Fast-multipole methods

This applies to explicit charge-charge interactions, while the error in multipoles remains controlled by the order at which the multipole expansions are truncated. The explicit pairwise interactions between Gaussian charges in the leaf boxes are still computed explicitly using the erf(f,yr,y)/r,y kernel. 

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White C A, Johnson B G, Gill P M W and Head-Gordon M 1994 The fast multipole method Chem. Phys. Lett. 230 8-16 White C A, Johnson B G, Gill P M W and Head-Gordon M 1996 Linear sealing density funetional ealeulations via the eontinuous fast multipole method Chem. Phys. Lett. 253 268-78... 

R. Zhou and B. J. Berne. A new molecular dynamics method combining the reference system propagator algorithm with a fast multipole method for simulating proteins and other complex systems. J. Phys. Chem., 103 9444-9459, 1995. 

To separate the non-bonded forces into near, medium, and far zones, pair distance separations are used for the van der Waals forces, and box separations are used for the electrostatic forces in the Fast Multipole Method,[24] since the box separation is a more convenient breakup in the Fast Multipole Method (FMM). Using these subdivisions of the force, the propagator can be factorized according to the different intrinsic time scales of the various components of the force. This approach can be used for other complex systems involving long range forces. 

Zhou R and B J Berne 1995. A New Molecular Dynamics Method Combining the Reference Sys Propagator Algorithm with a Fast Multipole Method for Simulating Proteins and Ol Complex Systems. Journal of Chemical Physics 103 9444-9459. 

One recent development in DFT is the advent of linear scaling algorithms. These algorithms replace the Coulomb terms for distant regions of the molecule with multipole expansions. This results in a method with a time complexity of N for sufficiently large molecules. The most common linear scaling techniques are the fast multipole method (FMM) and the continuous fast multipole method (CFMM). 

One of the major selling points of Q-Chem is its use of a continuous fast multipole method (CFMM) for linear scaling DFT calculations. Our tests comparing Gaussian FMM and Q-Chem CFMM indicated some calculations where Gaussian used less CPU time by as much as 6% and other cases where Q-Chem ran faster by as much as 43%. Q-Chem also required more memory to run. Both direct and semidirect integral evaluation routines are available in Q-Chem. 

Cartesian coordinates system for locating points in space based on three coordinates, which are usually given the symbols x, y, z or i, j, k CBS (complete basis set) an ah initio method CC (coupled cluster) a correlated ah initio method CFF (consistent force field) a class of molecular mechanics force fields CFMM (continuous fast multipole method) a method for fast DFT calculations on large molecules... 

The first simulation studies of full double layers with molecular models of ions and solvent were performed by Philpott and coworkers [51,54,158] for the NaCl solution, using the fast multipole method for the calculation of Coulomb interactions. The authors studied the screening of a negative surface charge by free ions in several highly concentrated NaCl solutions. A combination of (9-3) LJ potential and image charges was used to describe the metal surface. 

However, unlike point charges, the continuous charge distributions that occur in quantum chemistry have varying extents and the applicability of the multipole approximation is not only limited by the distance but also by the extent or diffuseness of the charge distribution. This additional complexity makes a transfer of the concepts of the fast multipole method to applications in quantum chemistry less straightforward. Therefore it should come as no surprise that several adaptations to extend the applicability of the FMM to the Coulomb problem with continuous charge distributions have been suggested. These lead to... 

White, C. A., Johnson, B. G., Gill, P. M. W, Head-Gordon, M., 1994, The Continuous Fast Multipole Method , Chem. Phys. Lett., 230, 8. 

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... 

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