On the time-dependent Born-Oppenheimer Approximation

TL;DR

This work develops a rigorous time-dependent Born-Oppenheimer framework for molecular Schrödinger dynamics by projecting onto electronic ground states and deriving an effective, nonlocal nuclear dynamics. Central to the approach is non-Abelian integration by parts (NAIP), which yields iterative higher-order corrections to the nuclear Hamiltonian in powers of the small mass-ratio parameter \kappa, including a nonlocal-in-time operator w^{\kappa}. The authors prove a first-order TD-BOA with h_{\rm eff}^{\kappa} = T + E + \kappa^2 v and extend it to arbitrary orders, culminating in a second-order autonomous effective Hamiltonian h_{\rm eff}^{(2)} = T + E + \kappa^2 v - \kappa^2 w_1, with explicit expansion formulas for the electronic feedback. The results provide a systematic, controllable expansion of the full molecular dynamics in terms of reduced nuclear variables, enabling tractable dynamics beyond the standard BOA and aligning with established second-order corrections in the literature.

Abstract

In this paper, we consider the time-dependent Born-Oppenheimer approximation (BOA) of a classical quantum molecule involving a possibly large number of nuclei and electrons, described by a Schrödinger equation. In the spirit of Born and Oppenheimer's original idea we study quantitatively the approximation of the molecular evolution. We obtain an iterable approximation of the molecular evolution to arbitrary order and we derive an effective equation for the reduced dynamics involving the nuclei equivalent to the original Schrödinger equation and containing no electron variables. We estimate the coefficients of the new equation and find tractable approximations for the molecular dynamics going beyond the one corresponding to the original Born and Oppenheimer approximation.

Theorems & Definitions (78)

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