Gradient Corrections to Exchange Energies within Density Functional Theory

Sammanfattning: The small-wavector expansions for the first- and second-order exchange energy kernels (second-and third-order functional density derivatives of the exchange energy) have been investigated. Expansion coefficients are found, both in the case of a statically screened Coulomb interaction and a 'bare' interaction. The ensuing series of gradient corrections to the local density approximation (LDA) is then given, with particular emphasis on the second-order and fourth-degree term. Exactly calculated exchange energies for model solids are compared to a truncated series of gradient terms. It is found that the first-principles gradient series can improve the LDA description of the exchange energy by two orders of magnitude for metallic s-p bonded systems. In the case of semi-conductors the results are less accurate. A scheme for obtaining exact Density-Functional orbitals and eigenvalues is tested on the Neon atom. These exact properties are compared to approximate ones within two so-called Generalized Gradient Approximation schemes. The schemes are found to give an accurate total energy, whereas the exchange-correlation potentials are of similar quality as within the LDA. A method for obtaining total energies of solids from third-order perturbation theory is tried out in the case of impurities in the electron gas. The method turns out to work well for singly charged impurities.