Distributed multipole methods

If the charge distribution is described by a set of distributed multipoles, as described in Section 4.2.3, the coulombic contributions to the intermolecular potential energy are calculated as multipole-multipole terms. The main disadvantage of even a rigorous distributed multipole model is that such a representation is still very localized, so that coulombic energies miss a large part of the penetration contribution. For use in a complete representation of intermolecular interactions, the dispersion, polarization, and repulsion terms must be evaluated separately by some semi-empirical or fiilly empirical method, for example the approximate atom-atom formulations of equations 4.38. 39. This approach has been extensively exploited by S. L. Price and coworkers over the years, in applications to molecular crystals [48]. 

A conceptually similar approach based on a completely different computational scheme has been developed by Leiserowitz and Berkovitch-Yellin [49] using experimental electron densities obtained from experimental electron density distributions from accurate X-ray experiments. TheLeiserowitz-Yellin potential is in the following atom-atom form  

In a related approach, Spackman has developed a force field [50] that includes an empirically derived part for the dispersion and repulsion terms, while coulombic energy terms are treated in the following maimer. The molecular electron distribution is divided into a promolecule term and a deformation term, as in the Hirshfeld definition. The product of the distributions in two interacting monomers A and B, as required in the evaluation of the coulombic energy, is then expanded as  

In the first applications the electron distributions were derived from ab initio molecular orbital calculations. In an interesting development, in further work [51] the experimental electron densities as obtained by X-ray diffraction work were used. In this sense the method is similar to the Leiserowitz-Yellin approach, but the analytical derivation is somewhat different. The electron density is written as 

Equations 4.46 and 4.47 open the way to a calculation of coulombic energies in crystals directly from experimental electron densities, and represent a connection between X-ray diffraction and energies, a step forward from the usual interpretation of diffraction data just in terms of molecular structure. A rich literature has developed along these lines [52]. 

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There have been many studies that contrast the accuracy of various atomic charge and distributed multipole models. These studies include the extensive tests provided when various distributed multipole methods were first proposed. For example, there are published contour plots of the potential around a water molecule, the amino acid histidine, and variations in the electrostatic energies of nucleic acid bases,which confirm the significance of the atomic anisotropy shown in the color three-dimensional displays of the electrostatic field around uracil and pyrimidine. It is clear that the difference... 

Stone A J and Alderton M 1985 Distributed multipole analysis—methods and applications Mol. Phys. 56 1047... 

Distributed multipole models for Nj and HF. (Figure adapted from Stone A j and M Alderton 19S5. ibuted Multipole Analysis Methods and Applications. Molecular Physics 56 3 047-1064.)... 

However, unlike point charges, the continuous charge distributions that occur in quantum chemistry have varying extents and the applicability of the multipole approximation is not only limited by the distance but also by the extent or diffuseness of the charge distribution. This additional complexity makes a transfer of the concepts of the fast multipole method to applications in quantum chemistry less straightforward. Therefore it should come as no surprise that several adaptations to extend the applicability of the FMM to the Coulomb problem with continuous charge distributions have been suggested. These lead to... 

Iversen, B.B., Larsen, F.K., Souhassou, M. and Takata, M. (1995) Experimental evidence forthe existence of non-nuclear maxima in the electron density distribution of metallic beryllium. A comparative study ofthe maximum entropy method and the multipole method, Acta Cryst., B51, 580-591. 

Cisneros GA, Piquemal J-P, Darden TA (2006) Generalization of the Gaussian electrostatic model extension to arbitrary angular momentum, distributed multipoles and speedup with reciprocal space methods. J Chem Phys 125 184101... 

Electron Stimulated Desorption Ion Angular Distributions Fast Multipole Method Generalized Gradient Approximation... 

Stone AJ, Alderton M (1985) Distributed multipole analysis methods and applications. Mol Phys... 

At this stage, when the nature of the basis is known, we return to the question, how the MME needed to evaluate (i > I FPerm. cj) and (i > I Fp0i. cj) is done in practice. We use a method that takes advantage of basic properties of Gaussian functions, which is also very similar to the distributed multipole analysis of Stone [123,124,125], For an arbitrary pair of basis functions we use the orbital expansion... 

A.J. Stone and M. Alderton, Distributed multipole analysis. Methods and applications. Mol. Phys., 56 (1985) 1047-1064. 

The method that is used in most of the work described in this chapter is the distributed multipole analysis (DMA) of Stone,which is implemented in the CADPAC ab initio suite. DMA is based on the density matrix p,y of the ab initio wavefunction of the molecule, expressed in terms of the Gaussian primitives q that comprise the atomic orbital basis set ... 

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