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When I am looking into the Hartree-Fock solution to the many bodies electronic system, I came across the one electron integral, which is defined as:

$$h_{pq}=\int{\psi_p^\ast\left(r\right)\left(-\frac{1}{2}\nabla^2-\sum_{I}\frac{Z_I}{R_I-r}\right)\psi_q\left(r\right)dr}$$

Where, $\psi$ are wave functions expressed in the second quantized forms as slatter determinants:

$$\psi\left(r\right)=\left|x_1x_2\right\rangle=\frac{1}{\sqrt2}\left|\begin{matrix}X_i\left(x_1\right)&X_j\left(x_1\right)\\X_i\left(x_2\right)&X_j\left(x_2\right)\\\end{matrix}\right|$$

I am wondering what mathematically rigorous steps are to apply the Hamiltonian to the wave functions to solve the integral? In particular, how do we apply the laplacian operator on wave functions? Are there any resources online that show a step-by-step solution to this integral? Thanks for helping!

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