, where d j αβ ≡ ⟨α|∇ j |β⟩ is the well known derivative coupling. Besides these adiabatic derivative couplings, the light-matter interaction also induces additional "derivative"-type couplings, -∇ j μ Â and Pj , regardless of the electronic representation used in constructing P. When using the Mulliken-Hush diabatic states which are the eigenstates of the μ ≡ P μ P operator, such that μ = ϕ µ ϕϕ |ϕ⟩⟨ϕ|, one can prove that Pj = 0 for all nuclei. This is because that ∇ j μ = ϕ ∇ j µ ϕϕ |ϕ⟩⟨ϕ|, thus both μ Â and [ μ Â, pj ] become purely diagonal matrices, hence all of the higher order commutators in Û † pj Û become zero, resulting in Pj = 0 for j ∈ R. Unfortunately, Ĥp•A no longer remains in a minimum coupling form in Eq. 45 (except when approaching the complete electronic states limit), by only involving charges but not higher multipole moments. Nevertheless, Ĥp•A is invariant from Ĥd•E through the Û transformation, thus resolving the gauge ambiguity between them. Additionally, this method explains why the proposed resolution of ambiguities in Ref. 42 only works for matter systems that can be well approximated by two-level systems without a permanent dipole. For those types of systems, the truncated dipole operator is proportional to the Pauli σx matrix, and P μ P = µ eg σx , where µ eg is the transition dipole from the ground state to the excited state. In this special case, P μn P ≈ ( P μ P) n . Then, the properly truncated PZW operator is Û ≈ P Û P.