Dirac Exchange

Shortly after the introduction of Thomas-Fermi theory, Dirac [16] developed an approximation for the exchange interaction based on the homogeneous electron gas. The resulting formula is simple, and is also a local functional of the density,

$$E_{\hbox{{X}}}[n({\bf r})] = - \frac{3}{4} \left(\frac{3}{{\pi}}\right)^{1/3} \int n({\bf r})^{4/3} \, d{\bf r} \, .$$ (1.15)

Relation (1.16) is usually written in terms of the exchange energy density $\varepsilon_{\hbox{{X}}}[n({\bf r})]$ as,

$$E_{\hbox{{XC}}}^{\hbox{{LDA}}}[n({\bf r})] = \int \, n({\bf r}) \, \varepsilon_{\hbox{{XC}}}[n({\bf r})] \, d{\bf r}\, ,$$ (1.16)

where $\varepsilon_{\hbox{{XC}}}[n({\bf r})]$ can be given simply in terms of the Seitz radius $r_s$,

$$\varepsilon_{\hbox{{X}}}[n({\bf r})] \; = \; - \frac{3}{4} \left(... ...9}{4 \pi^2} \right)^{1/3} \frac{1}{r_s} \; \approx \; - \frac{0.4582}{r_s} \, .$$ (1.17)

The Dirac exchange term was naturally incorporated into to Thomas-Fermi theory by simply adding (1.16) to (1.8), and including the term, $4/3 \, \varepsilon[n({\bf r})]$, in the corresponding Euler-Lagrange equation (1.15). The inclusion of local exchange did not improve the Thomas-Fermi method [6].