Density-functional errors in ionization potential with increasing system size

Correcting Dispersion Corrections with Density-Corrected DFT

Almost all empirical parametrizations of dispersion corrections in DFT use only energy errors, thereby mixing functional and density-driven errors. We introduce density and dispersion-corrected DFT (D2C-DFT), a dual-calibration approach that accounts for density delocalization errors when parametrizing dispersion interactions. We simply exclude density-sensitive reactions from the training data. We find a significant reduction in both errors and variation among several semilocal functionals and their global hybrids when tailored dispersion corrections are employed with Hartree-Fock densities.

First and Second Reductions in an Aprotic Solvent: Comparing Computational and Experimental One-Electron Reduction Potentials for 345 Quinones

Using reference reduction potentials of quinones recently measured relative to the saturated calomel electrode (SCE) in N,N-dimethylformamide (DMF), we benchmark absolute one-electron reduction potentials computed for 345 Q/Q•- and 265 Q•-/Q2- half-reactions using adiabatic electron affinities computed with density functional theory and solvation energies computed with four continuum solvation models: IEF-PCM, C-PCM, COSMO, and SM12. Regression analyses indicate a strong linear correlation between experimental and absolute computed Q/Q•- reduction potentials with Pearson's correlation coefficient (r) between 0.95 and 0.96 and the mean absolute error (MAE) relative to the linear fit between 83.29 and 89.51 mV for different solvation methods when the slope of the regression is constrained to 1. The same analysis for Q•-/Q2- gave a linear regression with r between 0.74 and 0.90 and MAE between 95.87 and 144.53 mV, respectively. The y-intercept values obtained from the linear regressions are in good agreement with the range of absolute reduction potentials reported in the literature for the SCE but reveal several sources of systematic error. The y-intercepts from Q•-/Q2- calculations are lower than those from Q/Q•- by around 320-410 mV for IEF-PCM, C-PCM, and SM12 compared to 210 mV for COSMO. Systematic errors also arise between molecules having different ring sizes (benzoquinones, naphthoquinones, and anthraquinones) and different substituents (titratable vs nontitratable). SCF convergence issues were found to be a source of random error that was slightly reduced by directly optimizing the solute structure in the continuum solvent reaction field. While SM12 MAEs were lower than those of the other solvation models for Q/Q•-, SM12 had larger MAEs for Q•-/Q2- pointing to a larger error when describing multiply charged anions in DMF. Altogether, the results highlight the advantages of, and further need for, testing computational methods using a large experimental data set that is not skewed (e.g., having more titratable than nontitratable substituents on different parent groups or vice versa) to help further distinguish between sources of random and systematic errors in the calculations.

Outstanding improvement in removing the delocalization error by global natural orbital functional

This work assesses the performance of the recently proposed global natural orbital functional (GNOF) against the charge delocalization error. GNOF provides a good balance between static and dynamic electronic correlations leading to accurate total energies while preserving spin, even for systems with a highly multi-configurational character. Several analyses were applied to the functional, namely, (i) how the charge is distributed in super-systems of two fragments, (ii) the stability of ionization potentials while increasing the system size, and (iii) potential energy curves of a neutral and charged diatomic system. GNOF was found to practically eliminate the charge delocalization error in many of the studied systems or greatly improve the results obtained previously with PNOF7.

One-electron self-interaction error and its relationship to geometry and higher orbital occupation

Density Functional Theory (DFT) sees prominent use in computational chemistry and physics; however, problems due to the self-interaction error (SIE) pose additional challenges to obtaining qualitatively correct results. As an unphysical energy an electron exerts on itself, the SIE impacts most practical DFT calculations. We conduct an in-depth analysis of the one-electron SIE in which we replicate delocalization effects for simple geometries. We present a simple visualization of such effects, which may help in future qualitative analysis of the one-electron SIE. By increasing the number of nuclei in a linear arrangement, the SIE increases dramatically. We also show how molecular shape impacts the SIE. Two- and three-dimensional shapes show an even greater SIE stemming mainly from the exchange functional with some error compensation from the one-electron error, which we previously defined [D. R. Lonsdale and L. Goerigk, Phys. Chem. Chem. Phys. 22, 15805 (2020)]. Most tested geometries are affected by the functional error, while some suffer from the density error. For the latter, we establish a potential connection with electrons being unequally delocalized by the DFT methods. We also show how the SIE increases if electrons occupy higher-lying atomic orbitals; seemingly one-electron SIE free methods in a ground are no longer SIE free in excited states, which is an important insight for some popular, non-empirical density functional approximations (DFAs). We conclude that the erratic behavior of the SIE in even the simplest geometries shows that robust DFAs are needed. Our test systems can be used as a future benchmark or contribute toward DFT development.

