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WorksPhysics and Astronomy

Machine-Learned NMR Shieldings in Molecular Solids with Built-In Hybrid-Functional Molecular Corrections

Matthias Kellner, Ruben Rodriguez-Madrid, Jacob B. Holmes, Pablo A. Unzueta, Gregory J. O. Beran

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Abstract

Fast and accurate chemical shielding estimators are essential for shielding-driven Nuclear Magnetic Resonance (NMR) crystallography. Machine-learning models for shielding predictions have matured significantly and today are primarily limited by the electronic structure reference data they are trained on. Here, we introduce ShiftML4, a shielding-tensor model trained directly on monomer-corrected calculations that approximate PBE0, rather than the PBE reference targeted by earlier ShiftML models. ShiftML4 is trained on a diverse set of structures containing 12 of the most common NMR nuclei in molecular organic solids. On experimental benchmark sets, the 13C isotropic RMSE against experiment is 1.67 ppm, compared with 2.34 ppm for GIPAW-PBE on the same geometries. ShiftML4 gives a similar 1H prediction RMSE to ShiftML3 (0.5 ppm) and improves the 15N RMSE from 7.24 to 6.08 ppm. The model also reduces errors in the shielding-tensor anisotropy, with an RMSE of 4.63 ppm on 13C CSA principal components against 5.85 ppm for GIPAW. The improvements in prediction accuracy are retained on better geometries. Basing shift predictions on structures relaxed with PET-MOLS, a recent machine-learned interatomic potential that reaches approximate hybrid-DFT geometries in seconds, lowers the ShiftML4 errors further to 0.48 ppm (1H), 1.49 ppm (13C) and 3.66 ppm (15N).

Chemical-shift-driven NMR crystallography determines structure by enumerating candidate geometries and selecting those whose predicted shifts best match experiment. [1] Chemical shifts are particularly informative for determining the structure of both crystalline and disordered materials, because the shift of an environment is sensitive to local geometric changes and does not depend strongly on long-range ordering of the lattice. The resolution of shift-driven NMR crystallography and confidence in the structure determination critically depend on the accuracy of the predicted shifts. [2, 4] For this reason, shifts are usually computed with ab-initio electronic structure calculations at the DFT level of theory. [5, 7] NMR crystallography has determined, or helped determine, structures across a wide range of compositions, including crystalline organic molecular solids, [8, 17] inorganic materials, [18, 22] zeolites, [23, 24] ceramics, [25, 26] amorphous drugs, [27, 30] enzymatic sites, [31, 32] cementitious materials, [33, 36] and hybrid-perovskites. [37, 38] The extent of this list of successful structure-determination campaigns motivates continued development of the accompanying computational tools used for chemical shift prediction.

Shielding calculations must also scale to the large models needed to study disordered or amorphous systems. Periodic-DFT GIPAW calculations are ill-suited to study such systems since their computational cost scales cubically with system size. GIPAW is also limited in accuracy, because standard implementations rely on GGA functionals, which have proved insufficient for confident structure selection in challenging polymorphic systems. [39] In this letter we address both limitations with ShiftML4, a transferable machine-learning surrogate for chemical shielding tensors in organic crystals with built-in hybrid-DFT molecular corrections.

Over recent years numerous Machine Learning (ML) models have been developed to predict chemical shifts directly from geometry, [40] with materials of interest ranging from organic crystals [41, 43] to inorganic materials. [44, 48] In a similar spirit, a series of ML models have been developed for other relevant quantities in the nuclear spectroscopies, including zero-field splitting tensors [49] and electric field gradients. [50, 51] Other ML models have instead aimed to correct low-fidelity DFT calculations, as a post-hoc correction step. [52, 53] For crystalline materials there are no experimental databases comparable to those for proteins or solvated small molecules, so ML models for shielding predictions in condensed matter are typically trained on ab-initio reference calculations instead. Focusing specifically on transferable models for organic solids, Paruzzo et al. established the ShiftML family of models, trained on a wide corpus of reference GIPAW-DFT calculations. [41] The family has developed steadily: from the initial model which was only parametrized to predict isotropic shifts for crystals containing up to four elements, to models equipped with uncertainty estimators and additional coverage of sulfur-containing crystals, [3] to the inclusion of thermally distorted structures and the extension of ShiftML to 12 elements, ShiftML2, [54] and ShiftML3 [55] which predicts shielding tensors directly.

Against experimental benchmark data, the latest iteration ShiftML3 is almost as accurate as its GIPAW-DFT training reference. Recent benchmark studies also indicate that ShiftML3 can be used as a drop-in replacement for many NMR-crystallographic tasks, whether it is used for isotropic chemical shielding predictions, [56] or for chemical shielding anisotropy. [57] The fast evaluation speed of ShiftML also enables the study of systems that require extensive sampling of configurational space, with repeated evaluation of chemical shieldings for an ensemble of structures, either because of thermal and quantum nuclear fluctuations or because of intrinsic structural disorder. This is particularly relevant for computing the shifts of acidic or labile protons, which are inherently ensemble averages, as shown in solids by our recent quantum-nuclei-corrected NMR (QNC-NMR) protocol, [58] amorphous structure determination from MD trajectories, [27] and in solution by combining ShiftML3 with machine-learned interatomic potential (MLIP) molecular dynamics. [59]

The accuracy of ShiftML is now mostly limited by the reference data itself rather than by the model. Improving the electronic-structure reference to a higher level of theory, such as hybrid DFT, is essential to improve prediction accuracies of ShiftML against experiment. However, to the best of our knowledge, periodic hybrid-DFT shielding calculations are not routinely available for molecular solids of the size considered here, which makes hybrid-quality training sets hard to generate. More practical schemes, which apply chemical-shift computations from molecular DFT codes up to the coupled-cluster level of theory [60] on organic crystals, have therefore focused on fragment based approaches, [39, 61] or on computing the shifts of molecular clusters with point-charge embeddings of the surrounding molecules. [62] Other shielding computation schemes focus instead on improving periodic shielding computations by computing correction terms on model structures that have been cut from the bulk. Early efforts by Nakajima and coworkers towards correcting chemical shielding tensor calculations in thymine and inorganic compounds relied on QM/MM ONIOM-type [63, 64] calculations, cutting fragments from the bulk and capping cut bonds with hydrogen atoms. [65] Later, Dračíńský and co-workers refined these molecular corrections, by separating entire molecules from the bulk, which reduced shift prediction errors by 50% and 32% on 13C^{13}\mathrm{C} and 15N^{15}\mathrm{N} experimental benchmarks with respect to reference periodic PBE calculations. [66] In this approach, the hybrid-level crystalline shieldings σcrystPBE0\sigma_{\mathrm{cryst}}^{\mathrm{PBE0}} are approximated as:

