BIPOLAR EXPANSION FOR R12NYLM(THETA12, PHI12)

Explicit formulas for the radial functions Vnι 1ι2ι3ι(r1, r 2, R) in the bipolar expansion for r12nYι m(δ12, Φ12), r12 nYιm(δ12, Φ12) = ∑(2λ+1)1/2(2l3+1)1/2c λ(lm;l1m1)cιs(λ, m-m1; l2m2) × Yι1 m1 (δ1, Φ1) Yι2 m2(δ2, Φ2) Yι8 m-m1-m2(δR, ΦR) V(n) ι1ι2ι3ι(r1, r2, R), where r12=r1-r2-R, are derived with the use of the theory of generalized functions andFourier 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 (n)ι1ι2ι3ι are given.