Ligand-Field-Dependent Behavior of Meta-GGA Exchange in Transition-Metal Complex Spin-State Ordering

Redox-active ligands as a challenge for electronic structure methods

To improve the catalytic activity of 3d transition metal catalysts, redox-active ligands are a promising tool. These ligands influence the oxidation state of the metal center as well as the ground spin-state and make the experimental determination of both properties challenging. Therefore, first-principles calculations, in particular employing density functional theory with a proper choice of exchange-correlation (xc) functional, are crucial. Common xc functionals were tested on a simple class of metal complexes: homoleptic, octahedral tris(diimine) iron(II) complexes. The spin-state energy splittings for most of these complexes showed the expected linear dependence on the amount of exact exchange included in the xc functionals. Even though varying redox-activity affects the electronic structure of the complexes considerably, the sensitivity of the spin-state energetics to the exact exchange admixture is surprisingly small. For iron(II) complexes with highly redox-active ligands and for a broad range of ligands in the reduced tris(diimine) iron(I) complexes, self-consistent field convergence to local minima was observed, which differ from the global minimum in the redox state of the ligand. This may also result in convergence to a molecular structure that corresponds to an energetically higher-lying local minimum. One criterion to detect such behavior is a change in the sign of the slope for the dependence of the spin-state energy splittings on the amount of exact exchange. We discuss possible protocols for dealing with such artifacts in cases in which a large number of calculations makes checking by hand unfeasible.

Spin-State Splittings in 3d Transition-Metal Complexes Revisited: Benchmarking Approximate Methods for Adiabatic Spin-State Energy Differences in Fe(II) Complexes

The CASPT2+δMRCI composite approach reported in a companion paper has been extended and used to provide high-quality reference data for a series of adiabatic spin gaps (defined as ΔE = E_quintet - E_singlet) of [Fe^IIL₆]²⁺ complexes (L = CNH, CO, NCH, NH₃, H₂O), either at nonrelativistic level or including scalar relativistic effects. These highly accurate data have been used to evaluate the performance of various more approximate methods. Coupled-cluster theory with singles, doubles, and perturbative triples, CCSD(T), is found to agree well with the new reference data for Werner-type complexes but exhibits larger underestimates by up to 70 kJ/mol for the π-acceptor ligands, due to appreciable static correlation in the low-spin states of these systems. Widely used domain-based local CCSD(T) calculations, DLPNO-CCSD(T), are shown to depend very sensitively on the cutoff values used to construct the localized domains, and standard values are not sufficient. A large number of density functional approximations have been evaluated against the new reference data. The B2PLYP double hybrid gives the smallest deviations, but several functionals from different rungs of the usual ladder hierarchy give mean absolute deviations below 20 kJ/mol. This includes the B97-D semilocal functional, the PBE0* global hybrid with 15% exact-exchange admixture, as well as the local hybrids LH07s-SVWN and LH07t-SVWN. Several further functionals achieve mean absolute errors below 30 kJ/mol (M06L-D4, SSB-D, B97-1-D4, LC-ωPBE-D4, LH12ct-SsirPW92-D4, LH12ct-SsifPW92-D4, LH14t-calPBE-D4, LHJ-HFcal-D4, and several further double hybrids) and thereby also still overall outperform CCSD(T) or uncorrected CASPT2. While exact-exchange admixture is a crucial factor in favoring high-spin states, the present evaluations confirm that other aspects can be important as well. A number of the better-performing functionals underestimate the spin gaps for the π-acceptor ligands but overestimate them for L = NH₃, H₂O. In contrast to a previous suggestion, non-self-consistent density functional theory (DFT) computations on top of Hartree-Fock orbitals are not a promising path to produce accurate spin gaps in such complexes.

