Very fast multipole methods

Petersen H G, D Soelvaso, J W Perram and E R Smith 1994 The Very Fast Multipole Method. Journal of Chemical Physics 101.8870-8876... 

Pavese A, Catti M, Price GD, Jackson RA (1992) Interatomic potentials for the CaCOs polymorphs (calcite and aragonite) fitted to elastic and vibrational data. Phys Chem Miner 19 80-87 Perram JW, Petersen HG, Leenw SW de (1988) An algorithm for the simulation of condensed matter which grows as the 3/2 power of the nurnber of particles. Mol Phys 65 875-893 Petersen HG, Soelvason D, Perram JW, Smith ER (1994) The very fast multipole method. J Chem Phys 101 8870-8876... 

Very fast multipole methods have been developed in order to calculate these electron repulsion integralsl . The near field is determined by analytical Gaussian calculations. The far field is calculated usiug multipole expansions to treat the distant charges and their interactions. The scahng for this approach has been reduced to Fast quadrature... 

Calculation of the Coulomb matrix element J s in (15.79) involves not point charges (as in the FMM method) but continuous distributions of charge defined by the basis functions. Therefore, quantum chemists modified the FMM method to deal with interactions involving continuous charge distributions. One such modification for rapid evaluation of the Coulomb matrix elements for large molecules is the continuous fast multipole method (CFMM) [C. A. White et al., Chem. Phys. Lett., 253,268 (1996)]. Another is the Gaussian very fast multipole method (GvFMM) [M. C. Strain, G. E. Scuseria, and M. J. Frisch, Science, 271, 51 (1996)]. 

The original FMM has been refined by adjusting the accuracy of the multipole expansion as a function of the distance between boxes, producing the very Fast Multipole Moment (vFMM) method. Both of these have been generalized tc continuous charge distributions, as is required for calculating the Coulomb interactioi between electrons in a quantum description. The use of FMM methods in electronic structure calculations enables the Coulomb part of the electron-electron interaction h be calculated with a computational effort which depends linearly on the number of basi functions, once the system becomes sufficiently large. 

With very, very large systems, fast-multipole methods analogous to those described in Section 2.4.2 can be used to reduce the scaling of Coulomb integral evaluation to linear... 

The original FMM has been refined by also adjusting the accuracy of the multipole expansion as a function of the distance between boxes, producing the very Fast Multipole Moment (vFMM) method. ... 

Starting from this convergence factor approach along the lines of the Lekner sum, R. Strebel and R. Sperb constructed a method of computational order O (N log N), MMM [54]. The favorable scaling is obtained, very much as in the Ewald case, by technical tricks in the calculation of the far formula. The far formula has a product decomposition and can be evaluated hierarchically, similar in spirit to the fast multipole methods. 

Recently there has been increasing interest in many techniques which achieve linear or MogN scaling for the evaluation of the electrostatic contributions, such as the fast multipole method (Petersen et al. 1994) and particle mesh approaches (Essmann et al. 1995). These methods are clearly beneficial for very large systems, but have a larger prefactor and there is some debate as to where the crossover point with the Ewald sum occurs. The best estimates indicate that this happens at close to 10,000 ions. Since we are currently largely concerned with crystalline materials, most systems to be studied will be considerably smaller than this and so the Ewald technique represents the most efficient solution. However, in large-scale molecular dynamics other approaches will often be the method of choice. 

The second DFT LCAO linear-scaling method by Scuseria and Kudin (SK method) [379] uses Gaussian atomic orbitals and a fast multipole method, which achieves not only linear-scaling with system size, but also very high accuracy in aU infinite summations [397]. This approach allows both all-electron and pseudopotential calculations and can be applied also with hybrid HF-DFT exchange-correlation functionals. 

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