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

TL;DR

This paper extends a mathematical method to compute higher-order terms in the van der Waals interaction between two hydrogen atoms, highlighting the importance of scattering states in the interaction's leading coefficient.

Contribution

It introduces an extended approach solving modified PDEs to obtain more terms in the van der Waals asymptotic expansion, validated by numerical comparisons.

Findings

Higher-order terms improve accuracy of van der Waals models.

Scattering states significantly influence the C₆ coefficient.

Numerical results align well with existing literature.

Abstract

We extend a method (E. Canc\`es and L.R. Scott, SIAM J. Math. Anal., 50, 2018, 381--410) 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 C$_6$ coefficient of the van der Waals expansion.