Integration grids

Some convergence problems are due to numerical accuracy problems. Many programs use reduced accuracy integrals at the beginning of the calculation to save CPU time. However, this can cause some convergence problems for difficult systems. A course DFT integration grid can also lead to accuracy problems, as can an incremental Fock matrix formation procedure. 

These calculations employ a grid of points in space in order to perform the numerical integration. Grids are specified as a number of radial shells around each atom, each of which contains a set number of integration points. For example, in the (75,302) grid, 75 radial shells each contain 302 points, resulting in a total of 22,650 integration points. 

An equidistant integration grid may not be the best choice. Let us therefore consider that we perform a variable transformation in the integral before we discretize. 

The ultrafine pruned (99,590) integration grid was used in all DFT calculations. 

On the practical side, neither the ADA nor the WDA was widely applied in general to many-electron, realistic systems, due to their complicated functional forms.98,136-141,260,314-318,329-345 Only very recently did efficient implementations of the WDA become available.338 342 Even today, the ADA is still a museum artifact, which has been applied only io spherical atomic species and the spherical jellium model.31 316-346 The main obstacle lies in Eq. (110), where in addition to the recursion problem, one needs to do the integration over all space of r for every point r, yielding a numerical cost scaling quadratically, 0( M2), with respect to the integration grid size M. A straightforward application of the... 

C. S. Pomelli, A tessellationless integration grid for the polarizable continuum model reaction field, J. Comput. Chem., 25 (2004) 1532-1541. 

Kuppens T, Langenaeker W, Tollenaere JP et al (2003) Determination of the stereochemistry of 3-hydroxymethyl-2, 3-dihydro-[l, 4]dioxino[2, 3-b]pyridine by vibrational circular dichroism and the effect of DFT integration grids. J Phys Chem A 107 542-553... 

Standardization refers to the fact that complete specifications of the integration grids used to produce many published results are generally not given. This is unsatisfactory for two reasons. Firstly and fundamentally, this prevents strict reproducibility of the results by other workers (this is also true of the fitting... 

When (r — c)/a > 1, p(r) 0 and V)y(r) -> —Z/r, and the simple Coulomb potential may be used for r > lOR. Care is required in selecting an integration grid which is sufficiently dense over the nuclear skin region where the charge density is changing rapidly. An analytic Fourier-Bessel expansion which may be used in finite-difference calculations is presented in [75]. However, the complexity of (109) makes the calculation of the potential matrix required in (99)... 

The self-consistent field procedure in Kohn-Sham DFT is very similar to that of the conventional Hartree-Fock method [269]. The main difference is that the functional Exc[p] and matrix elements of Vxc(r) have to be evaluated in Kohn-Sham DFT numerically, whereas the Hartree-Fock method is entirely analytic. Efficient formulas for computing matrix elements of Vxc(r) in finite basis sets have been developed [270, 271], along with accurate numerical integration grids [272-277] and techniques for real-space grid integration [278,279]. 

When, propagated down column E. the mesh size is doubled and there are only 1600 Integration grid points, because of the actions of the second logical condition. You repeat this procedure in columns F to I but using the divisors 4, 8, 16 and 32 in the second MOD() function to provide for integration grids of 800, 400, 200, 100 and 50 points. 

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