Integrals via Multipole Expansion

As mentioned, it is clear that the MBIE estimates require the validity of the multipole expansion for two-electron integrals, similar to the requirements for the fast multipole methods (these multipole expansions will be presented in detail in the next section). For the near-field part of the integrals, i.e., for charge distributions that are so close that the multipole expansion is not applicable, MBIE cannot be used. Here one can resort to, for example, the Schwarz bounds. 

As we have seen, the number of Coulomb integrals scales as O(M ) for sufficiently large molecules. To overcome this potential bottleneck, the naive pair-wise summation over electron-electron interactions has to be circumvented. We will see that the multipole expansion of the two-electron integrals can be used, thus allowing us to achieve an overall 0(M) scaling for calculating the Coulomb matrix. 

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In this Section I review our work on force fleld design, which is based on the transferabihty of atomic multipole moments. We adopt Stone s spherical tensor formahsm and the idea of a multi-centre multipole expansion, one site for each nucleus. However, the moments are obtained via integration over an atomic basin of the appropriate tensor times the electron density, rather than via the DMA route. Starting with the electrostatic potential I move on... 

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