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.

del Campo J. Charge delocalization error in Piris Natural Orbital Functionals

Piris Natural Orbital Functionals (PNOF) have been recognized as a low-scaling alternative to study strong correlated systems. In this work, we address the performance of the fifth functional (PNOF5) and the seventh functional (PNOF7) to deal with another common problem, the charge delocalization error. The effects of this problem can be observed in charged systems of repeated well-separated fragments, where the energy should be the sum of the charged and neutral fragments, regardless of how the charge is distributed. In practice, an energetic overstabilization of fractional charged fragments leads to a preference for having the charge delocalized throughout the system. To establish the performance of PNOF functionals regarding charge delocalization error, charged chains of helium atoms and the W4-17-MR set molecules were used as base fragments and their energy, charge distribution and correlation regime were studied. It was found that PNOF5 prefers localized charge distributions, while PNOF7 improves the treatment of interpair static correlation and tends to the correct energetic limit for several cases, although a preference for delocalized charge distributions may arise in highly strong correlation regimes. Overall, it is concluded that PNOF functionals can simultaneously deal with static correlation and charge delocalization errors, resulting in a promising choice to study charge-related problems.

Poisoning density functional theory with benchmark sets of difficult systems

Large benchmark sets like GMTKN55 [Goerigk et al., Phys. Chem. Chem. Phys., 2017, 19, 32184] let us analyse the performance of density functional theory over a diverse range of systems and bonding types. However, assessing over a large and diverse set can miss cases where approaches fail badly, and can give a misleading sense of security. To this end we introduce a series of 'poison' benchmark sets, P30-5, P30-10 and P30-20, comprising systems with up to 5, 10 and 20 atoms, respectively. These sets represent the most difficult-to-model systems in GMTKN55. We expect them to be useful in developing new approximations, identifying weak points in existing ones, and to aid in selecting appropriate DFAs for computational studies involving difficult physics, e.g. catalysis.

Linear fractional charge behavior in density functional theory through dielectric tuning of conductor-like polarizable continuum model

A property of exact density functional theory is linear fractional charge behavior as electrons are added or removed from a molecule. Typical density functional approximations (DFAs) exhibit delocalization error, which overstabilizes this fractional charge. Conversely, solvent corrections have been shown to erroneously destabilize this fractional charge. This work will show that an implicit solvent correction with a tuned dielectric can be used as an ad hoc correction to offset the delocalizing character of DFAs and achieve linear fractional charge behavior. While desirable, in principle, we find that this linear charge behavior degrades the vertical ionization energies reported by DFAs. Our results reveal that the localizing character of the solvent correction and the Hartree-Fock (HF) exchange offset each other. This helps explain the decreased ratios of HF exchange to DFA exchange in long-range hybrid tuning studies that use a solvent correction.

Correcting the Charge Delocalization Error of Density Functional Theory

The charge delocalization error, besides nondynamic correlation, has been a major challenge to density functional theory. Contemporary functionals undershoot the dissociation of symmetric charged dimers A₂⁺, a simple but stringent test, predict a spurious barrier, and improperly delocalize charges for charged molecular clusters. We extend a functional designed for nondynamic correlation to treat the charge delocalization error by modifying the nondynamic correlation for parallel spins. The modified functional eliminates those problems and reduces the multielectron self-interaction error. Furthermore, its results are the closest to those of CCSD(T) in the whole range of the dissociation compared with contemporary functionals. It correctly localizes the net positive charge in (CH₄)_n⁺ clusters and predicts a nearly constant ionization potential as a result. Testing of the SIE4x4 set shows that the new functional outperforms a wide variety of functionals assessed for this set in the literature. Overall, we show the feasibility of treating charge delocalization together with nondynamic correlation.

Solvation energies of ions with ensemble cluster-continuum approach

An alternative cluster-continuum approach for the calculation of solvation free energies of ions.