σ^crystPBE0≈σ^crystGIPAW,PBE+(σmolGIAO,PBE0−σmolGIAO,PBE)(1)\hat{\sigma}_{\mathrm{cryst}}^{\mathrm{PBE0}} \approx\hat{\sigma}_{\mathrm{cryst}}^{\mathrm{GIPAW,PBE}} + \left(\sigma_{\mathrm{mol}}^{\mathrm{GIAO,PBE0}}-\sigma_{\mathrm{mol}}^{\mathrm{GIAO,PBE}}\right) \tag*{(1)}

where σcrystGIPAW,PBE\sigma_{\mathrm{cryst}}^{\mathrm{GIPAW,PBE}} is the shielding tensor of an environment computed by GIPAW-PBE periodic reference calculations. σmolGIAO,PBE\sigma_{\mathrm{mol}}^{\mathrm{GIAO,PBE}} and σmolGIAO,PBE0\sigma_{\mathrm{mol}}^{\mathrm{GIAO,PBE0}} are the corresponding tensors for the same site in the isolated molecule cut from the crystal, computed with a molecular DFT code using the GGA functional matching GIPAW and the PBE0 hybrid, with the GIAO approach [67], respectively.

Molecular correction schemes were later evaluated on various experimental benchmarks using both the monomer correction in the gas phase, [68, 69] and using implicit PCM solvation corrections [70, 71] further increasing 13C^{13}\mathrm{C} and 15N^{15}\mathrm{N} prediction accuracies. Iuliucci et al. examined how the functional choice influences the prediction accuracy of molecular corrections, finding hybrid functionals sufficient to match the accuracy of double-hybrid ones. [71] Ramos and co-workers then showed that hybrid-DFT-relaxed geometries remove the accuracy limitations imposed by GGA geometries, with double-hybrid corrections slightly improving 13C^{13}\mathrm{C} predictions over hybrid corrections, achieving isotropic shift prediction RMSEs of 1.1 ppm using hybrid PBE0 corrections and 0.9 ppm using double-hybrid DSD-PBEP86 corrections. [72]

Molecular correction and fragment based computation schemes have been successfully used to study challenging systems in which GIPAW computations were not accurate enough. Hartman et al. demonstrated that a fragment-based approach on the hybrid-DFT level of theory enabled the discrimination of pharmaceutically relevant organic crystal forms, [39] including acetaminophen, phenobarbital and testosterone polymorphs. Dračíńský and coworkers reported similar gains in discriminating polymorphs using molecular correction schemes and applied molecular corrections to study isocytosine tautomers. [66] Dračíńský also used molecular corrections up to coupled cluster CCSD level of theory to investigate pyridinium fumarates. [73] Further applications of molecular corrections or shielding computations from molecular clusters, ranged from determining tautomers in active sites of enzymes, [32] to computing corrected shift-tensors, [74] additional verification of shift-driven structure refinement, [75] and even shielding computations of 51V^{51}\mathrm{V} nuclei in molecular crystals. [76]

Chaloupecká and coworkers applied molecular corrections directly to ShiftML predictions to improve their accuracy. [77] They concluded that a sufficiently accurate baseline model is needed for the corrections to take full effect. In fact, correcting ShiftML2 reduced the 13C^{13}\mathrm{C} isotropic shift errors from 3.0 to 2.5 ppm, [77] whereas the more accurate ShiftML3 could be corrected from 2.4 ppm to 1.6 ppm. [78] Correcting ShiftML predictions this way remains laborious, and it would be preferable if the molecular corrections were absorbed directly into the model predictions.

Here, we present ShiftML4, a new shielding prediction model trained directly on the monomer-corrected shielding tensors σcrystPBE0\sigma_{\mathrm{cryst}}^{\mathrm{PBE0}} of (1). The model takes the periodic crystal structure as input and directly predicts the corrected shielding tensors. The monomer correction, including implicit cutting of the molecular fragments, is added directly to the periodic GIPAW-PBE targets, and neither isolated-molecule GIAO calculations nor semiempirical calculations are required during inference. Because ShiftML4 uses the same architecture as ShiftML3, its evaluation cost and linear scaling with the number of atoms remain unchanged.

Computing molecular corrections for the over 15,000 crystals in the ShiftML2/ShiftML3 training, validation and test sets by hand would be tedious and likely induce biases in separating molecules from the solid. We therefore built a pipeline that takes the crystal structure, cuts out molecules, prepares and runs the low- and high-level molecular DFT calculations, removes outliers, and assembles the training data. We automate the cutting of molecular fragments based on the bond connectivity between atoms in the crystals. For every molecular crystal in the ShiftML2/3 datasets, we compute Mayer-Wiberg bond orders [79, 80] between all pairs of atoms at the semiempirical GFN2-xTB level of theory. [81] We consider pairs of atoms bonded if their bond order exceeds 0.5. We represent the crystal as a graph, with atoms as nodes and bonded atom pairs as edges. The connected components of this graph define the molecular fragments. Both steps explicitly consider periodic boundary conditions, so molecules crossing the cell boundary are reconstructed across images. After cutting, each fragment is assigned a charge and a spin multiplicity. Atomic graphs and bookkeeping of atomic indices of the molecular fragments are handled with the pymatgen [82], Atomic Simulation Environment (ASE) [83] and networkx [84] libraries. Atomic GFN2-xTB Mulliken charges [85] from the periodic calculation are rounded to integers, constrained to sum to zero over the cell, and then summed within each fragment to give its net charge. Each fragment is then assigned the lowest spin multiplicity consistent with its resulting electron count. The procedure required handling of selected edge cases, such as spurious bonds between π\pi-stacked sulfur-containing heterocycles and fractional protonation states, which could be resolved by selectively increasing bond-order thresholds between selected nuclei. GFN2-xTB calculations use the tblite integration. [86] Molecular GIAO shieldings [87] are computed with ORCA 6.1.0 [88] interfaced with the ORCA QPI Python API [89] to orchestrate shielding computations. We use ORCA input parameters close to the ones selected by Ramos et al. [72] ORCA DFT input parameters are listed in the SI section S10.

