I'm trying to understand the Hartree-Fock mathematical formulation I understand the Coulomb operator, but I dont understand the exchange operator: [tex] \hat{K_{j}}[\Psi](\textbf{x})=\Phi_{j}(\textbf{x})\int d\textbf{x}'\frac{\Phi_{j}^{*}(\textbf{x}')\Psi(\textbf{x}')}{|\textbf{r}-\textbf{r}'|}[/tex] Can any one explain me why this operator is like this. I understand that it is the interaction of the j-th electron with the electrons' cloud but... how it come to be like that thanks in advance
I don't know anything about that formulation, but this book on Google Print might help: Google Print You can't see all the relevant information, but I think it might help you understand where the idea's headed.
I guess the integral on the rhs is simply the matrix element <j|K|phi> of the electrostatic potential. Therefore K|phi> is indeed given by |j><j|K|ph> .