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Over the past decade we have developed Koopmans functionals, a computationally efficient approach for predicting spectral properties with an orbital-density-dependent functional framework. These functionals impose a generalized piecewise linearity condition to the entire electronic manifold, ensuring that orbital energies match the corresponding electron removal/addition energy differences (in contrast to semilocal DFT, where a mismatch between the two lies at the heart of the band gap problem and, more generally, the unreliability of Kohn-Sham orbital energies). This strategy has proven to be very powerful, yielding molecular orbital energies and solid-state band structures with comparable accuracy to many-body perturbation theory but at greatly reduced computational cost while preserving a functional formulation. This paper reviews the theory of Koopmans functionals, discusses the algorithms necessary for their implementation, and introduces koopmans, an open-source package that contains all of the code and workflows needed to perform Koopmans functional calculations and obtain reliable spectral properties of molecules and materials.

Obtaining a precise theoretical description of the spectral properties of liquid water poses challenges for both molecular dynamics (MD) and electronic structure methods. The lower computational cost of the Koopmans-compliant functionals with respect to Green's function methods allows the simulations of many MD trajectories, with a description close to the state-of-art quasi-particle self-consistent GW plus vertex corrections method (QSGW + f xc ). Thus, we explore water spectral properties when different MD approaches are used, ranging from classical MD to first-principles MD, and including nuclear quantum effects. We have observed that different MD approaches lead to up to 1 eV change in the average band gap; thus, we focused on the band gap dependence with the geometrical properties of a system to explain such spread. We have evaluated the changes in the band gap due to variations in the intramolecular O-H bond distance and HOH angle, as well as the intermolecular hydrogen bond O···O distance and the OHO angles. We have observed that the dominant contribution comes from the O-H bond length; the O···O distance plays a secondary role, and the other geometrical properties do not significantly influence the gap. Furthermore, we analyze the electronic density of states (DOS), where the KIPZ functional shows good agreement with the DOS obtained with state-of-art approaches employing quasi-particle self-consistent GW plus vertex corrections. The O-H bond length also significantly influences the DOS. When nuclear quantum effects are considered, broadening of the peaks driven by the broader distribution of the O-H bond lengths is observed, leading to a closer agreement with the experimental photoemission spectra.

Koopmans-compliant (KC) functionals have been shown to provide accurate spectral properties through a generalized condition of piecewise linearity of the total energy as a function of the fractional addition/removal of an electron to/from any orbital. We analyze the performance of different KC functionals on a large and standardized set of 100 molecules, the GW100 test set, comparing vertical ionization potentials (taken as opposite of the orbital energies) to those obtained from accurate quantum chemistry methods, and to experimental results. We find excellent agreement, with a mean absolute error of 0.20 eV for the KIPZ functional on the first ionization potential, which is state-of-the-art for both density functional theory (DFT)-based calculations and many-body perturbation theory. We highlight similarities and differences between KC functionals and other electronic-structure approaches, such as dielectric-dependent hybrid functionals and Green's function methods, both from a theoretical and from a practical point of view, arguing that KC potentials can be considered as local and orbital-dependent approximations to the electronic self-energy, already including approximate vertex corrections.

Electronic-structure functionals that include screening effects, such as Hubbard or Koopmans' functionals, are required to describe the response of a system to the fractional addition or removal of an electron from an orbital or a manifold. Here, we present a general method to incorporate screening based on linear-response theory, and we apply it to the case of orbital-by-orbital screening of Koopmans' functionals. We illustrate the importance of such generalization when dealing with challenging systems containing orbitals with very different chemical character, also highlighting the simple dependence of the screening on the localization of the orbitals. We choose a set of 46 transition-metal complexes for which experimental data and accurate many-body perturbation theory calculations are available. When compared to experiment, results for ionization potentials show a very good performance, with a mean absolute error of 0.2 eV, comparable to the most accurate many-body perturbation theory approaches. These results reiterate the role of Koopmans-compliant functionals as simple and accurate quasiparticle approximations to the exact spectral functional, bypassing diagrammatic expansions and relying only on the physics of the local density or generalized-gradient approximation.

The need to interpret ultraviolet photoemission data strongly motivates the refinement of first-principles techniques that are able to accurately predict spectral properties. In this work, we employ Koopmans-compliant functionals, constructed to enforce piecewise linearity in approximate density functionals, to calculate the structural and electronic properties of DNA and RNA nucleobases. Our results show that not only ionization potentials and electron affinities are accurately predicted with mean absolute errors of <0.1 eV, but also that calculated photoemission spectra are in excellent agreement with experimental ultraviolet photoemission spectra. In particular, the role and contribution of different tautomers to the photoemission spectra are highlighted and discussed in detail. The structural properties of nucleobases are also investigated, showing an improved description with respect to local and semilocal density-functional theory. Methodologically, our results further consolidate the role of Koopmans-compliant functionals in providing, through orbital-density-dependent potentials, accurate electronic and spectral properties.