# Effective non-adiabatic Hamiltonians for the quantum nuclear motion over coupled electronic states

## TL;DR

This paper develops a systematic method to derive effective non-adiabatic Hamiltonians for nuclear motion in molecules, including higher-order corrections and mass-correction terms, improving the accuracy of quantum nuclear dynamics models.

## Contribution

It introduces a general framework for block-diagonalizing the electron-nuclear Hamiltonian up to arbitrary order, deriving explicit formulas for second- and third-order corrections, and incorporating previously neglected outlying electronic states.

## Key findings

- Derived explicit second- and third-order effective Hamiltonians.
- Revealed generalized mass-correction terms in multi-dimensional coupled electronic states.
- Unified treatment of adiabatic and non-adiabatic nuclear motion.

## Abstract

The quantum mechanical motion of the atomic nuclei is considered over a single- or a multi-dimensional subspace of electronic states which is separated by a gap from the rest of the electronic spectrum over the relevant range of nuclear configurations. The electron-nucleus Hamiltonian is block-diagonalized up to $\mathcal{O}(\varepsilon^{n+1})$ through a unitary transformation of the electronic subspace and the corresponding $n$th-order effective Hamiltonian is derived for the quantum nuclear motion. Explicit but general formulae are given for the second- and the third-order corrections. As a special case, the second-order Hamiltonian corresponding to an isolated electronic state is recovered which contains the coordinate-dependent mass-correction terms in the nuclear kinetic energy operator. For a multi-dimensional, explicitly coupled electronic band, the second-order Hamiltonian contains the usual BO terms and non-adiabatic corrections but generalized mass-correction terms appear as well. These, earlier neglected terms, perturbatively account for the outlying (discrete and continuous) electronic states not included in the explicitly coupled electronic subspace.