van der Waals forces, sum rules, and pseudostate expansions

We reexamine the pseudostate expansion used previously to evaluate van der Waals (vdW) coefficients and finite-mass corrections. In particular, we wish to understand the very rapid convergence of the expansion, which for the case of two hydrogenic atoms requires only ten terms to achieve an accuracy of 1×${10}^{\mathrm{-}10}$. The analysis proceeds from the representation of the vdW coefficients as integrals of single-atom frequency-dependent polarizabilities. These in turn can be expanded in power series with coefficients of the form ${\mathcal{J}}_n$ $M^2$(nl)/($E_{\mathrm{nl}}$-$E_{1s}$ ${)}^k$, where M(nl) is a multipole matrix element between the ground (1s) and excited (nl) states. Although defined as sums over an infinite set of hydrogenic states, the coefficients of each order k are reproduced exactly by a finite sum over suitable pseudostates. By analytic continuation, the frequency-dependent polarizabilities computed using pseudostates are expected to be very close to the exact values, and hence the rapid convergence of the vdW coefficients is understood.