Revisiting the divergent multipole expansion of atom-surface interactions: Hydrogen and positronium, $\alpha$-quartz, and physisorption

We revisit the derivation of multipole contributions to the atom-wall interaction previously presented in Łach et al. [G. Łach, M. DeKieviet, and U. D. Jentschura, Phys. Rev. A 81, 052507 (2010)]. A careful reconsideration of the angular momentum decomposition of the second-, third-, and fourth-rank tensors composed of the derivatives of the electric-field modes leads to a modification for the results for the quadrupole, octupole, and hexadecupole contributions to the atom-wall interaction. Asymptotic results are given for the asymptotic long-range forms of the multipole terms, in both the short-range and long-range limits. Calculations are carried out for hydrogen and positronium in contact with $\alpha$-quartz; a reanalysis of analytic models of the dielectric function of $\alpha$-quartz is performed. Analytic results are provided for the multipole polarizabilities of hydrogen and positronium. The quadrupole correction is shown to be numerically significant for atom-surface interactions. The expansion into multipoles is shown to constitute a divergent, asymptotic series. Connections to van der Waals corrected density-functional theory and applications to physisorption are described.