Linear scaling techniques

The advantage of using electron density is that the integrals for Coulomb repulsion need be done only over the electron density, which is a three-dimensional function, thus scaling as. Furthermore, at least some electron correlation can be included in the calculation. This results in faster calculations than FIF calculations (which scale as and computations that are a bit more accurate as well. The better DFT functionals give results with an accuracy similar to that of an MP2 calculation. 

Density functionals can be broken down into several classes. The simplest is called the Xa method. This type of calculation includes electron exchange but not correlation. It was introduced by J. C. Slater, who in attempting to make an approximation to Hartree-Fock unwittingly discovered the simplest form of DFT. The Xa method is similar in accuracy to HF and sometimes better. 

The simplest approximation to the complete problem is one based only on the electron density, called a local density approximation (LDA). For high-spin systems, this is called the local spin density approximation (LSDA). LDA calculations have been widely used for band structure calculations. Their performance is less impressive for molecular calculations, where both qualitative and quantitative errors are encountered. For example, bonds tend to be too short and too strong. In recent years, LDA, LSDA, and VWN (the Vosko, Wilks, and Nusair functional) have become synonymous in the literature. 

A more complex set of functionals utilizes the electron density and its gradient. These are called gradient-corrected methods. There are also hybrid methods that combine functionals from other methods with pieces of a Hartree-Fock calculation, usually the exchange integrals. 

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). 

DFT is generally faster than Hartree Fock for systems with more than 10 

BLYP Becke correlation functional with Lee, Yang, Parr exchange Gradient-corrected 

B3P86 Becke exchange, Perdew correlation Hybrid 

The original FMM has been refined by adjusting the accuracy of the multipole expansion as a function of the distance between boxes, producing the very Fast Multipole Moment (vFMM) method. Both of these have been generalized tc continuous charge distributions, as is required for calculating the Coulomb interactioi between electrons in a quantum description. The use of FMM methods in electronic structure calculations enables the Coulomb part of the electron-electron interaction h be calculated with a computational effort which depends linearly on the number of basi functions, once the system becomes sufficiently large. 

Although the exchange term in principle is short-ranged, and thus should benel significantly from integral screening, this is normally not observed in practic  

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Many of these fitting schemes were derived before linear scaling techniques (Section 3.8.6) were fully developed, and it is not clear whether they have any advantages. For calculation of energy derivatives, they acmally seem counterproductive, since the fitting procedures seriously complicate the computational expressions. ... 

Except for occasional discussions of the basis set dependence of the results, the numerical implementation issues such as grid integration techniques, electron-density fitting, frozen-cores, pseudopotentials, and linear-scaling techniques, are omitted. 

Linear Scaling Techniques Semi-Empirical Methods 3.9.1 Neglect of Diatomic Differential Overlap Approximation (NDDO) 3.9.2 Intermediate Neglect of Differential Overlap Approximation (INDO) 80 81 82 83 4.13 Locahzed Orbital Methods 4.14 Summary of Electron Correlation Methods 4.15 Excited States References 5 Basis Sets 144 144 147 148 150... 

Linear-scaling techniques for solving Poisson s equation include multigrid approaches [58-61] and fast Poisson solvers [62]. 

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. 

It is fair to note that the oscillator strengths of both the TDDFT and CASPT2 results still leave much room for improvement. The deficiency for the TDDFT case for this molecule is not shared by other choices for the xc functionals, and also does not show up in similar studies on metal-porphyrin spectra, where excellent agreement with experiment is usually obtained. A calculation on a system of this size can nowadays be performed in a few minutes. So-called linear scaling techniques make TDDFT calculations on molecules with hundreds of atoms feasible. 

Most parts of ADF have been efficiently parallelized. Because of the exponential spatial decay of the STO basis functions, linear scaling techniques reduce the computational complexity from 0 NI to O(Vat) for the most time-consuming parts of the calculation. " A density-fit procedure and the possibility of making a frozen core approximation " further reduce the cost of the calculations. 

Hence, in this chapter, we proceed further on our way from the fundamental theory to different representations of first-quantized relativistic quantum chemistry — now guided mostly by questions of algorithmic technique and feasibility. For the sake of compactness, the focus in this chapter must be on techniques that are specific to the relativistic realm. In nonrelativistic theory numerous approaches have been devised to reduce the computational effort of quantum chemical calculations. Apart from the just mentioned density-fitting approach, specific linear-scaling techniques have been devised [715-717] that ensure a linear increase in the computational effort with system size (measured by the atom or electron number or directly by the number of basis functions). These employ, for example, localized orbitals or sparse-matrix operations. All these techniques apply directly to the relativistic variants. 

Zalesny R, Papadopoulos MG, Mezey PG, Leszczynski J (eds) (2011) Linear-scaling techniques in computational chemistry and physics. In Challenges advances computational chemistry physics, vol 13. Springer, Dordrecht. doi 10.1007/978-90-481-2853-2... 

R, Mezey PG, Leszczynski J (eds) Linear-scaling techniques in 45. computational chemistry and physics methods and applications. 46. 

His research interests are in both theoretical developments and applications of Quantum Chemistry. The range of the theoretical developments is wide, including non-Born Oppenheimer theory, linear-scaling technique, density functional theory, ab initio molecular dynamics, and relativistic quantum chemistry. He also contributed in discovering novel chemical concepts, such as hyperconjugation of methyl group in the excited state and symmetry rules for degenerate excitations. 

Finely, we illustrate the efficiency of the hierarchical method in Figure 1 (data are taken from the original work, to which we refer the reader for the technical details). The O(N ) series corresponds to the use of equation (21) without the benefit of any of the linear-scaling techniques just described (or equivalently, threshold r = 0). Series with r = 1.0 X 10 and r = 1.0 x 10" use the hierarchical method to achieve near-linear scaling with an accuracy similar to the magnitude of r. The savings in CPU time are spectacular. 

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