Combining Local Range Separation and Local Hybrids: A Step in the Quest for Obtaining Good Energies and Eigenvalues from One Functional.

Some of the most successful exchange-correlation approximations in density functional theory are "hybrids", i.e., they rely on combining semilocal density functionals with exact nonlocal Fock exchange. In recent years, two classes of hybrid functionals have emerged as particularly promising: range-separated hybrids on the one hand, and local hybrids on the other hand. These functionals offer the hope to overcome a long-standing "observable dilemma", i.e., the fact that density functionals typically yield either a good description of binding energies, as obtained, e.g., in global and local hybrids, or physically interpretable eigenvalues, as obtained, e.g., in optimally tuned range-separated hybrids. Obtaining both of these characteristics from one and the same functional with the same set of parameters has been a long-standing challenge. We here discuss combining the concepts of local range separation and local hybrids as part of a constraint-guided quest for functionals that overcome the observable dilemma.

Predicting fundamental gaps accurately from density functional theory with non-empirical local range separation.

We present an exchange-correlation approximation in which the Coulomb interaction is split into long- and short-range components and the range separation is determined by a non-empirical density functional. The functional respects important constraints, such as the homogeneous and slowly varying density limits, leads to the correct long-range potential, and eliminates one-electron self-interaction. Our approach is designed for spectroscopic purposes and closely approximates the piecewise linearity of the energy as a function of the particle number. The functional's accuracy for predicting the fundamental gap in generalized Kohn-Sham theory is demonstrated for a large number of systems, including organic semiconductors with a notoriously difficult electronic structure.

Meta-generalized gradient approximations in time dependent generalized Kohn-Sham theory: Importance of the current density correction.

We revisit the use of Meta-Generalized Gradient Approximations (mGGAs) in time-dependent density functional theory, reviewing conceptual questions and solving the generalized Kohn-Sham equations by real-time propagation. After discussing the technical aspects of using mGGAs in combination with pseudopotentials and comparing real-space and basis set results, we focus on investigating the importance of the current-density based gauge invariance correction. For the two modern mGGAs that we investigate in this work, TASK and r2SCAN, we observe that for some systems, the current density correction leads to negligible changes, but for others, it changes excitation energies by up to 40% and more than 0.8 eV. In the cases that we study, the agreement with the reference data is improved by the current density correction.

Exact exchange-like electric response from a meta-generalized gradient approximation: A semilocal realization of ultranonlocality.

We review the concept of ultranonlocality in density functional theory and the relation between ultranonlocality, the derivative discontinuity of the exchange energy, and the static electric response in extended molecular systems. We present the construction of a new meta-generalized gradient approximation for exchange that captures the ultranonlocal response to a static electric field in very close correspondence to exact exchange, yet at a fraction of its computational cost. This functional, in particular, also captures the dependence of the response on the system size. The static electric polarizabilities of hydrogen chains and oligo-acetylene molecules calculated with this meta-GGA are quantitatively close to the ones obtained with exact exchange. The chances and challenges associated with the construction of meta-GGAs that are intended to combine a substantial derivative discontinuity and ultranonlocality with an accurate description of electronic binding are discussed.

Delocalized electronic excitations and their role in directional charge transfer in the reaction center of Rhodobacter sphaeroides.

In purple bacteria, the fundamental charge-separation step that drives the conversion of radiation energy into chemical energy proceeds along one branch-the A branch-of a heterodimeric pigment-protein complex, the reaction center. Here, we use first principles time-dependent density functional theory (TDDFT) with an optimally-tuned range-separated hybrid functional to investigate the electronic and excited-state structure of the six primary pigments in the reaction center of Rhodobacter sphaeroides. By explicitly including amino-acid residues surrounding these six pigments in our TDDFT calculations, we systematically study the effect of the protein environment on energy and charge-transfer excitations. Our calculations show that a forward charge transfer into the A branch is significantly lower in energy than the first charge transfer into the B branch, in agreement with the unidirectional charge transfer observed experimentally. We further show that the inclusion of the protein environment redshifts this excitation significantly, allowing for energy transfer from the coupled Qx excitations. Through analysis of transition and difference densities, we demonstrate that most of the Q-band excitations are strongly delocalized over several pigments and that both their spatial delocalization and charge-transfer character determine how strongly affected they are by thermally-activated molecular vibrations. Our results suggest a mechanism for charge-transfer in this bacterial reaction center and pave the way for further first-principles investigations of the interplay between delocalized excited states, vibronic coupling, and the role of the protein environment in this and other complex light-harvesting systems.

Hybrid functionals with local range separation: Accurate atomization energies and reaction barrier heights.

