Self-consistency

Quantum chemical calculations require self-consistent solution, and we illustrate this here with a standard Kohn-Sham DFT calculation. We make 

A second strategy would be to update the effective potential on coarse levels during the V-cycle so that the eigenfunctions and effective potential evolve together. This approach fits nicely into the overall MG philosophy 

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This equation is usually solved self-consistently . An approximate charge is assumed to estimate the exchange-correlation potential and to detennine the Flartree potential from equation Al.3.16. These approximate potentials are inserted in the Kolm-Sham equation and the total charge density is obtained from equation A 1.3.14. The output charge density is used to construct new exchange-correlation and Flartree potentials. The process is repeated nntil the input and output charge densities or potentials are identical to within some prescribed tolerance. 

The work done increases the energy of the total system and one must now decide how to divide this energy between the field and the specimen. This separation is not measurably significant, so the division can be made arbitrarily several self-consistent systems exist. The first temi on the right-hand side of equation (A2.1.6) is obviously the work of creating the electric field, e.g. charging the plates of a condenser in tlie absence of the specimen, so it appears logical to consider the second temi as the work done on the specimen. 

Weeks J D, Selinger R L B and Broughton J Q 1995 Self consistent treatment of attractive forces in... 

This example illustrates how the Onsager theory may be applied at the macroscopic level in a self-consistent maimer. The ingredients are the averaged regression equations and the entropy. Together, these quantities pennit the calculation of the fluctuating force correlation matrix, Q. Diffusion is used here to illustrate the procedure in detail because diffiision is the simplest known case exlribiting continuous variables. 

On the ordinate, two quantities are plotted (i) the mean-field potential between the second electron and the other 1 s electron computed, via the self-consistent field (SCF) process (described later), as the interaction of... 

Flehre W J, Ditchfieid R and Popie J A 1972 Self-consistent molecular-orbital methods XII. Further extension of Gaussian-type basis sets for use in molecular orbital studies of organic molecules J. Chem. Phys. 56 2257-61 Flariharan P C and Popie J A 1973 The influence of polarization functions on molecular orbital hydrogenation energies Theoret. Chim. Acta. 28 213-22... 

Krishnan R, Binkley J S, Seeger R and Popie J A 1980 Self-consistent molecular orbital methods XX. A basis set for correlated wave functions J. Chem. Phys. 72 650-4... 

Kohn W and Sham L J 1965 Self-consistent equations including exchange and correlation effects Phys. Rev A 140 1133-8... 

Roos B O 1987 The complete active space self-consistent field method and its applications in electronic structure calculations Adv. Chem. Phys. 69 399-445... 

Olsen J and J0rgensen P 1995 Time-dependent response theory with applications to self-consistent field and multiconfigurational self-consistent field wave functions Modern Electronic Structure Theory vo 2, ed D R Yarkony (Singapore World Scientific) pp 857-990... 

B3.2.2.1 TIGHT BINDING FROM EMPIRICAL TO SELF-CONSISTENT... 

Frequent approximations made in TB teclmiques in the name of achieving a fast method are the use of a minimal basis set, the lack of a self-consistent charge density, the fitting of matrix elements of the potential. 

Most TB approaches are not charge self-consistent. This means that they do not ensure that the charge derived from the wavefiinctions yields the effective potential assumed in their calculation. Some methods have been developed which yield charge densities consistent with the electronic potential [14, H and 16]. 

The accuracy of most TB schemes is rather low, although some implementations may reach the accuracy of more advanced self-consistent LCAO methods (for examples of the latter see [18,19 and 20]). However, the advantages of TB are that it is fast, provides at least approximate electronic properties and can be used for quite large systems (e.g., thousands of atoms), unlike some of the more accurate condensed matter methods. TB results can also be used as input to detennine other properties (e.g., photoemission spectra) for which high accuracy is not essential. 

One of the first cluster embedding schemes was put forth by Ellis and co-workers [172]. They were interested in studying transition metal impurities in NiAl alloys, so they considered a TMAl cluster embedded in a periodic self-consistent crystal field appropriate for bulk p -NiAl. The field was calculated via calculations, as was the cluster itself The idea was to provide a relatively inexpensive alternative to supercell DET calculations. 

Elstner M, Porezag D, Jungnickel G, Eisner J, Flaugk M, Frauenheim Th, Suhai S and Seifert G 1998 Self-consistent-charge density-functional tight-binding method for simulations of complex materials properties Phys. Rev. B 58 7260... 

Yussouff M 1987 Fast self-consistent KKR method Electronic Band Structure and Its Applications (Lecture Notes in Physics vol 283) ed M Yussouff (Berlin Springer) pp 58-76... 

Pisani C 1978 Approach to the embedding problem in chemisorption in a self-consistent-field-molecular-orbital formalism Phys. Rev. B 17 3143... 

Ellis D E, Benesh G A and Byrom E 1978 Self-consistent embedded-cluster model for magnetic impurities p -NiAl J. Appl. Phys. 49 1543... 

Inglesfield J E and Benesh G A 1988 Surface electronic structure embedded self-consistent calculations Phys. Rev. 

Scheffler M, Droste Ch, Fleszar A, Maca F, Wachutka G and Barzel G 1991 A self-consistent surface-Green-function (SSGF) method Physica B 172 143... 

Wachutka G, Fleszar A, Maca F and Scheffler M 1992 Self-consistent Green-function method for the calculation of electronic properties of localized defects at surfaces and in the bulk J. Phys. Condens Matter A 2831 Bormet J, Neugebauer J and Scheffler M 1994 Chemical trends and bonding mechanisms for isolated adsorbates on Al(111) Phys. Rev. B 49 17 242... 

Cortona P 1991 Self-consistently determined properties of solids without band structure calculations Phys. Rev. B 44 8454... 

Cortona P 1992 Direct determination of self-consistent total energies and charge densities of solids A study of the cohesive properties of the alkali halides Phys. Rev. B 46 2008... 

Vanderbilt D 1990 Soft self-consistent pseudopotentials in a generalized eigenvalue formalism Phys. Rev. B 41 7892-5... 

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