# Integral of relative distance–dependent potential

Isotropicaf
Homework Statement:
Hello,
Imagine the Hamiltonian of a two-atom molecule, you have the kinetic energy of one+the other and then you have a relative distance potential ( V(|r2-r1|^2), ri is 3 dimensional).
How to change the variable to solve the following integral with infinite limits?
Relevant Equations:
Intg( exp( |r2-r1|^2) dr1dr2)
I think its going to be intg(dr2)intg(exp(r^2) dr) or something like that.

Abhishek11235
Try putting one variable on the axis

Isotropicaf
Try putting one variable on the axis
Im sorry, i don't know what you mean by that, you mean i should assume r1 as a constant and analyse how |r2-r1|^2 behaves in this condition ?

Abhishek11235
you mean i should assume r1 as a constant and analyse how |r2-r1|^2 behaves in this condition ?
No!

##|r_2-r_1|=\sqrt{r_2^2+r_1^2-2r_2 r_1cos\theta}##

Now the integrations are easy!

Isotropicaf
No!

##|r_2-r_1|=\sqrt{r_2^2+r_1^2-2r_2 r_1cos\theta}##

Now the integrations are easy!
Oh thanks that totally solved my problem, seems obvious now ahah just to check, the integral of the sum is the sum of the integrals?