Molecular DFT calculations are performed for PBE [90] and PBE0 [91] functionals employing the cc-pVTZ basis set [92, 93], and the resolution of identity approximation. [94, 95] Hybrid computations are performed with the accelerated “chain-of-spheres” algorithm. [96] All molecular calculations are performed with the implicit CPCM solvation correction [97] using dichloromethane as the medium. The molecular corrections are then added to the GIPAW shielding calculations whose parameters are described in the ShiftML2 reference publication. [54] We perform calculations for the entire ShiftML2/ShiftML3 training set and after discarding unconverged calculations and outliers we obtain a database with a total of 12,613 structures in the training set, 495 structures in the validation set and 1,218 structures in the test set. We discard crystals for which the splitting and spin-state assignment produce doublet fragments. Outliers are removed by discarding entire crystal structures containing any atom with an anomalous absolute shielding or an unusually large molecular correction relative to its element’s typical value. We then train a committee of eight nanoPET models, a variant of the graph neural network (GNN) Point-Edge-Transformer (PET) architecture, [98] on the cleaned training sets. nanoPET is trained on, and predicts, the shielding tensors in their irreducible spherical tensor (IST) representation: a scalar (λ=0\lambda= 0), a pseudovector (λ=1\lambda= 1) and a proper five-vector (λ=2\lambda= 2), sharing the same rotational behaviour as the corresponding spherical harmonics YλY^\lambda. nanoPET converts atomic environments into local representations through one message-passing sequence of unconstrained graph transformer blocks. The IST components are predicted from these representations via linear readouts. The ShiftML implementation converts the IST components back to the Cartesian basis before returning the tensors. Model training was performed with the metatrain library and using the metatomic integration for model deployment. [99] The training hyperparameters are identical to those of the ShiftML3 training. Each model is initialized with a different random seed and trained by mini-batch gradient descent with the Adam optimizer [100], a learning rate of 3⋅10−43 \cdot10^{-4}, a batch size of 4, and 1,500 epochs. Note that we remove one model of the committee due to convergence issues and the final ShiftML4 model is made up by an ensemble of 7 nanoPET models. The ShiftML3 paper discusses the benefits of a committee over a single PET model. [55] Importantly, using a committee, higher prediction accuracies are achieved and the spread of the committee predictions can be interpreted as an uncertainty estimate of the model predictions.

We first evaluate ShiftML4 and ShiftML3 against the DFT reference they are trained on: ShiftML4 against monomer-corrected isotropic shieldings, ShiftML3 against GIPAW/PBE. On the same hold-out test structures, both reach comparable accuracy. For ${}^{1}\mathrm{H}

ShiftML4 achieves prediction RMSEs of 0.39 ppm and ShiftML3 0.42 ppm against DFT and for 13C{}^{13}\mathrm{C}, 2.20 ppm for ShiftML4 and 2.24 ppm for ShiftML3. In Table 1 we compare prediction error metrics of ShiftML4 and ShiftML3 for isotropic chemical shieldings σiso\sigma_{\mathrm{iso}} on the most commonly studied spin-1/2 nuclei in the test set. We report the full prediction errors against DFT in the SI, tbl:S4. Across the remaining elements, both models show similar errors. This suggests that the cutting procedure and the GIAO calculations add little noise to the targets, and that the corrected shieldings can be learned directly from the periodic structure.

nucleusσiso\sigma_{\mathrm{iso}}σij\sigma_{ij}σPAS\sigma_{\mathrm{PAS}}
sML3sML4sML3sML4sML3sML4
1H{}^{1}\mathrm{H}0.420.390.810.780.850.82
13C{}^{13}\mathrm{C}2.242.203.433.313.773.68
15N{}^{15}\mathrm{N}10.2011.1111.5311.8315.6716.41
19F{}^{19}\mathrm{F}5.554.716.085.557.546.83
31P{}^{31}\mathrm{P}18.6816.3019.7019.4223.9821.62

Table 1. RMSE (in ppm) of ShiftML3 (sML3) and ShiftML4 (sML4) predictions of the isotropic shieldings σiso\sigma_{\mathrm{iso}}, the tensor components σij\sigma_{ij} and the principal tensor components σPAS\sigma_{\mathrm{PAS}}, evaluated on a hold-out test set. ShiftML3 is evaluated against GIPAW-PBE reference calculations and ShiftML4 against PBE0 molecular-corrected GIPAW-PBE calculations.

Ref. CodeExp. shift [ppm]PBE geometriesPBE0 geometries
DFTsML3sML4sML3sML4
2.9028.0028.1128.3928.2328.54
1.7829.3228.8029.2128.9529.39
1.8828.8829.4729.6729.6929.88
1.8828.4528.4728.8128.6028.88
1.8029.1428.8629.1429.0329.34
1.6029.2929.5829.8929.8630.18
0.4430.5130.2930.4930.4830.74
1.5429.1829.0129.3529.0829.42
1.8828.2028.6228.8329.0229.20
1.8828.8729.5429.6129.6829.81
0.8030.3629.8330.0129.9730.17
0.8029.5430.1130.2730.1830.34
1.0030.1629.1729.3629.3029.54
1.7428.6629.0329.2129.2929.55
1.7429.6629.7530.0929.9230.29
0.7329.9530.0930.0530.3730.37
0.7329.9530.0930.0530.3730.37
0.7329.9530.0930.0530.3730.37
0.7329.5929.9029.9430.0830.14
0.7329.5929.9029.9430.0830.14
0.7329.5929.9029.9430.0830.14
0.7330.7930.7630.8431.1231.17
0.7330.7930.7630.8431.1231.17
0.7330.7930.7630.8431.1231.17
Penicillin4.1026.4626.7226.4526.7826.65
6.4024.3624.5324.3524.9024.87
5.7025.6325.4525.5525.3825.36
1.7029.1329.4929.5129.6829.73
1.7029.1329.4929.5129.6829.73
1.7029.1329.4929.5129.6829.73
0.9030.3230.3930.3830.4930.55
0.9030.3230.3930.3830.4930.55
0.9030.3230.3930.3830.4930.55
6.2024.7624.9124.8625.6425.44
3.9026.8627.2227.3626.8027.13
4.7025.7926.3226.2825.9626.05
7.1023.3723.5723.4123.7123.59
7.1023.3723.5723.4123.7123.59
7.1023.3723.5723.4123.7123.59
7.1023.3723.5723.4123.7123.59
7.1023.3723.5723.4123.7123.59
Phenylphosphonic acid12.7016.9416.8416.6417.7517.66
11.5017.9218.2017.7019.3818.93
6.5024.4924.7424.3224.9424.54
6.8024.0224.1623.9124.3824.14
8.5022.1922.2821.8222.5422.09
7.1023.9624.1123.5524.3223.76
6.2024.7224.8224.4924.9824.64
Cocaine3.7627.0127.0126.9927.1727.16
3.7826.7527.2427.4527.4727.70
5.6325.1925.4425.6025.6325.87
3.3227.1227.3327.4627.5327.66
3.0628.2028.3228.6528.4428.79
3.4927.6928.0628.2128.3628.51
2.9128.6728.9829.1229.1929.37
3.3827.6428.3328.1528.6028.41
2.5628.3928.6428.7228.8328.88
2.1228.7028.9728.9029.1329.09
1.0429.5829.4729.3729.6229.54
1.0429.5829.4729.3729.6229.54
1.0429.5829.4729.3729.6229.54
8.0122.5522.5322.5322.7522.71
8.0122.9522.9922.8823.1323.03
8.0122.4822.7922.6422.9422.82
8.0122.3322.7322.4122.9122.60
8.0122.5622.8622.6523.0422.84
3.7826.7726.8126.8327.0327.06
3.7826.7726.8126.8327.0327.06
3.7826.7726.8126.8327.0327.06

