Curl-free phase-space coupling in three dimensions

Construct a meaningful phase-space electronic structure coupling operator \(\hat{\Gamma}\) for three-dimensional molecular systems that both improves energy predictions and maintains zero non-Abelian curl.

Background

The paper establishes that, for one-dimensional nuclear motion, the proposed phase-space electronic structure operator Γ^\hat{\Gamma} can be constructed with zero non-Abelian curvature, enabling the phase-space curvature-free formulation (PS-CF) and an exact correspondence with the full vibronic Hamiltonian. In two- and three-dimensional systems, however, the authors state that their existing forms of Γ^\hat{\Gamma}, which satisfy relevant translational and rotational constraints, generally have nonzero curvature.

For three-dimensional diatomic molecules, non-Abelian curvature arises from the noncommutativity of rotations, and the authors note that reproducing the exact curvature may itself be physically desirable. The unresolved problem is therefore not merely to impose zero curvature, but to determine whether a physically meaningful three-dimensional Γ^\hat{\Gamma} can simultaneously be curl-free and yield improved electronic and vibronic energies.

References

Moreover, as shown by Bhati et al.. [39], our choice of ˆΓ mimics the exact non-Abelian curvature and is also nonzero for a diatomic molecule in 3D system, so that constructing a curl-free ˆΓ may not even be desirable. That being said, one must wonder if it is possible to construct a meaningful ˆΓ operator in 3D that offers improved energies while also maintaining a zero non-Abelian curl. This question will need to be addressed as soon as possible, and help from mathematicians may be necessary.

— An Exact Formulation of Phase-Space Electronic Structure Theory in One Dimension by Exploiting the Zero Curvature Condition  (2609.13120 - Zaidi et al., 11 Sep 2026) in Section VII.B, “Three Dimensions,” p. 33