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On the Mass of Atoms in Molecules: Beyond the Born-Oppenheimer Approximation

Arne Scherrer, Federica Agostini, Daniel Sebastiani, E. K. U. Gross, and Rodolphe Vuilleumier
Phys. Rev. X 7, 031035 – Published 25 August 2017

Abstract

Describing the dynamics of nuclei in molecules requires a potential energy surface, which is traditionally provided by the Born-Oppenheimer or adiabatic approximation. However, we also need to assign masses to the nuclei. There, the Born-Oppenheimer picture does not account for the inertia of the electrons, and only bare nuclear masses are considered. Nowadays, experimental accuracy challenges the theoretical predictions of rotational and vibrational spectra and requires the participation of electrons in the internal motion of the molecule. More than 80 years after the original work of Born and Oppenheimer, this issue has still not been solved, in general. Here, we present a theoretical and numerical framework to address this problem in a general and rigorous way. Starting from the exact factorization of the electron-nuclear wave function, we include electronic effects beyond the Born-Oppenheimer regime in a perturbative way via position-dependent corrections to the bare nuclear masses. This maintains an adiabaticlike point of view: The nuclear degrees of freedom feel the presence of the electrons via a single potential energy surface, whereas the inertia of electrons is accounted for and the total mass of the system is recovered. This constitutes a general framework for describing the mass acquired by slow degrees of freedom due to the inertia of light, bounded particles; thus, it is applicable not only in electron-nuclear systems but in light-heavy nuclei or ions as well. We illustrate this idea with a model of proton transfer, where the light particle is the proton and the heavy particles are the oxygen atoms to which the proton is bounded. Inclusion of the light-particle inertia allows us to gain orders of magnitude in accuracy. The electron-nuclear perspective is adopted, instead, to calculate position-dependent mass corrections using density functional theory for a few polyatomic molecules at their equilibrium geometry. These data can serve as input for the computation of high-precision molecular spectra.

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  • Received 23 May 2017

DOI:https://doi.org/10.1103/PhysRevX.7.031035

Published by the American Physical Society under the terms of the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to the author(s) and the published article’s title, journal citation, and DOI.

Published by the American Physical Society

Physics Subject Headings (PhySH)

General Physics

Authors & Affiliations

Arne Scherrer1,2,3, Federica Agostini4,5,*, Daniel Sebastiani1, E. K. U. Gross4,6, and Rodolphe Vuilleumier2,3,†

  • 1Martin-Luther-Universität Halle-Wittenberg, von-Danckelmann-Platz 4, D-06120 Halle, Germany
  • 2UMR 8640 ENS-CNRS-UPMC, Département de Chimie, 24 rue Lhomond, École Normale Supérieure, 75005 Paris, France
  • 3UPMC Université Paris 06, 4, Place Jussieu, 75005 Paris, France
  • 4Max-Planck-Institut für Mikrostrukturphysik, Weinberg 2, D-06120 Halle, Germany
  • 5Laboratoire de Chimie Physique, UMR 8000 CNRS/University Paris-Sud, University Paris-Saclay, 91405 Orsay, France
  • 6Fritz Haber Center for Molecular Dynamics, Institute of Chemistry, The Hebrew University of Jerusalem, Jerusalem 91904, Israel

  • *federica.agostini@u-psud.fr
  • rodolphe.vuilleumier@ens.fr

Popular Summary

In quantum mechanics, exact solutions to problems are difficult (if not impossible) to come by for all but the simplest situations. Physicists often must turn to mathematical approximations to describe the internal workings of atoms and molecules. One of the central approximations is known as the Born-Oppenheimer approximation, which is the assumption that the motion of nuclei in molecules can be separated from that of the surrounding electrons. When making this assumption, one needs to assign a mass to the nucleus, which is, in this approximation, the bare nuclear mass. But the inertia of the electrons, which alters the effective mass, is completely neglected, resulting in fundamental and practical issues when theoretical predictions are required to meet experimental accuracy. We have developed a theoretical technique for solving this problem.

We take as a starting point the exact factorization of the electron-nuclear wave function. This allows us to develop a rigorous procedure where the effects of the electrons on the nuclear mass are handled as position-dependent corrections to the bare nuclear mass. We demonstrate an orders-of-magnitude increase in accuracy by applying this technique to the situation where a proton, taken as the light particle, forms a linear bond with two oxygen atoms.

Our scheme resolves a decades-old inconsistency in the Born-Oppenheimer approximation. This theory naturally leads to a numerical scheme efficient enough to provide the first predictions of nonadiabatic corrections to vibrational frequencies beyond diatomic and triatomic molecules.

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Vol. 7, Iss. 3 — July - September 2017

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