Table 1.

Table 1 also compares ShiftML4 and ShiftML3 prediction errors on tensor components σij\sigma_{ij} and principal components σPAS\sigma_{\mathrm{PAS}}. We find that, as for the isotropic shieldings, ShiftML4 and ShiftML3 perform comparably on tensorial quantities. More detailed evaluation metrics of principal components and tensor components are listed in tables S5 and S6. Due to the cutting and charge assignment procedure, the alkali and alkaline-earth nuclei (23Na{}^{23}\mathrm{Na}, 25Mg{}^{25}\mathrm{Mg}, 39K{}^{39}\mathrm{K}, 43Ca{}^{43}\mathrm{Ca}) are represented by only tens to hundreds of environments in the training set, so their reported metrics rest on limited statistics and predictions of ShiftML4 on metal-ion containing structures should be treated with care. However, including them is important to cover prediction on ionic molecular solids of particular relevance for pharmaceutical salt formulations, for example.

Next, we evaluate ShiftML4 on representative experimental benchmark sets of organic crystals and assess the prediction accuracy against assigned experimental 1H{}^{1}\mathrm{H}, 13C{}^{13}\mathrm{C} and 15N{}^{15}\mathrm{N} shifts. We evaluate ShiftML4 and ShiftML3 on published geometries relaxed with PBE-D2 (1H{}^{1}\mathrm{H} benchmark set) and PBE-D3(BJ) (13C{}^{13}\mathrm{C} and 15N{}^{15}\mathrm{N} sets) and compare against GIPAW-PBE reference. In Figure 1 we show parity plots of ShiftML4, ShiftML3 and GIPAW-DFT predictions against experimental reference values. We find that ShiftML4 prediction errors decrease across all species. On identical PBE-geometries the 13C{}^{13}\mathrm{C} RMSE drops from 2.44 ppm (ShiftML3) to 1.67 ppm, a ≈30%\approx30\% reduction, and a ≈20%\approx20\% reduction for the 15N{}^{15}\mathrm{N} RMSE, from 7.24 ppm (ShiftML3) to 6.08 ppm (ShiftML4). The regression slopes converting shieldings to shifts are systematically smaller in magnitude for ShiftML4 than for ShiftML3 across all three nuclei, and closely match the ab initio PBE0 molecular-corrected slopes of Ramos et al. [72] on the two nuclei taken from their benchmark using identical geometries (-0.913 against -0.930 for 13C{}^{13}\mathrm{C}, -0.967 against -0.975 for 15N{}^{15}\mathrm{N}). The deviation from the original slope is inherited from the reference level of the theory rather than introduced by the ML model. In line with the observations from Ramos et al., [72] molecular corrections and ShiftML4 improve prediction accuracies predominantly for 13C{}^{13}\mathrm{C} and 15N{}^{15}\mathrm{N}, whose shieldings are dominated by intramolecular interactions, whereas 1H{}^{1}\mathrm{H} predictions, which are governed to a much greater extent by intermolecular contributions, change only marginally. In SI Figure S5 we plot differences between ShiftML4 and ShiftML3 isotropic shift predictions against experimental reference shifts, for 1H{}^{1}\mathrm{H}, 13C{}^{13}\mathrm{C} and 15N{}^{15}\mathrm{N}, visualizing the effects of molecular corrections across the observed shift range of the reference nucleus. Evaluating all 49 benchmark geometries with ShiftML4 took under a minute on a laptop (MacBook Pro M1). The GIPAW reference calculations took 80 hours on a 32-core compute node.

Parity plots comparing ShiftML4, ShiftML3, and DFT-PBE predicted chemical shifts with experimental reference values for $^{1}\mathrm{H}$, $^{13}\mathrm{C}$, and $^{15}\mathrm{N}$

Figure 1. Parity plots between ShiftML4, ShiftML3 and DFT predicted shifts and experimental reference values, for 1H^{1}\mathrm{H} (a,b,c), 13C^{13}\mathrm{C} (d,e,f), and 15N^{15}\mathrm{N} (g,h,i). Predictions and GIPAW-PBE calculations (DFT) are performed based on PBE geometries (in blue) from Ramos et al. [72] (relaxed with PBE-D3(BJ)) and for 1H^{1}\mathrm{H} on 13 structures from the ShiftML2 publication (relaxed with PBE-D2) [54] and PBE0 geometries (in red) relaxed using PET-MOLS. The black line shows the linear relation, converting computational isotropic shieldings into shifts. The slope and intercept for each method and nucleus are indicated in the respective subplot.

Delta shifts correction plots for $^{1}$H, $^{13}$C and $^{15}$N experimental benchmarks

Figure S5. Delta shifts correction evaluated on the 1^{1}H, 13^{13}C and 15^{15}N isotropic chemical shift benchmarks as a function of the experimental chemical shifts. The delta (Δ\Delta-shifts) is defined as the difference ShiftML4 - ShiftML3 predicted shifts (i.e. removing constant offsets and skew through linear regression). In blue and red, PBE and PBE0 (PET-MOLS) geometries were taken as starting points for shielding calculations.

