Linear scaling methods applications

The present review has been very selective, stressing the rationale behind density-functional methods above their applications and excluding many important topics (both theoretical and computational). The interested reader may refer to anyone of the many books [91-93] or review articles [94-101] on density-functional theory for more details. Of special importance is the extension of density-functional theory to time-dependent external potentials [102-105], as this enables the dynamical behavior of molecules, including electronic excitation, to be addressed in the context of DFT [106-108]. As they are particularly relevant to the present discussion, we cite several articles related to the formal foundations of density-functional theory [85,100,109-111], linear-scaling methods [63,112-116], exchange-correlation energy functionals [25, 117-122], and qualitative tools for describing chemical reactions [123-126,126-132]. 

Finally, we mention an approach that treats the whole (or a large fraction) of a protein by QM using linear-scaling methods. The ability to treat whole molecules the size of proteins using quantum chemical methods is a significant achievement and may obviate the need for the MM region. Thus, for a moderately sized solute molecule, hundreds of explicit water molecules could easily be treated quantum mechanically. However, at present these models have only limited application [108], as they remain too costly for the purpose of MD simulation and are restricted to semiempirical QM methods which are prone to unpredictable errors. 

Some recent developments concerning macromolecular quantum chemistry, especially the first linear-scaling method applied successfully for the ab initio quality quantum-chemistry computation of the electron density of proteins, have underlined the importance and the applicability of quantum chemistry-based approaches to molecular similarity. These methods, the linear-scaling numerical Molecular Electron Density Lego Approach (MEDLA) method [6 9] and the more advanced and more generally applicable linear-scaling macromolecular density matrix method called Adjustable Density Matrix Assembler or ADMA method [10,11], have been employed for the calculation of ab initio quality protein electron densities and other... 

Traditional wavefunction and density functional theories are applicable to large systems, including nanomaterials however, their inqilementation often involves different algorithms. These include the various linear scaling methods, hybrid (often referred to as QM/MM) mediods, and sparse matrix mediods. 

The applications we have chosen to present here are just few examples of what these methods are able to tackle. However, one has to bear in mind that one can only study systems in which the electronic phenomenon is locahzed in a small portion of the total space. This is perhaps the area where linear scaling methods will find their most exciting applications. 

Basis Sets Correlation Consistent Sets Configuration Interaction Coupled-cluster Theory Density Functional Applications Density Functional Theory Applications to Transition Metal Problems G2 Theory Integrals of Electron Repulsion Integrals Overlap Linear Scaling Methods for Electronic Structure Calculations Localized MO SCF Methods Mpller-Plesset Perturbation Theory Monte Carlo Quantum Methods for Electronic Structure Numerical Hartree-Fock Methods for Molecules Pseudospectral Methods in Ab Initio Quantum Chemistry Self-consistent Reaction Field Methods Symmetry in Hartree-Fock Theory. 

Linear-scaling methods for electronic structure calculations are still in the developing stage, with their practical applications emerging. The development so far is encouraging. For quantum mechanical description of large systems, linear-scaling methods are the best possible approach. 

Much work has been done by many scientists over the last decades to bring quantum chemistry to the impressive stage it is today. Thinking back to just a bit more than 15 years ago, computing a non-symmetric molecule with, say, 10-20 atoms at the Hartree-Fock level was painful. Today molecules with more than 1000 atoms can be tackled at the HF or DFT level on one-processor computers, and widespread applicability to a multitude of chemical and biochemical problems has been achieved. Although advances in quantum chemistry certainly go hand in hand with the fast-evolving increase of computer speed, it is clear that the introduction of linear-scaling methods over the last ten years, or so, has made important contributions to this success. 

This section describes the main methodological advances that will be used in subsequent selected applications, including (1) Development of fast semiempirical methods for multiscale quantum simulations, (2) Directions for development of next-generation QM/MM models, and (3) Linear-scaling electrostatic and generalized solvent boundary methods. 

As discussed, an alternative to direct diagonalization is by recursion. The recursive diagonalization approach has several attractive features, including more favorable scaling laws, which make it ideally suited for large eigenproblems. For example, some applications of linear-scaling recursive methods in... 

If affordable, there is a range of very accurate coupled-cluster and symmetry-adapted perturbation theories available which can approach spectroscopic accuracy [57, 200, 201]. However, these are only applicable to the smallest alcohol cluster systems using currently available computational resources. Near-linear scaling algorithms [192] and explicit correlation methods [57] promise to extend the applicability range considerably. Furthermore, benchmark results for small systems can guide both experimentalists and theoreticians in the characterization of larger molecular assemblies. 

Like SNV, this pretreatment method [1,21] is a sample-wise scaling method, which has been effectively used in many spectroscopic applications where multiplicative variations are present. However, unlike SNV, the MSC correction parameters are not the mean and standard deviation of the variables in the spectrum x, but rather the result of a linear lit of a reference spectrum x f to the spectrum. The MSC model is given by the following equation ... 

In this chapter, we would like to demonstrate that one of the very old MVA tools, nonmetric multidimensional scaling (nMDS) [3], can work well as an unsupervised truly data-driven method for data reduction. We first explain an efficient maximally nonmetric algorithm [4] and then demonstrate its superiority to linear MVA methods. We also demonstrate that the subsequent application of linear MVA after data reduction by nMDS can often be a powerful data mining technique. 

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