Density Functional Theory Errors Research Articles

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.

Going for Gold(-Standard): Attaining Coupled Cluster Accuracy in Oxide-Supported Nanoclusters.

The structure of oxide-supported metal nanoclusters plays an essential role in their sharply enhanced catalytic activity over that of bulk metals. Simulations provide the atomic-scale resolution needed to understand these systems. However, the sensitive mix of metal-metal and metal-support interactions, which govern their structure, puts stringent requirements on the method used, requiring calculations beyond standard density functional theory (DFT). The method of choice is coupled cluster theory [specifically CCSD(T)], but its computational cost has so far prevented its application to these systems. In this work, we showcase two approaches to make CCSD(T) accuracy readily achievable in oxide-supported nanoclusters. First, we leverage the SKZCAM protocol to provide the first benchmarks of oxide-supported nanoclusters, revealing that it is specifically metal-metal interactions that are challenging to capture with DFT. Second, we propose a CCSD(T) correction (ΔCC) to the metal-metal interaction errors in DFT, reaching accuracy comparable to that of the SKZCAM protocol at significantly lower cost. This approach forges a path toward studying larger systems at reliable accuracy, which we highlight by identifying a ground-state structure in agreement with experiments for Au20 on MgO, a challenging system where DFT models have yielded conflicting predictions.

AFLOW-CCE for the thermodynamics of ionic materials.

Accurate thermodynamic stability predictions enable data-driven computational materials design. Standard density functional theory (DFT) approximations have limited accuracy with average errors of a few hundred meV/atom for ionic materials, such as oxides and nitrides. Thus, insightful correction schemes as given by the coordination corrected enthalpies (CCE) method, based on an intuitive parametrization of DFT errors with respect to coordination numbers and cation oxidation states, present a simple, yet accurate solution to enable materials stability assessments. Here, we illustrate the computational capabilities of our AFLOW-CCE software by utilizing our previous results for oxides and introducing new results for nitrides. The implementation reduces the deviations between theory and experiment to the order of the room temperature thermal energy scale, i.e., ∼25 meV/atom. The automated corrections for both materials classes are freely available within the AFLOW ecosystem via the AFLOW-CCE module, requiring only structural inputs.

Electronic structure analysis and DFT benchmarking of Rydberg-type alkali-metal-crown ether, -cryptand, and -adamanzane complexes.

Density functional theory (DFT) and electron propagator theory (EPT) calculations were performed to study ground and excited electronic structures of alkali-metal (M) coordinated 9-crown-3, 24-crown-8, [2.1.1]cryptand, o-Me2-1.1.1, and 36Adamanzane complexes. Each complex bears an expanded electron in the periphery and occupies diffuse 1p-, 1d-, 1f-type molecular orbitals (or superatomic 1P, 1D, 1F orbitals) in excited electronic states. The calculated superatomic shell model of the M(9-crown-3)2 is 1S, 1P, 1D, 1F, 2S, 2P, 2D, 1G and it is held by all other complexes up to the studied 1F level. Due to the highly diffuse nature of the electron, the ionization energies of these complexes are significantly lower (1.6-2.0 eV) and hence these complexes belong to the superalkali category. The ab initio EPT ionization energy and the excitation energies of the Li(9-crown-3)2 were used to evaluate DFT errors associated with a series of exchange correlation functionals that span multiple rungs of Jacob's ladder (i.e., GGA, meta-GGA, global GGA hybrid, meta-GGA hybrid, range-separated hybrid, double-hybrid). Among these, the best performing functional is the range-separated hybrid CAM-B3LYP and the errors are within 6% of high-level ab initio EPT results. The accuracy of CAM-B3LYP is indeed transferable to similar complexes and hence the findings are expected to accelerate the progression of studies of Rydberg-type systems.

Interpretable delta-learning of GW quasiparticle energies from GGA-DFT

Accurate prediction of the ionization potential and electron affinity energies of small molecules are important for many applications. Density functional theory (DFT) is computationally inexpensive, but can be very inaccurate for frontier orbital energies or ionization energies. The GW method is sufficiently accurate for many relevant applications, but much more expensive than DFT. Here we study how we can learn to predict orbital energies with GW accuracy using machine learning (ML) on molecular graphs and fingerprints using an interpretable delta-learning approach. ML models presented here can be used to predict quasiparticle energies of small organic molecules even beyond the size of the molecules used for training. We furthermore analyze the learned DFT-to-GW corrections by mapping them to specific localized fragments of the molecules, in order to develop an intuitive interpretation of the learned corrections, and thus to better understand DFT errors.

