Thus, the momentum operator behaves in a "non-local" fashion in the energy representation, because (E α -E β ) can get very large even when the corresponding x αβ is small. This behavior can be seen from the the middle panels of Fig. , where the large amplitudes of p αβ exist among states |α⟩ and |β⟩, even when (E α -E β ) is large. Rabl and co-workers 72 argue that this why Ĥ′ p•A behaves less accurately upon matter state truncation due to the non-local behavior of the coupling term zj mj P pj P Â in Eq. 66. Thus, the large energy gaps in molecular systems do not guarantee small matrix elements of the p operator, hence a finite-level truncation in the "p • A" Hamiltonian often leads to large numerical errors. Hence, it is often more convenient to use the dipole gauge when applying the finite-level approximation for the matter DOFs. Note that such an asymmetry in the x and p operators disappears for the quantized electromagnetic mode or for a harmonically bound dipole, where momentum and position operators are interchangeable. However, when the molecular potential is highly anharmonic, the gauge invariance is explicitly broken under the finite-state approximation, for Ĥ′ p•A (Eq. 66) and Ĥd•E (Eq. 67), due to the lack of a complete basis.