Kohn—Sham theory

The foundation for the use of DFT methods in computational chemistry is the introduction of orbitals, as suggested by Kohn and Sham (KS). The main flaw in orbital-free models is the poor representation of the kinetic energy, and the idea in the KS formalism is to split the kinetic energy functional into two parts, one which can be calculated exactly, and a small correction term. The price to be paid is that orbitals are re-introduced, thereby increasing the complexity from 3 to 3N variables, and that electron correlation re-emerges as a separate term. The KS model is closely related to the HF method, sharing identical formulas for the kinetic, electron-nuclear and Coulomb electron-electron energies. 

The division of the electron kinetic energy into two parts, with the major contribution being equivalent to the HF kinetic energy, can be justified as follows. Assume for the moment a Hamiltonian operator of the form in eq. (6.5) with 0 A 1. 

The external potential operator Vext is equal to Vne for A = 1, but for intermediate A values it is assumed that Vext(A) is adjusted such that the same density is obtained for A = 1 (the real system), for A = 0 (a hypothetical system with non-interacting electrons) and for all intermediates A values. For the A = 0 case, the electrons are noninteracting, and the exact solution to the Schrodinger equation is given as a Slater determinant composed of (molecular) orbitals, and the exact kinetic energy functional is given in eq. (6.6). 

The subscript S denotes that it is the kinetic energy calculated from a Slater determinant. The A = 1 case corresponds to interacting electrons, and eq. (6.6) is therefore only an approximation to the real kinetic energy, but a substantial improvement over the TF formula (eq. (6.2)). 

Another way of justifying the use of eq. (6.6) for calculating the kinetic energy is by reference to natural orbitals (eigenvectors of the density matrix. Section 9.5). The exact kinetic energy can be calculated from the natural orbitals (NO) arising from the exact density matrix. 

To find the ground-state energy corresponding to the energy functional in eq.(l), the following KS equation needs to be solved self-consistently with eqs. (5), (6), and (8). 

The expression in the bracket of eq.(7) is the KS Hamiltonian. No spin-polarization is considered in eqs (1) - (8), but extension to spin-polarized case is trivial. It is possible to expand pt( r ) as a linear combination of atomic orbitals (LCAO) 

To get the coefficients in eq. (9), two N N matrices (the Hamiltonian matrix H and the overlap matrix S) have to be constructed and diagonalized here N is the number of basis functions. To construct the density from eqs. (6) and (9), the summations have to run all over the atoms in the molecule for each molecular orbital. This is too computationally demanding and can be avoided as will be shown later in this section and section 7. 

Two years ago Yang developed a divide-and-conquer strategy to do the DFT computations [43], By projecting the Hamiltonian into subspaces, he was able to divide a large system into smaller subsystems, and solve a KS-like equation for each subsystem. Recently, Zhou has provided an alternative construction of the divide-and-conquer method by projecting the solutions of KS equations with different basis sets associated with each subsystem [44]. The divide-and-conquer method is shown to be as rigorous as the conventional KS method. As far as electron density and energy density are concerned the divide-and-conquer method is different from the conventional KS method only by the way in which the basis sets are truncated. The rest of this section will review the divide-and-conquer method and try to provide some reasons why the method should work. 

Three known facts are essentially important in the development of a divide-and-conquer strategy. First, the KS Hamiltonian is a single particle operator that depends only on the total density, not on individual orbitals. This enables one to project the energy density in real space in the same manner in which one projects the density (see below). Second, any complete basis set can solve the KS equation exactly no matter where the centers of the basis functions are. Thus, one has the freedom to select the centers. It is well known that for a finite basis set the basis functions can be tailored to better represent wavefunctions, and thus the density, of a particular region. The inclusion of basis functions at the midpoint of a chemical bond is the best known example. Finally, the atomic centered basis functions used in almost all quantum chemistry computations decay exponentially. Hence both the density and the energy density contributed by atomic centered basis functions also decrease rapidly. All these 

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Bartolotti LJ (1987) Absolute Electronegatives as Determined from Kohn-Sham Theory. 66 27-40... 

Gazquez JL, Vela A, Galvan M (1987) Fukui Function, Electronegativity and Hardness in the Kohn-Sham Theory. 66 79-98... 

Towards Linear Scaling Kohn-Sham Theory... 

Handy, N. C., Tozer, D. J., 1999, Excitation Energies of Benzene from Kohn-Sham Theory , J. Comput. Chem., 20, 106. 

Harris, J., 1984, Adiabatic-Connection Approach to Kohn-Sham Theory , Phys. Rev. A, 29, 1648. 

Murray, C. W., Handy, N. C., Amos, R. D., 1993, A Study of 03, S3, CH2, and Be2 Using Kohn-Sham Theory With Accurate Quadrature and Large Basis Sets , J. Chem. Phys., 98, 7145. 

The Kohn-Sham theory made a dramatic impact in the field of ab initio molecular dynamics. In the 1985, Car and Parrinello38 introduced a new formalism to study dynamics of molecular systems in which the total energy functional defined as in the Kohn-Sham formalism proved to be instrumental for practical applications. In the Car-Parrinello method (CP), the equations of motion are based on a Lagrangian (Lcp) which includes fictitious degrees of freedom associated with the electronic state. It is defined as ... 

In this short review, a brief overview of the underlying principles of TDDFT has been presented. The formal aspects for TDDFT in the presence of scalar potentials with periodic time dependence as well as TD electric and magnetic fields with arbitrary time dependence are discussed. This formalism is suitable for treatment of interaction with radiation in atomic and molecular systems. The Kohn-Sham-like TD equations are derived, and it is shown that the basic picture of the original Kohn-Sham theory in terms of a fictitious system of noninteracting particles is retained and a suitable expression for the effective potential is derived. 

J Exchange-Correlation Potential of Kohn-Sham Theory A Physical Perspective... 

Galvan, M. and Vargas, R. 1992. Spin potential in Kohn-Sham theory. J. Phys. Chem. 96 1625-1630. 

The universal functional in Kohn-Sham theory is chosen to be... 

The basic concepts of the one-electron Kohn-Sham theory have been presented and the structure, properties and approximations of the Kohn-Sham exchange-correlation potential have been overviewed. The discussion has been focused on the most recent developments in the theory, such as the construction of from the correlated densities, the methods to obtain total energy and energy differences from the potential, and the orbital dependent approximations to v. The recent achievements in analysis of the atomic shell and molecular bond midpoint structure of have been... 

The analysis of the asymptotic density profile of HOMO s in Kohn-Sham theory [29] allows one to make the identification [30] ... 

ROKS Restricted open-shell Kohn-Sham theory... 

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