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Predicting Electronic Structure Properties of Transition Metal Complexes with Neural Networks

Jon Paul Janet, Heather J. Kulik

arXiv:1702.05771v1cond-mat.mtrl-sciphysics.chem-ph

TL;DR

Direct DFT calculations are costly for screening, and transition-metal-complex properties are sensitive to the exchange-correlation functional used. The authors trained ANN models using 2,690 DFT geometry optimizations and a simple descriptor set including ligand connection atom and electronegativity. Spin-state splitting predictions were around 3 kcal/mol and metal-ligand bond distances around 0.02–0.03 Å; on CSD cases, MUE improved to 5 kcal/mol across the full 35 molecule set.

  • Problem

    Direct DFT calculations are costly for screening, and transition-metal-complex properties are sensitive to the exchange-correlation functional used.

    Source: §1 · pp. 2–3

  • Method

    The authors trained ANN models using 2,690 DFT geometry optimizations and a simple descriptor set including ligand connection atom and electronegativity.

    Source: §4 · p. 25

  • Results

    Spin-state splitting predictions were around 3 kcal/mol and metal-ligand bond distances around 0.02–0.03 Å; on CSD cases, MUE improved to 5 kcal/mol across the full 35 molecule set.

    Source: §4 · pp. 25–26

  • Takeaways & Limitations

    A relatively sparse feature space can predict electronic-structure properties of transition-metal complexes for screening without precise structural information.

    Source: §4 · p. 26

    Highly negatively charged complexes should be avoided in screening, and high-fitness candidates should be checked with single-point DFT calculations.

    Source: §2.1 · p. 7

Abstract

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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 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.

1. Introduction

Direct first-principles simulation is costly for high-throughput screening, while transition-metal-complex predictions need transferable descriptors and must address uncertainty about the appropriate amount of Hartree-Fock exchange. This work trains ANNs to predict complex properties across exchange fractions using descriptors designed for inorganic chemistry.

1. Introduction

  • Direct first-principles simulation limits exploration of chemical space because it is computationally costly.High-throughput screening is important for discovery, but direct simulation can cover only a small fraction of feasible compounds.Source: §1 · p. 2
  • ANNs for transition-metal complexes require descriptors that transfer beyond training molecules and account for spin-state- and coordination-dependent bonding.Descriptors should balance metal-proximal and metal-distant features because ligand-field strength can vary even when the direct metal-bonding atom is the same.Source: §1 · pp. 2–3
  • Recommendations for Hartree-Fock exchange in transition-metal complexes range from none to low or high amounts, and spin-state splitting is especially exchange-sensitive.The paper identifies predicting both spin-state ordering and its sensitivity across exchange fractions as useful for translating literature predictions or assessing computed data.Source: §1 · pp. 3–4
  • The ANN approach combines cheminformatics-inspired complex-structure generation with structure-functional sensitivity relationships to predict transition-metal-complex properties.The stated aims are descriptors balancing metal-proximal and metal-distant features and predictions across exchange-correlation mixing.Source: §1 · p. 4
  • The paper evaluates ANN predictions of spin-state ordering, exchange sensitivity, and spin-state-specific bond lengths on representative and diverse experimental complexes.The introduction frames rapid spin-state evaluation as relevant to spin-crossover complexes, dye-sensitizers, and open-shell catalysts.Source: §1 · p. 4

2. Methods

The methods describe DFT calculations on constructed transition-metal complexes and the selection and comparison of empirical descriptors for predicting spin-state splitting and sensitivity to Hartree–Fock exchange. They also cover ANN training, dropout-based uncertainty estimation, comparisons with kernel ridge and support vector regression, and data-quality and charge-related considerations for high-throughput screening.

2.1 Test Set Construction and Simulation Details

  • The dataset included 90 homoleptic and 114 heteroleptic complexes generated with molSimplify, plus 35 molecules selected from the Cambridge Structural Database.The homoleptic structures were generated from ten ligands; heteroleptic complexes paired compatible axial and equatorial ligands.Source: §2.1 · p. 5
  • DFT geometry optimizations used B3LYP with HF-exchange fractions from 0.00 to 0.30 and considered high- and low-spin states, yielding 2,690 optimizations.The calculations targeted adiabatic electronic spin-state splittings, omitting entropic and solvent effects.Source: §2.1 · pp. 5–7
  • For screening, the authors advise avoiding highly negatively charged complexes and using single-point DFT to check high-fitness candidates.Entropic and solvent effects were omitted from these calculations.Source: §2.1 · pp. 6–7

