## Abstract

We analyse the perturbatively calculated adsorption potential U{/i = 2covzf/3) of a neutral molecule a distance z from a flat metal surface, adopting for the metal the simplest non-trivial spatially dispersive model, nam ely a continuous fluid confined by impenetrable barriers and with assigned plasm a frequency wp and sound velocity /?. Then U emerges as a weighted sum , over molecular states with virtual excitations En0 = of a function G(en0,y) which is the Laplace transform of a function g{en0, y), where k = (vpy /3is the tangential component of a plasmon wavenum ber. For large y, Watson’s lemma applies because the Taylor series for g converges for small y, the lemma yields an asymptotic series for G{e,y) which in general has a leading term of order For positive e, this expansion was previously obtained by Mahanty & Paranjape (1977). It must be modified if e is negative, and in particular if e is close to the surface-plasmon branch point at — when is large b u t; + A l is n o t> the effective leading term is of order If e is close to the bulk-plasmon branch point at —1, similar modifications are needed in some next-toleading term s. G is singular at y = 0, where its behaviour is governed by the fact that, for large y, g has a convergent expansion in powers of y~x. For the Laplace transform G of such a function we prove a theorem which is a natural complement to W atson’s lemma but which, surprisingly, seems to be new. I t allows G to be expressed in term s of power series in y converging everywhere, with all singularities displayed explicitly: G = A n{e) y n + lnX ”=0 Bn(e)yn.The coefficients and B n are determined by the coefficients in the expansion of g(e, y); rem arkably, they are entire functions of e, whence those parts of G which are singular in y are entire in e and vice versa. Sum rules and closure relations allow the sum over molecular states to be found in closed form for the parts of G proportional to y~x and In y. In the special case where all en0 are negligible (wp > | En0), U(y) reduces to the far simpler yet still nontrivial classical limit Ue{y), expressible in closed form in term s of Bessel and Struve functions. The non-dispersive limit (/? = 0) is obtainable trivially, but is so special a case that it can shed no light on the small-distance behaviour of U, nor on the large-distance behaviour when e is close to the branch point at —

## Footnotes

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- Received January 7, 1980.

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