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

Extended N-centered ensemble density functional theory of double electronic excitations

A recent work (arXiv:2401.04685) has merged N-centered ensembles of neutral and charged electronic ground states with ensembles of neutral ground and excited states, thus providing a general and in-principle exact (so-called extended N-centered) ensemble density functional theory of neutral and charged electronic excitations. This formalism made it possible to revisit the concept of density-functional derivative discontinuity, in the particular case of single excitations from the highest occupied Kohn-Sham (KS) molecular orbital, without invoking the usual "asymptotic behavior of the density" argument. In this work, we address a broader class of excitations, with a particular focus on double excitations. An exact implementation of the theory is presented for the two-electron Hubbard dimer model. A thorough comparison of the true physical ground- and excited-state electronic structures with that of the fictitious ensemble density-functional KS system is also presented. Depending on the choice of the density-functional ensemble as well as the asymmetry of the dimer and the correlation strength, an inversion of states can be observed. In some other cases, the strong mixture of KS states within the true physical system makes the assignment "single excitation" or "double excitation" irrelevant.

Tight-binding model predicts exciton energetics and structure for photovoltaic molecules

Conjugated molecules and polymers are being designed as acceptor and donor materials for organic photovoltaic (OPV) cells. OPV performance depends on generation of free charge carriers through dissociation of excitons, which are electron-hole pairs created when a photon is absorbed. Here, we develop a tight-binding model to describe excitons on homo-oligomers, alternating co-oligomers, and a non-fullerene acceptor - IDTBR. We parameterize our model using density functional theory (DFT) energies of neutral, anion, cation, and excited states of constituent moieties. A symmetric molecule like IDTBR has two ends where an exciton can sit; but the product wavefunction approximation for the exciton breaks symmetry. So, we introduce a tight-binding model with full correlation between electron and hole, which allows the exciton to coherently explore both ends of the molecule. Our approach predicts optical singlet excitation energies for oligomers of varying length as well as IDTBR in good agreement with time-dependent DFT and spectroscopic results.

Basis Electronic Activity of Molecular Systems. A Theory of Bond Reactivity

In this paper, we present a new finding, the basis electronic activity (BEA) of molecular systems; it corresponds to the significant, although nonreactive, vibrationally induced electronic activity that takes place in any molecular system. Although the molecule's BEA is composed of an equal number of local contributions as the vibrational degrees of freedom, our results indicate that only stretching modes contribute to it. To account for this electronic activity, a new descriptor, the bond electronic flux (BEF), is introduced. The BEF combined with the force constant of the potential well hosting the electronic activity gives rise to the effective bond reactivity index (EBR), which turns out to be the first density functional theory-based descriptor that simultaneously accounts for structural and electronic effects. Besides quantifying the bond reactivity, EBR provides a basis to compare the reactivities of bonds inserted in different chemical environments and paves the way for the exertion of selective control to enhance or inhibit their reactivities. The new concepts formulated in this paper and the associated computational tools are illustrated with characterization of the BEA of a set of representative molecules. In all cases, the BEFs follow the same linear pattern, whose slopes indicate the intensity of the electronic activity and quantify the reactivity of chemical bonds.

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.

Role of Exact Exchange in Difference Projected Double-Hybrid Density Functional Theory for Treatment of Local, Charge Transfer, and Rydberg Excitations

Difference approaches to the study of excited states have undergone a renaissance in recent years, with the development of a plethora of algorithms for locating self-consistent field approximations to excited states. Density functional theory is likely to offer the best balance of cost and accuracy for difference approaches, and yet there has been little investigation of how the parametrization of density functional approximations affects performance. In this work, we aim to explore the role of the global Hartree-Fock exchange parameter in tuning accuracy of different excitation types within the framework of the recently introduced difference projected double-hybrid density functional theory approach and contrast the performance with conventional time-dependent double-hybrid density functional theory. Difference projected double-hybrid density functional theory was demonstrated to give vertical excitation energies with average error and standard deviation with respect to multireference perturbation theory comparable to more expensive linear-response coupled cluster approaches ( J. Chem. Phys.2020, 153, 074103). However, despite benchmarking of local excitations, there has been no investigation of the methods performance for charge transfer or Rydberg excitations. In this work we report a new benchmark of charge transfer, Rydberg, and local excited state vertical excitation energies and examine how the exact Hartree-Fock exchange affects the benchmark performance to provide a deeper understanding of how projection and nonlocal correlation balance differing sources of error in the ground and excited states.

