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

The bipolar angle average of a two-center, two-particle function f(ra1, rb2, r12; R) is 〈f〉=(4π)−2 ∫ ∫ fdωa1dωb2.A bipolar angle average weight function L0 is derived from geometrical considerations such that 〈f〉= ∫ r12minr12maxfL0dr12.The L0 is independent of f and has a different, although simple, functional form in each of 42 regions of ra1—rb2—r12 space. However, the 〈f〉 have different functional forms in only four regions of ra1—rb2 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(ra1, rb1, rb2, r12; R) which involve the additional variable rb1.