High-Throughput Condensed-Phase Hybrid Density Functional Theory for Large-Scale Finite-Gap Systems: The SeA Approach.

High-throughput electronic structure calculations (often performed using density functional theory (DFT)) play a central role in screening existing and novel materials, sampling potential energy surfaces, and generating data for machine learning applications. By including a fraction of exact exchange (EXX), hybrid functionals reduce the self-interaction error in semilocal DFT and furnish a more accurate description of the underlying electronic structure, albeit at a computational cost that often prohibits such high-throughput applications. To address this challenge, we have constructed a robust, accurate, and computationally efficient framework for high-throughput condensed-phase hybrid DFT and implemented this approach in the PWSCF module of Quantum ESPRESSO (QE). The resulting SeA approach (SeA = SCDM + exx + ACE) combines and seamlessly integrates: (i) the selected columns of the density matrix method (SCDM, a robust noniterative orbital localization scheme that sidesteps system-dependent optimization protocols), (ii) a recently extended version of exx (a black-box linear-scaling EXX algorithm that exploits sparsity between localized orbitals in real space when evaluating the action of the standard/full-rank operator), and (iii) adaptively compressed exchange (ACE, a low-rank approximation). In doing so, SeA harnesses three levels of computational savings: pair selection and domain truncation from SCDM + exx (which only considers spatially overlapping orbitals on orbital-pair-specific and system-size-independent domains) and low-rank approximation from ACE (which reduces the number of calls to SCDM + exx during the self-consistent field (SCF) procedure). Across a diverse set of 200 nonequilibrium (H2O)64 configurations (with densities spanning 0.4-1.7 g/cm3), SeA provides a 1-2 order-of-magnitude speedup in the overall time-to-solution, i.e., ≈8-26× compared to the convolution-based PWSCF(ACE) implementation in QE and ≈78-247× compared to the conventional PWSCF(Full) approach, and yields energies, ionic forces, and other properties with high fidelity. As a proof-of-principle high-throughput application, we trained a deep neural network (DNN) potential for ambient liquid water at the hybrid DFT level using SeA via an actively learned data set with ≈8,700 (H2O)64 configurations. Using an out-of-sample set of (H2O)512 configurations (at nonambient conditions), we confirmed the accuracy of this SeA-trained potential and showcased the capabilities of SeA by computing the ground-truth ionic forces in this challenging system containing >1,500 atoms.

Graphene at Liquid Copper Catalysts: Atomic-Scale Agreement of Experimental and First-Principles Adsorption Height.

Liquid metal catalysts have recently attracted attention for synthesizing high-quality 2D materials facilitated via the catalysts' perfectly smooth surface. However, the microscopic catalytic processes occurring at the surface are still largely unclear because liquid metals escape the accessibility of traditional experimental and computational surface science approaches. Hence, numerous controversies are found regarding different applications, with graphene (Gr) growth on liquid copper (Cu) as a prominent prototype. In this work, novel in situ and in silico techniques are employed to achieve an atomic-level characterization of the graphene adsorption height above liquid Cu, reaching quantitative agreement within 0.1 Å between experiment and theory. The results are obtained via in situ synchrotron X-ray reflectivity (XRR) measurements over wide-range q-vectors and large-scale molecular dynamics simulations based on efficient machine-learning (ML) potentials trained to first-principles density functional theory (DFT) data. The computational insight is demonstrated to be robust against inherent DFT errors and reveals the nature of graphene binding to be highly comparable at liquid Cu and solid Cu(111). Transporting the predictive first-principles quality via ML potentials to the scales required for liquid metal catalysis thus provides a powerful approach to reach microscopic understanding, analogous to the established computational approaches for catalysis at solid surfaces.

