Numerical Quadrature Techniques to Handle the Exchange-Correlation Potential

4 Numerical Quadrature Techniques to Handle the Exchange-Correlation Potential 

What we have not discussed so far is how the contribution of the final components of the Kohn-Sham matrix in equation (7-12), i. e., the exchange-correlation part, can be computed. What we need to solve are terms such as 

The straightforward numerical integration of V c maps equation (7-33) onto 

In words, the integral of equation (7-33) for the exchange-correlation potential is approximated by a sum of P terms. Each of these is computed as the product of the numerical values of the basis functions r M and r v with the exchange-correlation potential Vxc at each point ip on the grid. Each product is further weighted by the factor Wp, whose value depends on the actual numerical technique used. 

The atomic integrands FA are chosen such that their sum over all nuclei returns the original function, 

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