Numerical quadrature techniques

Numerical Quadrature Techniques to Handle the Exchange-Correlation Potential... 

Families of numerical integration methods have been suggested which use the standard (Gaussian) numerical quadrature techniques within each of a set of mutually exclusive polyhedra formed by the planes which bisect each bond emanating from a given atom. Integration proceeds outwards from the nucleus at the centre of each polyhedron with some suitable choice for tiic boundaries of the atoms on the periphery of the molecule. 

In our present PS-DFT module, the numerical integration is performed using the procedures proposed by Becke which are based upon classical numerical quadratures. Other approaches include fitting F to an expansion of Gaussian functions and evaluating the resulting three center overlap integrals analytically. However, PS methods can profitably be applied to this term as well, and one can expect substantial accelerations in DFT calculations via this technique. 

Numerical integration of a known function. To illustrate the Gauss quadrature integration technique and compare it to classical integration techniques, we will evaluate the following integral... 

A method of numerical integration (or quadrature, as it is also called) is required to evaluate I in any of these cases. The specific techniques we will present are algebraic, but the general approach to the problem is best visualized graphically. For the moment, we will suppose that all we have relating x and y is a table of data points, which we may graph on a plot of y versus x. 

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