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Non-adiabatic perturbation theory of the exact factorisation

Matisse Wei-Yuan Tu, E. K. U. Gross

TL;DR

This work develops a nonadiabatic perturbation theory (NAPT) within the exact factorisation (EF) framework to address correlated electron–nuclear dynamics beyond the Born–Oppenheimer (BO) approximation. By exploiting the small mass ratio $\mu=\frac{m_e}{M}$, the electronic equation of motion is perturbed with the electron–nuclear correlation term, yielding a systematic expansion whose finite-order truncations preserve the conditional electronic wavefunction normalization and gauge covariance. The method is validated through a Berry-phase calculation in the $E\otimes e$ Jahn–Teller model, where first-order NAPT captures nonadiabatic corrections to the BO geometric phase and correctly reproduces the quadratic scaling of the phase as the conical-intersection gap is varied. This EF-based perturbative framework offers a transparent, gauge-consistent route to include nonadiabatic effects in ab initio-like calculations and can be adapted with flexible nuclear treatments (quantum or classical), with potential impact on nonadiabatic dynamics and geometric-phase studies.

Abstract

We present a novel nonadiabatic perturbation theory (NAPT) for correlated systems of electrons and nuclei beyond the Born-Oppenheimer (BO) approximation. The essence of the method is to exploit the smallness of the electronic-to-nuclear mass ratio by treating the electron-nuclear correlation terms in the electronic equation of motion of the exact factorisation (EF) framework as perturbation. We prove that any finite-order truncation of the NAPT preserves the normalisation of the conditional electronic factor as well as the gauge covariance of the resulting perturbative equations of motion. We illustrate the usefulness of NAPT by obtaining nonadiabatic corrections to the BO Berry phase in Jahn--Teller systems with a conical intersection. It well captures the departure of the exact Berry phase from being topological via the lowest-order NAPT. By removing the conical intersection with a constant gap, it further yields the correct scaling of the Berry phase toward zero.

Non-adiabatic perturbation theory of the exact factorisation

TL;DR

This work develops a nonadiabatic perturbation theory (NAPT) within the exact factorisation (EF) framework to address correlated electron–nuclear dynamics beyond the Born–Oppenheimer (BO) approximation. By exploiting the small mass ratio , the electronic equation of motion is perturbed with the electron–nuclear correlation term, yielding a systematic expansion whose finite-order truncations preserve the conditional electronic wavefunction normalization and gauge covariance. The method is validated through a Berry-phase calculation in the Jahn–Teller model, where first-order NAPT captures nonadiabatic corrections to the BO geometric phase and correctly reproduces the quadratic scaling of the phase as the conical-intersection gap is varied. This EF-based perturbative framework offers a transparent, gauge-consistent route to include nonadiabatic effects in ab initio-like calculations and can be adapted with flexible nuclear treatments (quantum or classical), with potential impact on nonadiabatic dynamics and geometric-phase studies.

Abstract

We present a novel nonadiabatic perturbation theory (NAPT) for correlated systems of electrons and nuclei beyond the Born-Oppenheimer (BO) approximation. The essence of the method is to exploit the smallness of the electronic-to-nuclear mass ratio by treating the electron-nuclear correlation terms in the electronic equation of motion of the exact factorisation (EF) framework as perturbation. We prove that any finite-order truncation of the NAPT preserves the normalisation of the conditional electronic factor as well as the gauge covariance of the resulting perturbative equations of motion. We illustrate the usefulness of NAPT by obtaining nonadiabatic corrections to the BO Berry phase in Jahn--Teller systems with a conical intersection. It well captures the departure of the exact Berry phase from being topological via the lowest-order NAPT. By removing the conical intersection with a constant gap, it further yields the correct scaling of the Berry phase toward zero.

Paper Structure

This paper contains 12 sections, 33 equations, 2 figures.

Table of Contents

  1. Introduction
  2. An electronic-nonadiabatic perturbation theory
  3. The EF and the essential properties of the NAPT
  4. Indepedent post-perturbation approximations for the nuclear subsystem
  5. partial normalisation condition and gauge covariance
  6. The Berry phase of the $E\otimes e$ Jahn-Teller model
  7. Berry phase of the $E\otimes e$ Jahn-Teller model in the adiabatic limit
  8. Berry phase of the $E\otimes e$ Jahn-Teller model in first-order NAPT
  9. discussion and outlook
  10. Details of Berry-phase correction calculations
  11. Case of finite $Q$
  12. Limit of $Q\rightarrow0$

Figures (2)

  • Figure 1: Overview of interrelated perturbation methods exploiting the smallness of the electronic-over-nuclear mass parameter $\mu$. Here "EF" abbreviates exact factorisation and "PT" denotes perturbation theory, while BOPT, EF-NVPT, and NAPT refer to Born-Oppenheimer, EF-based nuclear-velocity, and electronic-nonadiabatic PT, respectively.
  • Figure 2: The main findings of applying NAPT developed here to the Berry phase problem in Jahn-Teller systems for degenerate (left panel) and gapped (right panel) cases.