2.4 Integral <AB||AB> over hyperbolic Prolate Spheroidal Coordinates

Place the integral in parentheses as I'. I and I' looks like follows.

① Integrate with electron 1. And we find I'.

Substitute rA1 and r12. Set β=rA2/a0. Solve the integral in parentheses. MyLibrary 2-2-1 MyLibrary 2-2-2 MyLibrary 1-1-2 MyLibrary 1-1-1 Introducing these, we obtain I' below. We get I' below. Introducing I', we obtain I below. ② Integrate with electron 2. And we find I.

rA2 will be substituted later. Set α=rAB/a0. Here, we set I''1, I''2, and I''3 as follows. We obtain I below. And we will integrate I''1, I''2, and I''3 individually. ②-1　Integrate I''1. MyLibrary 2-2-1 MyLibrary 2-2-2 MyLibrary 1-1-2 MyLibrary 1-1-1 ②-2　Integrate I''2. Here we use the next volume element. MyLibrary 2-3-1 MyLibrary 1-1-1 MyLibrary 1-1-3 ②-3　Integrate I''3. MyLibrary 2-3-2 MyLibrary 1-1-2 Getting I''1, I''2, and I''3 list as follows. Introducing I''1, I''2, and I''3, we obtain I below.