Numerical integration Gaussian quadrature

Obara-Saika or McMurchie-Davidson schemes. For more general charge distributions, representing, for example, a large part of a molecular system, it may be better to carry out the integration numerically, using Gaussian quadrature or some other numerical scheme. 

Numerical integration methods are widely used to solve these integrals. The Gauss-Miihler method [28] is employed in all of the calculations used here. This method is a Gaussian quadrature [29] which gives exact answers for Coulomb scattering. 

The inner-integral of Equation (2) was numerically integrated using a four-point Gaussian quadrature. The mean bubble length was calculated from the first moment of the frequency distribution function given in Equation (2). 

In this subsection, a simple trapezoidal rule is used to derive a numerical scheme to solve Fredholm equations of the second kind. The use of a Gaussian quadrature to approximate the integral is discussed by Press et al. (1992). 

To say it in other words, without loosing accmacy of the finite difference scheme, the number of nodes of the optimal grid along each direction can be reduced to just a few ones, if the steps are arranged in a specific way. It can be considered as an extension of the concept of the Gaussian quadrature rule for the numerical integration to the finite differences. 

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. 

It Is best to choose the well-known Gaussian quadrature points. With knowledge of the endpoint values, the optimum locations are the so-called "Lobatto points" (4), N interior points permitting exact numerical integration of a polynomial of degree (2W 1). The ease... 

Tlie integral in Eq. (4.54) must be evaluated numerically. Once the expansion coefficients aik(t,) are known, Gaussian quadrature with Nquad points can be used to obtain the energy dependent coefficients exactly. 

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. 

The integrals over the particle surface are usually computed by using appropriate quadrature formulas. For particles with piecewise smooth surfaces, the numerical stability and accuracy of the T-matrix calculations can be improved by using separate Gaussian quadratures on each smooth section [8,170]. 

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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