The KWIK algorithm for Coulomb interactions and its applications : a thesis presented in partial fulfilment of the requirements for the degree of Doctor of Philosophy, Department of Chemistry, Massey University
The KWIK algorithm is introduced, generalised and applied to the problem of determining the Coulomb energy of N localised charge distributions. Coulomb interactions are typical of N-body problems which require the exhaustive pairing of all distributions, which leads to prohibitive computational cost scaling characteristics for large N. The KWIK algorithm for Coulomb interactions begins by optimally separating the Coulomb operator into rapidly decaying real- and Fourier- space partitions yielding a hybrid technique not dissimilar in concept to other approximations methods. KWIK's superiority lies is that its efficiency increases with distribution size, so that large distributions become computationally advantageous for increasing accuracy. Model calculations on a distribution consisting of one million particles using KWIK afforded energies, to high accuracy, within minutes compared with days for quadratic methods. The extension of such a feat to even larger distributions is now limited by machine hardware configurations. Particular emphasis is placed on the application of the algorithm to Molecular Quantum Mechanics where it is illustrated that the algorithm may be applied to linearise single-point self consistent field calculations. In particular, KWIK can be used to form the Exchange matrix in linear computational cost. This has previously only been achieved by crude approximation techniques and cannot be achieved using Coulomb multipole based methods.