Molecular orbital projectors in non-empirical jmDFT recover exact conditions in transition-metal chemistry

Low-cost, non-empirical corrections to semi-local density functional theory are essential for accurately modeling transition-metal chemistry. Here, we demonstrate the judiciously modified density functional theory (jmDFT) approach with non-empirical U and J parameters obtained directly from frontier orbital energetics on a series of transition-metal complexes. We curate a set of nine representative Ti(III) and V(IV) d1 transition-metal complexes and evaluate their flat-plane errors along the fractional spin and charge lines. We demonstrate that while jmDFT improves upon both DFT+U and semi-local DFT with the standard atomic orbital projectors (AOPs), it does so inefficiently. We rationalize these inefficiencies by quantifying hybridization in the relevant frontier orbitals. To overcome these limitations, we introduce a procedure for computing a molecular orbital projector (MOP) basis for use with jmDFT. We demonstrate this single set of d1 MOPs to be suitable for nearly eliminating all energetic delocalization error and static correlation error. In all cases, MOP jmDFT outperforms AOP jmDFT, and it eliminates most flat-plane errors at non-empirical values. Unlike DFT+U or hybrid functionals, jmDFT nearly eliminates energetic delocalization error and static correlation error within a non-empirical framework.

On the Investigation of Effective Factors on Electronic Structure Properties of Transition Metal Complexes: Robust Modeling Using GPR Approach

Materials discovery is usually done using high-throughput computational screening. The use of costly and complex direct density functional theory (DFT) simulation methods has been commonly used to determine subtle trends in spin-state ordering and inorganic bonding of inorganic materials and, in general, to predict the electronic structure properties of transition metal complexes. A Gaussian process regression (GPR) framework consisting of four kernel functions is introduced for spin-state splitting estimation through inorganic chemistry-appropriate empirical inputs. To this end, the present study reviewed an extensive range of data values from earlier works. According to statistical analysis, the GPR model showed very good performance. The coefficients of determination were calculated to be 0.986 for the exponential and Matern kernel functions, suggesting the highest predictive power of these methods. Moreover, the sensitivity of output to inputs was measured. Artificial intelligence (AI) helped accurately predict the target values through various input ranges.

Computational Scaling Relationships Predict Experimental Activity and Rate-Limiting Behavior in Homogeneous Water Oxidation

While computational screening with first-principles density functional theory (DFT) is essential for evaluating candidate catalysts, limitations in accuracy typically prevent the prediction of experimentally relevant activities. Exemplary of these challenges are homogeneous water oxidation catalysts (WOCs) where differences in experimental conditions or small changes in ligand structure can alter rate constants by over an order of magnitude. Here, we compute mechanistically relevant electronic and energetic properties for 19 mononuclear Ru transition-metal complexes (TMCs) from three experimental water oxidation catalysis studies. We discover that 15 of these TMCs have experimental activities that correlate with a single property, the ionization potential of the Ru(II)-O₂ catalytic intermediate. This scaling parameter allows the quantitative understanding of activity trends and provides insight into the rate-limiting behavior. We use this approach to rationalize differences in activity with different experimental conditions, and we qualitatively analyze the source of distinct behavior for different electronic states in the other four catalysts. Comparison to closely related single-atom catalysts and modified WOCs enables rationalization of the source of rate enhancement in these WOCs.

Machine learning to tame divergent density functional approximations: a new path to consensus materials design principles

Virtual high-throughput screening (VHTS) with density functional theory (DFT) and machine-learning (ML)-acceleration is essential in rapid materials discovery. By necessity, efficient DFT-based workflows are carried out with a single density functional approximation (DFA). Nevertheless, properties evaluated with different DFAs can be expected to disagree for cases with challenging electronic structure (e.g., open-shell transition-metal complexes, TMCs) for which rapid screening is most needed and accurate benchmarks are often unavailable. To quantify the effect of DFA bias, we introduce an approach to rapidly obtain property predictions from 23 representative DFAs spanning multiple families, “rungs” (e.g., semi-local to double hybrid) and basis sets on over 2000 TMCs. Although computed property values (e.g., spin state splitting and frontier orbital gap) differ by DFA, high linear correlations persist across all DFAs. We train independent ML models for each DFA and observe convergent trends in feature importance, providing DFA-invariant, universal design rules. We devise a strategy to train artificial neural network (ANN) models informed by all 23 DFAs and use them to predict properties (e.g., spin-splitting energy) of over 187k TMCs. By requiring consensus of the ANN-predicted DFA properties, we improve correspondence of computational lead compounds with literature-mined, experimental compounds over the typically employed single-DFA approach.

