On exchange-correlation energy in DFT scenarios

Motivated by the considerable importance of material properties in modern condensed matter physics research, and using techniques of the $N_{e}$ -electron systems in terms of the electron density $n_{\sigma e}\left( r\right) $ needed to obtain the ground-state energy $E_{e0}$ in Density Functional theory scenarios, we approach the exchange-correlation energy $ E_{xc}\left[ n_{\sigma e}(r)\right] $ by considering the interelectronic position corrections $\Delta r_{x}^{\uparrow \uparrow ,\uparrow \downarrow }=\lambda _{x}\left\vert \delta r^{\uparrow \uparrow }-\delta r^{\uparrow \downarrow }\right\vert $ and $\Delta r_{c}^{e_{i}e_{j\neq i}}=\lambda _{c}\left\vert r-r^{\prime }\right\vert ^{-\left( N_{e}-1\right) ^{-1}}$ corresponding to the spin and the Coulomb correlation effects, respectively, through the electron-electron potential energy. Exploiting such corrections, we get approximate expressions for the exchange $E_{x}\left[ n_{\sigma e}\right]$ and the correlation $E_{c}\left[ n_{\sigma e}\right]$ functional energies which could be interpreted in terms of magnetic and electric dipole potential energies associated with the charge density $n_{\sigma e}(r)$ described by inverse-square potential behaviors. Based on these arguments, we expect that such obtained exchange-correlation functional energy could be considered in the Local Density Approximation functional as an extension to frame such interelectronic effects.