Kohn equations

Kohn and Sham rewrote the Hohenberg-Kohn equation (15.109) as follows. Let AT be defined by... 

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... 

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

Kohn W and Rostoker N 1954 Soiution of the Sohrddinger equation in periodio iattioes with an appiioation to metaiiio iithium Phys. Rev. 94 1111-20... 

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 ... 

Equation (3.74) is the exact exchange energy (obtained from the Slater determinant the Kohn-Sham orbitals), is the exchange energy under the local spin densit) ... 

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... 

Kohn W and L J Sham 1965. Self-consistent Equations Including Exchange and Correlation Effects. Physical Review A140 1133-1138. 

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... 

Self-Consistent Equations Including Exchange and Correlation Effects W. Kohn and L. J. Sham Physical Review 140 (1965) All33... 

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 LSDA approach requires simultaneous self-consistent solutions of the Schrbdinger and Poisson equations. This was accomplished using the Layer Korringa-Kohn-Rostoker technique which has many useful features for calculations of properties of layered systems. It is, for example, one of only a few electronic structure techniques that can treat non-periodic infinite systems. It also has the virtue that the computational time required for a calculation scales linearly with the number of different layers, not as the third power as most other techniques. 

W.Kohn and Rostoker, Solution of the Schrodinger equation in periodic lattices with an apphcation to metaUic hthium , Phys.Rev.94 1111 (1954). 

---