96 Density functional theory
which we have already encountered at the end ofSection 4.5.1. The value for the
constant is found as−( 3 / 4 )( 3 /π )^1 /^3.
For open-shell systems the spin-up and -down densitiesn+andn−are usu-
ally taken into account as two independent densities in the exchange correlation
energy according to a natural extension of the DFT formalism [4]. In local dens-
ity approximation (now called local spin density approximation), the exchange is
given as
Ex[n+,n−]=−Const.∫
d^3 r[n^4 +/^3 (r)+n^4 −/^3 (r)], (5.25)with Const.=( 3 / 2 )( 3 / 4 π)^1 /^3 in accordance with the closed-shell prefactor in
(5.24), as can be checked by puttingn+=n−=n/2. As is to be expected for
an exchange coupling, this expression contains interactions between parallel spin
pairs only.
In addition to exchange, there is a contribution from the dynamic correlation
effects (due to the Coulomb interaction between the electrons) present in the
exchange correlation potential, and several local density parametrisations of this
interaction have been proposed. A successful one is a parametrised version of the
correlation energy obtained in quantum Monte Carlo simulations of the homogen-
eous electron gas at different densities [ 11 , 9]. Other parametrisations have been
presented by von Barth and Hedin [12], and Gunnarson and Lundqvist [13]. These
dynamic correlations represent couplings between both parallel and opposite spin
pairs.
In calculations of the electronic structure, the DFT–LDA approach has turned
out very successfully. In some systems, however, it leads to noticeable deviations
or even failures – for examples some stable negative ions such as H−,O−and
F−are predicted to be unstable. Many improvements on LDA have therefore been
proposed. One of these consists of including gradients of the density in the exchange
correlation functional (we will come back to this in the second part of the next
section), whose form is motivated by calculations taking many-electron effects into
account[8].
Another approach focuses on the self-interaction present in the Hartree energy
which contains Coulomb couplings between an electron and its own charge distribu-
tion. This overestimation of the electron–electron interaction should be cancelled by
the exchange correlation term, which – in LDA – succeeds only partially (although
in the hydrogen atom for example, 95% of the self-interaction is cancelled by
the exchange correlation). It is possible to add these corrections afterward to the
exchange correlation potential[9], but this introduces a dependence of the exchange
correlation on the individual orbitals,ψk, instead of a dependence on the density
only. Both the gradient-correction and self-interaction methods lead to important
improvements in calculations of physical properties[4].