Grid function techniques

Werpetinski, K. S., Cook, M., 1997, A New Grid-Free Density Functional Technique Application to hie Torsional Energy Surfaces of Ethane, Hydrazine, and Hydrogen Peroxide , J. Chem. Phys., 106, 7124. 

In this section, the accuracy of the sparse grid collocation technique for computing numerically multidimensional integrals is examined. A test function is examined whose exact numerical integral is available. For the one-dimensional quadrature rules used in the Smolyak s algorithm, only the nested quadrature rules, namely, the Clenshaw-Curtis(CC) rule and the Gauss-Patterson(GP) rule, have been used. 

A finite difference formula is used to estimate the second derivatives of the coordinate vector with respect to time and S is now a function of all the intermediate coordinate sets. An optimal value of S can be found by a direct minimization, by multi-grid techniques, or by an annealing protocol [7]. We employed in the optimization analytical derivatives of S with respect to all the Xj-s. 

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 and rp, with the exchange-correlation potential Vxc at each point rp on the grid. Each product is further weighted by the factor Wp, whose value depends on the actual numerical technique used. 

Figure 15.2. Region of interest for computing potential based on Laplace or Poisson equations, where (a) a complete rectangular grid is established to cover the region, which may be adapted to finite-difference techniques using (b) a five-point method, or (c) a finite-element approach based on sampling functions.

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