New method for the direct calculation of electron density in many‐electron systems. I. Application to closed‐shell atoms

A new density‐functional equation is suggested for the direct calculation of electron density ρ( r ) in many‐electron systems. This employs a kinetic energy functional T 2 + f ( r ) T 0 , where T 2 is the original Weizsäcker correction, T 0 is the Thomas–Fermi term, and f ( r ) is a correction factor that depends on both r and the number of electrons N . Using the Hartree–Fock relation between the kinetic and the exchange energy density, and a nonlocal approximation to the latter, the kinetic energy–density functional is written (in a.u.)\documentclass{article}\pagestyle{empty}\begin{document}$$ t[\rho] = {\raise0.7ex\hbox{$1$} \!\mathord{\left/ {\vphantom {1 4}}\right.\kern-\nulldelimiterspace}\!\lower0.7ex\hbox{$4$}}\nabla ^2 \rho + {\raise0.7ex\hbox{$1$} \!\mathord{\left/ {\vphantom {1 8}}\right.\kern-\nulldelimiterspace}\!\lower0.7ex\hbox{$8$}}(\nabla \rho \cdot \nabla \rho)/\rho + C_k f({\bf r})\rho ^{5/3}, $$\end{document} where \documentclass{article}\pagestyle{empty}\begin{document}$ C_k = {\raise0.7ex\hbox{$2$} \!\mathord{\left/ {\vphantom {2 {10}}}\right.\kern-\nulldelimiterspace}\!\lower0.7ex\hbox{${10}$}}(3\pi ^2)^{2/3} $\end{document} . Incorporating the above expression in the total energy density functional and minimizing the latter subject to N representability conditions for ρ( r ) result in an Euler–Lagrange nonlinear second‐order differential equation\documentclass{article}\pagestyle{empty}\begin{document}$$ \left[{ - {\raise0.7ex\hbox{$1$} \!\mathord{\left/ {\vphantom {1 2}}\right.\kern-\nulldelimiterspace}\!\lower0.7ex\hbox{$2$}}\nabla ^2 + v_{{\rm nuc}} ({\bf r}) + v_{{\rm cou}} ({\bf r}) + v_{XC} ({\bf r}) + {\raise0.7ex\hbox{$5$} \!\mathord{\left/ {\vphantom {5 3}}\right.\kern-\nulldelimiterspace}\!\lower0.7ex\hbox{$3$}}C_k g({\bf r})\rho ^{2/3}} \right]\phi ({\bf r}) = \mu \phi ({\bf r}) $$\end{document} where μ is the chemical potential, we have ρ( r ) = |ϕ( r )| 2 , and g ( r ) is related to f ( r ). Numerical solutions of the above equation for Ne, Ar, Kr, and Xe, by modeling f ( r ) and g ( r ) as simple sums over Gaussians, show excellent agreement with the corresponding Hartree–Fock ground‐state densities and energies, indicating that this is likely to be a promising method for calculating fairly accurate electron densities in atoms and molecules.