Ramos et al. [72] recognized that the quality of reference geometries contributes significantly to the chemical shift prediction accuracies. They showed that moving from PBE-D3 to PBE0-D3 geometries removes the bias in bond lengths relative to experiment and improves shift predictions when computing either GIPAW-PBE, or molecular-corrected shieldings from PBE0-D3 geometries. The steep computational cost of ab-initio geometry relaxations forms another bottleneck in the NMR-crystallography workflow. We therefore employ the recently developed PET-MOLS machine-learned interatomic potential (MLIP) [58] to relax the benchmark crystal structures, cutting the cost for relaxations from CPU-hours to seconds. PET-MOLS covers the same chemical space as ShiftML4 and ShiftML3 and is trained on PBE0+MBD reference data. A similar approach making use of gas-phase molecule trained MLIPs for geometry relaxation in the context of NMR crystallography has been proposed by Gunaga et al. [56]. In Figure 1 we also show parity plots of ShiftML4 and ShiftML3 predictions from PET-MOLS relaxed geometries against experimental reference values. Basing chemical shielding predictions on approximate hybrid-DFT quality geometries further decreases prediction errors of both ShiftML4 and ShiftML3. On PET-MOLS relaxed geometries the prediction errors of ShiftML4 on 13C{}^{13}\mathrm{C} decrease from 1.67 ppm to 1.49 ppm, for 15N{}^{15}\mathrm{N} from 6.08 ppm to 3.66 ppm and slightly for 1H{}^{1}\mathrm{H} from 0.51 ppm to 0.48 ppm over the ShiftML4 predictions on PBE-geometries. The 13C{}^{13}\mathrm{C} isotropic shift RMSE of ShiftML3 also improves significantly on PET-MOLS geometries, from 2.44 ppm to 1.76 ppm. For comparison, Gunaga and coworkers [56] report 1.78 ppm and 1.84 ppm for ShiftML3 predictions on the same crystals relaxed with MACE-POLAR-1 [101] and UMA-OMOL, [102] respectively, albeit with looser relaxation convergence criteria. Taken together, moving from ShiftML3 on PBE geometries to ShiftML4 on PET-MOLS geometries reduces the isotropic RMSE against experiment by 39% for 13C^{13}\mathrm{C} (from 2.44 ppm to 1.49 ppm) and by 49% for 15N^{15}\mathrm{N} (from 7.24 ppm to 3.66 ppm). We observe smaller improvements for 1H^{1}\mathrm{H} (0.53 ppm, to 0.48 ppm) and note in passing that 1H^{1}\mathrm{H} chemical shifts are particularly sensitive to nuclear quantum effects, which are not treated here relying only on static geometry-relaxed structures. [58, 103]

Finally, we evaluate ShiftML4 chemical shift anisotropy (CSA) predictions against experiment on a benchmark of 13C^{13}\mathrm{C} principal tensor components σPAS\sigma_{\mathrm{PAS}} of the symmetric shift tensor, from Hartman et al. [61]. In Figure 2 we show parity plots of GIPAW-DFT, ShiftML3 and ShiftML4 σPAS\sigma_{\mathrm{PAS}} predictions against experimental benchmark values. We find that the gains in isotropic prediction accuracies of ShiftML4 against ShiftML3 also translate to gains in prediction accuracy of σPAS\sigma_{\mathrm{PAS}}. ShiftML4 achieves a joint component RMSE of 4.63 ppm against experiment, compared with 5.20 ppm for ShiftML3 and 5.85 ppm for DFT. That ShiftML3 already outperforms its own DFT reference here is likely fortuitous.

Parity plots comparing ShiftML4, ShiftML3, and DFT-PBE predictions with experimental $^{13}\mathrm{C}$ principal shielding components

Figure 2. Parity plots between ShiftML4 (a), ShiftML3 (b), DFT (c) predicted principal components σPAS\sigma_{\mathrm{PAS}} of the shielding tensor and experimental reference values. The black line shows the linear relation, converting computational principal components into experimental components. The slope and intercept for each method are indicated in the respective subplot. In blue and red, PBE and PBE0 (PET-MOLS) geometries were taken as starting points for shielding calculations.

In the Supporting Information Section S7, we also present an evaluation of ShiftML3 and ShiftML4 on an experimental benchmark set of 15N^{15}\mathrm{N} shielding tensor principal components. Both models achieve comparable prediction RMSEs of 9.49 ppm (ShiftML3) and 9.22 ppm (ShiftML4). On PET-MOLS geometries, the prediction accuracies of ShiftML4 and ShiftML3 further improve to 7.89 ppm and 8.12 ppm, respectively, comparable to those of PBE0-fragment based calculations using PBE geometries (7.95 ppm). Kalakewich and coworkers [104] report experimental measurement errors of 5.5 ppm for the 15N^{15}\mathrm{N} shift tensor in single crystalline histidine, which may explain the smaller improvement in prediction accuracy achieved with ShiftML4 compared to 13C^{13}\mathrm{C} σPAS\sigma_{\mathrm{PAS}} predictions.

In summary, here we have introduced ShiftML4, a transferable deep learning model that predicts chemical shielding tensors in organic solids at near-hybrid-DFT quality, with the monomer corrections absorbed directly into the model so that no molecular DFT calculation is needed at inference.

On experimental benchmarks it lowers the isotropic 13C^{13}\mathrm{C} RMSE from 2.44 ppm to 1.67 ppm with respect to ShiftML3, below the 2.34 ppm of established GIPAW/PBE workflows. 1H^{1}\mathrm{H} shifts, which are dominated by intermolecular contributions, are largely unaffected by the monomer correction, with the ShiftML4 RMSE only slightly lower than that of ShiftML3 (0.53 ppm to 0.51 ppm), in line with previous observations. [66, 72] For 15N^{15}\mathrm{N} the isotropic RMSE is reduced from 7.24 ppm to 6.08 ppm, comparable to the 5.85 ppm of GIPAW-PBE. The improved prediction accuracies extend to the anisotropy, with a 13C^{13}\mathrm{C} σPAS\sigma_{\mathrm{PAS}} RMSE of 4.63 ppm against 5.85 ppm for GIPAW-PBE. Combined with the PET-MOLS MLIP for fast geometry relaxation at approximate hybrid-DFT quality, it brings the 13C^{13}\mathrm{C} RMSE against experiment to 1.49 ppm and the 15N^{15}\mathrm{N} RMSE to 3.66 ppm, at machine-learning cost throughout. ShiftML4 is available in the ShiftML Python package.

