On the errors of local density (LDA) and generalized gradient (GGA) approximations to the Kohn-Sham potential and orbital energies

The Amsterdam Modeling Suite

In this paper, we present the Amsterdam Modeling Suite (AMS), a comprehensive software platform designed to support advanced molecular and materials simulations across a wide range of chemical and physical systems. AMS integrates cutting-edge quantum chemical methods, including Density Functional Theory (DFT) and time-dependent DFT, with molecular mechanics, fluid thermodynamics, machine learning techniques, and more, to enable multi-scale modeling of complex chemical systems. Its design philosophy allows for seamless coupling between components, facilitating simulations that range from small molecules to complex biomolecular and solid-state systems, making it a versatile tool for tackling interdisciplinary challenges, both in industry and in academia. The suite also emphasizes user accessibility, with an intuitive graphical interface, extensive scripting capabilities, and compatibility with high-performance computing environments.

Going beyond the Computational Tool: Fermi Potential from DFT as an Electron (De)Localization Descriptor for Correlated Wave Functions

The Fermi potential, appearing in the basic equations of density functional theory (DFT), has been found to be an indispensable tool for the measurement of electron localization intensity in molecules and crystals. The regions of the most intensive electron localization appear there as negative wells, while the positive barriers of the Fermi potential prevent the electron concentration there to some extent. The shape of the Fermi potential distribution in covalent bonds reflects the bond order, while the structure of its components is able to provide valuable information about the bonding nature, e.g., helping to draw the line between covalent and noncovalent bonds. The accuracy of the Fermi potential's estimates of electron (de)localization stems from the ability of its components to preserve all the main features of the exchange-correlation hole behavior within the one-electron functions, while other popular descriptors can easily fail in this task. Such analysis is not restricted to DFT calculations; when applied to post-Hartree-Fock wave functions, it unravels details of how instantaneous Coulomb correlation prevents the overestimation of electron localization intensity in strongly correlated and ordinary systems. Generally, the slight decrease in localization intensity is achieved by the intensified response of electron correlation to variations in electron density, while in systems where instantaneous Coulomb correlation is particularly important, it also comes from the growth in the exchange-correlation hole mobility; the average hole depth increases in all cases.

Simple Linear Regression Models for Prediction of Ionization Energies, Electron Affinities, and Fundamental Gaps of Atoms and Molecules

Linear regression equations were developed for different density functionals using data from the CCCBDB, along with a test set of 89 ionization energies (IE) and 76 electron affinities (EA) so that experimental IE and EA can be predicted from orbital energies. Separate equations were determined for different classes of atoms and molecules. These relationships were also applied to all occupied orbitals to simulate the photoemission spectra of organic molecules with accuracy similar to that of other computational methods at a fraction of the cost. The error for large molecules (up to 200 atoms) can be below 0.2 eV with many functionals for the prediction of the IE and EA.

Electronic Vector Potential from the Exact Factorization of a Complex Wavefunction

We generalize the definitions of local scalar potentials namedυ kin ${\upsilon _{{\rm{kin}}} }$ andυ N - 1 ${\upsilon _{N - 1} }$ , which are relevant to properly describe phenomena such as molecular dissociation with density-functional theory, to the case in which the electronic wavefunction corresponds to a complex current-carrying state. In such a case, an extra term in the form of a vector potential appears which cannot be gauged away. Both scalar and vector potentials are introduced via the exact factorization formalism which allows us to express the given Schrödinger equation as two coupled equations, one for the marginal and one for the conditional amplitude. The electronic vector potential is directly related to the paramagnetic current density carried by the total wavefunction and to the diamagnetic current density in the equation for the marginal amplitude. An explicit example of this vector potential in a triplet state of two non-interacting electrons is showcased together with its associated circulation, giving rise to a non-vanishing geometric phase. Some connections with the exact factorization for the full molecular wavefunction beyond the Born-Oppenheimer approximation are also discussed.