Structural evolution and electronic properties of pure and semiconductor atom doped in clusters: Inn - , Inn Si- , and Inn Ge- (n = 3-16)

The bonding and electronic properties of In_n ^- , In_n Si^- , and In_n Ge^- (n = 3-16) clusters have been computationally investigated. An intensive global search for the ground-state structures of these clusters were conducted using the genetic algorithm coupled with density functional theory (DFT). The ground-state structures of these clusters have been identified through the comparison between simulated photoelectron spectra (PES) of the found lowest-energy isomers and the experimentally measured ones. Doping semiconductor atom (Si or Ge) can significantly change the structures of the In clusters in most sizes, and the dopant prefers to be surrounded by In atoms. There are three structural motifs for In_n X^- (X = Si, Ge, n = 3-16), and the transition occurs at sizes n = 5 and 13. All In_n Si^- and In_n Ge^- share the same configurations and similar electronic properties except for n = 8. Among all above studied clusters, In₁₃ ^- stands out with the largest vertical detachment energy (VDE), HOMO-LUMO gap, (E_b ) and second order energy difference Δ² E due to its closed electronic shell of (1S)² (1P)⁶ (1D)¹⁰ (2S)² (1F)¹⁴ (2P)⁶ . Similarly, the neutral In₁₂ X (X = Si, Ge) clusters are also identified as superatoms but with electronic configuration of (1S)² (1P)⁶ (2S)² (1D)¹⁰ (1F)¹⁴ (2P)⁶ .

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.

Recommendation of Orbitals for G0W0 Calculations on Molecules and Crystals

The many-body GW approximation, especially the G₀W₀ method, has been widely used for condensed matter and molecules to calculate quasiparticle energies for ionization, electron attachment, and band gaps. Because G₀W₀ calculations are well-known to have a strong dependence on the orbitals, the goal of the present work is to provide guidance on the choice of density functional used to generate orbitals and to recommend a choice that gives the most broadly accurate results. We have systematically investigated the dependence of G₀W₀ calculations on the orbitals for 100 molecules and 8 crystals by considering orbitals obtained with a diverse set of Kohn-Sham (KS) and generalized KS (GKS) functionals (63 functionals plus Hartree-Fock). The percentage of Hartree-Fock exchange employed in density functionals has been found to have strong influence on the predicted molecular ionization energy and crystal fundamental band gaps (with optimum values between 40 and 56%), but to have less effect on predicting molecular electron affinities. The low cost of the Gaussian implementation, even with hybrid functionals in periodic calculations, the better performance of global hybrids as compared to range-separated hybrids of either than screened exchange or long-range-corrected type, and the relatively low cost of global-hybrid-functional periodic calculations using Gaussians means that one can employ global-hybrid functionals at a very reasonable cost and obtain more accurate band gaps of semiconductors than are obtained by the methods currently widely employed, namely local gradient approximations. We single out three global-hybrid functionals that give especially good results for both molecules (100 in the test set) and crystals (8 in the test set, for all of which our benchmark data are the proper band gap rather than an optical band gap uncorrected for exciton effects).

Machine learning the derivative discontinuity of density-functional theory

Machine learning is a powerful tool to design accurate, highly non-local, exchange-correlation functionals for density functional theory. So far, most of those machine learned functionals are trained for systems with an integer number of particles. As such, they are unable to reproduce some crucial and fundamental aspects, such as the explicit dependency of the functionals on the particle number or the infamous derivative discontinuity at integer particle numbers. Here we propose a solution to these problems by training a neural network as the universal functional of density-functional theory that (a) depends explicitly on the number of particles with a piece-wise linearity between the integer numbers and (b) reproduces the derivative discontinuity of the exchange-correlation energy. This is achieved by using an ensemble formalism, a training set containing fractional densities, and an explicitly discontinuous formulation.

d- and s-orbital populations in the d block: unbound atoms in physical vacuum versus chemical elements in condensed matter. A Dronskowski-population analysis

The electron configurations of Ca, Zn and the nine transition elements M in between (and their heavier homologs) are reviewed on the basis of density functional theory and experimental facts. The d-s orbital energy and population patterns are systematically diverse. (i) The dominant valence electron configuration of most free neutral atoms M0 of groups g = 2–12 is 3d g−2 4s 2 (textbook rule), or 3d g−14s 1. (ii) Formal M q+ cations in chemical compounds have the dominant configuration 3d g−q 4s 0 (basic concept of transition metal chemistry). (iii) M0 atoms in metallic phases [M∞] of hcp, ccp(fcc) and bcc structures have intermediate populations near 3d g−1 4s 1 (lower d populations for Ca (ca. ½) and Zn (ca. 10)). Including the 4p valence orbitals, the dominant metallic configuration is 3d g−δ 4(sp) δ with δ ≈ 1.4 (±0.2) throughout (except for Zn). (iv) The 3d,4s population of atomic clusters M m varies for increasing m smoothly from single-atomic 3d g−24s 2 toward metallic 3d g−14s 1. – The textbook rule for the one-electron energies, i.e., ns < (n−1)d, holds ‘in a broader sense’ for the s block, but in general not for the d block, and never for the p block. It is more important to teach realistic atomic orbital (AO) populations such as the ones given above.