Machine learning (ML)-based feature analysis reveals universal design rules regardless of density functional choices. Using the consensus among multiple functionals, we identify robust lead complexes in ML-accelerated chemical discovery.

Computational Discovery of Transition-metal Complexes: From High-throughput Screening to Machine Learning

Transition-metal complexes are attractive targets for the design of catalysts and functional materials. The behavior of the metal-organic bond, while very tunable for achieving target properties, is challenging to predict and necessitates searching a wide and complex space to identify needles in haystacks for target applications. This review will focus on the techniques that make high-throughput search of transition-metal chemical space feasible for the discovery of complexes with desirable properties. The review will cover the development, promise, and limitations of "traditional" computational chemistry (i.e., force field, semiempirical, and density functional theory methods) as it pertains to data generation for inorganic molecular discovery. The review will also discuss the opportunities and limitations in leveraging experimental data sources. We will focus on how advances in statistical modeling, artificial intelligence, multiobjective optimization, and automation accelerate discovery of lead compounds and design rules. The overall objective of this review is to showcase how bringing together advances from diverse areas of computational chemistry and computer science have enabled the rapid uncovering of structure-property relationships in transition-metal chemistry. We aim to highlight how unique considerations in motifs of metal-organic bonding (e.g., variable spin and oxidation state, and bonding strength/nature) set them and their discovery apart from more commonly considered organic molecules. We will also highlight how uncertainty and relative data scarcity in transition-metal chemistry motivate specific developments in machine learning representations, model training, and in computational chemistry. Finally, we will conclude with an outlook of areas of opportunity for the accelerated discovery of transition-metal complexes.

Putting Density Functional Theory to the Test in Machine-Learning-Accelerated Materials Discovery

Accelerated discovery with machine learning (ML) has begun to provide the advances in efficiency needed to overcome the combinatorial challenge of computational materials design. Nevertheless, ML-accelerated discovery both inherits the biases of training data derived from density functional theory (DFT) and leads to many attempted calculations that are doomed to fail. Many compelling functional materials and catalytic processes involve strained chemical bonds, open-shell radicals and diradicals, or metal-organic bonds to open-shell transition-metal centers. Although promising targets, these materials present unique challenges for electronic structure methods and combinatorial challenges for their discovery. In this Perspective, we describe the advances needed in accuracy, efficiency, and approach beyond what is typical in conventional DFT-based ML workflows. These challenges have begun to be addressed through ML models trained to predict the results of multiple methods or the differences between them, enabling quantitative sensitivity analysis. For DFT to be trusted for a given data point in a high-throughput screen, it must pass a series of tests. ML models that predict the likelihood of calculation success and detect the presence of strong correlation will enable rapid diagnoses and adaptation strategies. These "decision engines" represent the first steps toward autonomous workflows that avoid the need for expert determination of the robustness of DFT-based materials discoveries.

Improved Spin-State Energy Differences of Fe(II) Molecular and Crystalline Complexes via the Hubbard U-Corrected Density

We recently showed that the DFT+U approach with a linear-response U yields adiabatic energy differences biased toward high spin [Mariano et al. J. Chem. Theory Comput. 2020, 16, 6755-6762]. Such bias is removed here by employing a density-corrected DFT approach where the PBE functional is evaluated on the Hubbard U-corrected density. The adiabatic energy differences of six Fe(II) molecular complexes computed using this approach, named PBE[U] here, are in excellent agreement with coupled cluster-corrected CASPT2 values for both weak- and strong-field ligands resulting in a mean absolute error (MAE) of 0.44 eV, smaller than that of the recently proposed Hartree-Fock density-corrected DFT (1.22 eV) and any other tested functional, including the best performer TPSSh (0.49 eV). We take advantage of the computational efficiency of this approach and compute the adiabatic energy differences of five molecular crystals using PBE[U] with periodic boundary conditions. The results show, again, an excellent agreement (MAE = 0.07 eV) with experimentally extracted values and a superior performance compared with the best performers M06-L (MAE = 0.08 eV) and TPSSh (MAE = 0.31 eV) computed on molecular fragments.

