Communications in Mathematical Sciences

Volume 21 (2023)

Number 4

Van der Waals interactions between two hydrogen atoms: The next orders

Pages: 915 – 948



Eric Cancès (CERMICS, Ecole des Ponts and Inria Paris, Marne-la-Vallée, France)

Rafaël Coyaud (CERMICS, Ecole des Ponts and Inria Paris, Marne-la-Vallée, France)

L. Ridgway Scott (University of Chicago, Illinois, U.S.A.)


We extend a method [E. Cancès and L.R. Scott, SIAM J. Math. Anal., 50:381–410, 2018] to compute more terms in the asymptotic expansion of the van der Waals attraction between two hydrogen atoms. These terms are obtained by solving a set of modified Slater–Kirkwood partial differential equations. The accuracy of the method is demonstrated by numerical simulations and comparison with other methods from the literature. It is also shown that the scattering states of the hydrogen atom, that are the states associated with the continuous spectrum of the Hamiltonian, have a major contribution to the $\mathrm{C}_6$ coefficient of the van der Waals expansion.


electronic structure, Van der Waals force, hydrogen molecule

2010 Mathematics Subject Classification

35Q40, 81Q05, 81Q15

Received 29 October 2020

Received revised 14 October 2021

Accepted 25 August 2022

Published 24 March 2023