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arXiv:2603.26151

Geometric Phase Effect in Thermodynamic Properties and in the Imaginary-Time Multi-Electronic-State Path Integral Formulation

Jian Liu

physics.chem-phphysics.comp-phquant-phSubmitted Mar 27, 2026• Announced Mar 30, 2026
v1 · Mar 27, 2026v2 · Mar 30, 2026
arXivHTMLPDF

Abstract

The geometric phase (GP) is a fundamental quantum effect arising from conical intersections (CIs), with profound consequences for vibronic energy levels. Standard imaginary-time path integral molecular dynamics (PIMD) based on the Born-Oppenheimer approximation does not account for the GP, potentially leading to significant errors in low-temperature thermodynamic properties. In this Perspective, we demonstrate that the multi-electronic-state path integral (MES-PI) formulation in imaginary time (developed in J. Chem. Phys. 2018, 148, 102319) naturally captures the GP effect through the electronic trace of the product of statistically weighted overlap matrices between successive imaginary-time slices. This crucial capability was already implicit in the benchmark MES-PIMD simulations in that foundational work. To isolate this topological effect from other nonadiabatic effects, we introduce a geometric signature matrix (for the CI) and a winding-number-induced phase factor, constructing an ad hoc GP-excluded MES-PI method. Comparing this ad hoc baseline against the rigorous MES-PI approach allows us to unambiguously quantify the impact of the GP on thermodynamic properties. While simpler approximations exist when only the ground electronic-state is considered, MES-PIMD is the most general and accurate approach applicable to real complex systems where the location and topology of CI seams are often not known a priori.

Journal reference
The Journal of Physical Chemistry Letter 2026, 17, 15, 4274-4291
DOI
10.1021/acs.jpclett.6c00429

