Exact Definition of Exchange and Correlation

An exact definition for the exchange-correlation energy $E_{\hbox{{XC}}}[n({\bf r})]$ in DFT can be derived using a method known as adiabatic connection, and is presented in Appendix A. The basic concept is that while keeping the density fixed, the non-interacting system is connected to the interacting system via a coupling-constant $\lambda$, which represents the strength of the electron-electron interaction: $\lambda=0$ implies the non-interacting system and $\lambda=1$ is the fully interacting system. The result is elegantly simple:

$$E_{\hbox{{XC}}}[n({\bf r})] = \frac{1}{2} \int \, n({\bf r}) \, d... ...ox{{XC}}}({\bf r},{\bf r'})} {\vert\,{\bf r}-{\bf r'}\,\vert} \, d{\bf r'} \, .$$ (2.8)

The exchange-correlation hole $n_{\hbox {{XC}}}({\bf r},{\bf r'})$ is actually averaged over a coupling-constant dependent hole $n_{\hbox{{XC}}}^{\lambda}({\bf r},{\bf r'})$, given by,

$$n_{\hbox{{XC}}}({\bf r},{\bf r'}) = \int_0^1 n_{\hbox{{XC}}}^{\lambda}({\bf r},{\bf r'}) \, d\lambda \, .$$ (2.9)

A useful quantity to define from (2.9) is the exchange-correlation energy per particle, otherwise known as the energy density, $\varepsilon_{\hbox{{XC}}}[n({\bf r})]$,

$$\varepsilon_{\hbox{{XC}}}[n({\bf r})] = \frac{1}{2} \int \frac{n_... ...ox{{XC}}}({\bf r},{\bf r'})} {\vert\,{\bf r}-{\bf r'}\,\vert} \, d{\bf r'} \, .$$ (2.10)

Simply put, the electron many-body problem would be solved if $n_{\hbox {{XC}}}({\bf r},{\bf r'})$ were known exactly in analytic form.

The adiabatic connection method also provides other significant results. Most prominent is the fact that the difference between the interacting and non-interacting kinetic energy, $T_c[n({\bf r})] = T[n({\bf r})] - T_s[n({\bf r})]$, is included within the definition (2.9). So it transforms an energy contribution that is kinetic in origin, into quantity that resembles a potential energy. It also provides the link between the electron density in DFT, and the many-body wavefunction $\Psi$, through the exchange-correlation hole in relations (2.3), (2.7) and (2.9).