  We revisit the derivation of multipole contributions to the atom-wall
interaction previously presented in [G. Lach et al., 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 decribed.