Putting error bars on density functional theory

Predicting the error in density functional theory (DFT) calculations due to the choice of exchange-correlation (XC) functional is crucial to the success of DFT, but currently, there are limited options to estimate this a priori. This is particularly important for high-throughput screening of new materials. In this work, the structure and elastic properties of binary and ternary oxides are computed using four XC functionals: LDA, PBE-GGA, PBEsol, and vdW-DF with C09 exchange. To analyze the systemic errors inherent to each XC functional, we employed materials informatics methods to predict the expected errors. The predicted errors were also used to better the DFT-predicted lattice parameters. Our results emphasize the link between the computed errors and the electron density and hybridization errors of a functional. In essence, these results provide "error bars" for choosing a functional for the creation of high-accuracy, high-throughput datasets as well as avenues for the development of XC functionals with enhanced performance, thereby enabling the accelerated discovery and design of new materials.

Performance of Density Functionals and Semiempirical 3c Methods for Small Gold-Thiolate Clusters

In light of the recent surge in computational studies of gold thiolate clusters, we present a comparison of popular density functionals (DFAs) and three-part corrected methods (3c-methods) on their performance by taking a data set named as AuSR18 consisting of 18 isomers of Au_n(SCH₃)_m (m ≤ n = 1-3). We have compared the efficiency and accuracy of the DFAs and 3c-methods in geometry optimization with RI-SCS-MP2 as the reference method. Similarly, the performance for accurate and efficient energy evaluation was compared with DLPNO-CCSD(T) as the reference method. The lowest energy structure among the isomers of the largest stoichiometry from our data set, AuSR18, i.e., Au₃(SCH₃)₃, is considered to evaluate the computational time for SCF and gradient evaluations. Alongside this, the numbers of optimization steps to locate the most stable minima of Au₃(SCH₃)₃ are compared to assess the efficiency of the methods. A comparison of relevant bond lengths with the reference geometries was made to estimate the accuracy in geometry optimization. Some methods, such as LC-BLYP, ωB97M-D3BJ, M06-2X, and PBEh-3c, could not locate many of the minima found by most of the other methods; thus, the versatility in locating various minima is also an important criterion in choosing a method for the given project. To determine the accuracy of the methods, we compared the relative energies of the isomers in each stoichiometry and the interaction energy of the gold core with the ligands. The dependence of basis set size and relativistic effects on energies are also compared. The following are some of the highlights. TPSS has shown accuracy, while mPWPW shows comparable speed and accuracy. For the relative energies of the clusters, the hybrid range-separated DFAs are the best option. CAM-B3LYP excels, whereas B3LYP performs poorly. Overall, LC-BLYP is a balanced performer considering both the geometry and relative stability of the structures, but it lacks diversity. The 3c-methods, although fast, are less impressive in relative stability.

Interexcited State Photophysics I: Benchmarking Density Functionals for Computing Nonadiabatic Couplings and Internal Conversion Rate Constants

We present the first benchmarking study of nonadiabatic matrix coupling elements (NACMEs) calculated using different density functionals. Using the S₁ → S₀ transition in perylene solvated in toluene as a case study, we calculate the photophysical properties and corresponding rate constants for a variety of density functionals from each rung of Jacob's ladder. The singlet photoluminescence quantum yield (sPLQY) is taken as a measure of accuracy, measured experimentally here as 0.955. Important quantum chemical parameters such as geometries, absorption, emission, and adiabatic energies, NACMEs, Hessians, and transition dipole moments were calculated for each density functional basis set combination (data set) using density functional theory based multireference configuration interaction (DFT/MRCI) and compared to experiment where possible. We were able to derive simple relations between the TDDFT and DFT/MRCI photophysical properties; with semiempirical damping factors of ∼0.843 ± 0.017 and ∼0.954 ± 0.064 for TDDFT transition dipole moments and energies to DFT/MRCI level approximations, respectively. NACMEs were dominated by out-of-plane derivative components belonging to the center-most ring atoms with weaker contributions from perturbations along the transverse and longitudinal axes. Calculated theoretical spectra compared well to both experiment and literature, with fluorescence lifetimes between 7.1 and 12.5 ns, agreeing within a factor of 2 with experiment. Internal conversion (IC) rates were then calculated and were found to vary wildly between 10⁶-10¹⁶ s^-1 compared with an experimental rate of the order 10⁷ s^-1. Following further testing by mixing data sets, we found a strong dependence on the method used to obtain the Hessian. The 5 characterized data sets ranked in order of most promising are PBE0/def2-TZVP, ωB97XD/def2-TZVP, HCTH407/TZVP, PBE/TZVP, and PBE/def2-TZVP.

