Understanding Coulombic Operator: J Equation

[greisen]

Hi,

I am trying to understand this equation where the coulombic operator is given by

J = \int d\tau \phi(2) \frac{1}{r_{12}}\phi(2)

so I integrate over \tau but what is tau and the number I get from the equation is the energy I pressume?
Any hints or help appreciated.

Thanks in advance

 

[Meir Achuz]

$$d\tau$$ just represents the differential volume element.
In spherical coordinates, it is $$r^2 dr d\Omega$$.

 

[greisen]

what is that more exactly? I know from spherical geometry that the volume is calculate
dV = \rho^2 sin \phi d\phi d\rho d\theta

If I integrate over two wave functions representing two electrons how to interpreted it?

Any help or advice appreciated

Thanks in advance

 

[Jack111]

Are you trying to integrate the expectation of the coulombic interaction operator over two wavefunctions? <a(r1)|J|b(r2)> ? (with r1, r2 position vectors so in general dependent on r, theta and phi).

If so, which wavefunctions a and b are you using? You'd also need to rexpress the 1/r term in J as some function r1-r2, which will be fairly complicated in 3D. I'd guess this problem is best solved computationally...