Charting C–C coupling pathways in electrochemical CO2 reduction on Cu(111) using embedded correlated wavefunction theory

The electrochemical CO2 reduction reaction (CO2RR) powered by excess zero-carbon-emission electricity to produce especially multicarbon (C2+) products could contribute to a carbon-neutral to carbon-negative economy. Foundational to the rational design of efficient, selective CO2RR electrocatalysts is mechanistic analysis of the best metal catalyst thus far identified, namely, copper (Cu), via quantum mechanical computations to complement experiments. Here, we apply embedded correlated wavefunction (ECW) theory, which regionally corrects the electron exchange-correlation error in density functional theory (DFT) approximations, to examine multiple C-C coupling steps involving adsorbed CO (*CO) and its hydrogenated derivatives on the most ubiquitous facet, Cu(111). We predict that two adsorbed hydrogenated CO species, either *COH or *CHO, are necessary precursors for C-C bond formation. The three kinetically feasible pathways involving these species yield all three possible products: *COH-CHO, *COH-*COH, and *OCH-*OCH. The most kinetically favorable path forms *COH-CHO. In contrast, standard DFT approximations arrive at qualitatively different conclusions, namely, that only *CO and *COH will prevail on the surface and their C-C coupling paths produce only *COH-*COH and *CO-*CO, with a preference for the first product. This work demonstrates the importance of applying qualitatively and quantitatively accurate quantum mechanical method to simulate electrochemistry in order ultimately to shed light on ways to enhance selectivity toward C2+ product formation via CO2RR electrocatalysts.

Correcting π-delocalisation errors in conformational energies using density-corrected DFT, with application to crystal polymorphs

We consider several molecules characterised by π-electron conjugation whose extent changes along a flexible torsional coordinate, and which represent the monomer units of polymorphic molecular crystals. Delocalisation error in density functional theory (DFT) adversely impacts conformational energies in these species, overstabilising the conformation that maximises conjugation length and leading to incorrect relative energies for the corresponding crystal polymorphs. We demonstrate that density-corrected (DC-)DFT, in which a DFT exchange-correlation functional is evaluated using a Hartree–Fock density, significantly reduces these conformational energy errors. When DC-DFT monomer energies are used as a low-cost intramolecular correction to a periodic DFT calculation of the molecular crystal, the resulting relative polymorph energies are within 1 kJ/mol of benchmark results.

Role of "S" Substitution on C-H Activation Reactivity of Iron(IV)-Oxo Cyclam Complexes: a Computational Investigation.

A comprehensive density functional theory (DFT) investigation has been presented in this article to address the role of equatorial sulfur ligation in C-H activation. A non-heme iron-oxo compound with four nitrogen atoms constituting the equatorially connected macrocyclic framework (represented as N4) [Fe(IV)═O(THC)(CH3CN)]2+(THC = 1,4,8,11-tetrahydro1,4,8,11-tetraazacyclotetradecane) has been considered as the base compound. Other complexes have been anticipated by the sequential replacement of this nitrogen by sulfur, that is, N4, N3S1, N2S2, N1S3, and S4. Counterions, as always, have been considered to avoid the self-interaction error in DFT. Generally, the anti-conformers (with respect to equatorial N-H and Fe═O) turned out to be the most stable. It was found that with the enrichment of the equatorial sulfur atom, reactivity increases successively, that is, we get the trend N4 < N3S1 < N2S2 < N1S3 < S4. Our investigations have also verified the available experimental results where it has been reported that N2S2 is more reactive than N4 in their mixed conformation. In search of insights into this typical pattern of reactivity, the interplay of several factors has been recognized, such as the distortion energy which decreases for the transition states with the addition of sulfur; the spin density on the oxygen atom which increases implying that the radical character of abstractor increases on sulfur ligation; the energy of the electron acceptor orbital (the lowest unoccupied molecular orbital (σz2*)) which decreases continuously with the sulfur substitution; and the triplet-quintet oxidant energy gap which decreases consistently with S enrichment in the equatorial position. The computational predictions reported here, if further validated by experiments, will definitely encourage the synthesis of sulfur-ligated bio-inspired complexes instead of the ones constituting nitrogen exclusively.

Modeling Hydrogen and Oxygen Evolution Reactions on Single Atom Catalysts with Density Functional Theory: Role of the Functional

AbstractThe most widely used approach to predict catalytic activity is density functional theory, whose results however depend on the adopted exchange‐correlation functional. In this work, the role played by the functional in predicting the activity of single atom catalysts (SAC) in the hydrogen and oxygen evolution reactions (HER and OER) is studied. 16 transition metal (TM) atoms embedded in N‐doped graphene are simulated and the performance of the widely adopted Perdew–Burke–Ernzerhof (PBE) functional against the hybrid PBE0 functional is assessed. The PBE+U approach, a computationally less complex way to correct for the self‐interaction error in density functional theory, is also considered. The predictions obtained with PBE have a substantial deviation from PBE0 for first row TMs, i.e., 3d systems, while smaller deviations are found for the 4d and 5d series. The PBE+U results represent an improvement with respect to PBE, although some differences from PBE0 remain. This study underlines the importance of the choice of the DFT functional in screening new catalysts and in predicting catalytic activities. The use of PBE appears acceptable for 4d and 5d metals, while in the case of 3d systems PBE+U or PBE0 approaches are recommended, in particular for magnetic ground states.

