Rawlinson and Tronci (Phys. Rev. A 102, 032811, 2020) introduced a regularized Born-Oppenheimer molecular dynamics scheme that captures geometric-phase effects near conical intersections: a Bohmian closure of the nuclear Ehrenfest dynamics replaces the singular Berry connection by its convolution with a smooth reference nuclear density, yielding nuclear trajectory equations with a Lorentz-type force from the regularized Berry curvature and a loop-dependent geometric phase. The scheme was formulated and demonstrated only for the linear vibronic-coupling E⊗eE\otimes e Jahn-Teller model, where symmetry makes the Berry connection available analytically. The authors left open how to compute the Berry connection systematically for realistic polyatomic systems (e.g. by fitting to a two-state diabatic representation) and whether the regularization survives quadratic vibronic coupling - which multiplies the conical-intersection structure - while noting that the practical application of the scheme is left for future studies and that the scheme, being adiabatic by construction, cannot reproduce the exactly vanishing Berry phase that the exact electron-nuclear factorization yields when the full wavefunction is real and single-valued. The adjacent research lines (exact-factorization gauge theory, diabatization, phase-space electronic structure) contain no work carrying out these extensions, and the problem of extending the scheme to realistic polyatomic adiabatic dynamics remains open.

Background

The Born-Oppenheimer (BO) approximation separates electronic and nuclear motion in molecules: the electronic Schrodinger equation is solved at fixed nuclear positions, and the nuclei then move on the resulting potential energy surface. At a conical intersection (CI) - a geometry where two electronic energy surfaces meet in a double-cone shape - the adiabatic electronic eigenstates are undefined at the crossing point and the surfaces are not smooth. When nuclear motion encircles a CI, the electronic wavefunction changes sign, equivalent to a geometric (Berry) phase of π\pi in the nuclear motion; Mead and Truhlar showed that this is captured by endowing the nuclear BO equation with a vector potential (the Berry connection) A(R)=i⟨ϕ(R)∣∇Rϕ(R)⟩\mathbf{A}(\mathbf{R}) = i\langle\phi(\mathbf{R})|\nabla_{\mathbf{R}}\phi(\mathbf{R})\rangle built from the parametric dependence of the electronic state ϕ\phi on the nuclear coordinates R\mathbf{R}. The Berry connection is singular at the CI, and the diagonal BO correction (the Born-Huang term, the kinetic-energy coupling ⟨∇ϕ∣∇ϕ⟩\langle\nabla\phi|\nabla\phi\rangle) diverges there, so naive classical-trajectory versions of adiabatic dynamics with geometric phase are ill-behaved near intersections.

The archetypal example is the E⊗eE\otimes e Jahn-Teller model: a doubly degenerate electronic E state coupled to a doubly degenerate vibrational e mode. With linear vibronic coupling, the symmetric configuration hosts a CI whose encirclement produces the π\pi Berry phase, and the model's rotational symmetry makes the Berry connection available in closed analytic form. Adding quadratic vibronic coupling warps the circular symmetry of the lower energy trough to threefold symmetry and produces additional conical intersections (four in total: the central one plus symmetry-related ones), a situation studied by Zwanziger and Grant for the topological phase of the bound states.

The exact factorization (XF) of the full electron-nuclear wavefunction Ψ(r,R,t)=χ(R,t) ϕR(r,t)\Psi(\mathbf{r},\mathbf{R},t) = \chi(\mathbf{R},t)\,\phi_{\mathbf{R}}(\mathbf{r},t) into a nuclear marginal amplitude and a conditional electronic amplitude gives an exact gauge theory of the same physics: its nuclear equation has the same structure as the BO equation with a Berry-type gauge potential, but without any adiabatic approximation. Within XF, Requist, Tandetzky, Gross and co-workers computed the exact Berry curvature and geometric phase in the E⊗eE\otimes e model, and Min, Abedi, Kim and Gross showed that when the full molecular wavefunction can be chosen real and single-valued, the exact Berry phase vanishes even though the adiabatic (BO) Berry phase is π\pi - that is, the molecular Berry phase can be an artifact of the BO approximation.

