Scaled σ-functionals for the Kohn-Sham correlation energy with scaling functions from the homogeneous electron gas

The recently introduced σ-functionals constitute a new type of functionals for the Kohn-Sham (KS) correlation energy. σ-Functionals are based on the adiabatic-connection fluctuation-dissipation theorem, are computationally closely related to the well-known direct random phase approximation (dRPA), and are formally rooted in many-body perturbation theory along the adiabatic connection. In σ-functionals, the function of the eigenvalues σ of the Kohn-Sham response matrix that enters the coupling constant and frequency integration in the dRPA is replaced by another function optimized with the help of reference sets of atomization, reaction, transition state, and non-covalent interaction energies. σ-Functionals are highly accurate and yield chemical accuracy of 1 kcal/mol in reaction or transition state energies, in main group chemistry. A shortcoming of σ-functionals is their inability to accurately describe processes involving a change of the electron number, such as ionizations or electron attachments. This problem is attributed to unphysical self-interactions caused by the neglect of the exchange kernel in the dRPA and σ-functionals. Here, we tackle this problem by introducing a frequency- and σ-dependent scaling of the eigenvalues σ of the KS response function that models the effect of the exchange kernel. The scaling factors are determined with the help of the homogeneous electron gas. The resulting scaled σ-functionals retain the accuracy of their unscaled parent functionals but in addition yield very accurate ionization potentials and electron affinities. Moreover, atomization and total energies are found to be exceptionally accurate. Scaled σ-functionals are computationally highly efficient like their unscaled counterparts.