Equation Kohn-Sham

Once a solution of the Kohn-Sham equation is obtained, the total energy can be computed from... 

The wavevector is a good quantum number e.g., the orbitals of the Kohn-Sham equations [21] can be rigorously labelled by k and spin. In tln-ee dimensions, four quantum numbers are required to characterize an eigenstate. In spherically syimnetric atoms, the numbers correspond to n, /, m., s, the principal, angular momentum, azimuthal and spin quantum numbers, respectively. Bloch s theorem states that the equivalent... 

By introducing this expression for the electron density and applying the appropriate variational condition the following one-electron Kohn-Sham equations result ... 

To. solve the Kohn-Sham equations a self-consistent approach is taken. An initial guess of the density is fed into Equation (3.47) from which a set of orbitals can be derived, leading to an improved value for the density, which is then used in the second iteration, and so on until convergence is achieved. 

The total electron density is just the sum of the densities for the two types of electron. The exchange-correlation functional is typically different for the two cases, leading to a set of spin-polarised Kohn-Sham equations ... 

In addition to the energy terms for the exchange-correlation contribution (which enables the total energy to be determined) it is necessary to have corresponding terms for the potential, Vxc[p(i )]/ which are used to solve the Kohn-Sham equations. These are obtained as the appropriate first derivatives using Equation (3.52). 

To solve the Kohn-Sham equations a number of different approaches and strategies have been proposed. One important way in which these can differ is in the choice of basis set for expanding the Kohn-Sham orbitals. In most (but not all) DPT programs for calculating the properties of molecular systems (rather than for solid-state materials) the Kohn-Sham orbitals are expressed as a linear combination of atomic-centred basis functions ... 

The application of density functional theory to isolated, organic molecules is still in relative infancy compared with the use of Hartree-Fock methods. There continues to be a steady stream of publications designed to assess the performance of the various approaches to DFT. As we have discussed there is a plethora of ways in which density functional theory can be implemented with different functional forms for the basis set (Gaussians, Slater type orbitals, or numerical), different expressions for the exchange and correlation contributions within the local density approximation, different expressions for the gradient corrections and different ways to solve the Kohn-Sham equations to achieve self-consistency. This contrasts with the situation for Hartree-Fock calculations, wlrich mostly use one of a series of tried and tested Gaussian basis sets and where there is a substantial body of literature to help choose the most appropriate method for incorporating post-Hartree-Fock methods, should that be desired. 

Kohn-Sham equations of the density functional theory then take on the following... 

Density Eunctional Methods. The Kohn-Sham equations are... 

In this equation Exc is the exchange correlation functional [46], is the partial charge of an atom in the classical region, Z, is the nuclear charge of an atom in the quantum region, is the distance between an electron and quantum atom q, r, is the distance between an electron and a classical atom c is the distance between two quantum nuclei, and r is the coordinate of a second electron. Once the Kohn-Sham equations have been solved, the various energy terms of the DF-MM method are evaluated as... 

The Kohn-Sham equations look like standard HF equations, except that the exchange term is replaced with an exchange-correlation potential whose form is unknown. 

As mentioned above, a KS-LCAO calculation adds one additional step to each iteration of a standard HF-LCAO calculation a quadrature to calculate the exchange and correlation functionals. The accuracy of such calculations therefore depends on the number of grid points used, and this has a memory resource implication. The Kohn-Sham equations are very similar to the HF-LCAO ones and most cases converge readily. 

We may again chose a unitary transfonnation which makes tlie matrix of the Lagrange multiplier diagonal, producing a set of canonical Kohn-Sham (KS) orbitals. The resulting pseudo-eigenvalue equations are known as the Kohn-Sham equations. 

The reader should note that no restrictions were placed on the form of the density expansion Eq. (3.26) in particular there is no limit on the number of terms. As already noted, therefore Eqs. (3.29) are not conventional Kohn-Sham equations. Rather they are an exact one-particle form of the Hohenberg-Kohn variation procedure and use Hohenberg-Kohn potentials in the definition of the... 

Note that in all current implementations of TDDFT the so-called adiabatic approximation is employed. Here, the time-dependent exchange-correlation potential that occurs in the corresponding time-dependent Kohn-Sham equations and which is rigorously defined as the functional derivative of the exchange-correlation action Axc[p] with respect to the time-dependent electron-density is approximated as the functional derivative of the standard, time-independent Exc with respect to the charge density at time t, i. e.,... 

Bauernschmitt, R., Ahlrichs, R., 1996a, Stability Analysis for Solutions of the Closed Shell Kohn-Sham Equation , J. Chem. Phys., 104, 9047. 

Morrison, R. C., Zhao, Q., 1995, Solution to the Kohn-Sham Equations Using Reference Densities from Accurate, Correlated Wave Functions for the Neutral Atoms Hehum Through Argon , Phys. Rev. A, 51, 1980. 

Nagy, A., 1998b, Kohn-Sham Equations for Multiplets , Phys. Rev. A, 57, 1672. 

Scheiner, A. C., Baker, J., Andzelm, J. W., 1997, Molecular Energies and Properties from Density Functional Theory Exploring Basis Set Dependence of Kohn-Sham Equation Using Several Density Functionals , J. Comput. 

The Kohn-Sham equations have been applied to study gas-phase properties of many systems of biological interest. Most of such studies have been made for relatively small mol-... 

Since surface charges depend on the electrostatic potential (Eq. 4.20), Eqs. 4.20-4.22 are solved in an iterative way leading to self-consistent surface charges. At the end of this procedure, surface charges and the electrostatic potential satisfy the boundary condition specified in Eq. 4.21. In practical applications, this self-consistent procedure for calculating reaction field potential is coupled to self-consistent procedure which governs solving the Kohn-Sham equations. A special case for infinite dielectric constant outside the cavity... 

Although the Kohn-Sham equations form the quantum mechanical core of the DFT/SCRF methods, the final energetical results obtained by these methods also depend on other features of a particular DFT/SCRF implementation. It is important, therefore, to stress that the DFT/SCRF represents a whole family of methods. Each particular implementation may differ in the following details concerning ... 

Wesolowski, T. A. and J. Weber. 1996. Kohn-Sham equations with constrained electron density an iterative evaluation of the ground-state electron density of interacting molecules. Chem. Phys. Lett. 248,71. 

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