Cellular decomposition

Next, we note that since the q are bond-orbitals, they are localized functions, and if we write V in terms of a cellular decomposition... 

The cellular decomposition of the local part of V is trivially exact and allows a complete correspondence between the geometric arrangement of the ligands in space, and their additive contributions to V. The decomposition of the non-local part is however not so obvious and is worth looking at in more detail. To do this it is convenient to recall Eq. (6-13 b) for the screened exchange operator A typical matrix element of can be put in the form, using (6-27)... 

For basis sets derived from a cellular decomposition of space, e.g. the muflSn-tin decomposition used in LAPW and LMTO methods, the difficulty can be sidestepped by constructing the basis in terms of fully relativistic atomic-like functions determined by the miiflfin-tin potential. By restricting the selection to the electron states, the desired decomposition is automatically enforced. This is the essence of the Koelling-Harmon scalar relativistic scheme widely used in cellular-based modified plane-wave basis methods [45]. 

Choice of Optimal Regimes in Cellular Decomposition, Diffusion-Induced Crain Boundary Migration, and the Inverse Diffusion Problem... 

Kaur, I. and Gust, W. (1989) Fundamentals of Grain and Interphase Boundary Diffusion, Ziegler Press, Stuttgart. Larikov, L.N. and Shmatko, O.A. (1976) Cellular Decomposition of Supersaturated Solid Solutions, Naukova Dumka, Kyiv (in Russian). 

W. A. Goddard, R. Biswas, and D. Srivastava, Comput. Mater. Sci., 38,642-652 (2007). A Divide-and-Conquer/Cellular-Decomposition Framework for Million-to-Billion Atom Simulations of Chemical Reactions. 

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