DFT exchange: sharing perspectives on the workhorse of quantum chemistry and materials science

In this paper, the history, present status, and future of density-functional theory (DFT) is informally reviewed and discussed by 70 workers in the field, including molecular scientists, materials scientists, method developers and practitioners. The format of the paper is that of a roundtable discussion, in which the participants express and exchange views on DFT in the form of 302 individual contributions, formulated as responses to a preset list of 26 questions. Supported by a bibliography of 777 entries, the paper represents a broad snapshot of DFT, anno 2022.

E/Z photoisomerization pathway in pristine and fluorinated di(3-furyl)ethenes

We report an XMCQDPT2 study of the E/Z photoisomerization in a series of fluorinated di(3-furyl)ethenes (3DFEs). Upon excitation, pristine and low-fluorinated 3DFE show conventional behavior of many diarylethenes: unhindered twisting motion toward the pyramidalized zwitterionic state where relaxation to the ground state occurs. However, deep fluorination of 3DFEs can hamper E-to-Z isomerization by giving rise to an alternative excited-state relaxation pathway: an out-of-plane motion of a ring fluorine atom. Importantly, the case of fluorinated 3DFEs reveals serious deficiencies of the popular TDDFT approach. With some commonly used exchange-correlation functionals, the alternative relaxation pathway is not reproduced and, moreover, an irrelevant ring rotation coordinate is predicted instead. Nevertheless, TDDFT remains qualitatively adequate for the E-to-Z twisting coordinate taken alone.

Can We Merge the Weak and Strong Tetrel Bonds? Electronic Features of Tetrahedral Molecules Interacted with Halide Anions

Using the orbital-free quantum crystallography approach, we have disclosed the quantitative trends in electronic features for bonds of different strengths formed by tetrel (Tt) atoms in stable molecular complexes consisting of electrically neutral tetrahedral molecules and halide anions. We have revealed the role of the electrostatic and exchange-correlation components of the total one-electron static potential that are determined by the equilibrium atomic structure and by kinetic Pauli potential, which reflects the spin-dependent electron motion features of the weak and strong bonds. The gap between the extreme positions in the electrostatic and total static potentials along the line linking the Tt atom and halide anion is wide for weak bonds and narrow for strong ones. It is in very good agreement with the number of minima in the Pauli potential between the bounded atoms. This gap exponentially correlates with the exchange-correlation potential in various series with a fixed nucleophilic fragment. A criterion for categorizing the noncovalent tetrel bonds (TtB) based on the potential features is suggested.

Chemical potential, derivative discontinuity, fractional electrons, jump of the Kohn-Sham potential, atoms as thermodynamic open systems, and other (mis)conceptions of the density functional theory of electrons in molecules

