On the 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 \(\Delta r_{x}^{{ \uparrow \uparrow , \uparrow \downarrow }} = \) \({{\lambda }_{x}}\left| {\delta {{r}^{{ \uparrow \uparrow }}} - \delta {{r}^{{ \uparrow \downarrow }}}} \right|\) and \(\Delta r_{c}^{{{{e}_{i}}{{e}_{{j \ne i}}}}} = \) \({{\lambda }_{c}}{{\left| {r - r{\kern 1pt} '{\kern 1pt} } \right|}^{{ - {{{\left( {{{N}_{e}} - 1} \right)}}^{{ - 1}}}}}}\) 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 \({{n}_{{\sigma e}}}\left( r \right)\) 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.