Molecular DFT+U: A Transferable, Low-Cost Approach to Eliminate Delocalization Error

While density functional theory (DFT) is widely applied for its combination of cost and accuracy, corrections (e.g., DFT+U) that improve it are often needed to tackle correlated transition-metal chemistry. In principle, the functional form of DFT+U, consisting of a set of localized atomic orbitals (AOs) and a quadratic energy penalty for deviation from integer occupations of those AOs, enables the recovery of the exact conditions of piecewise linearity and the derivative discontinuity. Nevertheless, for practical transition-metal complexes, where both atomic states and ligand orbitals participate in bonding, standard DFT+U can fail to eliminate delocalization error (DE). Here, we show that by introducing an alternative valence-state (i.e., molecular orbital or MO) basis to the DFT+U approach, we recover exact conditions in cases for which standard DFT+U corrections have no error-reducing effect. This MO-based DFT+U also eliminates DE where standard AO-based DFT+U is already successful. We demonstrate the transferability of our approach on representative transition-metal complexes with a range of ligand field strengths, electron configurations (i.e., from Sc to Zn), and spin states.

Seeing Is Believing: Experimental Spin States from Machine Learning Model Structure Predictions

Determination of ground-state spins of open-shell transition-metal complexes is critical to understanding catalytic and materials properties but also challenging with approximate electronic structure methods. As an alternative approach, we demonstrate how structure alone can be used to guide assignment of ground-state spin from experimentally determined crystal structures of transition-metal complexes. We first identify the limits of distance-based heuristics from distributions of metal–ligand bond lengths of over 2000 unique mononuclear Fe(II)/Fe(III) transition-metal complexes. To overcome these limits, we employ artificial neural networks (ANNs) to predict spin-state-dependent metal–ligand bond lengths and classify experimental ground-state spins based on agreement of experimental structures with the ANN predictions. Although the ANN is trained on hybrid density functional theory data, we exploit the method-insensitivity of geometric properties to enable assignment of ground states for the majority (ca. 80–90%) of structures. We demonstrate the utility of the ANN by data-mining the literature for spin-crossover (SCO) complexes, which have experimentally observed temperature-dependent geometric structure changes, by correctly assigning almost all (>95%) spin states in the 46 Fe(II) SCO complex set. This approach represents a promising complement to more conventional energy-based spin-state assignment from electronic structure theory at the low cost of a machine learning model.

Large-scale comparison of 3d and 4d transition metal complexes illuminates the reduced effect of exchange on second-row spin-state energetics

The origin of distinct 3d vs. 4d transition metal complex sensitivity to exchange is explored over a large data set.

Improving the Light-Induced Spin Transition Efficiency in Ni(II)-Based Macrocyclic-Ligand Complexes

The structural stability and photoabsorption properties of Ni(II)-based metal-organic complexes with octahedral coordination having different planar ligand ring structures were investigated employing density functional theory (DFT) and its time-dependent extension (TD-DFT) considering the M06 exchange-correlation functional and the Def2-TZVP basis set. The results showed that the molecular composition of different planar cyclic ligand structures had significant influences on the structural stability and photoabsorption properties of metal-organic complexes. Only those planar ligands that contained aromatic rings met the basic criteria (thermal stability, structural reversibility, and appropriate excitation frequency domain) for light-induced excited spin state trapping, but their spin transition efficiencies were very different. While, in all three aromatic cases, the singlet electronic excitations induced charge distribution that could help in the singlet-to-triplet spin transition, and triplet excitations, which could assist in the backward (triplet-to-singlet) spin transition, was found only for one complex.

Spin-State Ordering in Metal-Based Compounds Using the Localized Active Space Self-Consistent Field Method

Quantitatively accurate calculations for spin-state ordering in transition-metal complexes typically demand a robust multiconfigurational treatment. The poor scaling of such methods with increasing size makes them impractical for large, strongly correlated systems. Density matrix embedding theory (DMET) is a fragmentation approach that can be used to specifically address this challenge. The single-determinantal bath framework of DMET is applicable in many situations, but it has been shown to perform poorly for molecules characterized by strong correlation when a multiconfigurational self-consistent field solver is used. To ameliorate this problem, the localized active space self-consistent field (LASSCF) method was recently described. In this work, LASSCF is applied to predict spin-state energetics in mono- and di-iron systems, and we show that the model offers an accuracy equivalent to that of CASSCF but at a substantially lower computational cost. Performance as a function of basis set and active space is also examined.

