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Homework 5: Overlap Integral

1. Problem Statement: \(2p_z\) Orbital Overlap

The Slater-type orbital function for a \(2p_z\) orbital is given by:

$$ \psi_{2p_z}(\mathbf{r}) = \sqrt{\frac{\zeta^5}{\pi}} z e^{-\zeta r} $$

The overlap integral for the \(p_z\) orbitals of two atoms located at \(-\mathbf{R}/2\) and \(\mathbf{R}/2\) is:

$$ S(\mathbf{R}) = \int \psi_{2p_z}\left(\mathbf{r} - \frac{1}{2}\mathbf{R}\right) \psi_{2p_z}\left(\mathbf{r} + \frac{1}{2}\mathbf{R}\right) d^3r $$

2pz Orbitals Interaction

Red: (+) Phase | Blue: (-) Phase

2. High-Performance Numerical Integration (O(N²) Cylindrical Symmetry)

Adjust the parameters below to evaluate the integral numerically. By exploiting cylindrical symmetry around the Z-axis, the 3D integral is reduced to a highly optimized 2D computation. You can safely crank up the Mesh Steps to 1000+!

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