Range-separated hybrid approximations to the exchange-correlation density functional mix exact and semi-local exchange in a position-dependent manner. In their conventional form, the range separation is controlled by a constant parameter. Turning this constant into a density functional leads to a locally space-dependent range-separation function and thus a more powerful and flexible range-separation approach. In this work, we explore the self-consistent implementation of a local range-separated hybrid, taking into account a one-electron self-interaction correction and the behavior under uniform density scaling. We discuss different forms of the local range-separation function that depend on the electron density, its gradient, and the kinetic energy density. For test sets of atomization energies, reaction barrier heights, and total energies of atoms, we demonstrate that our best model is a clear improvement over common global range-separated hybrid functionals and can compete with density functionals that contain multiple empirical parameters. Promising results for equilibrium bond lengths, harmonic vibrational frequencies, and vertical ionization potentials further underline the potential and flexibility of our approach.

Piecewise linearity, freedom from self-interaction, and a Coulomb asymptotic potential: three related yet inequivalent properties of the exact density functional.

The exact energy functional of density functional theory (DFT) is well known to obey various constraints. Three conditions that must be obeyed by the exact energy functional, but may or may not be obeyed by approximate ones, are often pointed out as important in general and for accurate computation of spectroscopic observables in particular. These are: (1) piecewise linearity as a function of the fractional particle number, (2) freedom from one-electron self-interaction, and (3) for a finite system, the functional derivative with respect to the density results in an asymptotic -1/r potential (in Hartree atomic units), where r is the distance from the system center. In this overview, we explain what these conditions are, what they address, and why each one is of importance for spectroscopy. We then show, using specific examples from the literature, that these three properties are related, but are not equivalent and need to be assessed individually.

Molecular excitations from meta-generalized gradient approximations in the Kohn-Sham scheme.

Meta-Generalized Gradient Approximations (meta-GGAs) can, in principle, include spatial and temporal nonlocality in time-dependent density functional theory at a much lower computational cost than functionals that use exact exchange. We here test whether a meta-GGA that has recently been developed with a focus on capturing nonlocal response properties and the particle number discontinuity can realize such features in practice. To this end, we extended the frequency-dependent Sternheimer formalism to the meta-GGA case. Using the Krieger-Li-Iafrate (KLI) approximation, we calculate the optical response for the selected paradigm molecular systems and compare the meta-GGA Kohn-Sham response to the one found with exact exchange and conventional (semi-)local functionals. We find that the new meta-GGA captures important properties of the nonlocal exchange response. The KLI approximation, however, emerges as a limiting factor in the evaluation of charge-transfer excitations.

Self-interaction correction, electrostatic, and structural influences on time-dependent density functional theory excitations of bacteriochlorophylls from the light-harvesting complex 2.

First-principles calculations offer the chance to obtain a microscopic understanding of light-harvesting processes. Time-dependent density functional theory can have the computational efficiency to allow for such calculations. However, the (semi-)local exchange-correlation approximations that are computationally most efficient fail to describe charge-transfer excitations reliably. We here investigate whether the inexpensive average density self-interaction correction (ADSIC) remedies the problem. For the systems that we study, ADSIC is even more prone to the charge-transfer problem than the local density approximation. We further explore the recently reported finding that the electrostatic potential associated with the chromophores' protein environment in the light-harvesting complex 2 beneficially shifts spurious excitations. We find a great sensitivity on the chromophores' atomistic structure in this problem. Geometries obtained from classical molecular dynamics are more strongly affected by the spurious charge-transfer problem than the ones obtained from crystallography or density functional theory. For crystal structure geometries and density-functional theory optimized ones, our calculations confirm that the electrostatic potential shifts the spurious excitations out of the energetic range that is most relevant for electronic coupling.

Exploring local range separation: The role of spin scaling and one-electron self-interaction.

Range-separated hybrid functionals with a fitted or tuned global range-separation parameter are frequently used in density functional theory. We here explore the concept of local range separation, i.e., of turning the range-separation parameter into an explicit semilocal density functional. We impose three simple constraints on the local range-separation parameter that are frequently used in density functional construction: uniform density scaling, the homogeneous electron gas limit, and freedom from one-electron self-interaction. We further discuss different ways of how to model the spin dependence in combination with local range separation. We evaluate our local range-separation energy functionals exactly for closed-shell atoms using the previously suggested hypergeneralized gradient approximation for molecules and assess the quality of this approximation. We find a local range-separated hybrid functional that yields accurate binding energies for a set of small molecules.

Assessing density functional theory in real-time and real-space as a tool for studying bacteriochlorophylls and the light-harvesting complex 2.