Acknowledgements

The authors thank Frédéric Mentink-Vigier for stimulating discussions on higher-level reference corrections for machine-learned chemical shieldings. This work has been supported by Swiss National Science Foundation Grant No. 200020_212046, and by the NCCR MARVEL, funded by the Swiss National Science Foundation (grant number 182892). MC also acknowledges funding from the ERC Horizon 2020 Grant No. 101001890-FIAMMA. GJOB acknowledges funding from the U.S. National Science Foundation (CHE-1955554). PAU thanks the U.S. Department of Energy, Office of Science, Office of Workforce Development for Teachers and Scientists, Office of Science Graduate Student Research (SCGSR) program. The SCGSR program is administered by the Oak Ridge Institute for Science and Education (ORISE) for the DOE. ORISE is managed by ORAU under contract number DE-SC0014664. All opinions expressed in this paper are the author’s and do not necessarily reflect the policies and views of DOE, ORAU, or ORISE.

Supporting Information

The supporting information is provided below, containing additional details on the dataset composition, detailed model evaluations and the usage of the ShiftML Python package, which is available under an open-source license at https://github.com/lab-cosmo/shiftml. The Python scripts for cutting, evaluation against experimental reference data, training, validation and test data in the xyz format as well as raw ORCA computations for the molecular computations can be accessed at the following link DOI:10.5281/zenodo.XXXXXXX [will be completed on acceptance] and is available under the CC BY-SA 4.0 (Creative Commons Attribution-ShareAlike 4.0 International) license.

Experimental shifts and predicted shieldings for the 1^{1}H benchmark dataset

Assigned chemical shift values were used from the previously curated 1^{1}H benchmark data set. [54] Methyl rotational dynamics were accounted for by averaging the chemical shifts of the three proton positions into a single effective value per methyl group, which was assigned a weight of three in the linear regression. The same treatment is applied to other magnetically equivalent protons, such as those in methylene groups. In cases of pairwise ambiguities of the experimental assignments, experimental chemical shifts were resolved by selecting the assignment yielding the lowest shift RMSE. (e.g. Ha_{\mathrm{a}} and Hb_{\mathrm{b}} of CH2_{2} groups)

Ref. CodeExp. shift [ppm]PBE geometriesPBE geometriesPBE geometriesPBE0 geometriesPBE0 geometries
DFTsML3sML4sML3sML4
Dimethylimidazole4.8025.9125.9125.7525.9225.66
0.7029.5529.5329.4729.8729.75
0.7029.5529.5329.4729.8729.75
0.7029.5529.5329.4729.8729.75
1.4029.4829.4129.4629.5729.62
1.4029.4829.4129.4629.5729.62
1.4029.4829.4129.4629.5729.62
13.0015.5916.3015.6516.1015.59
1.4028.8829.0828.9629.3129.25
1.4028.8829.0828.9629.3129.25
1.4028.8829.0828.9629.3129.25
1.5028.8728.9728.8729.1529.06
1.5028.8728.9728.8729.1529.06
1.5028.8728.9728.8729.1529.06
15.0014.3414.2314.1814.5414.52
5.2024.7324.8324.8224.9625.05
AZD57181.2029.2029.3229.2929.4929.47
1.2029.2029.3229.2929.4929.47
1.2029.2029.3229.2929.4929.47
10.6018.4518.1118.1618.1518.16
5.8023.8123.7823.7024.0323.98
6.9023.1723.8823.8223.9923.93
7.3023.0423.0622.8423.2323.03
6.7023.6724.3424.2124.5724.49
7.0022.5122.7822.6622.9422.95
3.9026.2127.0126.7527.2627.02
1.6028.8528.6728.8228.9029.13
0.0030.3430.3530.6230.5230.79
1.7028.4028.5528.6028.7528.79
1.6028.4629.0128.9729.3729.21
1.6029.0528.9029.0529.0629.37
-0.5029.9630.0230.2630.1930.43
0.8029.3629.5129.6229.7229.84
-0.5030.7930.5730.9130.7231.07
0.8029.3029.2629.2729.4129.42
7.7021.3321.3621.7221.7222.14
7.6022.4022.5822.3722.8822.55
1.7028.8729.6329.7129.6129.62
2.7027.4927.8028.0827.9428.19
6.9022.5922.4622.1823.4123.15
1.9028.0328.0727.8628.2428.03
2.7027.2327.5027.5927.6427.79
Uracil7.5022.1521.9921.8422.0521.98
10.8017.9518.3218.2319.0619.01
11.2017.1617.9817.4819.1918.75
6.0023.8723.9923.6824.3724.06
Naproxen7.0022.7723.0922.9323.2923.18
6.1024.1924.2424.0624.3224.14
3.8027.1326.9626.6927.0726.80
4.5025.7725.6925.5825.9725.86
4.1026.1825.8525.5025.9525.64
5.9024.8724.9324.6025.0524.73
3.2026.7627.7327.6827.7927.78
1.8028.6429.3629.3029.5729.53
1.8028.6429.3629.3029.5729.53
1.8028.6429.3629.3029.5729.53
2.3028.1628.2528.0928.4428.26
Ref. CodeExp. shift [ppm]PBE geometriesPBE0 geometries
DFTsML3sML4sML3sML4
2.3028.1628.2528.0928.4428.26
2.3028.1628.2528.0928.4428.26
11.5015.8316.3216.2817.3617.28
Furosemide8.4022.2122.4722.1322.7622.41
12.7016.6516.7316.5017.1016.95
6.5023.3423.8023.3524.0523.65
6.5023.3423.8023.3524.0523.65
7.9022.7123.2622.9023.2622.94
8.7021.8622.4121.9622.5422.12
5.6025.3925.5925.3425.5425.49
3.8026.5826.5726.5326.6026.57
6.0024.1224.2624.0524.5524.38
6.0023.9624.4024.5824.6424.85
7.7022.1322.4722.5222.6422.67
8.4021.8022.3922.0122.5922.17
4.3026.3625.7025.7625.8125.91
12.7016.2716.4316.1217.1316.90
6.7022.6622.9822.6023.4523.09
6.7022.6622.9822.6023.4523.09
6.2024.9625.2324.6325.4724.85
8.6021.9722.4522.0222.5822.13
4.3025.8725.6425.5825.8825.65
5.7024.8624.6424.6524.9124.94
6.4023.4523.4823.4923.6723.77
6.5023.6023.7623.8824.1024.26
Theophylline14.6014.2214.3213.8615.4615.10
7.7022.7822.8022.5523.0322.73
3.4027.1327.4927.2827.7227.50
3.4027.1327.4927.2827.7227.50
3.4027.1327.4927.2827.7227.50
3.4027.1327.4927.2827.7227.50
3.4027.1327.4927.2827.7227.50
3.4027.1327.4927.2827.7227.50
Flutamide7.1022.9123.1823.0623.4223.17
9.9020.4920.6620.4720.6320.51
8.0022.6323.0822.7223.3322.93
8.0020.8421.0320.8221.8221.44
2.0028.3028.3928.5928.5328.71
1.2029.4629.5129.4929.7129.67
1.2029.4629.5129.4929.7129.67
1.2029.4629.5129.4929.7129.67
1.2029.4629.5129.4929.7129.67
1.2029.4629.5129.4929.7129.67
1.2029.4629.5129.4929.7129.67
Indomethacin-nicotinamide7.3023.5523.2023.1623.3923.33
5.5025.1525.1424.7925.3124.98
6.8024.0423.9523.7724.1323.91
0.9030.1530.0430.0430.1530.15
0.9030.1530.0430.0430.1530.15
0.9030.1530.0430.0430.1530.15
2.9028.1628.0227.6428.1927.84
2.9028.1628.0227.6428.1927.84
2.9028.1628.0227.6428.1927.84
3.4027.4427.6927.6727.8327.91
3.4027.4427.6927.6727.8327.91
16.3012.8613.4712.7814.4313.89
6.4025.2825.0724.5825.2824.76
6.0023.5523.4623.1523.9223.60
6.0024.9525.0924.8325.2425.05
6.4024.6924.6124.6024.7824.76
9.8021.1821.7121.4021.9521.68
9.8020.9822.2921.9022.5422.11
8.3022.3723.2722.8323.5923.18
7.7023.1623.3523.3923.6123.62
7.3022.2122.3622.3722.6222.69
9.0020.9721.1321.0621.5821.52
Flufenamic acid8.3022.3022.3622.2322.3622.31
6.0024.3524.5824.3824.7924.62
5.4025.1925.4625.0025.6525.22
6.8023.1123.3323.4523.6323.81
12.4016.1316.0415.7916.9416.67
9.6020.0219.5919.5419.9719.92
6.9023.8223.5923.4723.6623.52
6.2024.2124.0823.8024.2523.88
5.9024.6324.6224.6224.9324.86
7.3023.1523.8523.8124.0024.00
AZD8329 Form 46.9223.7124.1323.7624.3524.01
8.6921.9622.3021.9022.5422.14
9.0121.3821.5021.3521.4921.33
8.4722.2922.8022.6122.8422.79
15.3712.9113.6512.9515.9015.18
7.7322.5722.4322.2322.5922.45
9.6418.9219.6019.4620.6520.52