Relationships between Orbital Energies, Optical and Fundamental Gaps, and Exciton Shifts in Approximate Density Functional Theory and Quasiparticle Theory

The relationships between Kohn-Sham (KS) and generalized KS (GKS) density functional orbital energies and fundamental gaps or optical gaps raise many interesting questions including the physical meanings of KS and GKS orbital energies when computed with currently available approximate density functionals (ADFs). In this work, by examining three diverse databases with various ADFs, we examine such relations from the point of view of the exciton shift of quasiparticle theory. We start by calculating a large number of excitation energies by time-dependent density functional theory (TDDFT) with a large number of ADFs. To relate the exciton shift implicit in TDDFT to the exciton shift that is explicit in Green's function theory, we define the exciton shift in TDDFT as the difference of the response shift and the quasiparticle shift. We found a strong correlation between the response shift and the amount of Hartree-Fock exchange included in the density functional, with the response shift varying between -1 and 5 eV. This range is an order of magnitude larger than the mean errors of the TDDFT excitation energies. This result suggests that, with currently available functionals, the KS or GKS orbital energies should be treated as intermediate mathematical variables in the calculation of excitation energies rather than as the energies of independent-particle reference states for quasiparticle theory.

Nanocrystals of Lead Chalcohalides: A Series of Kinetically Trapped Metastable Nanostructures

We report the colloidal synthesis of a series of surfactant-stabilized lead chalcohalide nanocrystals. Our work is mainly focused on Pb4S3Br2, a chalcohalide phase unknown to date that does not belong to the ambient-pressure PbS-PbBr2 phase diagram. The Pb4S3Br2 nanocrystals herein feature a remarkably narrow size distribution (with a size dispersion as low as 5%), a good size tunability (from 7 to ∼30 nm), an indirect bandgap, photoconductivity (responsivity = 4 ± 1 mA/W), and stability for months in air. A crystal structure is proposed for this new material by combining the information from 3D electron diffraction and electron tomography of a single nanocrystal, X-ray powder diffraction, and density functional theory calculations. Such a structure is closely related to that of the recently discovered high-pressure chalcohalide Pb4S3I2 phase, and indeed we were able to extend our synthesis scheme to Pb4S3I2 colloidal nanocrystals, whose structure matches the one that has been published for the bulk. Finally, we could also prepare nanocrystals of Pb3S2Cl2, which proved to be a structural analogue of the recently reported bulk Pb3Se2Br2 phase. It is remarkable that one high-pressure structure (for Pb4S3I2) and two metastable structures that had not yet been reported (for Pb4S3Br2 and Pb3S2Cl2) can be prepared on the nanoscale by wet-chemical approaches. This highlights the important role of colloidal chemistry in the discovery of new materials and motivates further exploration into metal chalcohalide nanocrystals.

Band Alignment as the Method for Modifying Electronic Structure of Metal-Organic Frameworks

Electronic-level ordering in metal-organic frameworks (MOFs) is a route to modulate their electronic properties such as optical absorption, band alignment, work function, charge separation, charge carrier lifetimes, and ground- or excited-state conductivity. A systematic application of this approach requires the knowledge on how a MOF chemical composition affects its electronic structure. In this work, the fundamental principles for selecting MOF components to achieve targeted level alignment are considered. Correlations between the electronic parameters of building blocks and MOF band structure are analyzed. The factors affecting the energy position of constituents are discussed. In particular, the impact of the chemical composition of ligands, including the structure of its scaffold and side groups, on their energy positions in MOFs is addressed. Besides, the effect of the choice of reference potential and surface termination on the band alignment is investigated. The performance of several density functionals in the computation of absolute band positions is assessed. Finally, general principles for the modification of the MOF electronic structure are formulated and the routes to achieve an appropriate band alignment with carrier-transporting materials, co-catalysts, and redox reaction potentials are suggested.

