Efficient construction of exchange and correlation potentials by inverting the Kohn–Sham equations

Given a set of canonical Kohn–Sham orbitals, orbital energies, and an external potential for a many-electron system, one can invert the Kohn–Sham equations in a single step to obtain the corresponding exchange-correlation potential, \documentclass[12pt]{minimal}\begin{document}$v_\text{XC}(\mathbf {r})$\end{document}vXC(r). For orbitals and orbital energies that are solutions of the Kohn–Sham equations with a multiplicative \documentclass[12pt]{minimal}\begin{document}$v_\text{XC}(\mathbf {r})$\end{document}vXC(r) this procedure recovers \documentclass[12pt]{minimal}\begin{document}$v_\text{XC}(\mathbf {r})$\end{document}vXC(r) (in the basis set limit), but for eigenfunctions of a non-multiplicative one-electron operator it produces an orbital-averaged potential. In particular, substitution of Hartree–Fock orbitals and eigenvalues into the Kohn–Sham inversion formula is a fast way to compute the Slater potential. In the same way, we efficiently construct orbital-averaged exchange and correlation potentials for hybrid and kinetic-energy-density-dependent functionals. We also show how the Kohn–Sham inversion approach can be used to compute functional derivatives of explicit density functionals and to approximate functional derivatives of orbital-dependent functionals.