The Becke Exchange Functional

The exchange energy is an order of magnitude larger than correlation energy, therefore, the 10% error of E_x^D30 is the major problem of the LSDA. One reason for this could be the incorrect asymptotic behaviour of E_x^D30.

The exchange energy density $\varepsilon_{x}({\bf r}_{1})$ may be defined as
$$\begin{align}E_{x}[\rho] &= \int \rho({\bf r}_{1}) \varepsilon_{x}({\bf r}_{1}) ... ...nt \frac{\rho_{x}({\bf r}_{1},{\bf r}_{2})}{r_{12}} \, d{\bf r}_{2}. \end{align}$$
Using equation (2.32) the following constraint for $E_{x}[\rho]$ is obtained:
$$\begin{align}\lim_{r_{1}\to\infty} \varepsilon_{x}({\bf r}_{1}) = - \frac{1}{2r_{1}}. \end{align}$$
The long range behaviour of the electron density is
$$\begin{align}\lim_{r\to\infty} \rho({\bf r}) = \exp \left[ -2 \sqrt{2I_{\mathrm{min}}} r \right]. \end{align}$$
where I_min is the exact first ionization potential [101]. Therefore the LSDA $\varepsilon_{x}$ will have the asymptotic form
$$\begin{align}\lim_{r\to\infty} \varepsilon_{x}^{LSDA}({\bf r}_{1}) = \exp \left[ -\frac{2}{3} \sqrt{2I_{\mathrm{min}}} r \right] \end{align}$$

In 1988 Becke [102] introduced a correction to the Dirac exchange functional which gives the exchange energy density the correct asymptotic behaviour. The functional form is
$$\begin{align}E_{x}^{B88}[\rho_{\sigma}] = E_{x}^{D30}[\rho_{\sigma}] - b \int \r... ...{x_{\sigma}^{2}}{1+6 b x_{\sigma} \sinh^{-1} x_{\sigma}}\, d{\bf r}, \end{align}$$
with the parameter b=0.0042 determined by fitting the exchange energies of the first six noble gas atoms. One deficiency of Becke's functional is that the potential decays asymptotically as [103,104]
$$\begin{align}\lim_{r\to\infty} v_{x}^{B88}({\bf r}) = \frac{1}{r^{2}}. \end{align}$$
instead of the correct [105,104]
$$\begin{align}\lim_{r\to\infty} v_{x}({\bf r}) = \frac{1}{r} \end{align}$$

Despite this drawback, E_x^B88 is an extremely accurate density functional. For predicting atomic exchange energies it is 1-2 orders of magnitude better than the Sham-Kleinman functional [106].