We use real-time density functional theory on a real-space grid to calculate electronic excitations of bacteriochlorophyll chromophores of the light-harvesting complex 2 (LH2). Comparison with Gaussian basis set calculations allows us to assess the numerical trust range for computing electron dynamics in coupled chromophores with both types of techniques. Tuned range-separated hybrid calculations for one bacteriochlorophyll as well as two coupled ones are used as a reference against which we compare results from the adiabatic time-dependent local density approximation (TDLDA). The tuned range-separated hybrid calculations lead to a qualitatively correct description of the electronic excitations and couplings. They allow us to identify spurious charge-transfer excitations that are obtained with the TDLDA. When we take into account the environment that the LH2 protein complex forms for the bacteriochlorophylls, we find that it substantially shifts the energy of the spurious charge-transfer excitations, restoring a qualitatively correct electronic coupling of the dominant excitations also for TDLDA.

Linear response time-dependent density functional theory without unoccupied states: The Kohn-Sham-Sternheimer scheme revisited.

The Sternheimer approach to time-dependent density functional theory in the linear response regime is attractive because of its computational efficiency. The latter results from avoiding the explicit calculation of unoccupied orbitals and from the basic structure of the Sternheimer equations, which naturally lend themselves to far-reaching parallelization. In this article, we take a fresh look at the frequency-dependent Sternheimer equations. We first give a complete, self-contained derivation of the equations that complements previous derivations. We then discuss several aspects of an efficient numerical realization. As a worked example, we compute the photoabsorption spectra of small hydrogenated silicon clusters and confirm that for these the quality of the Kohn-Sham eigenvalues is more important than the effects of the exchange-correlation kernel. Finally, we demonstrate how triplet excitations can readily be computed from the Sternheimer approach.

Investigating the electronic structure of a supported metal nanoparticle: Pd in SiCN.

We investigate the electronic structure of a Palladium nanoparticle that is partially embedded in a matrix of silicon carbonitride. From classical molecular dynamics simulations we first obtain a representative atomic structure. This geometry then serves as input to density-functional theory calculations that allow us to access the electronic structure of the combined system of particle and matrix. In order to make the computations feasible, we devise a subsystem strategy for calculating the relevant electronic properties. We analyze the Kohn-Sham density of states and pay particular attention to d-states which are prone to be affected by electronic self-interaction. We find that the density of states close to the Fermi level is dominated by states that originate from the Palladium nanoparticle. The matrix has little direct effect on the electronic structure of the metal. Our results contribute to explaining why silicon carbonitride does not have detrimental effects on the catalytic properties of palladium particles and can serve positively as a stabilizing mechanical support.

Hydrogen binding energies and electronic structure of Ni-Pd particles: a clue to their special catalytic properties.

We investigate the electronic structure of nickel-palladium systems with first-principles density functional theory (DFT). Our study is motivated by the experimental observation that nickel-palladium nanoalloys containing approximately equal amounts of nickel and palladium show a higher catalytic activity in hydrogenation reactions than pure particles of either metal. Our calculations show that the energy with which hydrogen is bound to the metal surface, a decisive factor in catalytic activity, strongly depends on the nickel-palladium ratio. The weakest binding and Gibbs free energies of hydrogen adsorption close to zero are found for systems with roughly equal amounts of nickel and palladium. While for extended Ni-Pd surfaces this observation can be explained by the well-established d-band model, for Ni-Pd clusters analysis of the electronic density of states shows a complex interplay of s- and d-orbital contributions. Studying extended surfaces further reveals that the formation of nanostructures on the surface can influence hydrogen adsorption pronouncedly. We discuss the important role that the exchange-correlation functional approximation plays in DFT calculations for nickel-palladium systems.

Tuned range separated hybrid functionals for solvated low bandgap oligomers.

The description of charge transfer excitations has long been a challenge to time dependent density functional theory. The recently developed concept of "optimally tuned range separated hybrid (OT-RSH) functionals" has proven to describe charge transfer excitations accurately in many cases. However, describing solvated or embedded systems is yet a challenge. This challenge is not only computational but also conceptual, because the tuning requires identifying a specific orbital, typically the highest occupied one of the molecule under study. For solvated molecules, this orbital may be delocalized over the solvent. We here demonstrate that one way of overcoming this problem is to use a locally projected self-consistent field diagonalization on an absolutely localized molecular orbital expansion. We employ this approach to determine ionization energies and the optical gap of solvated oligothiophenes, i.e., paradigm low gap systems that are of relevance in organic electronics. Dioxane solvent molecules are explicitly represented in our calculations, and the ambiguities of straightforward parameter tuning in solution are elucidated. We show that a consistent estimate of the optimal range separated parameter (ω) at the limit of bulk solvation can be obtained by gradually extending the solvated system. In particular, ω is influenced by the solvent beyond the first coordination sphere. For determining ionization energies, a considerable number of solvent molecules on the first solvation shell must be taken into account. We demonstrate that accurately calculating optical gaps of solvated systems using OT-RSH can be done in three steps: (i) including the chemical environment when determining the range-separation parameter, (ii) taking into account the screening due to the solvent, and (iii) using realistic molecular geometries.

