  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.