Stable Surfaces That Bind Too Tightly: Can Range-Separated Hybrids or DFT+U Improve Paradoxical Descriptions of Surface Chemistry?

Approximate, semilocal density functional theory (DFT) suffers from delocalization error that can lead to a paradoxical model of catalytic surfaces that both overbind adsorbates yet are also too stable. We investigate the effect of two widely applied approaches for delocalization error correction, (i) affordable DFT+U (i.e., semilocal DFT augmented with a Hubbard U) and (ii) hybrid functionals with an admixture of Hartree-Fock (HF) exchange, on surface and adsorbate energies across a range of rutile transition metal oxides widely studied for their promise as water-splitting catalysts. We observe strongly row- and period-dependent trends with DFT+U, which increases surface formation energies only in early transition metals (e.g., Ti and V) and decreases adsorbate energies only in later transition metals (e.g., Ir and Pt). Both global and local hybrids destabilize surfaces and reduce adsorbate binding across the periodic table, in agreement with higher-level reference calculations. Density analysis reveals why hybrid functionals correct both quantities, whereas DFT+U does not. We recommend local, range-separated hybrids for the accurate modeling of catalysis in transition metal oxides at only a modest increase in computational cost over semilocal DFT.

Structure of Electronically Reduced N-Donor Bidentate Ligands and Their Heteroleptic Four-Coordinate Zinc Complexes: A Survey of Density Functional Theory Results

The role of Hartree-Fock exchange in describing the structural changes occurring upon reduction of bipyridine-based ligands and their complexes is investigated within the framework of density functional theory (DFT) calculations. A set of four free ligands in their neutral and radical anionic forms, and two of their zinc complexes in their dicationic and monocationic radical forms, is used to compare a large panel of pure, conventional, and long-range corrected hybrid DFT functionals; coupled cluster single and double calculations are used alongside experimental results as benchmarks. Particular attention has been devoted to the magnitude of the change, upon reduction, of the Δ-parameter, which measures the difference between the C_py-C_py and the C-N bond lengths in bipyridine ligand and is known to experimentally correlate with the charge of the ligands. Our results indicate that the structural changes significantly depend on the amount of exact exchange included in the functional. A progressive evolution is observed for the free ligands, whereas two distinct sets of results are obtained for the complexes. Functionals with a small degree of HF exchange, e.g., B3LYP, do not adequately describe geometric changes for the considered species, and, quite surprisingly, the same holds for the CC2 method. The best agreement to experimental and CCSD values is obtained with functionals that include a significant but not excessive part of exact exchange, e.g., CAM-B3LYP, M06-2X, and ωB97X-D. The calculated localization of the added electron after reduction, which depends on the self-interaction error, is used to rationalize these outcomes. Static correlation is also shown to play a role in the accurate description of the electronic structure.

Non-empirical, low-cost recovery of exact conditions with model-Hamiltonian inspired expressions in jmDFT

Density functional theory (DFT) is widely applied to both molecules and materials, but well known energetic delocalization and static correlation errors in practical exchange-correlation approximations limit quantitative accuracy. Common methods that correct energetic delocalization errors, such as the Hubbard U correction in DFT+U or Hartree-Fock exchange in global hybrids, do so at the cost of worsening static correlation errors. We recently introduced an alternate approach [Bajaj et al., J. Chem. Phys. 147, 191101 (2017)] known as judiciously modified DFT (jmDFT), wherein the deviation from exact behavior of semilocal functionals over both fractional spin and charge, i.e., the so-called flat plane, was used to motivate functional forms of second order analytic corrections. In this work, we introduce fully nonempirical expressions for all four coefficients in a DFT+U+J-inspired form of jmDFT, where all coefficients are obtained only from energies and eigenvalues of the integer-electron systems. We show good agreement for U and J coefficients obtained nonempirically as compared with the results of numerical fitting in a jmDFT U+J/J' correction. Incorporating the fully nonempirical jmDFT correction reduces and even eliminates the fractional spin error at the same time as eliminating the energetic delocalization error. We show that this approach extends beyond s-electron systems to higher angular momentum cases including p- and d-electrons. Finally, we diagnose some shortcomings of the current jmDFT approach that limit its ability to improve upon DFT results for cases such as weakly bound anions due to poor underlying semilocal functional behavior.

