Gaussian very fast multipole methods

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)]. 

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... 

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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