Bipolar Angle Averages and Two-Center, Two-Particle Integrals Involving r12

The bipolar angle average of a two‐center, two‐particle function f(r_a1, r_b2, r₁₂; R) is

$${\text{〈f〉=(4π)}}^{\text{−2}}{\displaystyle \int }{\displaystyle \int }{\mathrm{fd\omega }}_{\text{a1}}{\mathrm{d\omega }}_{\text{b2}}.$$

A bipolar angle average weight function L₀ is derived from geometrical considerations such that

$$\mathrm{〈f〉=}{\displaystyle {\int }_{r_{12}\min }^{r_{12}\max }}{\mathrm{fL}}_0{\mathrm{dr}}_{12}.$$

The L₀ is independent of f and has a different, although simple, functional form in each of 42 regions of r_a1—r_b2—r₁₂ space. However, the 〈f〉 have different functional forms in only four regions of r_a1—r_b2 space. The expressions which we derive for the bipolar angle average are surprisingly simple and general, requiring only the evaluation of integrals of the form ∫fr₁₂dr₁₂ and ∫fr₁₂²dr₁₂. The bipolar angle averages are very useful in the evaluation of two‐center, two‐particle integrals. Many of our relations are greatly simplified by the use of homogeneous coordinates. Bipolar angle averages are also developed for functions f(r_a1, r_b1, r_b2, r₁₂; R) which involve the additional variable r_b1.