Bridging the Homogeneous-Heterogeneous Divide: Modeling Spin for Reactivity in Single Atom Catalysis

Single atom catalysts (SACs) are emergent catalytic materials that have the promise of merging the scalability of heterogeneous catalysts with the high activity and atom economy of homogeneous catalysts. Computational, first-principles modeling can provide essential insight into SAC mechanism and active site configuration, where the sub-nm-scale environment can challenge even the highest-resolution experimental spectroscopic techniques. Nevertheless, the very properties that make SACs attractive in catalysis, such as localized d electrons of the isolated transition metal center, make them challenging to study with conventional computational modeling using density functional theory (DFT). For example, Fe/N-doped graphitic SACs have exhibited spin-state dependent reactivity that remains poorly understood. However, spin-state ordering in DFT is very sensitive to the nature of the functional approximation chosen. In this work, we develop accurate benchmarks from correlated wavefunction theory (WFT) for relevant octahedral complexes. We use those benchmarks to evaluate optimal DFT functional choice for predicting spin state ordering in small octahedral complexes as well as models of pyridinic and pyrrolic nitrogen environments expected in larger SACs. Using these guidelines, we determine Fe/N-doped graphene SAC model properties and reactivity as well as their sensitivities to DFT functional choice. Finally, we conclude with broad recommendations for computational modeling of open-shell transition metal single-atom catalysts.

Designing in the Face of Uncertainty: Exploiting Electronic Structure and Machine Learning Models for Discovery in Inorganic Chemistry

Recent transformative advances in computing power and algorithms have made computational chemistry central to the discovery and design of new molecules and materials. First-principles simulations are increasingly accurate and applicable to large systems with the speed needed for high-throughput computational screening. Despite these strides, the combinatorial challenges associated with the vastness of chemical space mean that more than just fast and accurate computational tools are needed for accelerated chemical discovery. In transition-metal chemistry and catalysis, unique challenges arise. The variable spin, oxidation state, and coordination environments favored by elements with well-localized d or f electrons provide great opportunity for tailoring properties in catalytic or functional (e.g., magnetic) materials but also add layers of uncertainty to any design strategy. We outline five key mandates for realizing computationally driven accelerated discovery in inorganic chemistry: (i) fully automated simulation of new compounds, (ii) knowledge of prediction sensitivity or accuracy, (iii) faster-than-fast property prediction methods, (iv) maps for rapid chemical space traversal, and (v) a means to reveal design rules on the kilocompound scale. Through case studies in open-shell transition-metal chemistry, we describe how advances in methodology and software in each of these areas bring about new chemical insights. We conclude with our outlook on the next steps in this process toward realizing fully autonomous discovery in inorganic chemistry using computational chemistry.