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The geometric phase (GP) is a fundamental quantum effect arising from conical intersections (CIs), with profound consequences for vibronic energy levels. Standard imaginary-time path integral molecular dynamics (PIMD) based on the Born-Oppenheimer approximation does not account for the GP, potentially leading to significant errors in low-temperature thermodynamic properties. In this Perspective, we demonstrate that the multi-electronic-state path integral (MES-PI) formulation in imaginary time (developed in J. Chem. Phys. 2018, 148, 102319) naturally captures the GP effect through the electronic trace of the product of statistically weighted overlap matrices between successive imaginary-time slices. This crucial capability was already implicit in the benchmark MES-PIMD simulations in that foundational work. To isolate this topological effect from other nonadiabatic effects, we introduce a geometric signature matrix (for the CI) and a winding-number-induced phase factor, constructing an ad hoc GP-excluded MES-PI method. Comparing this ad hoc baseline against the rigorous MES-PI approach allows us to unambiguously quantify the impact of the GP on thermodynamic properties. While simpler approximations exist when only the ground electronic-state is considered, MES-PIMD is the most general and accurate approach applicable to real complex systems where the location and topology of CI seams are often not known a priori.
The Born-Oppenheimer electronic wave function ${\mathrm{\ensuremath{\Phi}}}_{R}^{\text{BO}}(r)$ picks up a topological phase factor $\ifmmode\pm\else\textpm\fi{}1$, a special case of Berry phase, when it is transported around a conical intersection of two adiabatic potential energy surfaces in RRR space. We show that this topological quantity reverts to a geometric quantity ${e}^{i\ensuremath{\gamma}}$ if the geometric phase $\ensuremath{\gamma}=\ensuremath{\oint}\mathrm{Im}\ensuremath{\langle}{\mathrm{\ensuremath{\Phi}}}_{R}|{\mathbf{\ensuremath{\nabla}}}_{\ensuremath{\mu}}{\mathrm{\ensuremath{\Phi}}}_{R}\ensuremath{\rangle}\ifmmode\cdot\else\textperiodcentered\fi{}d{\mathbf{R}}_{\ensuremath{\mu}}$ is evaluated with the conditional electronic wave function ${\mathrm{\ensuremath{\Phi}}}_{R}(r)$ from the exact electron-nuclear factorization ${\mathrm{\ensuremath{\Phi}}}_{R}(r)\ensuremath{\chi}(R)$ instead of the adiabatic function ${\mathrm{\ensuremath{\Phi}}}_{R}^{\text{BO}}(r)$. A model of a pseudorotating triatomic molecule, also applicable to dynamical Jahn-Teller ions in bulk crystals, provides examples of nontrivial induced vector potentials and molecular geometric phase from the exact factorization. The induced vector potential gives a contribution to the circulating nuclear current that cannot be removed by a gauge transformation. The exact potential energy surface is calculated and found to contain a term depending on the Fubini-Study metric for the conditional electronic wave function.
The role of the geometric phase effect in chemical reaction dynamics has long been a topic of active experimental and theoretical investigations. The topic has received renewed interest in recent years in cold and ultracold chemistry where it was shown to play a decisive role in state-to-state chemical dynamics. We provide a brief review of these developments focusing on recent studies of O + OH and hydrogen exchange in the H + H 2 and D + HD reactions at cold and ultracold temperatures. Non-adiabatic effects in ultracold chemical dynamics arising from the conical intersection between two electronic potential energy surfaces are also briefly discussed. By taking the hydrogen exchange reaction as an illustrative example it is shown that the inclusion of the geometric phase effect captures the essential features of non-adiabatic dynamics at collision energies below the conical intersection.
Quantum eigenstates undergoing cyclic changes acquire a phase factor of geometric origin. This phase, known as the Berry phase, or the geometric phase, has found applications in a wide range of disciplines throughout physics, including atomic and molecular physics, condensed matter physics, optics, and classical dynamics. In this article, the basic theory of the geometric phase is presented along with a number of representative applications. The article begins with an account of the geometric phase for cyclic adiabatic evolutions. An elementary derivation is given along with a worked example for two-state systems. The implications of time-reversal are explained, as is the fundamental connection between the geometric phase and energy level degeneracies. We also discuss methods of experimental observation. A brief account is given of geometric magnetism; this is a Lorenz-like force of geometric origin which appears in the dynamics of slow systems coupled to fast ones. A number of theoretical developments of the geometric phase are presented. These include an informal discussion of fibre bundles, and generalizations of the geometric phase to degenerate eigenstates (the nonabelian case) and to nonadiabatic evolution. There follows an account of applications. Manifestations in classical physics include the Hannay angle and kinematic geometric phases. Applications in optics concern polarization dynamics, including the theory and observation of Pancharatnam's phase. Applications in molecular physics include the molecular Aharonov-Bohm effect and nuclear magnetic resonance studies. In condensed matter physics, we discuss the role of the geometric phase in the theory of the quantum Hall effect.
The level crossing problem and associated geometric terms are neatly formulated by the second-quantized formulation. This formulation exhibits a hidden local gauge symmetry related to the arbitrariness of the phase choice of the complete orthonormal basis set. By using this second-quantized formulation, which does not assume adiabatic approximation, a convenient exact formula for the geometric terms including off-diagonal geometric terms is derived. The analysis of geometric phases is then reduced to a simple diagonalization of the Hamiltonian, and it is analyzed both in the operator and path-integral formulations. If one diagonalizes the geometric terms in the infinitesimal neighborhood of level crossing, the geometric phases become trivial (and thus no monopole singularity) for arbitrarily large but finite time interval TTT. The integrability of Schr\"odinger equation and the appearance of the seemingly nonintegrable phases are thus consistent. The topological proof of the Longuet-Higgins' phase-change rule, for example, fails in the practical Born-Oppenheimer approximation where a large but finite ratio of two time scales is involved and TTT is identified with the period of the slower system. The difference and similarity between the geometric phases associated with level crossing and the exact topological object such as the Aharonov-Bohm phase become clear in the present formulation. A crucial difference between the quantum anomaly and the geometric phases is also noted.
Conical intersections (CIs) between energy surfaces arise in various physical contexts. A rich example, involving the interaction of a qubit with a bosonic field, is the energy landscape of the asymmetric quantum Rabi model (AQRM). Conventional approximations for this model, including the slow-qubit approximation, fail to capture the correct location of the CIs. To go beyond this limitation, we introduce a relatively simple solvable light-matter interaction model, which by construction, shares the same CIs as the AQRM. As a demonstration of the use of this simplified model we calculate the geometric phases around CIs in the energy landscape. We find that the geometric phases acquired by closed loops in parameter space around CIs are multiples of π\piπ, depending on the CIs encircled by the trajectories considered. These results are seen to be in agreement with numerical results for the corresponding geometric phases of the AQRM.
The H3 system has served as a prototype for geometric phase (GP) effects in bimolecular chemical reactions for over three decades. Despite a large number of theoretical and experimental efforts, no conclusive evidence of GP effects in the integral cross section or reaction rate has been presented until recently [B. Kendrick et al., Phys. Rev. Lett. 115, 153201 (2015)]. Here we report a more detailed account of GP effects in the H + H2(v = 4, j = 0) → H + H2(v′, j′) (para-para) reaction rate coefficients for temperatures between 1 μK (8.6 × 10−11 eV) and 100 K (8.6 × 10−3 eV). The GP effect is found to persist in both vibrationally resolved and total rate coefficients for collision energies up to about 10 K. The GP effect also appears in rotationally resolved differential cross sections leading to a very different oscillatory structure in both energy and scattering angle. It is shown to suppress a prominent shape resonance near 1 K and enhance a shape resonance near 8 K, providing new experimentally verifiable signatures of the GP effect in the fundamental hydrogen exchange reaction. The GP effect in the D + D2 and T + T2 reactions is also examined in the ultracold limit and its sensitivity to the potential energy surface is explored.