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Efficient construction of exchange and correlation potentials by inverting the Kohn–Sham equations
3. E. Engel and R. M. Dreizler, Density Functional Theory: An Advanced Course (Springer, Berlin, 2011).
16.The right-hand sides of Eqs. (5) and (7) are real despite the presence of complex-valued terms. This is because the eigenfunctions ϕi of a static Kohn–Sham Hamiltonian are either real or occur in degenerate pairs which are complex conjugates of one another, so the terms are also either real or occur in pairs of complex conjugates.
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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, . For orbitals and orbital energies that are solutions of the Kohn–Sham equations with a multiplicative this procedure recovers (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.
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