Iron(II) and Iron(III) Spin Crossover: Toward an Optimal Density Functional

Spin crossover (SCO) plays a major role in biochemistry, catalysis, materials, and emerging technologies such as molecular electronics and sensors, and thus accurate prediction and design of SCO systems is of high priority. However, the main tool for this purpose, density functional theory (DFT), is very sensitive to applied methodology. The most abundant SCO systems are Fe(II) and Fe(III) systems. Even with average good agreement, a functional may be significantly more accurate for Fe(II) or Fe(III) systems, preventing balanced study of SCO candidates of both types. The present work investigates DFT's performance for well-known Fe(II) and Fe(III) SCO complexes, using various design types and customized versions of GGA, hybrid, meta-GGA, meta-hybrid, double-hybrid, and long-range-corrected hybrid functionals. We explore the limits of DFT performance and identify proficient Fe(II)-Fe(III)-balanced functionals. We identify and quantify remarkable differences in the DFT description of Fe(II) and Fe(III) systems. Most functionals become more accurate once Hartree-Fock exchange is adjusted to 10-17%, regardless of the type of functionals involved. However, this typically introduces a clear Fe(II)-Fe(III) bias. The most accurate functionals measured by mean absolute errors <10 kJ/mol are CAMB3LYP-17, B3LYP*, and B97-15 with 15-17% Hartree-Fock exchange, closely followed by CAMB3LYP and CAMB3LYP-15, OPBE, rPBE-10, and B3P86-15. While GGA functionals display a small Fe(II)-Fe(III) bias, they are generally inaccurate, except the O exchange functional. Hybrid functionals (including B2PLYP double hybrids and meta hybrids) tend to favor HS too much in Fe(II) vs Fe(III), which is important in many studies where the oxidation state of iron can vary, e.g. rational SCO design and studies of catalytic processes involving iron. The only functional with a combined bias <5 kJ/mol and a decent MAE (15 kJ/mol) is our customized PBE0-12 functional. Alternatively one has to sacrifice Fe(II)-Fe(III) balance to use the best functionals for each group separately. We also investigated the precision (measured as the standard deviation of errors) and show that the target accuracy for iron SCO is 10 kJ/mol for accuracy and 5 kJ/mol for precision, and DFT is probably not going to break this limit in the near future. Importantly, all four types of functional behavior (accurate/precise, accurate/imprecise, inaccurate/precise, inaccurate/imprecise) are observed. More generally, our work illustrates the importance not only of overall accuracy but also of balanced accuracy for systems likely to occur in context.

Accelerating Chemical Discovery with Machine Learning: Simulated Evolution of Spin Crossover Complexes with an Artificial Neural Network

Machine learning (ML) has emerged as a powerful complement to simulation for materials discovery by reducing time for evaluation of energies and properties at accuracy competitive with first-principles methods. We use genetic algorithm (GA) optimization to discover unconventional spin-crossover complexes in combination with efficient scoring from an artificial neural network (ANN) that predicts spin-state splitting of inorganic complexes. We explore a compound space of over 5600 candidate materials derived from eight metal/oxidation state combinations and a 32-ligand pool. We introduce a strategy for error-aware ML-driven discovery by limiting how far the GA travels away from the nearest ANN training points while maximizing property (i.e., spin-splitting) fitness, leading to discovery of 80% of the leads from full chemical space enumeration. Over a 51-complex subset, average unsigned errors (4.5 kcal/mol) are close to the ANN's baseline 3 kcal/mol error. By obtaining leads from the trained ANN within seconds rather than days from a DFT-driven GA, this strategy demonstrates the power of ML for accelerating inorganic material discovery.

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.

Resolving Transition Metal Chemical Space: Feature Selection for Machine Learning and Structure-Property Relationships

Machine learning (ML) of quantum mechanical properties shows promise for accelerating chemical discovery. For transition metal chemistry where accurate calculations are computationally costly and available training data sets are small, the molecular representation becomes a critical ingredient in ML model predictive accuracy. We introduce a series of revised autocorrelation functions (RACs) that encode relationships of the heuristic atomic properties (e.g., size, connectivity, and electronegativity) on a molecular graph. We alter the starting point, scope, and nature of the quantities evaluated in standard ACs to make these RACs amenable to inorganic chemistry. On an organic molecule set, we first demonstrate superior standard AC performance to other presently available topological descriptors for ML model training, with mean unsigned errors (MUEs) for atomization energies on set-aside test molecules as low as 6 kcal/mol. For inorganic chemistry, our RACs yield 1 kcal/mol ML MUEs on set-aside test molecules in spin-state splitting in comparison to 15-20× higher errors for feature sets that encode whole-molecule structural information. Systematic feature selection methods including univariate filtering, recursive feature elimination, and direct optimization (e.g., random forest and LASSO) are compared. Random-forest- or LASSO-selected subsets 4-5× smaller than the full RAC set produce sub- to 1 kcal/mol spin-splitting MUEs, with good transferability to metal-ligand bond length prediction (0.004-5 Å MUE) and redox potential on a smaller data set (0.2-0.3 eV MUE). Evaluation of feature selection results across property sets reveals the relative importance of local, electronic descriptors (e.g., electronegativity, atomic number) in spin-splitting and distal, steric effects in redox potential and bond lengths.

