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arxiv: 2605.27053 · v1 · pith:6FPHI3JVnew · submitted 2026-05-26 · ⚛️ physics.chem-ph

Electronic Structure in a Phase Space, non-Born-Oppenheimer Framework: Geometric Forces and Moody-Shapere-Wilzcek Revisited

Mansi Bhati , D. Vale Cofer-Shabica , Jonathan I. Rawlinson , Robert G. Littlejohn , Joseph Subotnik , Nadine C. Bradbury This is my paper

Pith reviewed 2026-07-01 15:48 UTC · model grok-4.3

classification ⚛️ physics.chem-ph
keywords phase space electronic structurenon-Born-OppenheimerCoriolis forcecentrifugal forceBerry curvaturethree-body problemgeometric phases
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0 comments X

The pith

Phase space electronic structure calculations parameterized by nuclear position and momentum correctly incorporate non-inertial Coriolis and centrifugal forces.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper tests methods for separating nuclear and electronic motion by solving the three-body quantum problem exactly and with various approximations. It shows that electronic structure calculations can be performed using both nuclear position and nuclear momentum rather than position alone. This phase space approach captures forces felt by electrons due to the moving nuclear frame that standard methods miss. The resulting energies and angular momenta match exact results more closely. The same framework extends earlier geometric phase ideas to systems where nuclei can vibrate.

Core claim

Performing electronic structure calculations parameterized by both the nuclear position vector X and the nuclear momentum vector P incorporates the non-inertial Coriolis and centrifugal forces felt by electrons in a moving nuclear frame, leading to more accurate eigenenergies and electronic angular momenta than Born-Oppenheimer methods allow, while also generalizing the Moody-Shapere-Wilczek magnetic monopole to permit vibrational motion.

What carries the argument

Phase space electronic structure, in which electronic calculations depend on both nuclear position X and nuclear momentum P.

If this is right

  • Eigenenergies become more accurate once non-inertial forces are included.
  • Electronic angular momenta are recovered with higher fidelity than in position-only calculations.
  • Geometric forces such as Coriolis and centrifugal terms appear automatically in the electronic problem.
  • The Berry curvature monopole is extended from fixed-length diatomics to vibrating systems.
  • Dynamics calculations can track angular momentum flow between nuclei and electrons.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The approach may allow practical modeling of spin selectivity effects when nuclear and electronic motions couple through vibrations.
  • If the three-body accuracy gains hold for larger systems, the method could reduce errors in simulations of rotating and vibrating molecules.
  • Connections to other non-adiabatic dynamics techniques become natural once momentum dependence is explicit.

Load-bearing premise

That electronic structure calculations can be effectively parameterized by nuclear momentum in addition to position and that three-body test results will generalize to realistic multi-nuclei molecules.

What would settle it

Direct numerical comparison of eigenenergies and electronic angular momenta obtained from phase space calculations against exact diagonalization of the full three-body Hamiltonian.

Figures

Figures reproduced from arXiv: 2605.27053 by D. Vale Cofer-Shabica, Jonathan I. Rawlinson, Joseph Subotnik, Mansi Bhati, Nadine C. Bradbury, Robert G. Littlejohn.

Figure 1
Figure 1. Figure 1: FIG. 1: A comparison of the symmetries of the relevant electron-nuclear couplings within Born- Oppenheimer [PITH_FULL_IMAGE:figures/full_fig_p002_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: FIG. 2: Angles that define the notion of the body-frame [PITH_FULL_IMAGE:figures/full_fig_p003_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: FIG. 3: Vibrational (top left) and rotational energy (top right) transitions for the model system as plotted as a [PITH_FULL_IMAGE:figures/full_fig_p013_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: FIG. 4: Error in the ground state energy as a function of mass ratio (left) and ground state energy as total angular [PITH_FULL_IMAGE:figures/full_fig_p014_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: FIG. 5: Data for linear observables. Absolute errors vs mass ratio for matrix elements of the electronic and nuclear [PITH_FULL_IMAGE:figures/full_fig_p015_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: FIG. 6: Electronic angular momentum in the ground state for different kinds of phase space calculations vs the log of [PITH_FULL_IMAGE:figures/full_fig_p016_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: FIG. 7: Electronic angular momentum data for [PITH_FULL_IMAGE:figures/full_fig_p017_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: FIG. 8: Representative data from the 2D calculations. In the top row we plot error in the vibrational state gap [PITH_FULL_IMAGE:figures/full_fig_p022_8.png] view at source ↗
read the original abstract

