Bipolar Angle Averages and Twohyphen;Center, Twohyphen;Particle Integrals Involvingr12

摘要

The bipolar angle average of a twohyphen;center, twohyphen;particle functionf(ra1,rb2,r12;R) islang;frang;equals;lpar;4pgr;rpar;minus;2fdohgr;a1dohgr;b2.A bipolar angle average weight functionL0is derived from geometrical considerations such thatlang;frang;equals;r12minr12maxfL0dr12.TheL0is independent offand has a different, although simple, functional form in each of 42 regions ofra1mdash;rb2mdash;r12space. However, the lang;frang; have different functional forms in only four regions ofra1mdash;rb2space. The expressions which we derive for the bipolar angle average are surprisingly simple and general, requiring only the evaluation of integrals of the form int;fr12dr12and int;fr122dr12. The bipolar angle averages are very useful in the evaluation of twohyphen;center, twohyphen;particle integrals. Many of our relations are greatly simplified by the use of homogeneous coordinates. Bipolar angle averages are also developed for functionsf(ra1,rb1,rb2,r12;R) which involve the additional variablerb1.