2.2. Descriptor Selection

  • The descriptor sets included metal identity, oxidation state, ligand formal charge, and denticity; LASSO regression was used to select descriptors and reduce overfitting.Continuous variables were normalized, while discrete variables used binary coding.Source: §2.2 · pp. 8–9
  • Explicitly encoding axial or equatorial ligand identity produced one of the lowest RMSEs for spin-state splitting and its HF-exchange sensitivity, but limited model applicability.The authors compared seven descriptor sets and preferred broader applicability when sets performed comparably.Source: §2.2 · p. 9
  • Adding set c pairwise electronegativity-difference descriptors slightly reduced ΔE_H-L MSE and gave the lowest ∂ΔE_H-L MSE; removing min(Δχ) slightly increased both errors.Source: §2.2 · pp. 9–10
  • Set f replaced molecule size and bond order with a truncated Kier shape index, yielding comparable or slightly higher MSEs while eliminating system-size dependence.Source: §2.2 · pp. 10–11
  • Set g achieved the lowest MSEs except sets a or c, yet was more transferable; its performance comparable to full-ligand set a supports combined metal- and ligand-centric descriptors.The final feature space has 15 dimensions, with five per-complex descriptors and five per-ligand descriptors for each equatorial or axial ligand.Source: §2.2 · p. 11

2.3 Training and Uncertainty Quantification of ML models

  • The ANN used two hidden layers with tanh activations and linear output, plus dropout and L2 regularization; data were split 60/40 and λ selected by 10-fold cross-validation.Source: §2.3 · pp. 11–12
  • Averaging J=100 dropout-sampled networks provides uncertainty estimates, justified by the loss's equivalence to approximate Gaussian-process variational optimization; variance separates baseline data noise from model variability.Source: §2.3 · p. 13
  • The study compared ANN predictions with KRR and SVR using square-exponential kernels, tuning by grid search and 10-fold cross-validation, and also tested Coulomb-matrix KRR.Source: §2.3 · p. 14

3. Results and Discussion

The results evaluate ANNs for predicting spin-state energetics, exchange sensitivity, and spin-state-specific bond lengths in transition-metal complexes, including transfer to experimental complexes. They also examine factors that shape spin-state ordering and prediction reliability, including ligand effects, Hartree-Fock exchange, descriptor-space distance, and geometry.

3.1 Overview of Data Set Spin-State Energetics

  • The spectrochemical tree identified direct carbon ligands (no Cl, N, O or S) as the root division, producing low-spin ground states for 92% of all Mn, Fe and Co complexes.Source: §3.1 · p. 14
  • Replacing two axial chlorides with pisc ligands increased Fe(III) spin-state splitting by 15.1 kcal/mol per ligand, but subsequent ligand additions had diminishing effects.The increases were 10.4 kcal/mol per additional ligand at n=4 and 7.5 kcal/mol per additional ligand beyond n=4, showing that an additive model cannot reproduce the trend precisely.Source: §3.1 · p. 15

3.2 Spin-State Splittings from an ANN

  • The ANN achieved a test RMSE of 3.0 kcal/mol, predicted splittings within 3 kcal/mol for 72% (389 of 538) of test cases, and classified ground states correctly in 98% (528 of 538).Training RMSE was 3.1 kcal/mol.Source: §3.2 · p. 16
  • Test RMSEs (maximum unsigned errors) were 3.8 (15.7) kcal/mol for Co and 3.3 (13.0) kcal/mol for Fe, attributed to the train/test partition and variable ligand dependence.Errors above 5 kcal/mol occurred for 49 test molecules, and errors above 10 kcal/mol for 4; these were strong-field Co and Cr complexes.Source: §3.2 · p. 16
  • Estimated ±1 and ±2 standard-deviation intervals captured 83% and 98% of test-set computed values, respectively, but these intervals are not rigorous confidence intervals.Estimated standard deviations were highest, around 5 kcal/mol, for Fe(II) and Mn(II), and lowest, around 3 kcal/mol, for Cr and Co.Source: §3.2 · p. 17
  • The ANN outperformed KRR with either the study’s descriptors or a sorted Coulomb matrix, performed slightly better than SVR on test data, and surpassed LASSO.KRR comparisons covered the full dataset and fixed HF exchange.Source: §3.2 · p. 18

3.3 Predicting Exchange Sensitivity with an ANN

  • Spin-state splittings exhibit linear exchange sensitivity that depends strongly on direct ligand identity and field strength, as previously identified [40, 52].Source: §3.3 · p. 18
  • Exchange-sensitivity RMSE was 13 kcal/mol.HFX^-1 on training data and 22 kcal/mol.HFX^-1 on test data; test errors were below 20 (40) kcal/mol.HFX^-1 for 65% (95%) of cases.Source: §3.3 · p. 19
  • The ANN reproduced nonmonotonic and nonconvex ligand exchange sensitivity in heteroleptic compounds, including a Fe(III) complex more sensitive than its respective homoleptic complexes.Source: §3.3 · p. 19
  • Exchange-sensitivity uncertainty intervals ranged from 14 to 17 kcal/mol.HFX^-1; they spanned computed sensitivity for 23 of 29 Fe(III) complexes (80%), and captured 70% (83%) of all data within ±1 (±2) standard deviations.Source: §3.3 · p. 19