Effect of ensemble generalization on the highest-occupied Kohn-Sham eigenvalue.

There are several approximations to the exchange-correlation functional in density-functional theory, which accurately predict total energy-related properties of many-electron systems, such as binding energies, bond lengths, and crystal structures. Other approximations are designed to describe potential-related processes, such as charge transfer and photoemission. However, the development of a functional which can serve the two purposes simultaneously is a long-standing challenge. Trying to address it, we employ in the current work the ensemble generalization procedure proposed by Kraisler and Kronik [Phys. Rev. Lett. 110, 126403 (2013)]. Focusing on the prediction of the ionization potential via the highest occupied Kohn-Sham eigenvalue, we examine a variety of exchange-correlation approximations: the local spin-density approximation, semi-local generalized gradient approximations, and global and local hybrid functionals. Results for a test set of 26 diatomic molecules and single atoms are presented. We find that the aforementioned ensemble generalization systematically improves the prediction of the ionization potential, for various systems and exchange-correlation functionals, without compromising the accuracy of total energy-related properties. We specifically examine hybrid functionals. These depend on a parameter controlling the ratio of semi-local to non-local functional components. The ionization potential obtained with ensemble-generalized functionals is found to depend only weakly on the parameter value, contrary to common experience with non-generalized hybrids, thus eliminating one aspect of the so-called "parameter dilemma" of hybrid functionals.

Gas-phase valence-electron photoemission spectroscopy using density functional theory.

We present a tutorial overview of the simulation of gas-phase valence-electron photoemission spectra using density functional theory (DFT), emphasizing both fundamental considerations and practical applications, and making appropriate links between the two. We explain how an elementary quantum mechanics view of photoemission couples naturally to a many-body perturbation theory view. We discuss a rigorous approach to photoemission within the framework of time-dependent DFT. Then we focus our attention on ground-state DFT. We clarify the extent to which it can be used to mimic many-body perturbation theory in principle, and then provide a detailed discussion of the accuracy one can and cannot expect in practice with various approximate DFT forms.

One-electron self-interaction and the asymptotics of the Kohn-Sham potential: an impaired relation.

One-electron self-interaction and an incorrect asymptotic behavior of the Kohn-Sham exchange-correlation potential are among the most prominent limitations of many present-day density functionals. However, a one-electron self-interaction-free energy does not necessarily lead to the correct long-range potential. This is shown here explicitly for local hybrid functionals. Furthermore, carefully studying the ratio of the von Weizsäcker kinetic energy density to the (positive) Kohn-Sham kinetic energy density, τW/τ, reveals that this ratio, which frequently serves as an iso-orbital indicator and is used to eliminate one-electron self-interaction effects in meta-generalized-gradient approximations and local hybrid functionals, can fail to approach its expected value in the vicinity of orbital nodal planes. This perspective article suggests that the nature and consequences of one-electron self-interaction and some of the strategies for its correction need to be reconsidered.

Charge-transfer excitations in low-gap systems under the influence of solvation and conformational disorder: exploring range-separation tuning.

Charge transfer excitations play a prominent role in the fields of molecular electronics and light harvesting. At the same time they have developed a reputation for being hard to predict with time-dependent density functional theory, which is the otherwise predominant method for calculating molecular structure and excitations. Recently, it has been demonstrated that range-separated hybrid functionals, in particular with an "optimally tuned" range separation parameter, describe charge-transfer excitations reliably for different molecules. Many of these studies focused on molecules in vacuum. Here we investigate the influence of solvation on the electronic excitations of thiophene oligomers, i.e., paradigm low gap systems. We take into account bulk solvation using a continuum solvation model and geometrical distortions from molecular dynamics. From our study, three main findings emerge. First, geometrical distortions increase absorption energies by about 0.5 eV for the longer thiophene oligomers. Second, combining optimal tuning of the range separation parameter with a continuum solvation method is not straightforward and has to be approached with great care. Third, optimally tuned range-separated hybrids without a short-range exchange component tend to inherit undesirable characteristics of semi-local functionals: with increasing system size the range separation parameter takes a smaller value, leading to a functional of effectively more semi-local nature and thus not accurately capturing, e.g., the saturation of the optical gap with increasing system size.