Probing the Partial Activation of Water by Open-Shell Interactions, Cl(H2O)1-4

The partial chemical activation of water by reactive radicals was examined computationally for small clusters of chlorine and water, Cl^•(H₂O)_n=1-4. Using an automated isomer-search procedure, dozens of unique, stable structures were computed. Among the resulting structural classes were intact, hydrated-chlorine isomers, as well as hydrogen-abstracted (HCl)(OH)(H₂O)_n-1 configurations. The latter showed increased stability as the degree of hydration increased, until n = 4, where a new class of structures was discovered with a chloride ion bound to an oxidized water network. The electronic structure of these three structural classes was investigated, and spectral signatures of this hydration-based evolution were connected to these electronic properties. An ancillary outcome of this detailed computational analysis, including coupled-cluster benchmarks, was the calibration of cost-effective quantum chemistry methods for future studies of these radical-water complexes.

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.

Accelerated long-range corrected exchange functional using a two-gaussian operator combined with one-parameter progressive correlation functional [LC-BOP(2Gau)]

Recently, we proposed a simple yet efficient method for the computation of a long-range corrected (LC) hybrid scheme [LC-DFT(2Gau)], which uses a modified two-Gaussian attenuating operator instead of the error function for the long-range HF exchange integral. This method dramatically reduced the computational time while maintaining the improved features of the LC density functional theory (DFT). Here, we combined an LC hybrid scheme using a two-Gaussian attenuating operator with one-parameter progressive correlation functional and Becke88 exchange functional with varying range-separation parameter values [LC-BOP(2Gau) with various μ values of 0.16, 0.2, 0.25, 0.3, 0.35, 0.4, and 0.42] and demonstrated that LC-BOP(2Gau) reproduces well the thermochemical and frontier orbital energies of LC-BOP. Additionally, we revised the scaling factors of the Gaussian multipole screening scheme for LC-DFT(2Gau) to correspond to the angular momentum of orbitals, which decreased the energy deviations from the energy with the no-screening scheme. © 2018 Wiley Periodicals, Inc.

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).

The effect of electronic excitation on London dispersion

Atomic and molecular dispersion coefficients can now be calculated routinely using density-functional theory. In this work, we present the first determination of how electronic excitation affects molecular C6 London dispersion coefficients from the exchange-hole dipole moment (XDM) dispersion model. Excited states are typically found to have larger dispersion coefficients than the corresponding ground states, due to their more diffuse electron densities. A particular focus is both intramolecular and intermolecular charge-transfer excitations, which have high absorbance intensities and are important in organic dyes, light-emitting diodes, and photovoltaics. In these classes of molecules, the increase in C6 for the electron-accepting moiety is largely offset by a decrease in C6 for the electron-donating moiety. As a result, the change in dispersion energy for a chromophore interacting with neighbouring molecules in the condensed phase is minimal.

Communication: Recovering the flat-plane condition in electronic structure theory at semi-local DFT cost

The flat-plane condition is the union of two exact constraints in electronic structure theory: (i) energetic piecewise linearity with fractional electron removal or addition and (ii) invariant energetics with change in electron spin in a half filled orbital. Semi-local density functional theory (DFT) fails to recover the flat plane, exhibiting convex fractional charge errors (FCE) and concave fractional spin errors (FSE) that are related to delocalization and static correlation errors. We previously showed that DFT+U eliminates FCE but now demonstrate that, like other widely employed corrections (i.e., Hartree-Fock exchange), it worsens FSE. To find an alternative strategy, we examine the shape of semi-local DFT deviations from the exact flat plane and we find this shape to be remarkably consistent across ions and molecules. We introduce the judiciously modified DFT (jmDFT) approach, wherein corrections are constructed from few-parameter, low-order functional forms that fit the shape of semi-local DFT errors. We select one such physically intuitive form and incorporate it self-consistently to correct semi-local DFT. We demonstrate on model systems that jmDFT represents the first easy-to-implement, no-overhead approach to recovering the flat plane from semi-local DFT.

Where Does the Density Localize in the Solid State? Divergent Behavior for Hybrids and DFT+U

