Hartree fock derivation problem

[georg gill]

Can anyone show the algebra that leads to the rewriting that I have underlined in the attachment. It is a derivation

It is taken from the start of a derivation of the roothans method of hartree fock from 1972 with the authors Snow and Bills.

 

[mfb]

Plugging in the Hamiltonian, you can group its terms into three parts:
One involving only $\phi_1$-related terms, one only involving $\phi_2$-related terms, and one with $r_{12}$. In the first two, you can take the other field out of the integral, where it then cancels the corresponding expression in the denominator.
Then use the symmetry (1<->2) to combine both simplified fractions. The third fraction from the original expansion just stays as it is.

 

[ChrisVer]

write down your Hamiltonian... some part of it is going to see only the \phi_1's and some other part will only see the \phi_2's (that's the reason of the 2* integral)...
So you can break the integral \int \int f(x_1,x_2) dx_1 dx_2 = \int \int f(x_1) f(x_2) dx_1 dx_2 = \int f(x_1) dx_1 \int f(x_2) dx_2...that's the first integral in your orange... The second integral in your orange comes from the Hamiltonian's part that mixes r_{1},r_2 \text{ into } r_{12} and so your double integral remains : \int \int f(x_1,x_2) dx_1 dx_2 which you cannot separate.

I'm pretty sure this should go to quantum physics XD

 

[georg gill]

Thanks I did get it from your explanation!