Kohn-Sham computational methodologies

There are a number of factors which contribute to the lack of consistency among current DFT programs. For example, many different basis representations of the KS orbitals are employed, including plane waves, Slater-type orbitals, numerically tabulated atomic orbitals, numerical functions generated from muffin-tin potentials, and delta functions. Gaussian basis functions, ubiquitous in the ab initio realm, were introduced into KS calculations in 1974 by Sambe and 

Felton [40], and we shall focus here only on implementations using Gaussian expansions. Then, there are primarily two general aspects which are responsible for this lack of consistency, which we shall briefly discuss. 

The purported N3 dependence of KS methods refers to procedures which reduce the integral evaluation work by fitting the computationally intensive terms in auxiliary basis sets. There are a number of different approaches which are used (and we shall not attempt to cover them all), but these are all more or less variations on a linear least-squares theme. The earliest work along these lines [21, 42], done in the context of Xa calculations, involved the replacement of the density in the Coulomb potential by a model 

It is clear that the least-squares equations for the model density require only the Coulomb integrals (pv I fj) and (fi I fj), which are 0(N3) and 0(N2) in number, respectively, and therefore the integral evaluation problem is formally reduced by one order to 0(N3). 

Second, fitted densities do not automatically conserve the number of electrons in the system. Though it is straightforward to enforce charge conservation via a Lagrange constraint, which is usually done, the fitted density 

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In actual practice, self-consistent Kohn-Sham DFT calculations are performed in an iterative manner that is analogous to an SCF computation. This simiBarity to the methodology of Hartree-Fock theory was pointed out by Kohn and Sham. 

During the past 10 - 15 years, Kohn-Sham density functional theory has been a major factor in a dramatic expansion of the scope of computational chemistry and its capability for treating systems of practical importance [45-51]. Density functional methodology includes electronic correlation, so that the energies are more accurate than Hartree-Fock however the Kohn-Sham formalism is similar to the latter, as are therefore the demands upon computer resources. It is therefore feasible to treat relatively large systems at a reasonably high (post-Hartree-Fock) level. 

For direct Af-electron variational methods, the computational effort increases so rapidly with increasing N that alternative simplified methods must be used for calculations of the electronic structure of large molecules and solids. Especially for calculations of the electronic energy levels of solids (energy-band structure), the methodology of choice is that of independent-electron models, usually in the framework of density functional theory [189, 321, 90], When restricted to local potentials, as in the local-density approximation (LDA), this is a valid variational theory for any A-electron system. It can readily be applied to heavy atoms by relativistic or semirelativistic modification of the kinetic energy operator in the orbital Kohn-Sham equations [229, 384],... 

In this review we have seen examples representing three main types of biomolecules and biomodels proteins, nucleic acids, and lipids. Although it is not always classified as a biomolecule, water represents a fourth major type and it has been center stage in many of the examples we have treated. For each of these classes of biomolecules, DFT has played a major role in our work and in that of other workers. But this is not a one-man show we have also shown how DFT can be combined with molecular dynamics, either in the Bom-Oppenheimer-MD approach or in hybrid QM/MM methodologies. And we have shown examples that go beyond strictly Kohn-Sham DFT in the use of constraints that allow coimections with other theories and concepts, notably the Marcus theory of electron transfer. Finally, we have given a glimpse of some of the tools that can be used to analyze and interpret the DFT-based computations. 

The outline of the computational setups are as follows. For more details we refer the readers to our previous paper [24]. AU the QM/MM simulations were performed with the code developed by Takahashi et al. [9-11], where the QM subsystem is described with the Kohn-Sham density functional theory utilizing the real-space grid method. The major methodological part of our grid approach was common to that provided in Sect. 6.2.1 except that the Hartree potential was constructed using FFT. For these calculations we utilized non-paraUelized version of our program. 

The last term in Eq. (2) represents the main problem of the DFT. In DFT methodologies the optimal electron density (p ) is computed following the variational principle. Kohn and Sham proposed that a real electron density can be represented by a fictitious noninteracting reference system. Electrons in the latter system do not interact, but its ground-state electron density distribution is exactly the same as corresponding to the real system under consideration. The deviation in the behavior of noninteracting electrons from that of the real ones is then taken into account by the unknown XC functional that is included in DFT methods in an approximate form. The development of such approximate functional is a very active research topic in theoretical chemistry. 

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