Coulomb linear scaling methods

The Fade function has a cusp at r = 0 that can be adjusted to match the Coulomb cusp conditions by adjusting the a parameter. The Sun form also has a cusp, but approaches its asymptotic value far more quickly than the Fade function, which is useful for the linear scaling methods. An exponential form proposed by Manten and Luchow is similar to the Sun form, but shifted by a constant. By itself, the shift affects only the normalization of the Slater-Jastrow function, but has other consequences when the function is used to construct more elaborate correlation functions. The polynomial Fade function does not have a cusp, but its value goes to zero at a finite distance. 

N. Linear scaling would thus enable one to examine much larger heterogeneous catalytic systems as well as biocatalytic systems. One of the inherent difficulties in developing linear scaling methods resides in the Coulomb electron-electron repulsion integrals shown in Elq. (A38) which formally scale as N. ... 

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). 

Often, the bottleneck in linear-scaling density-functional theory is the evaluation of the Coulomb potential the trade olf between the simple and direct method of integrating Eq. (94) and the more sophisticated linear-scaling approaches is evidenced by the fact that, for moderately large systems, linear-scaling density-functional techniques are often less efficient than direct solution to the Kohn-Sham system. As the size of the system increases beyond 10 to 20 A, however, linear-scaling techniques become essential. 

The work of Schutz, Hetzer and Wemer " begins a series of papers exploring the development of local electron correlation methods with low-order scaling by presenting a linear scaling local MP2 . They describe a novel multipole approximation based on a sphtting of the Coulomb operator into two terms... 

Low-order scaling local correlation methods II Splitting the Coulomb operator in linear scaling local second-order Moller-Plesset perturbation theory ... 

The quantum-chemical tree code (QCTC) [M. Challacombe and E. Schwegler, J. Chem. Phys., 106, 5526 (1997)] is a modification of the classical tree-code method. The QCTC method allows calculation of the matrix elements of the Coulomb matrix J for large molecules in a time that is proportional to the number of basis functions b this calculation is 0 b), and one says that the calculation exhibits linear scaling with size of the molecule. Challacombe and Schwegler used the QCTC method to do an ab initio SCE MO calculation on the 698-atom monomer of the P53 protein at a fixed geometry (obtained from a protein data bank) using the 3-21G basis set (3836 basis functions). They then calculated the molecular electrostatic potential (Section 15.7) of the P53 monomer. (The P53 protein is a tetramer and acts as a tumor suppressor. Mutations in the gene for this protein are found in half of human cancers.)... 

Order Scaling Local Correlation Methods If Sphtting the Coulomb Operator in Linear Scaling Local Second-Order MoUer-Plesset Perturbation Theory. 

In the present section, we shall discuss the application of the multipole method to the evaluation of Coulomb interactions in large systems, with special emphasis on aspects related to the linear scaling of the evaluation with the size of the system [25,28,29]. To simplify the presentation, we shall in Sections 9.14.1-9.14.3 assume that the system consists of a set of point charges contained in an equilateral cubic box. The modifications needed to treat Gaussian charge distributions in a molecular system are considered in Section 9.14.4. 

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