Table S1. Experimental isotropic 1^{1}H chemical shifts and predicted shieldings for the atomic sites in the benchmark dataset. Predicted shieldings are given for PBE and PBE0 (PET-MOLS) geometries, computed with GIPAW-DFT and with the ShiftML3 (sML3) and ShiftML4 (sML4) machine-learning models.

Experimental shifts and predicted shieldings for the 13C^{13}\mathrm{C} benchmark dataset

Ref. CodeExp. shift [ppm]PBE geometriesPBE0 geometries
DFTsML3sML4sML3sML4
ADENOS12154.1012.4013.0111.4817.4516.17
147.9019.5219.5419.3422.8622.68
119.3045.5746.1249.1549.7352.79
154.7015.0215.1513.1618.7116.54
137.4029.8628.1526.2632.4530.80
91.9072.0775.8881.4080.1085.54
74.6091.6092.1199.2396.35103.23
70.9095.2696.01103.1999.77106.17
84.5081.4681.5988.5786.7593.17
62.50106.28106.96114.53110.85118.13
ASPARM03176.40-12.88-12.11-14.23-4.34-6.04
51.80119.78119.42123.86122.90127.06
36.10137.08136.95142.76138.63144.49
177.10-8.05-7.22-9.61-1.44-3.24
FRUCTO0265.40102.89102.71109.21106.42112.75
99.7060.6362.6670.3468.4976.03
67.2099.9499.59106.42103.60110.12
69.0099.7499.14106.69102.16109.53
71.4094.2696.00103.47100.45107.53
64.90101.98103.54109.72107.97113.85
GLUTAM01173.00-9.35-9.86-12.01-2.40-3.97
53.30116.24115.99121.25119.47124.51
25.50146.35145.36152.14147.31153.97
28.50143.00143.23148.26145.42150.43
176.50-9.24-8.84-11.66-3.10-5.33
GLYCIN29176.20-11.16-10.94-13.59-3.91-6.40
43.50129.32128.90132.47131.08135.21
HXACAN09133.1035.8937.4938.1240.9541.59
123.4045.5946.8047.1848.0247.88
115.7054.2352.5353.9654.3855.60
152.3014.6914.4716.9317.4819.81
116.4052.4551.9953.4654.4655.76
120.6048.9849.3250.0750.2650.69
169.80-1.91-4.05-6.572.630.48
23.80148.06148.73153.17150.68155.03
LALNIN12176.80-14.33-13.47-15.54-6.38-7.88
50.90119.89119.84124.76123.41128.08
19.80153.62154.74159.37156.07160.73
LCYSTN21174.00-8.99-9.71-11.99-1.74-3.73
56.70114.47114.10118.93117.20121.47
28.80140.68139.78146.17144.15150.97
LSERIN01175.10-11.31-10.53-13.15-3.11-5.16
55.60114.92113.87118.35116.86121.11
62.90103.47104.28111.81108.32115.99
LSERMH10175.60-12.30-12.45-14.72-5.20-6.98
58.30111.17111.98116.55115.28119.59
61.80106.44106.96114.60110.45118.28
LTHREO01171.90-7.17-5.76-8.171.01-0.70
61.20109.64110.40115.40113.44118.16
66.8099.5098.49106.50102.68110.32
20.40153.78152.89157.91153.95158.91
LTYROS11175.40-11.60-12.34-14.82-5.35-7.29
130.3037.3836.9736.8939.0138.68
116.4052.8352.1854.0454.6456.21
56.40113.77114.49119.20117.87122.58
131.0038.3436.8436.8638.5138.28
118.0051.4651.9652.8251.4755.17
155.6010.4111.3013.3014.0515.64
36.80134.63134.78142.19136.58143.84
123.6046.2244.2346.3646.4048.70
MBDGAL02105.7056.6958.0465.7162.9770.44
71.2096.6597.59103.78101.15107.15
72.1094.3995.11101.8598.84105.35
69.3096.8796.52104.22100.54107.80
75.6091.8789.0696.0992.5799.27
Ref. CodeExp. shift [ppm]PBE geometries DFTPBE geometries sML3PBE geometries sML4PBE0 geometries sML3PBE0 geometries sML4
62.80105.35106.24113.18109.42116.22
57.60109.48111.18116.57116.16121.28
MEMANP1199.6063.1164.9472.5469.8377.06
71.3095.9295.11103.3298.63106.63
71.7094.9696.10103.7499.99107.47
64.80102.74102.86109.12106.82112.68
71.9094.3296.03101.82100.24105.90
58.90110.07108.85115.38112.28118.86
54.90114.21113.93119.83118.07123.65
MGALPY01100.4062.4963.5671.7069.1277.13
67.60100.65100.43107.36103.87110.46
72.6093.8493.10101.1697.32105.24
70.0097.2198.13105.79101.70108.90
72.9092.9393.84100.4498.44104.88
61.40106.83106.83113.68109.98116.85
55.20114.24113.94120.14118.69124.36
