Fast Multipole Moment method

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. 

The Fast Multipole Moment (FMM) method (Section 16.2.1) was originally developed for calculating interactions between point charges. A direct calculation Tn fr. ves i-5rnTnn ii inn over. i11 pairs, i.e. n computational effort which increases as -Tlie. itlea in F TM is to split the total interaction into near- and far-fields. The near-field... 

It is possible to treat the large molecular systems that are compact and three-dimensional with high-quality basis sets in contrast to the fast multipole moment (FMM) method. 

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. ... 

The Onsager model describes the system as a molecule with a multipole moment inside of a spherical cavity surrounded by a continuum dielectric. In some programs, only a dipole moment is used so the calculation fails for molecules with a zero dipole moment. Results with the Onsager model and HF calculations are usually qualitatively correct. The accuracy increases significantly with the use of MP2 or hybrid DFT functionals. This is not the most accurate method available, but it is stable and fast. This makes the Onsager model a viable alternative when PCM calculations fail. 

In this case, and perhaps for all robust fits, if the fit is robust then its LCAO coefficients can be determined by variation of the energy. In that case the fit is said to be variational. Quantum chemists are beginning to use variational fits, but they do not yet include robust energies, in a method that they call resolution of the identity [14,15]. Equation (6), with pab replaced by PlM where L and M are the usual multipole-moment quantum numbers, can also be used to remove the first order error from fast-multipole methods [16]. 

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