Bipolar Expansion for r12nYlm(θ12, φ12)

Explicit formulas for the radial functions Vl1l2l3l(n)(r1, r2, R) in the bipolar expansion for r12nYlm(θ12, φ12), r12nYlm(θ12, φ12) = Σ(2λ + 1)1/2(2l3 + 1)1/2cλ( l1m1)c3l(λ, m − m1; l2m2) × Yl1m1(θ1, φ1)Yl2m2(θ2,φ2) Yl3m−m1−m2(θR, φR)Vl1l2l3l(n)(r1, r2, R), where r12 = r1 − r2 − 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 Vl1l2l3l(n) are given.