Linear-scaling methods fast multipole method

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. 

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... 

Considering long-range MD, a variety of approximate methods have been developed to overcome the bottleneck that characterizes the forces treatment these include particle mesh algorithms, hierarchical methods, and fast multipole methods. One of the most promising developments is the cell multipole method, which scales linearly with N, requires only modest memory, and is well suited to highly parallel and vector computers. 

The effort to evaluate individual matrix element will not increase with the molecular size with this cutoff. The evaluation of all matrix elements scales linearly with the molecular size under this condition because the number of subsystems so the number of matrix elements increases linearly with the molecular size. The matrix size corresponding a subsystem does not increase with the molecular size. The Coulombic interactions can be evaluated by a fast multipole method which scales as NlnN. Direct evaluations of these terms... 

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. 

The geometry optimizations of molecular structures were performed using the semi-empirical PM3 Hamiltonian [67,68] and the B3LYP/6-3 lG(d) method. All stationary points were confirmed to be true minima by evaluation of hessian. In order to speed up HE and DFT calculations, the fast multipole method (EMM) [47, 48, 69] has been used as implemented in Gaussian suite of programs [56]. We also used linear scaling approaches for calculations of nonlinear optical properties as implemented in ADF package [58]. All the properties are expressed in atomic units. Conversion factors can be found elsewhere [28-31],... 

Korchowiec, J., Lewandowski, J., Makowski, M., Gu, E.L., Aoki, Y. Elongation cutoff technique armed with quantum fast multipole method for linear scaling. J. Comput. Chem. 30, 2515-2525 (2009)... 

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). 

White, C. A., B. G. Johnson, P. W. Gill, and M. Head-Gordon. 1996. Linear scaling density functional calculations via the continuous fast multipole method. Chem. Phys. Lett. 253, 268. 

---