Understanding the chemical bonding of ground and excited states of HfO and HfB with correlated wavefunction theory and density functional approximations.

Knowledge of the chemical bonding of HfO and HfB ground and low-lying electronic states provides essential insights into a range of catalysts and materials that contain Hf-O or Hf-B moieties. Here, we carry out high-level multi-reference configuration interaction theory and coupled cluster quantum chemical calculations on these systems. We compute full potential energy curves, excitation energies, ionization energies, electronic configurations, and spectroscopic parameters with large quadruple-ζ and quintuple-ζ quality correlation consistent basis sets. We also investigate equilibrium chemical bonding patterns and effects of correlating core electrons on property predictions. Differences in the ground state electron configuration of HfB(X4Σ-) and HfO(X1Σ+) lead to a significantly stronger bond in HfO than HfB, as judged by both dissociation energies and equilibrium bond distances. We extend our analysis to the chemical bonding patterns of the isovalent HfX (X = O, S, Se, Te, and Po) series and observe similar trends. We also note a linear trend between the decreasing value of the dissociation energy (De) from HfO to HfPo and the singlet-triplet energy gap (ΔES-T) of the molecule. Finally, we compare these benchmark results to those obtained using density functional theory (DFT) with 23 exchange-correlation functionals spanning multiple rungs of "Jacob's ladder." When comparing DFT errors to coupled cluster reference values on dissociation energies, excitation energies, and ionization energies of HfB and HfO, we observe semi-local generalized gradient approximations to significantly outperform more complex and high-cost functionals.

Density functional theory of waterwith the machine-learned DM21 functional.

The delicate interplay between functional-driven and density-driven errors in density functional theory (DFT) has hindered traditional density functional approximations (DFAs) from providing an accurate description of water for over 30 years. Recently, the deep-learned DeepMind 21 (DM21) functional has been shown to overcome the limitations of traditional DFAs as it is free of delocalization error. To determine if DM21 can enable a molecular-level description of the physical properties of aqueous systems within Kohn-Sham DFT, we assess the accuracy of the DM21 functional for neutral, protonated, and deprotonated water clusters. We find that the ability of DM21 to accurately predict the energetics of aqueous clusters varies significantly with cluster size. Additionally, we introduce the many-body MB-DM21 potential derived from DM21 data within the many-body expansion of the energy and use it in simulations of liquid water as a function of temperature at ambient pressure. We find that size-dependent functional-driven errors identified in the analysis of the energetics of small clusters calculated with the DM21 functional result in the MB-DM21 potential systematically overestimating the hydrogen-bond strength and, consequently, predicting a more ice-like local structure of water at room temperature.

Adiabatic projection: Bridging ab initio, density functional, semiempirical, and embedding approximations.

Modern electronic structure approximations routinely employ reference systems described by approximate Hamiltonians. This work introduces the adiabatic projection formalism for building formally exact corrections to such reference systems. Starting from the real Hamiltonian of a many-electron system, one constructs a reference system Hamiltonian by projecting the kinetic and electron-electron interaction operators onto "interesting" states. The reference system is corrected by density functionals for the difference between the projected and unprojected kinetic and electron-electron energies. These density functionals are constructed from adiabatic connections between the reference and real systems. The Hohenberg-Kohn theorems imply the existence of exact functionals, which can ensure that the reference system's ground-state energy and density match the real system. Adiabatic projection further generalizes Kohn-Sham density functional theory (DFT) and the generalized adiabatic connection [W. Yang, J. Chem. Phys. 109, 10107 (1998)] and recovers these methods for certain choices of projection operators. Other choices of projection operators offer new opportunities, including formally exact and systematically improvable analogues to wavefunction-in-DFT embedding, DFT+U, and semiempirical theories. Numerical results are presented for two representative choices: a projected exchange-correlation correction to small-basis-set coupled cluster theory and a projected kinetic energy density functional correcting basis set errors in DFT. The latter offers performance for dimerization energies approaching the Boys-Bernardi counterpoise correction while also correcting intramolecular basis set superposition errors.

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.

On the Use of Normalized Metrics for Density Sensitivity Analysis in DFT.