Negative Electron Affinities and Derivative Discontinuity Contribution from a Generalized Gradient Approximation Exchange Functional

Two methods to calculate negative electron affinities systematically from ground-state density functional methods are presented. One makes use of the lowest unoccupied molecular orbital energy shift provided by approximate inclusion of derivative discontinuity in the nearly correct asymptotic potential (NCAP) nonempirical, constraint-based generalized gradient approximation exchange functional. The other uses a second-order perturbation calculation of the derivative discontinuity based on the NCAP exchange-correlation potential. On a set of thirty-eight molecules, NCAP leads to a rather accurate description that is improved further through the perturbation correction. The results presented show the importance of the asymptotic behavior of the exchange-correlation potential in the calculation of negative electron affinities as well as demonstrating the versatility of the NCAP functional.

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.

Oganesson Is a Semiconductor: On the Relativistic Band-Gap Narrowing in the Heaviest Noble-Gas Solids

Oganesson (Og) is the most recent addition to Group 18. Investigations of its atomic electronic structure have unraveled a tremendous impact of relativistic effects, raising the question whether the heaviest noble gas lives up to its position in the periodic table. To address the issue, we explore the electronic structure of bulk Og by means of relativistic Kohn–Sham density functional theory and many‐body perturbation theory in the form of the GW method. Calculating the band structure of the noble‐gas solids from Ne to Og, we demonstrate excellent agreement for the band gaps of the experimentally known solids from Ne to Xe and provide values of 7.1 eV and 1.5 eV for the unknown solids of Rn and Og. While this is in line with periodic trends for Rn, the band gap of Og completely breaks with these trends. The surprisingly small band gap of Og moreover means that, in stark contrast to all other noble‐gas solids, the solid form of Og is a semiconductor.

Feature Engineering for Materials Chemistry-Does Size Matter?

The effects of structural featurizers in the prediction of band gaps have been investigated through machine learning by application to a silver nanoparticle data set and 2254 potential light-harvesting materials with known band gaps. Elemental properties were extended with structural features via Voronoi polyhedra, allowing for neighbor effects and thus presumably giving a better representation of the extended system. However, we did not find any noticeably significant difference in the predictive performance of our model. The biggest improvement in our model was due to inclusion of band gaps calculated using density functional theory. This resulted in a model that could predict the band gaps of the 2254 light-harvesting materials in the data set with an accuracy reflected in a root-mean-square error of 0.232 eV and mean absolute error of 0.142 eV. Furthermore, the good performance of our model was transferable to the prediction of a set of 72 experimental band gaps that were independent of the training set, giving a root-mean-square error of 0.91 eV and mean absolute error of 0.76 eV.

Transport and Optical Gaps in Amorphous Organic Molecular Materials

The standard procedure to identify the hole- or electron-acceptor character of amorphous organic materials used in OLEDs is to look at the values of a pair of basic parameters, namely, the ionization potential (IP) and the electron affinity (EA). Recently, using published experimental data, the present authors showed that only IP matters, i.e., materials with IP > 5.7 (<5.7) showing electron (hole) acceptor character. Only three materials fail to obey this rule. This work reports ab initio calculations of IP and EA of those materials plus two materials that behave according to that rule, following a route which describes the organic material by means of a single molecule embedded in a polarizable continuum medium (PCM) characterized by a dielectric constant ε . PCM allows to approximately describe the extended character of the system. This "compound" system was treated within density functional theory (DFT) using several combinations of the functional/basis set. In the preset work ε was derived by assuming Koopmans' theorem to hold. Optimal ε values are in the range 4.4⁻5.0, close to what is expected for this material family. It was assumed that the optical gap corresponds to the excited state with a large oscillator strength among those with the lowest energies, calculated with time-dependent DFT. Calculated exciton energies were in the range 0.76⁻1.06 eV, and optical gaps varied from 3.37 up to 4.50 eV. The results are compared with experimental data.

Simple DFT Scheme for Estimating Negative Electron Affinities

A simple density functional theory (DFT) scheme is proposed for estimating negative vertical electron affinities of neutral systems, based on a consideration of the integer discontinuity and density scaling homogeneity. The key feature is the derivation of two system-dependent exchange-correlation functionals, one appropriate for the electron deficient side of the integer and one appropriate for the electron abundant side. The electron affinity is evaluated as a linear combination of frontier orbital energies from self-consistent Kohn-Sham calculations on the neutral system using these functionals. For two assessments comprising a total of 43 molecules, the scheme provides electron affinities that are in good agreement with experimental values and which are an improvement over those from the DFT method of Tozer and De Proft [ J. Phys. Chem. A 2005 , 109 , 8923 ]. The scheme is trivial to implement in any Kohn-Sham program, and the computational cost is that of a series of generalized gradient approximation Kohn-Sham calculations. More generally, the study provides a prescription for performing low-cost, self-consistent Kohn-Sham calculations that yield frontier orbital energies that approximately satisfy the appropriate Koopmans conditions, without the need for exact exchange.