Numerical quadrature

These methods, which probably deserve more attention than they have received to date, simultaneously optimize the positions of a number of points along the reaction path. The method of Elber and Karpins [91] was developed to find transition states. It fiimishes, however, an approximation to the reaction path. In this method, a number (typically 10-20) equidistant points are chosen along an approximate reaction path coimecting two stationary points a and b, and the average of their energies is minimized under the constraint that their spacing remains equal. This is obviously a numerical quadrature of the integral s f ( (.v)where... 

Maurits, N.M., Altevogt, P., Evers, O.A., Fraaije, J.G.E.M. Simple numerical quadrature rules for Gaussian Chain polymer density functional calculations in 3D and implementation on parallel platforms. Comput. Theor. Polymer Sci. 6 (1996) 1-8. 

We choose to use the primitive coupled basis, rather than the 712 coupled basis that is often employed, primarily because the primitive coupled basis has a direct product form. This form makes it easy to accomplish the transformation from ji, j2, fci to 0i, 02, - The potential matrix, V, may be written as a numerical quadrature over the angular grid points in the following matrix form ... 

Here, the grid contains P points, which are located at positions rp and the Wp represent the weights in the numerical quadrature scheme. We will encounter a more detailed... 

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

We have found that data interpretation other than simple plotting from texture studies requires a sophisticated array of software tools. These include least squares curve resolving, non-linear least squares fitting, 2-dimensional data smoothing, numerical quadrature, and high-speed interactive graphics, to mention only a few. 

A typical CFD model for acid base and equilibrium chemistry solves Eqs. (25)-(29), and then uses Eq. (55) to approximate ft. Once ft is known, the expected values of the reactant concentrations are computed by numerical quadrature from the formula... 

For example, (A) and (B) can be computed using Eqs. (49) and (50), respectively. Note that instead of Eq. (55), we could use the simpler expression for given by Eq. (33), which avoids the need for numerical quadrature. In both cases, the mean and variance of the mixture fraction are identical (and thus both models account for finite-rate mixing effects.) In practical applications, the differences in the predicted values of () can often be small (Wang and Fox, 2004). 

The variables of the differential equation are separable, but to find C as a function of t requires a numerical quadrature and application of "root solver" many times. A direct solution by ODE program is simpler. 

This expression looks complicated but can be computed with minimum effort by using numerical quadrature software. Allowing a size distribution for the outgassing of argon from minerals (Turner, 1968) makes it easier to understand why more argon is lost at low temperature from natural crystals than predicted by the uniform size distribution, regardless of mineral geometry, o... 

If F t) is not equal to zero, so that image effects are included, it does not seem possible to obtain any simple analytical expression for P although there is no difficulty in obtaining results by numerical quadrature. For the special case where F(z) is approximated as a linear function of z, however, an asymptotic result has been found . 

In the original formulation of the R12 method [23], which is often referred to as "the standard approximation" or SA, a large uncontracted OBS was needed as the RI basis, strongly limiting the applicability of the R12 method to relatively small systems. Klopper and Samson [26] later pioneered the use of a separate RI basis set and considerably improved the applicability and accuracy of MP2-R12. Valeev developed an alternative formulation of the RI procedure [27] known as the complementary auxiliary basis set (CABS) approach, which had smaller RI errors. Ten-no, on the other hand, proposed the use of a numerical quadrature for the same purpose [28]. Kedzuch et al. later used the CABS approach to develop a particularly elegant and practical formalism for the MP2-R12 energies [30]. 

Ten-no, S. Explicitly correlated second order perturbation theory introduction of a rational generator and numerical quadratures. J. Chem. Phys. 2004, 121, 117-29. 

A sequence of independent simulations, conducted at a set of points [X] spanning a path from X to Ao, then allows one to estimate the free-energy difference by numerical quadrature ... 

Such a normalization condition can be readily discretized by considering, for example, the simple trapezoidal rule for performing the numerical quadrature ... 

This expression was computed by the numerical inverse Laplace transform embedded in the numerical quadrature. As previously, we used p = 1,4, 6, E [A] = P2/A2 = 1, and A2 = 4h 1. Figure 9.23 illustrates the influence of the p parameter on the shape of the state probability profile the larger p, the most... 

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