koopmans: An Open-Source Package for Accurately and Efficiently Predicting Spectral Properties with Koopmans Functionals

Abstract

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 , 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.

Accurate and Efficient Computation of the Fundamental Bandgap of the Vacancy-Ordered Double Perovskite Cs2TiBr6

“…KS-DFT methods that correct for (or minimize) self-interaction offer a new approach that is complementary to MBPT and have been found to provide accurate spectral properties of many solids at reduced computational cost. − The motivation for these methods is that the total energy of a system, E ( N ), as a function of the total number N of electrons in that system, ought to be piecewise linear. This is the well-known piecewise linearity condition (PWL) …”

“…In order to satisfy the aforementioned PWL condition, it is sufficient (but not necessary) to satisfy the generalized PWL condition (GPWL), which states that the total energy should be a linear function of the orbital occupancy f i ∂ 2 E ∂ f i 2 = 0 or equivalently, via Janak’s theorem − ∂ ε i ∂ f i = 0 …”

“…The Koopmans integral (KI) functional corrects the KS-DFT energy of the orbital φ i by linear interpolation between integral values of the charge. The energy difference between fully occupied and empty orbitals is , η i normalK normalI = E normalD normalF normalT | f i = 1 − E normalD normalF normalT | f i = 0 where E normalD normalF normalT | f i = f is the DFT energy of an ( N – 1 + f )-electron system, with the occupancy of the i th orbital constrained to be f . By linear interpolation with fractional occupancy f i the orbital-dependent energy is then f i η i KI .…”

“…Hence, the KI functional orbital-dependent correction to the DFT energy is , Π i normalK normalI = f i η i normalK normalI − [ E D F T − E D F T true| f i = 0 ] In order to evaluate E normalD normalF normalT | f i = f straightforwardly, it is simplest to initially neglect the implicit dependence of the orbitals φ i on the occupations f i that is, one adopts a frozen orbital approximation. The orbital relaxation is instead accounted for post hoc via screening coefficients α i , that are applied to each unscreened orbital-density-dependent correction. , These screening coefficients are determined ab initio, and as a result, the KI functional remains parameter-free. The resulting KI functional is a generalization of KS-DFT as the total energy is now a functional of not only the total density but also the individual orbital densities.…”

“…Alternatively, this spurious energy can be canceled by a corresponding PZ-SIC one-electron exchange energy . While the PZ-SIC improves KS-DFT eigenvalues for atoms, it can worsen performance for thermochemistry, and significant discrepancies associated with many-electron SIE remain. , The PZ-SIC and KI corrections can be combined to obtain the KIPZ functional. , …”