Gaussian quadrature weighted integrals

GAUSSIAN QUADRATURE FOR EVEN POLYNOMIALS AND WEIGHT FUNCTIONS Consider the Gaussian quadrature for integrals such as... 

In the case of 3b, Gaussian quadrature can be used, choosing the weighting function to remove the singularities from the desired integral. A variable step size differential equation integration routine [38, Chapter 15] produces the only practicable solution to 3c. 

Two features of the trapezoidal method are that we use a uniform spacing between the positions where we evaluate fix) and that every evaluation of fix) (except the end points) is given equal weight. Neither of these conditions is necessary or even desirable. An elegant class of integration methods called Gaussian quadrature defines methods that have the form... 

In Gaussian quadrature theory the NDF is called the weight function or measure. The weight function must be nonnegative and non-null in the integration interval and all its moments. 

The definition of the Gaussian quadrature formula in Eq. E.30 implies that the determination of this formula is determined by the selection of N quadrature points and N quadrature weights that is, we have 2N parameters to be found. With these degrees of freedom (2N parameters), it is possible to fit a polynomial of degree 2N - 1. This means that if Xj and are properly chosen, the Gausssian quadrature formula can exactly integrate a polynomial of degree up to 2N — 1. 

In the following, we shall first show how Gaussian quadrature can be simplified for the special case of even polynomials and weight functions such as those in (9.11.3). The orthogonal polynomials needed for the calculation of Coulomb integrals (9.11.3) are then introduced, and finally we show how a Gaussian-quadrature scheme for the evaluation of Coulomb integrals can be developed based on the McMurchie-Davidson and Obara-Saika schemes. 

We have now developed a complete scheme for the evaluation of one-electron Coulomb integrals by Gaussian quadrature. First, we calculate (by some numerical scheme) the abscissae and weights... 

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