Many references exist in the density functional theory (DFT) literature to the chemical potential of the electrons in an atom or a molecule. The origin of this notion has been the identification of the Lagrange multiplier μ = ∂E/∂N in the Euler-Lagrange variational equation for the ground state density as the chemical potential of the electrons. We first discuss why the Lagrange multiplier in this case is an arbitrary constant and therefore cannot be a physical characteristic of an atom or molecule. The switching of the energy derivative ("chemical potential") from -I to -A when the electron number crosses the integer, called integer discontinuity or derivative discontinuity, is not physical but only occurs when the nonphysical noninteger electron systems and the corresponding energy and derivative ∂E/∂N are chosen in a specific discontinuous way. The question is discussed whether in fact the thermodynamical concept of a chemical potential can be defined for the electrons in such few-electron systems as atoms and molecules. The conclusion is that such systems lack important characteristics of thermodynamic systems and do not afford the definition of a chemical potential. They also cannot be considered as analogues of the open systems of thermodynamics that can exchange particles with an environment (a particle bath or other members of a Gibbsian ensemble). Thermodynamical (statistical mechanical) concepts like chemical potential, open systems, grand canonical ensemble etc. are not applicable to a few electron system like an atom or molecule. A number of topics in DFT are critically reviewed in light of these findings: jumps in the Kohn-Sham potential when crossing an integer number of electrons, the band gap problem, the deviation-from-straight-lines error, and the role of ensembles in DFT.

Keto-enol tautomerism from the electron delocalization perspective

The equilibrium between keto and enol forms in acetylacetone and its derivatives is studied using electron delocalization indices and delocalization tensor density. We demonstrate how electron delocalization governs the equilibrium between keto and enol forms. The less stable enols have more distinct double and single bond character in the CCC fragment, while electron delocalization in this fragment is more pronounced in more stable enols. Looking for the origin of such behavior, we considered the one-electron potentials entering the Euler equation for the electron density. We found that electron delocalization is mainly governed by the static exchange potential, which depends on the three-dimensional atomic structure. It, however, does not distinguish differences in electron delocalization in more and less stable enols, the effect arising from the kinetic exchange contribution, which reflects spin-dependent effects in the electron motion. The local depletion of kinetic exchange in the conjugated fragment yields the enhanced electron delocalization along the CCC bonds in more stable enols. Thus, a combination of considered descriptors allowed us to reveal the influence of electron delocalization on the equilibrium between keto and enol forms and showed the significant features of this phenomenon.

Taking Advantage of a Systematic Energy Non-linearity Error in Density Functional Theory for the Calculation of Electronic Energy Levels

We present an approximate approach for the calculation of ionization potential (IP) and electron affinity (EA) by exploiting the complementary energy non-linearity errors for a species M and its one-electron-ionized counterpart (M⁺). Reasonable IPs and EAs are thus obtained by averaging the orbital energies of M and M⁺, even with a low-level method such as BLYP/6-31G(d). By combining the corrected IPs and EAs, we can further obtain reasonable excitation energies. The errors in uncorrected valence IPs and uncorrected virtual-orbital energies show systematic trends. These characteristics provide a convenient and computationally efficient avenue for qualitative estimation of these properties with single corrections for multiple IPs and excitation energies.

Electronic Structure of Molecules, Surfaces, and Molecules on Surfaces with the Local Modified Becke-Johnson Exchange-Correlation Potential

The knowledge of electronic properties of matter is the key to the understanding of its properties and to propose useful applications. To model hybrid organic/inorganic systems with the plane-wave approach, large supercells with many atoms are usually necessary to minimize artificial interactions between periodic images. For such systems, accurate approximations to the exchange-correlation functional of density functional theory, such as hybrid functionals, become computationally expensive, and cheaper approaches need to be considered. Here, we apply the local modified Becke-Johnson exchange-correlation potential to free molecules and surfaces and study its accuracy for calculated ionization potentials. This quantity being important to understand the band alignment of composite heterogeneous systems, we demonstrate the application of the potential to the electronic structure calculation of an exemplary composite semiconductor/molecule system, namely, a F6-TCNNQ molecule adsorbed on a hydrogenated Si(111) surface.