We revisit the three-body problem in quantum mechanics in two and three dimensions, generating both exact eigenvalues and eigenvectors of the Hamiltonian and a series of approximate solutions as calculated with a variety of different schemes to separate heavy ("nuclear") and light ("electronic") particles. We show that, with minimal extra cost, one can go beyond the Born-Oppenheimer approximation by performing electronic structure calculations parameterized by both the nuclear position (${\mathbf X})$ and the nuclear momentum ($\mathbf{P}$), a so-called phase space theory of electronic structure. In particular, we demonstrate that such phase space electronic structure calculations correctly incorporate the non-inertial Coriolis and centrifugal forces felt by electrons in a moving nuclear frame, thus leading to far more accurate eigenenergies and electronic angular momenta than has been possible before. We also demonstrate that our approach naturally incorporates and generalizes the Moody-Shapere-Wilczek magnetic monopole for the non-abelian Berry curvature (now allowing for vibrational motion rather than a diatomic of fixed length). We argue that the resulting approach should be extremely useful for propagating dynamics where angular momentum flows between nuclei and electrons; in particular, if extended to include spin degrees of freedom, the present approach will offer a practical means to study chiral induced spin selectivity through the lens of chiral phonons and coupled nuclear-electronic motion.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

1 major / 2 minor

Summary. The manuscript revisits the three-body problem (one heavy nucleus + electrons) in 2D and 3D quantum mechanics. It generates exact eigenvalues/vectors of the full Hamiltonian and compares them to Born-Oppenheimer and other approximations. The central claim is that electronic structure calculations parameterized by both nuclear position X and momentum P (phase-space electronic structure) incorporate non-inertial Coriolis and centrifugal forces, yielding substantially more accurate eigenenergies and electronic angular momenta than prior methods. The approach is shown to generalize the Moody-Shapere-Wilczek magnetic monopole (now including vibrational motion), and the authors argue it will be useful for nuclear-electronic angular-momentum transfer dynamics, including potential extensions to spin and chiral-induced spin selectivity.

Significance. If the three-body numerical demonstrations hold, the work supplies a concrete, low-cost route to include non-inertial frame effects directly in the electronic Hamiltonian. This addresses a known limitation of standard Born-Oppenheimer treatments when nuclear velocities are appreciable and provides a natural generalization of Berry-phase monopoles to vibrating systems. The explicit comparison against exact three-body benchmarks is a strength; the method could become relevant for dynamics simulations that track angular-momentum flow between nuclei and electrons.

major comments (1)
  1. [Abstract] Abstract (final paragraph): the assertion that the resulting approach 'should be extremely useful for propagating dynamics where angular momentum flows between nuclei and electrons' is not supported by the calculations, which treat only a single nuclear momentum P. For molecules with multiple nuclei the electronic Hamiltonian would need to be parameterized by a set of independent nuclear momenta; the manuscript provides neither a derivation of the required frame transformations nor numerical tests that would establish whether cross terms between nuclear velocities are automatically captured by a single global P parameterization.
minor comments (2)
  1. [Abstract] The abstract and introduction would benefit from an explicit statement of the particle content of the three-body Hamiltonian (one nucleus + N electrons) to make the scope of the numerical demonstrations immediately clear.
  2. Notation for the phase-space electronic Hamiltonian (dependence on both X and P) should be introduced with an equation number in the main text rather than only in the abstract.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their careful reading of the manuscript and for highlighting this important point regarding the scope of our claims. We respond to the major comment below.

read point-by-point responses

  1. Referee: [Abstract] Abstract (final paragraph): the assertion that the resulting approach 'should be extremely useful for propagating dynamics where angular momentum flows between nuclei and electrons' is not supported by the calculations, which treat only a single nuclear momentum P. For molecules with multiple nuclei the electronic Hamiltonian would need to be parameterized by a set of independent nuclear momenta; the manuscript provides neither a derivation of the required frame transformations nor numerical tests that would establish whether cross terms between nuclear velocities are automatically captured by a single global P parameterization.

    Authors: We agree that the numerical demonstrations are restricted to the three-body problem with a single nuclear momentum P, and that the abstract statement regarding multi-nuclear dynamics is prospective rather than directly supported by the presented calculations. The phase-space framework is formulated in a manner that conceptually extends to multiple nuclei by treating each nuclear momentum independently, but we acknowledge the absence of an explicit multi-nucleus derivation or cross-term tests. We will revise the abstract to qualify the claim as applying to the single-nucleus case demonstrated here, with generalization to multiple nuclei noted as a natural direction for future work. A brief clarifying paragraph will also be added to the discussion section. revision: partial

Circularity Check

0 steps flagged

No circularity: results derived from explicit three-body Hamiltonian calculations

full rationale

The paper computes exact eigenvalues/eigenvectors of the three-body Hamiltonian and compares them to phase-space parameterized approximations. No load-bearing step reduces by construction to a fitted input, self-citation chain, or ansatz smuggled via prior work. Demonstrations are direct numerical comparisons for the stated cases; the central claim follows from the explicit incorporation of P in the electronic Hamiltonian rather than from redefinition or renaming of known results.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

Review is based solely on the abstract; the paper relies on standard quantum mechanical treatment of the three-body problem as a test case and the conventional Born-Oppenheimer separation as a baseline to extend.

axioms (1)
  • standard math The three-body quantum Hamiltonian in two and three dimensions admits exact eigenvalues and eigenvectors that can be computed for benchmarking.
    The abstract states that exact solutions are generated for comparison with approximate schemes.

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discussion (0)

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Reference graph

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