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

TL;DR

This paper demonstrates how the multi-electronic-state path integral formulation naturally incorporates the geometric phase effect, significantly impacting low-temperature thermodynamic predictions in quantum systems with conical intersections.

Contribution

It introduces a method to explicitly include the geometric phase in imaginary-time path integral simulations, improving accuracy for complex systems with conical intersections.

Findings

The MES-PI formulation captures the geometric phase effect through the electronic trace of overlap matrices.

Comparison with a phase-excluded baseline quantifies the geometric phase's impact on thermodynamics.

MES-PIMD provides a general, accurate approach for systems with unknown conical intersection topology.

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.