A direct optimization method for calculating density functionals and exchange–correlation potentials from electron densities

A direct optimization method is developed for the computation of the Kohn–Sham kinetic energy density functional Ts from a given electron density and the exchange–correlation potential vxc if this density is from a ground state. The method is based on the construction of a variational functional of the one-electron potential. This functional is derived from the conventional Levy constrained-search formulation and is shown to be closely related to the Lieb functional construction. The one-electron potential is expanded in terms of some fixed terms plus a linear expansion in a basis set. The determination of the Kohn–Sham kinetic energy for an input density is then turned into the maximization of this functional of potential. The analytic first and second derivatives of the variational functional with respect to the linear basis set expansion coefficients and also the nonlinear parameters in the basis set are derived. This enables very efficient iterative optimization of the potential and hence the calculation of Ts and vxc. The efficiency and accuracy of the method is shown in the numerical implementation for atomic and molecular calculations with Gaussian basis set expansions both for molecular orbitals and for one-electron potentials. Finally, this direct optimization method is extended to general density functionals and the analytic derivatives are also developed for use in optimization methods.