2.5 Integral <AA||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'.

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 2-3-2 MyLibrary 1-1-2

②-2　Integrate I''2. Here we use the next volume element. MyLibrary 2-1-1 MyLibrary 2-1-3 MyLibrary 1-1-3 MyLibrary 1-1-1

②-3　Integrate I''3. MyLibrary 2-1-1 MyLibrary 2-1-2 MyLibrary 1-1-2 MyLibrary 1-1-1

Getting I''1, I''2, and I''3 list as follows. Introducing I''1, I''2, and I''3, we obtain I below.