When trying to solve the Schrodinger equation for the electronic hamiltonian:

$$H_{el} = \sum_{i=1}^{N} \bigg( - \frac{1}{2}\nabla_i^2 - \sum_A \frac{Z}{r_{i_A}} \bigg) + \sum_{i>j=1}^{N}\frac{1}{r_{ij}}$$

we note that two prominent integrals arise:

These are the exchange integral:

$$ K_b(1)\chi_a(1) = \bigg[ \int d\tau_2 \chi_b(2)^* \frac{1}{r_{12}} \chi_a(2) \bigg]\chi_b(1)$$

and the coulomb integral:

$$ J_b(1)\chi_a(1) = \bigg[ \int d\tau_2 \chi_b(2)^* \frac{1}{r_{12}} \chi_b(2) \bigg]\chi_a(1)$$

$r_{12}$ is the distance between electrons 1 and 2 and the integrals are done over all space. $\chi_b$ is an orbital labelled b.

I was wondering whether someone could give me a brief physical interpretation of these two integrals.


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