Low-Order Scaling G0W0 by Pair Atomic Density Fitting

We derive a low-scaling G0W0 algorithm for molecules using pair atomic density fitting (PADF) and an imaginary time representation of the Green's function and describe its implementation in the Slater type orbital (STO)-based Amsterdam density functional (ADF) electronic structure code. We demonstrate the scalability of our algorithm on a series of water clusters with up to 432 atoms and 7776 basis functions and observe asymptotic quadratic scaling with realistic threshold qualities controlling distance effects and basis sets of triple-ζ (TZ) plus double polarization quality. Also owing to a very small prefactor, a G0W0 calculation for the largest of these clusters takes only 240 CPU hours with these settings. We assess the accuracy of our algorithm for HOMO and LUMO energies in the GW100 database. With errors of 0.24 eV for HOMO energies on the quadruple-ζ level, our implementation is less accurate than canonical all-electron implementations using the larger def2-QZVP GTO-type basis set. Apart from basis set errors, this is related to the well-known shortcomings of the GW space-time method using analytical continuation techniques as well as to numerical issues of the PADF approach of accurately representing diffuse atomic orbital (AO) products. We speculate that these difficulties might be overcome by using optimized auxiliary fit sets with more diffuse functions of higher angular momenta. Despite these shortcomings, for subsets of medium and large molecules from the GW5000 database, the error of our approach using basis sets of TZ and augmented double-ζ (DZ) quality is decreasing with system size. On the augmented DZ level, we reproduce canonical, complete basis set limit extrapolated reference values with an accuracy of 80 meV on average for a set of 20 large organic molecules. We anticipate our algorithm, in its current form, to be very useful in the study of single-particle properties of large organic systems such as chromophores and acceptor molecules.

Exchange-Correlation Energy Densities and Response Potentials: Connection between Two Definitions and Analytical Model for the Strong-Coupling Limit of a Stretched Bond

We analyze in depth two widely used definitions (from the theory of conditional probability amplitudes and from the adiabatic connection formalism) of the exchange-correlation energy density and of the response potential of Kohn–Sham density functional theory. We introduce a local form of the coupling-constant-dependent Hohenberg–Kohn functional, showing that the difference between the two definitions is due to a corresponding local first-order term in the coupling constant, which disappears globally (when integrated over all space), but not locally. We also design an analytic representation for the response potential in the strong-coupling limit of density functional theory for a model single stretched bond.

The Electron Affinity as the Highest Occupied Anion Orbital Energy with a Sufficiently Accurate Approximation of the Exact Kohn-Sham Potential

Negative ions are not accurately represented in density functional approximations (DFAs) such as (semi)local density functionals (LDA or GGA or meta-GGA). This is caused by the much too high orbital energies (not negative enough) with these DFAs compared to the exact Kohn–Sham values. Negative ions very often have positive DFA HOMO energies, hence they are unstable. These problems do not occur with the exact Kohn–Sham potential, the anion HOMO energy then being equal to minus the electron affinity. It is therefore desirable to develop sufficiently accurate approximations to the exact Kohn–Sham potential. There are further beneficial effects on the orbital shapes and the density of using a good approximation to the exact KS potential. Notably the unoccupied orbitals are not unduly diffuse, as they are in the Hartree–Fock model, with hybrid functionals, and even with (semi)local density functional approximations (LDFAs). We show that the recently developed B-GLLB-VWN approximation [Gritsenko et al. J. Chem. Phys.2016, 144, 204114] to the exact KS potential affords stable negative ions with HOMO orbital energy close to minus the electron affinity.

Beyond Koopmans' theorem: electron binding energies in disordered materials

The topical review focuses on calculating ionization energies (IE), or electronic polarons in quasi-particle terminology, in large disordered systems, e.g. for a solute dissolved in a molecular solvent. The simplest estimate of the ionization energy is provided by one-electron energies in the Hartree-Fock theory, but the calculated quantities are not accurate. Density functional theory as many-body theory provides a principal opportunity for calculating one-electron energies including correlation and relaxation effects, i.e. the true energies of electronic polarons. We argue that such a principal possibility materializes within the concept of optimally tuned range-separated hybrid functionals (OT-RSH). We describe various schemes for optimal tuning. Importantly, the OT-RSH scheme is investigated for systems capped with dielectric continuum, providing a consistent picture on the QM/dielectric boundary. Finally, some limitations and open issues of the OT-RSH approach are addressed.