In the past years, there has been a discussion about how the errors in density functional theory might be related to errors in the self-consistent densities obtained from different density functional approximations. This, in turn, brings up the discussion about the different ways in which we can measure such errors and develop metrics that assess the sensitivity of calculated energies to changes in the density. It is important to realize that there cannot be a unique metric in order to look at this density sensitivity, simultaneously needing size-extensive and size-intensive metrics. In this study, we report two metrics that are widely applicable to any density functional approximation. We also show how they can be used to classify different chemical systems of interest with respect to their sensitivity to small variations in the density.

Revisiting Competing Paths in Electrochemical CO2 Reduction on Copper via Embedded Correlated Wavefunction Theory.

We re-evaluate two key steps in the mechanism of CO2 reduction on copper at a higher level of theory capable of correcting inherent errors in density functional theory (DFT) approximations, namely, embedded correlated wavefunction (ECW) theory. Here, we consider the CO reduction step on Cu(111), which is critical to understanding reaction selectivity. We optimize embedding potentials at the periodic plane-wave DFT level using density functional embedding theory (DFET). All possible adsorption sites (adsites) for each adsorbate then are screened with ECW theory at the catalytically active site to refine the local electronic structure. Unsurprisingly, DFT and ECW theory predict different adsite preferences, largely because of DFT's inability to properly situate the CO 2π* level. Differing preferred adsites suggest that different reaction pathways could emerge from DFT versus ECW theory. Starting from these preferred ECW theory adsites, we then obtain reaction pathways at the plane-wave DFT level using the climbing-image nudged elastic band method to determine minimum energy paths. Thereafter, we perform ECW calculations at the catalytically active site to correct the energetics at each interpolated structure (image) along the reaction pathways. Via this approach, we confirm that the first step in CO reduction via hydrogen transfer on Cu(111) is to form hydroxymethylidyne (*COH) instead of formyl (*CHO). Although the prediction to preferentially form *COH is consistent with that of DFT, the two theories predict quite different structural and mechanistic behaviors, suggesting that verification is needed for other parts of the mechanism of CO2 reduction, which is the subject of ongoing work.

Comparing the performance of density functionals in describing the adsorption of atoms and small molecules on Ni(111)

Reaction networks calculated using density functional theory (DFT) play a crucial role in understanding and predicting heterogeneous catalyst performance. However, results obtained using different density functionals can vary significantly. Accordingly, it is important to assess the accuracy of available functionals in capturing adsorption energetics derived from single-crystal adsorption microcalorimetry experiments. Here, we evaluate how different density functionals describe the molecular and dissociative adsorption of CH3I, the adsorption of CH3, I, and H, and the dissociative adsorption of CH4 on Ni(111). First, we identify energetically-preferred adsorbed states for each adsorbate at three surface coverages and calculate the corresponding enthalpies of adsorption at 160 K. Then, we compare the results to experimental measurements published in the literature. For each adsorption process we find at least one functional that is quantitatively accurate within experimental error, but no functional is accurate (within experimental error) for all adsorption processes studied. Even when assuming an additional DFT error of ±20 kJ/mol on top of the experimental error, only the PBE-D3 and RPBE-D3 functionals are accurate for all considered adsorption systems. These results indicate that quantitative agreement between density functional theory calculations and experimental measurements is both system- and functional-dependent. Importantly, single-crystal adsorption microcalorimetry experiments will continue to play an important role for benchmarking density functionals.

Simultaneous Attenuation of Both Self-Interaction Error and Nondynamic Correlation Error in Density Functional Theory: A Spin-Pair Distinctive Adiabatic-Connection Approximation.

We present a spin-pair distinctive algorithm in the context of adiabatic-connection fluctuation-dissipation (ACFD) theorem, which enables to quantify the self-interaction error (SIE) and the nondynamic/strong correlation error (NCE) in the direct random-phase approximation (dRPA). Using this knowledge, we propose a spin-component scaled dRPA (scsRPA) correlation model with simultaneous attenuation of both the SIE and the NCE. Along with the exact exchange, scsRPA is shown to present a comprehensive improvement over dRPA, as well as the well-established PBE and PBE0 functionals, for bonding energies of pronounced multireference characters and transition-metal complexes of strongly correlated systems, while consistently providing an accurate description for reaction energies, reaction barriers, and noncovalent bond interactions of weakly correlated systems.

Study of stretched H2 reveals new class of error in density functional theory

Poor spin density localization in elongated single bonds can lead to qualitative failure in predicting properties such as static polarizability and force constants.