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Van der Waals–London interaction of atoms with pseudorelavistic kinetic energy

Jean-Marie Barbaroux, Michael C. Hartig, Dirk Hundertmark and Semjon Vugalter

Vol. 15 (2022), No. 6, 1375–1428
Abstract

We consider a multi-atomic system where the nuclei are assumed to be point charges at fixed positions. Particles interact via Coulomb potential and electrons have pseudorelativistic kinetic energy. We prove the van der Waals–London law, which states that the interaction energy between neutral atoms decays as the sixth power of the distance |D| between the atoms. In the many-atom case, we rigorously compute all the terms in the binding energy up to the order |D|9 with error term of order 𝒪(|D|10). This yields the first proof of the famous Axilrod–Teller–Muto three-body correction to the van der Waals–London interaction, which plays an important role in atomic physics. As intermediate steps we prove exponential decay of eigenfunctions of multiparticle Schrödinger operators with permutation symmetry imposed by the Pauli principle and new estimates of the localization error.

Keywords
van der Waals–London force, Axilrod–Teller–Muto correction, pseudorelativistic kinetic energy, exponential decay, localization error, Herbst operator
Mathematical Subject Classification 2010
Primary: 81Q10
Secondary: 46N50, 34L15, 47A10, 35P15
Milestones
Received: 24 January 2019
Revised: 10 January 2021
Accepted: 15 February 2021
Published: 10 November 2022
Authors
Jean-Marie Barbaroux
Aix-Marseille Université, Université de Toulon, CNRS, CPT
Marseille
France
Michael C. Hartig
Aix Marseille Université, Université de Toulon, CNRS, CPT
Marseille
France
Dirk Hundertmark
Institute for Analysis
Karlsruhe Institute of Technology
Karlsruhe
Germany
Department of Mathematics
University of Illinois at Urbana-Champaign
Urbana, IL
United States
Semjon Vugalter
Institute for Analysis
Karlsruhe Institute of Technology
Karlsruhe
Germany