In this setting, Rawlinson and Tronci (2020) proposed a regularized BO molecular dynamics scheme. Instead of the usual classical limit, they close the nuclear Ehrenfest equations through a variational Bohmian (quantum-hydrodynamic) ansatz restricted to rigid translations of the nuclear configuration, and then regularize: the singular Berry connection is replaced by its convolution with a smooth reference nuclear density, and the divergent diagonal BO correction is mollified by smoothing the corresponding density-gradient term. The resulting nuclear trajectory equations contain a smooth Lorentz-type force from the regularized Berry curvature and produce a loop-dependent geometric phase that approaches the topological value as loops grow, reproducing the phenomenology of the fully nonadiabatic XF computations of Requist et al. in the same model. The scheme was demonstrated only for the linear-coupling E⊗eE\otimes e Jahn-Teller model, and the paper closed by identifying what is needed to apply it to real molecules: a systematic way to compute the Berry connection for polyatomic systems (suggesting a fit to a two-state diabatic representation, as developed for geometric-phase-corrected adiabatic dynamics by Malbon et al. and Xie et al.), a demonstration for quadratic vibronic coupling where the intersection structure multiplies, and a clear recognition that the adiabatic character of the scheme excludes the exact-factorization cases with vanishing Berry phase; the practical application of the molecular dynamics scheme itself was left for future studies.

Problem Statement

Extend the regularized Born-Oppenheimer molecular dynamics scheme of Rawlinson and Tronci (2020) beyond its linear-coupling E⊗eE\otimes e Jahn-Teller demonstration so that it applies to realistic polyatomic molecules with conical intersections. In the formulation left open by the source paper, a solution must accomplish the following linked components, which jointly constitute the whole of this single objective - computing and regularizing the Berry connection for general systems, and demonstrating the regularization for quadratic vibronic coupling:

  1. Berry-connection construction for polyatomics. Provide a systematic construction of the Berry connection on the nuclear configuration space of a polyatomic molecule - for example by fitting ab initio data to a two-state diabatic representation in the spirit of Malbon et al. (2016) and Xie et al. (2017) - so that the convolution regularization of the scheme (the smoothing of the singular connection with a reference nuclear density, and the analogous mollification of the diagonal Born-Oppenheimer correction) can be applied to it, replacing the symmetry-based analytic computation used in the Jahn-Teller model.

  2. Quadratic vibronic coupling. Determine whether the regularization survives quadratic vibronic coupling in the E⊗eE\otimes e Jahn-Teller model, in which the conical-intersection structure multiplies (the central intersection plus the additional intersections produced by the warping): whether the regularized Berry connection and curvature keep the same local profile near each of the intersections, whether the regularized trajectory equations remain well posed, and whether the loop-dependent geometric phase is consistent with the underlying intersection topology. The source paper expected this to lead to no essential difficulties but left it undemonstrated. Either outcome - the regularization preserved, or shown to break down or to hold only partially under quadratic coupling - resolves this component, provided it is established by an explicit demonstration.

The source paper also noted that the practical application of its molecular dynamics scheme is left for future studies. A demonstrated application of the regularized trajectory equations to a realistic polyatomic system with a conical intersection is an optional extension beyond the two components above, not a requirement for resolving the objective; the paper's open formulation attaches no specific application deliverables, and none are prescribed here.

The scheme is adiabatic by construction: it cannot, and need not, reproduce the exactly vanishing Berry phase that the exact electron-nuclear factorization yields when the full wavefunction is real and single-valued (Min et al. 2014). This limitation is intrinsic to the adiabatic approach: such cases escape the treatment, and the scheme does not capture the exact nonadiabatic geometry.

