Order-N Methods

The CPA is supposed to be a simple and inexpensive way to calculate the electronic structure of alloys, and it is not consistent with the philosophy to use a massive order-N method like the LSMS to generate potentials for it. At the present time, no alternative method has been conclusively demonstrated to produce such good potentials. A method that has been suggested very recently seems very promising.It provides a method for calculating both Uailoy and Uc. 

Finally, Maximally localized orbitals have recently attracted a lot of attention, as they may be a route towards order N methods for electronic structure calculations. 

Finally, can we dare to ask what is the future of first-principle MD It would be hard to be highly predictive. However we would like to quote the following directions of research QM/MM methods to treat quantum systems in an environment [92-94,225,226,269-272], Gaussian basis sets [23,30,38, 63,110,172] or Gaussian augmented plane waves methods [168] in search for order N methods [273,274] etc. Also, in order to go beyond Density Functional Theory, Quantum-Monte Carlo techniques are very attractive [119]. Some of these topics are already well-advanced and are discussed here in this book. 

The method is presently under development, but the possible fields of applications are many and diverse, including order-N methods and alloy ordering problems outside of the quantum corrals. 

As expected, the performance of all competing order-N methods depends on the system under investigation, the accuracy needed, the amount of experimental information available, the questions that need to be answered, and also computer-related parameters (processors, parallel architectures, etc.). All approaches have their pluses and minuses. It is also clear that the increased mathematical effort will "pay off" if at all) only beyond a critical number of atoms (around 100-1000 or so) below that, the normal route with cubic scaling is faster. Nonetheless, the same locality arguments may be used to derive linear-scaling methods for the extremely efficient calculation of electronic correlation (see Section 2.13). 

The standard SCF iteration framework is used in CheFSI, and a self-consistent solution is obtained as with previous work, which means that CheFSI has the same accuracy as other standard DFT approaches. Unlike, some so-called order-N methods [16,17] CheFSI is equally applicable to metals and insulators. 

In the present section we shall briefly outline the basic ideas behind the order-N methods as well as some related methods, i.e. the divide-and-conquer method and the elongation method. For more details the reader is referred to refs. 28, 29 and references therein. 

Considering these two drawbacks, the use of Density Functional Theory (DFT) within its Kohn-Sham (KS) formulation is very appealing First, because the cost of a KS-DFT calculation is at most of the same order of magnitude as a Hartree-Fock one, whereas most of the description of electron correlation is taken into account, it is substantially less expensive than traditional correlated techniques. Moreover, most order n methods which are presently under development are all built on the DFT framework. Indeed, in a number of systematic validation studies , DFT has been shown successfully predicting various molecular properties, often giving results of quality comparable to or better than second order Mpller-Plesset (MP2) perturbation theory. It appears more and more evident that the use of DFT techniques could lead to reliable theoretical evaluation of ionization potentials. Moreover, the obtention of new exchange-correlation (XC) functionals is still under development, and one can expect that ionization potentials will be calculated more accurately in a close future. 

W. HierseandE. Stechel, Phys. Rev. B, 50,17811 (1994). Order-N Methods in Self-consistent... 

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