Predicting electronic structure properties of transition metal complexes with neural networks

High-throughput computational screening has emerged as a critical component of materials discovery. Direct density functional theory (DFT) simulation of inorganic materials and molecular transition metal complexes is often used to describe subtle trends in inorganic bonding and spin-state ordering, but these calculations are computationally costly and properties are sensitive to the exchange-correlation functional employed. To begin to overcome these challenges, we trained artificial neural networks (ANNs) to predict quantum-mechanically-derived properties, including spin-state ordering, sensitivity to Hartree-Fock exchange, and spin-state specific bond lengths in transition metal complexes. Our ANN is trained on a small set of inorganic-chemistry-appropriate empirical inputs that are both maximally transferable and do not require precise three-dimensional structural information for prediction. Using these descriptors, our ANN predicts spin-state splittings of single-site transition metal complexes (i.e., Cr-Ni) at arbitrary amounts of Hartree-Fock exchange to within 3 kcal mol-1 accuracy of DFT calculations. Our exchange-sensitivity ANN enables improved predictions on a diverse test set of experimentally-characterized transition metal complexes by extrapolation from semi-local DFT to hybrid DFT. The ANN also outperforms other machine learning models (i.e., support vector regression and kernel ridge regression), demonstrating particularly improved performance in transferability, as measured by prediction errors on the diverse test set. We establish the value of new uncertainty quantification tools to estimate ANN prediction uncertainty in computational chemistry, and we provide additional heuristics for identification of when a compound of interest is likely to be poorly predicted by the ANN. The ANNs developed in this work provide a strategy for screening transition metal complexes both with direct ANN prediction and with improved structure generation for validation with first principles simulation.

Reliable Estimation of Prediction Uncertainty for Physicochemical Property Models

One of the major challenges in computational science is to determine the uncertainty of a virtual measurement, that is the prediction of an observable based on calculations. As highly accurate first-principles calculations are in general unfeasible for most physical systems, one usually resorts to parameteric property models of observables, which require calibration by incorporating reference data. The resulting predictions and their uncertainties are sensitive to systematic errors such as inconsistent reference data, parametric model assumptions, or inadequate computational methods. Here, we discuss the calibration of property models in the light of bootstrapping, a sampling method that can be employed for identifying systematic errors and for reliable estimation of the prediction uncertainty. We apply bootstrapping to assess a linear property model linking the ⁵⁷Fe Mössbauer isomer shift to the contact electron density at the iron nucleus for a diverse set of 44 molecular iron compounds. The contact electron density is calculated with 12 density functionals across Jacob's ladder (PWLDA, BP86, BLYP, PW91, PBE, M06-L, TPSS, B3LYP, B3PW91, PBE0, M06, TPSSh). We provide systematic-error diagnostics and reliable, locally resolved uncertainties for isomer-shift predictions. Pure and hybrid density functionals yield average prediction uncertainties of 0.06-0.08 mm s^-1 and 0.04-0.05 mm s^-1, respectively, the latter being close to the average experimental uncertainty of 0.02 mm s^-1. Furthermore, we show that both model parameters and prediction uncertainty depend significantly on the composition and number of reference data points. Accordingly, we suggest that rankings of density functionals based on performance measures (e.g., the squared coefficient of correlation, r², or the root-mean-square error, RMSE) should not be inferred from a single data set. This study presents the first statistically rigorous calibration analysis for theoretical Mössbauer spectroscopy, which is of general applicability for physicochemical property models and not restricted to isomer-shift predictions. We provide the statistically meaningful reference data set MIS39 and a new calibration of the isomer shift based on the PBE0 functional.

Multiconfiguration Pair-Density Functional Theory Predicts Spin-State Ordering in Iron Complexes with the Same Accuracy as Complete Active Space Second-Order Perturbation Theory at a Significantly Reduced Computational Cost

The spin-state orderings in nine Fe(II) and Fe(III) complexes with ligands of diverse ligand-field strength were investigated with multiconfiguration pair-density functional theory (MC-PDFT). The performance of this method was compared to that of complete active space second-order perturbation theory (CASPT2) and Kohn-Sham density functional theory. We also investigated the dependence of CASPT2 and MC-PDFT results on the size of the active-space. MC-PDFT reproduces the CASPT2 spin-state ordering, the dependence on the ligand field strength, and the dependence on active space at a computational cost that is significantly reduced as compared to CASPT2.