References

  1. Rawlinson, J. I.; Tronci, C. Regularized Born-Oppenheimer molecular dynamics. Physical Review A 102, 032811 (2020). DOI: 10.1103/PhysRevA.102.032811; arXiv:2005.10784. https://arxiv.org/abs/2005.10784
  2. Mead, C. A.; Truhlar, D. G. On the determination of Born-Oppenheimer nuclear motion wave functions including complications due to conical intersections and identical nuclei. The Journal of Chemical Physics 70, 2284-2296 (1979). DOI: 10.1063/1.437734. https://doi.org/10.1063/1.437734
  3. Mead, C. A. The geometric phase in molecular systems. Reviews of Modern Physics 64, 51-85 (1992). DOI: 10.1103/RevModPhys.64.51. https://doi.org/10.1103/RevModPhys.64.51
  4. Zwanziger, J. W.; Grant, E. R. Topological phase in molecular bound states: application to the E⊗e system. The Journal of Chemical Physics 87, 2954-2964 (1987). DOI: 10.1063/1.453083. https://doi.org/10.1063/1.453083
  5. Min, S. K.; Abedi, A.; Kim, K. S.; Gross, E. K. U. Is the molecular Berry phase an artifact of the Born-Oppenheimer approximation? Physical Review Letters 113, 263004 (2014). DOI: 10.1103/PhysRevLett.113.263004. https://doi.org/10.1103/PhysRevLett.113.263004
  6. Requist, R.; Tandetzky, F.; Gross, E. K. U. Molecular geometric phase from the exact electron-nuclear factorization. Physical Review A 93, 042108 (2016). DOI: 10.1103/PhysRevA.93.042108. https://doi.org/10.1103/PhysRevA.93.042108
  7. Requist, R.; Proetto, C. R.; Gross, E. K. U. Asymptotic analysis of the Berry curvature in the E⊗e Jahn-Teller model. Physical Review A 96, 062503 (2017). DOI: 10.1103/PhysRevA.96.062503. https://doi.org/10.1103/PhysRevA.96.062503
  8. Malbon, C. L.; Zhu, X.; Guo, H.; Yarkony, D. R. On the incorporation of the geometric phase in general single potential energy surface dynamics: a removable approximation to ab initio data. The Journal of Chemical Physics 145, 234111 (2016). DOI: 10.1063/1.4971369. https://doi.org/10.1063/1.4971369
  9. Xie, C.; Malbon, C. L.; Yarkony, D. R.; Guo, H. Dynamic mapping of conical intersection seams: a general method for incorporating the geometric phase in adiabatic dynamics in polyatomic systems. The Journal of Chemical Physics 147, 044109 (2017). DOI: 10.1063/1.4990002. https://doi.org/10.1063/1.4990002
  10. Holm, D. D.; Rawlinson, J. I.; Tronci, C. The bohmion method in nonadiabatic quantum hydrodynamics. Journal of Physics A: Mathematical and Theoretical 54, 495201 (2021). DOI: 10.1088/1751-8121/ac2ae8; arXiv:2012.03569. https://arxiv.org/abs/2012.03569
  11. Gay-Balmaz, F.; Tronci, C. Evolution of hybrid quantum-classical wavefunctions. Physica D: Nonlinear Phenomena 440, 133450 (2022). DOI: 10.1016/j.physd.2022.133450; arXiv:2112.12144. https://arxiv.org/abs/2112.12144
  12. Gay-Balmaz, F.; Tronci, C. Dynamics of mixed quantum-classical spin systems. Journal of Physics A: Mathematical and Theoretical 56, 144002 (2023). DOI: 10.1088/1751-8121/acc145; arXiv:2210.00988. https://arxiv.org/abs/2210.00988
  13. Holm, D. D.; Hu, R.; Street, O. D. Lagrangian reduction and wave mean flow interaction. Physica D: Nonlinear Phenomena 454, 133847 (2023). DOI: 10.1016/j.physd.2023.133847; arXiv:2301.04121. https://arxiv.org/abs/2301.04121
  14. Requist, R. Gauge- and coordinate-invariant equations for two-component systems. arXiv:2207.02198 (2022). https://arxiv.org/abs/2207.02198
  15. Requist, R. Quantum covariant derivative: a tool for deriving adiabatic perturbation theory to all orders. Journal of Physics A: Mathematical and Theoretical 56, 465301 (2023). DOI: 10.1088/1751-8121/ad0349; arXiv:2206.01716. https://arxiv.org/abs/2206.01716
  16. Bergold, P.; Tronci, C. Madelung transform and variational asymptotics in Born-Oppenheimer molecular dynamics. In: Geometric Science of Information (Lecture Notes in Computer Science), 231-241 (2023). DOI: 10.1007/978-3-031-38299-4_25; arXiv:2305.18972. https://arxiv.org/abs/2305.18972
  17. Bauer, W.; Bergold, P.; Gay-Balmaz, F.; Tronci, C. Koopmon trajectories in nonadiabatic quantum-classical dynamics. Multiscale Modeling and Simulation 22(4), 1365-1401 (2024). DOI: 10.1137/24M1630955; arXiv:2312.13878. https://arxiv.org/abs/2312.13878
  18. Littlejohn, R.; Rawlinson, J.; Subotnik, J. The parallel-transported (quasi)-diabatic basis. The Journal of Chemical Physics 157, 184303 (2022). DOI: 10.1063/5.0122781; arXiv:2208.12287. https://arxiv.org/abs/2208.12287
  19. Tao, Z.; Qiu, T.; Bhati, M.; Bian, X.; Duston, T.; Rawlinson, J.; Littlejohn, R. G.; Subotnik, J. E. Practical phase-space electronic Hamiltonians for ab initio dynamics. The Journal of Chemical Physics 160, 124101 (2024). DOI: 10.1063/5.0192084; arXiv:2401.14327. https://arxiv.org/abs/2401.14327
  20. Bhati, M.; Cofer-Shabica, D. V.; Rawlinson, J. I.; Littlejohn, R. G.; Subotnik, J.; Bradbury, N. C. Electronic structure in a phase space, non-Born-Oppenheimer framework: geometric forces and Moody-Shapere-Wilzcek revisited. arXiv:2605.27053 (2026). https://arxiv.org/abs/2605.27053