Halogen and Chalcogen Binding Dominated by Density-Driven Errors

Dispersion corrections of various kinds usually improve DFT energetics of weak noncovalent interactions. However, in some cases involving molecules or halides, especially those with σ-hole interactions, the density-driven errors of uncorrected DFT are larger than the dispersion corrections. In these abnormal situations, HF-DFT (using Hartree-Fock densities instead of self-consistent densities) greatly improves bond energies, while dispersion corrections can even worsen the results. On the other hand, pnictogen bonds and the S22 data set are normal and are not improved by this procedure. Such effects should be accounted for when parametrizing dispersion interactions.

Response Potential in the Strong-Interaction Limit of Density Functional Theory: Analysis and Comparison with the Coupling-Constant Average

Using the formalism of the conditional amplitude, we study the response part of the exchange-correlation potential in the strong-coupling limit of density functional theory, analyzing its peculiar features and comparing it with the response potential averaged over the coupling constant for small atoms and for the hydrogen molecule. We also use a simple one-dimensional model of a stretched heteronuclear molecule to derive exact properties of the response potential in the strong-coupling limit. The simplicity of the model allows us to unveil relevant features also of the exact Kohn-Sham potential and its different components, namely the appearance of a second peak in the correlation kinetic potential on the side of the most electronegative atom.

Density functional approximations for orbital energies and total energies of molecules and solids

The relation of Kohn-Sham (KS) orbital energies to ionization energies and electron affinities is different in molecules and solids. In molecules, the local density approximation (LDA) and generalized gradient approximations (GGA) approximate the exact ionization energy (I) and affinity (A) rather well with self-consistently calculated (total energy based) I^LDFA and A^LDFA, respectively. The highest occupied molecular orbital (HOMO) energy and lowest unoccupied molecular orbital (LUMO) energy, however, differ significantly (by typically 4-6 eV) from these quantities, ϵ_H^LDFA(mol)>-I(mol)≈-I^LDFA(mol), ϵ_L^LDFA(mol)<-A(mol)≈-A^LDFA(mol). In solids, these relations are very different, due to two effects. The (almost) infinite extent of a solid makes the difference of orbital energies and (L)DFA calculated ionization energy and affinity disappear: in the solid state limit, ϵ_H^(L)DFA(solid)=-I^(L)DFA(solid) and ϵ_L^(L)DFA(solid)=-A^(L)DFA(solid). Slater's relation ∂E/∂n_i = ϵ_i for local density functional approximations (LDFAs) [and Hartree-Fock (HF) and hybrids] is useful to prove these relations. The equality of LDFA orbital energies and LDFA calculated -I^LDFA and -A^LDFA in solids does not mean that they are good approximations to the exact quantities. The LDFA total energies of the ions with a delocalized charge are too low, hence I^LDFA(solid) < I and A^LDFA(solid) > A, due to the local-approximation error, also denoted delocalization error, of LDFAs in extended systems. These errors combine to make the LDFA orbital energy band gap considerably smaller than the exact fundamental gap, ϵ_L^LDFA(solid)-ϵ_H^LDFA(solid)=I^LDFA(solid)-A^LDFA(solid)<I-A (the LDFA band gap problem). These results for density functional approximations are compared to exact KS and to HF and hybrids. For the exact KS HOMO energy, one has ϵ_H^KS=-I. The exact KS LUMO energy does not approximate the experimental -A (neither in molecules nor in solids), but is considerably below, which is the main reason for the exact KS HOMO-LUMO energy gap being considerably below the fundamental gap I - A (the exact KS band gap problem).