3.4 Predicting Equilibrium Geometries with an ANN

  • Bond-length RMSEs were 0.02 Å for both high- and low-spin training data and 0.02 Å for low-spin test data; 79% (81%) and 96% (96%) fell within one and two standard deviations, respectively.Source: §3.4 · p. 20, §0.03 · p. 20

3.5 Expanding the Test Set with Experimental Transition Metal Complexes

  • 15 of 35 CSD complexes were predicted within 5 kcal/mol (mean unsigned error 10 kcal/mol; RMSE 13 kcal/mol), while early-transition-metal cyclams were overestimated by about 30 kcal/mol.Source: §3.5 · p. 21
  • 90% of CSD test cases had correct ground-state assignments (96% excluding cyclams), and the ANN outperformed the next-best model, SVR, by 30% on dissimilar structures.Source: §3.5 · p. 22
  • Reported correlations to prediction error were R2 = 0.3 and R2 = 0.2, respectively; errors >15 kcal/mol occurred only when Euclidean norm difference exceeded 1.0, but high distance did not guarantee inaccurate prediction.Source: §3.5 · p. 22
  • CSD equilibrium bond-length predictions had RMSEs of 0.10 Å for high-spin and 0.07 Å for low-spin geometries.Source: §3.5 · p. 23
  • Despite a 0.2 Å high-spin Fe–N underprediction in test 8, its spin-state splitting was predicted within 3 kcal/mol, reflecting independent ANN outputs.Test 8 and test 35 had the largest high-spin bond-distance errors, exceeding 0.2 Å; interligand effects ignored by the descriptors can constrain bond-length extension.Source: §3.5 · p. 23
  • An interligand O–H…O− hydrogen bond in test 16 produced an unusually short 1.9 Å high-spin Fe–N bond, versus the ANN prediction of 2.1 Å.The authors identify extended rigidity metrics as future work to account for such effects.Source: §3.5 · p. 23
  • Nine of twelve experimental bond lengths on or above the predicted high-spin boundary corresponded to high-spin ground states, while eleven of fifteen on or below the low-spin boundary corresponded to low-spin ground states.Three remaining structures fell at intermediate distances; bond-length-based assignment offers corroboration of direct spin-state prediction.Source: §3.5 · pp. 23–24

3.6 Extrapolating Pure Exchange-Correlation Functionals to Hybrids with an ANN

  • The exchange-sensitivity ANN reduced the maximum error to 23 kcal/mol, mean unsigned error to 5 kcal/mol, and RMSE to 7 kcal/mol.These results were reported for the 35-complex CSD test set when extrapolating from semi-local DFT to hybrids.Source: §3.6 · p. 24
  • Excluding nine cyclams left RMSE/MUE unchanged, and average errors approached the roughly 4 kcal/mol uncertainty of slope predictions over a 20% exchange interval.The unchanged errors indicate good exchange-sensitivity predictions even when spin-state splitting predictions are poor.Source: §3.6 · p. 24
  • Extrapolation eliminated the largest outliers and improved predictions across the CSD test set, but required semi-local DFT geometries or a judicious bond length and offered limited benefit for training-set-represented cases.On the original test set, extrapolated sensitivities atop calculated aHF = 0.00 splittings had an RMSE around 4 kcal/mol, comparable to or slightly worse than direct prediction.Source: §3.6 · pp. 24–25

4. Conclusions

The conclusion presents sparse, non-3D descriptors for predicting transition-metal electronic properties, reports performance on experimentally characterized complexes, and demonstrates exchange-sensitivity extrapolation from semi-local GGAs to hybrid DFT.

4. Conclusions

  • The ANN series was trained using 2,690 DFT geometry optimizations for octahedral complexes generated from 16 candidate axial and equatorial ligands.Source: §4 · p. 25
  • Around 3 kcal/mol splitting-prediction errors and 0.02-0.03 Å metal-ligand bond-distance errors were achieved on unseen test cases.The reported unseen cases used a 60-40% train-test partition.Source: §4 · p. 25
  • Five transferable inputs encode ligand connection, coordinating-atom environment, formal charge, denticity, and metal identity and oxidation state, without requiring 3D structural input.Source: §4 · p. 25
  • 5 kcal/mol MUE for spin states within the proposed Euclidean-distance reliability criteria (10 kcal/mol for the full diverse set), with ANNs outperforming other machine-learning models.The test set comprised experimentally characterized complexes.Source: §4 · p. 25
  • Extrapolating from semi-local GGAs to hybrid DFT with HF-exchange-sensitivity predictions improved MUE to 5 kcal/mol across the full 35-molecule CSD set.Source: §4 · p. 26
  • The authors identify high-throughput screening and assessment of approximate-DFT error sources as potential uses, and note that the data and descriptors could support predictions of ionization potential, redox potential, and molecular orbital energies.The authors state that efforts to predict these additional properties are underway.Source: §4 · pp. 25–26
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