the MMST formalism and hence, does not introduce any ambiguity of the identity expression. To conveniently evaluate any operator Â( R) under a S-W transformation, one starts by decomposing it on the GGM basis (Eqs 2-4) as follows