Approximate density functional theory (DFT) is widely used in chemistry and physics, despite delocalization errors that affect energetic and density properties. DFT+U (i.e., semilocal DFT augmented with a Hubbard U correction) and global hybrid functionals are two commonly employed practical methods to address delocalization error. Recent work demonstrated that in transition-metal complexes both methods localize density away from the metal and onto surrounding ligands, regardless of metal or ligand identity. In this work, we compare density localization trends with DFT+U and global hybrids on a diverse set of 34 transition-metal-containing solids with varying magnetic state, electron configuration and valence shell, and coordinating-atom orbital diffuseness (i.e., O, S, Se). We also study open-framework solids in which the metal is coordinated by molecular ligands, i.e., MCO₃, M(OH)₂, M(NCNH)₂, K₃M(CN)₆ (M = V-Ni). As in transition-metal complexes, incorporation of Hartree-Fock exchange consistently localizes density away from the metal, but DFT+U exhibits diverging behavior, localizing density (i) onto the metal in low-spin and late transition metals and (ii) away from the metal in other cases in agreement with hybrids. To isolate the effect of the crystal environment, we extract molecular analogues from open-framework transition-metal solids and observe consistent localization of the density away from the metal in all cases with both DFT+U and hybrid exchange. These observations highlight the limited applicability of trends established for functional tuning on transition-metal complexes even to equivalent coordination environments in the solid state.

Intramolecular London Dispersion Interaction Effects on Gas-Phase and Solid-State Structures of Diamondoid Dimers

The covalent diamantyl (C₂₈H₃₈) and oxadiamantyl (C₂₆H₃₄O₂) dimers are stabilized by London dispersion attractions between the dimer moieties. Their solid-state and gas-phase structures were studied using a multitechnique approach, including single-crystal X-ray diffraction (XRD), gas-phase electron diffraction (GED), a combined GED/microwave (MW) spectroscopy study, and quantum chemical calculations. The inclusion of medium-range electron correlation as well as the London dispersion energy in density functional theory is essential to reproduce the experimental geometries. The conformational dynamics computed for C₂₆H₃₄O₂ agree well with solution NMR data and help in the assignment of the gas-phase MW data to individual diastereomers. Both in the solid state and the gas phase the central C-C bond is of similar length for the diamantyl [XRD, 1.642(2) Å; GED, 1.630(5) Å] and the oxadiamantyl dimers [XRD, 1.643(1) Å; GED, 1.632(9) Å; GED+MW, 1.632(5) Å], despite the presence of two oxygen atoms. Out of a larger series of quantum chemical computations, the best match with the experimental reference data is achieved with the PBEh-3c, PBE0-D3, PBE0, B3PW91-D3, and M06-2X approaches. This is the first gas-phase confirmation that the markedly elongated C-C bond is an intrinsic feature of the molecule and that crystal packing effects have only a minor influence.

An Ab Initio Exciton Model Including Charge-Transfer Excited States

The Frenkel exciton model is a useful tool for theoretical studies of multichromophore systems. We recently showed that the exciton model could be used to coarse-grain electronic structure in multichromophoric systems, focusing on singly excited exciton states [ Acc. Chem. Res. 2014 , 47 , 2857 - 2866 ]. However, our previous implementation excluded charge-transfer excited states, which can play an important role in light-harvesting systems and near-infrared optoelectronic materials. Recent studies have also emphasized the significance of charge-transfer in singlet fission, which mediates the coupling between the locally excited states and the multiexcitonic states. In this work, we report on an ab initio exciton model that incorporates charge-transfer excited states and demonstrate that the model provides correct charge-transfer excitation energies and asymptotic behavior. Comparison with TDDFT and EOM-CC2 calculations shows that our exciton model is robust with respect to system size, screening parameter, and different density functionals. Inclusion of charge-transfer excited states makes the exciton model more useful for studies of singly excited states and provides a starting point for future construction of a model that also includes double-exciton states.

The ionic versus metallic nature of 2D electrides: a density-functional description

The two-dimensional (2D) electrides are a highly unusual class of materials, possessing interstitial electron layers sandwiched between cationic atomic layers of the solid.

Density functional theory modeling of C–Au chemical bond formation in gold implanted polyethylene

We have studied processes of gold ion implantation in polyethylene (PE) by theoretical chemistry methods.

Interpretation of molecular device transport calculations

The field of molecular electronics will benefit from rational design approaches based on a complete understanding of the electronic structure of molecule-based devices. However, many computational approaches that are used to study molecular-scale devices are based on methods that have deficiencies that must be understood in order for those methods to be useful to the modeling and experimental community. Density-functional theory based methods have some well-known pitfalls that limit their application to the study of electron transport in models of molecular junction devices. Some of the impacts of these deficiencies are highlighted in this work through the use of a graphene model system and a variety of simple hydrocarbon molecules. Self-interaction error in simple functionals built from the local density approximation and the generalized gradient approximation results in very large errors in predicted absolute and relative ionization potentials. This demonstrates that electron transmission spectra predicted using these functionals should be considered with caution. We also demonstrate that care must be taken with the use of finite models for electrodes.