Bipolar Expansion for r12nYlm(θ12, φ12)

Abstract

Explicit formulas for the radial functions V l 1 l 2 l 3 l (n) (r 1 , r 2 , R) in the bipolar expansion for r 12 n Y l m (θ 12 , φ 12 ) , r 12 n Y l m (θ 12 , φ 12 ) = Σ(2λ + 1) 1/2 (2l 3 + 1) 1/2 c λ ( l 1 m 1 )c 3 l (λ, m − m 1 ; l 2 m 2 ) × Y l 1 m 1 (θ 1 , φ 1 )Y l 2 m 2 (θ 2 ,φ 2 ) Y l3 m−m 1 −m 2 (θ R , φ R )V l 1 l 2 l 3 l (n) (r 1 , r 2 , R) , where r 12 = r 1 − r 2 − R , are derived with the use of the theory of generalized functions and Fourier transforms. When n ≤ − 4 and n − l is odd, there are delta‐function terms. In this approach the delta‐function terms and the four‐region form of the expansion are obtained from a single, unified formula valid in all regions. Recurrence formulas for the V l 1 l 2 l 3 l (n) are given.