Translations of fields represented by spherical-harmonic expansions for molecular calculations

Abstract

For an arbitrary integerN, the expansion theorem

$$|r_ > - r_< |^N = r_ > ^N \Sigma _l \Sigma _k (2l + 1)T_{l,k}^N (r_< /r_ > )^k P_l (\cos \theta )$$

is derived by an induction method, which yields explicit expressions for the expansion coefficientT _l,k ^N. Such expansions are useful in molecular theory because functions (r′)^N withN′=|r _> −r _<| are contained in many operators. This investigation provides also a basis for the derivations of expansion theorems for more complicated functions which will be dealt with in later articles of this series.