CFMM

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... 

Q-Chem (www.q-chem.com/), first released in 1997, is an ab initio package that allows calculations on large molecules (several hundred atoms) and can do Hartree-Fock, MP2, and density-functional calculations. It incorporates methods such as CFMM and ONX to achieve linear scaling (Section 15.5) for large molecules. 

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)]. 

Figure 4 Typical timing behavior of the quadratic Fock matrix formation versus the cubically scaling diagonalization step (small prefactor) in SCF energy calculations. The timings for a conventional Fock matrix formation, the linear-scaling CFMM/LinK schemes (as explained later in this review), and Fock matrix diagonalization for a series of DNA molecules (A-T), = 1 — 16 are depicted. Integral threshold is 10 , basis set 6-31G. ...

Figure 18 Comparison of timings for standard O(M ) (STD JKg ad) linear-scaling energy gradients for DNA fragments (HF/6-31G ) using the LinK and CFMM methods.

In generalizing the FMM to continuous distributions, we must accomplish two objectives. First, we must ensure that only classical interactions are included in the calculations. Second, to reduce as much as possible the number of classical interactions that are evaluated explicitly (i.e. as NF interactions), we must introduce a flexible system for determining the FF region. In the CFMM [21,30], these objectives are achieved by introducing, at each level, a range of NF width parameters... 

Since and are even numbers, is an integer. Having thus sorted the Gaussian distributions at the deepest level into boxes and branches based on their position and extent, we are now ready to discuss the CFMM evaluation of Coulomb interactions for continuous charge distributions. 

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