Simple Fully Nonlocal Density Functionals for Electronic Repulsion Energy

From a simplified version of the mathematical structure of the strong coupling limit of the exact exchange-correlation functional, we construct an approximation for the electronic repulsion energy at physical coupling strength, which is fully nonlocal. This functional is self-interaction free and yields energy densities within the definition of the electrostatic potential of the exchange-correlation hole that are locally accurate and have the correct asymptotic behavior. The model is able to capture strong correlation effects that arise from chemical bond dissociation, without relying on error cancellation. These features, which are usually missed by standard density functional theory (DFT) functionals, are captured by the highly nonlocal structure, which goes beyond the "Jacob's ladder" framework for functional construction, by using integrals of the density as the key ingredient. Possible routes for obtaining the full exchange-correlation functional by recovering the missing kinetic component of the correlation energy are also implemented and discussed.

Testing Exact Upper Bounds to Exact Exchange

The exact exchange energy and its energy density are useful but computationally expensive ingredients in density functional approximations for Kohn-Sham density functional theory. We present detailed tests of some exact nonempirical upper bounds to exact exchange. These "Rung 3.5" upper bounds contract the Kohn-Sham one-particle density matrix with model density matrices used to construct semilocal model exchange holes and invoke the Cauchy-Schwarz inequality. The contraction automatically eliminates the computationally expensive long-range component of the exact exchange hole. Numerical tests show that the exchange upper bounds underestimate total exchange energies while predicting other properties with accuracy approaching standard hybrid approximations.

Effects of the locality of a potential derived from hybrid density functionals on Kohn-Sham orbitals and excited states

Density functional theory (DFT) has been an essential tool for electronic structure calculations in various fields. In particular, its hybrid method including the Hartree-Fock (HF) exchange term remarkably improves the reliability of DFT for chemical applications and computational material design. There are two different types of exchange-correlation potential that can be derived from hybrid functionals. The conventional approach adopts a non-multiplicative potential including the non-local HF exchange operator. Herein, we propose to use a local multiplicative potential as an alternative for accurate excited state calculations. We show that such a local potential can be derived from existing global hybrid functionals using the optimized effective potential method. As a proof-of-concept, we chose PBE0 and investigated its performance for the Caricato benchmark set. Unlike the conventional one, the local potential produced orbital energy gaps with no strong dependence on the mixing ratio as a good approximation for optical excitations. Furthermore, its time-dependent DFT resulted in a surprisingly small mean absolute error even with a local density approximation kernel, surpassing all reported values with various popular functionals. In particular, most excitations were dictated by single orbital transitions due to physically meaningful virtual orbitals, which is beneficial to clear interpretations in the molecular orbital picture.

HLE16: A Local Kohn-Sham Gradient Approximation with Good Performance for Semiconductor Band Gaps and Molecular Excitation Energies

Local exchange-correlation functionals have low cost and convenient portability but are known to seriously underestimate semiconductor band gaps and the energies of molecular Rydberg states. Here we present a new local approximation to the exchange-correlation functional called HLE16 that gives good performance for semiconductor band gaps and molecular excitation energies and is competitive with hybrid functionals. By the simultaneous increase of the local exchange and decrease of the local correlation, electronic excitation energies were improved without excessively degrading the ground-state solid-state cohesive energies, molecular bond energies, or chemical reaction barrier heights, although the new functional is not recommended for optimizing lattice constants or molecular bond lengths. The new functional can be useful as-is for calculations on semiconductors or excited states where it is essential to control the cost, and it can also be useful in establishing a starting point for developing even better new functionals that perform well for excited states.

Reducing density-driven error without exact exchange

PBE calculations, performed non-self-consistently on densities evaluated with Rung 3.5 density functionals, give improved performance for hydrogen transfer reaction barriers.

From the Kohn–Sham band gap to the fundamental gap in solids. An integer electron approach

The upshift Δ of the level at the bottom of the conduction band (the LUMO) from the neutral N-electron crystal to the negative N + 1 system, and therefore the fundamental gap ε_LUMO(N + 1) − ε_HOMO(N) = I − A, can be calculated simply and cheaply from the response part of v_xc.