Improved numerical integration scheme

The most time-consuming step in RDVM calculations is the numerical evaluation of multicentre matrix elements. Although the Baerends integration scheme used so far (te Velde and Baerends 1992) is rather accurate and efficient due to the partitioning 

The ZA(f) are chosen so that they tend to 1 in the vicinity of atom A but drop to zero in the direction of all other nuclei. Thus, even for integrals involving atomic basis functions of two different atoms the integrand of each contribution la has no more than one singular point. The integration can be further simplified by suitable transformations to intrinsic coordinates, e.g. elliptic-hyperbolic coordinates for diatomic molecules or spherical coordinates for polyatomic systems. 

We thank all the collaborators, I. Andrejkovics, J. Anton, T. Auth, T. Ba tug, P. Blaha, N. Chetty, A. Facco Bonetti, E. K. U. Gross, A. Heitmann, A. H6ck, D. Geschke, S. Keller, T. Kreibich, C. Mosch, H. Muller, R. N. Schmid, K. Schulze, K. Schwarz and W.-D. Sepp, who contributed to the work summarized in this review. We are grateful to the Deutsche Forschungsgemeinschaft for the financial support of our projects by grants Dr 113/20-1, -2, -3 and Fr 637/8-1, -2, -3. 

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For either the accurate or approximate computation of the high dimensional integrals occurring in many electron atomic and molecular problems, the most efficient numerical integration scheme developed hitherto is Conroy s recently reported closed Diophantine method. This procedure, an improvement over Haselgrove s open method shares the advantage with Monte Carlo methods of not suffering from the dimensional effect. Moreover, its associated error ideally decreases with the inverse square of the number of sample points, whereas that associated with Monte Carlo methods shows at most an inverse square root dependence upon this number. 

Since the DPD method is fairly new, there are areas for research in theoretical formulations especially as applied to multiscale problems and energy conserving schemes, formulation of adaptable boundary condition to minimize density fluctuations, numerical integration schemes which improve accuracy and efficiency, and a wide range of applications as illustrated by the examples above. 

Numerical Representation The theory should be systematically improvable with respect to basis sets or integration schemes. 

One of the drawbacks of the first iteration, however, is that computation of energy quantities, e.g. orbital and total energies, requires to evaluate the integrals occurring in Eq. 3 on the basis of the ( )il )(p)- Unfortunately, the transcendental functions in terms of which the (]>il Hp) are expressed at the end of the first iteration do not lead to closed form expressions for these integrals and a numerical procedure is therefore needed. This constitutes a barrier to carry out further iterations to improve the orbitals by approaching the HE limit. A compromise has been proposed between a fully numerical scheme and the simple first iteration approach based on the fact that at the end of each iteration the < )j(k)(p) s entail the main qualitative characteristics of the exact solution and most... 

Filler [116] evaluated the integrals in Eq. (2.78) numerically. The polarizability is obtained from Eq. (2.22) using and iFl yf, while Eq. (2.19) remains unchanged. Hence, the WD does not modify the general numerical scheme. A significant improvement of DDA accuracy due to the WD was shown both by theoretical analysis [104] and in sample simulations [103,116]. 

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