MGLUCP11101.0062.4063.3871.4468.8576.60
72.3097.2195.72102.7999.02105.85
74.6092.6893.19100.9897.36104.82
72.5094.9993.47101.6897.20105.17
75.3093.7494.69101.3197.94104.32
63.80104.75105.34112.91108.94116.48
56.50112.69114.09120.10118.56124.05
PERYTO1050.20119.47120.38127.85121.48129.25
58.40109.67109.49116.50113.24120.02
RHAMAH1294.5068.0871.1278.8976.5484.12
72.2094.2495.24102.9099.59106.93
71.0097.7298.13105.80101.46109.00
72.5094.5895.38102.5799.44106.29
69.8096.1698.22104.89102.69109.26
17.80156.45157.07160.88158.32162.23
SUCROS0493.3071.0071.4878.1576.7583.17
66.00104.25105.35112.26107.71114.58
73.7093.8393.45101.2096.69104.27
102.4059.0752.8361.5259.4767.82
72.8094.0793.67101.4298.29105.67
82.9083.6887.0894.5390.0397.29
67.9099.8699.91107.29103.09110.18
71.8096.4596.26104.2499.89107.60
73.6093.0693.69100.7197.28103.82
81.8086.3683.5090.7886.9493.82
60.00109.19108.80116.04112.42119.75
61.00107.13106.43112.69110.23116.44
SULAMD06127.1039.4239.8246.2945.9850.38
129.5040.3039.6339.4841.0940.82
117.1053.3554.6057.1456.9259.24
153.4017.4719.7117.7621.6219.76
112.3057.4057.5460.6160.7163.52
129.5040.3740.9939.9442.6141.74
TRIPHE11126.4043.5742.1043.5044.2645.84
129.5038.4239.3340.6942.4944.00
124.5043.6644.9445.4848.0448.44
125.9042.5441.2442.8843.5245.16
127.5042.2042.7844.0644.3845.55
122.3047.9948.9149.7250.6350.89
130.2039.4438.9240.2940.7742.17
129.5039.7540.7041.7141.5742.85
120.9049.5350.3150.8151.7452.26
125.9044.1944.7045.7845.0346.30
121.7048.8849.1949.9050.2550.80
129.5040.0040.6741.6942.2843.72
129.5041.3939.6740.8640.8742.16
122.3046.7248.5048.9250.0850.53
126.9041.6541.6642.5843.7944.91
126.9043.4142.1043.4743.0644.59
123.8042.9844.4745.1748.6549.11
129.8040.1140.0741.5441.0542.45

Table S2. Experimental isotropic 13C^{13}\mathrm{C} chemical shifts and predicted shieldings for the atomic sites in the benchmark dataset. Predicted shieldings are given for PBE and PBE0 (PET-MOLS) geometries, computed with GIPAW-DFT and with the ShiftML3 (sML3) and ShiftML4 (sML4) machine-learning models.

Experimental shifts and predicted shieldings for the 15N^{15}\mathrm{N} benchmark dataset

Ref. CodeExp. shift [ppm]PBE geometries: DFTPBE geometries: sML3PBE geometries: sML4PBE0 geometries: sML3PBE0 geometries: sML4
BITZAF249.50-60.89-68.97-76.47-62.01-71.92
GEHHEH187.409.6411.167.8413.259.28
261.00-64.05-51.89-59.59-58.21-67.06
GEHHIL268.50-87.11-81.30-91.49-74.39-84.34
261.20-73.65-64.26-67.18-69.87-72.16
LHISTD02210.80-31.50-30.80-35.62-20.79-25.73
132.6045.1145.8148.1552.4854.85
LHISTD13210.60-30.87-30.58-35.26-20.50-25.45
132.4043.8744.6647.3052.0354.64
TEJWAG143.9036.5238.0339.9342.8644.69
GLYCIN03-6.50197.55195.64202.54199.57206.92
FUSVAQ01183.20-2.80-0.190.338.099.30
174.208.678.879.5215.2315.71
192.20-10.82-9.81-11.35-1.04-2.35
120.2055.1556.2161.1362.6067.27
50.20131.61131.18138.63136.42143.25
CYTSIN110.2065.6665.7174.7674.8783.87
165.2015.5017.7822.3725.1430.28
54.20131.20133.13139.80137.10143.64
THYMIN0190.2085.1483.1291.9392.32100.74
119.2059.8058.6866.4465.9173.63
CIMETD130.5046.5845.4048.1752.0154.25
213.10-32.07-32.76-34.90-24.66-25.45
56.60122.91125.63131.98133.37139.45
43.50138.52140.32147.66149.20156.02
45.80138.05139.41139.58146.13144.57
149.9034.8834.3037.4536.3738.23
BAPLOT01114.7057.4859.8967.8568.3975.90
72.70100.91101.92110.19109.49117.81
122.7053.1653.7155.9463.5866.32
178.701.471.611.768.729.10
LTYRHC108.00167.71166.90175.27184.36192.27
CYSCLM1.50186.09186.34193.96190.64198.19
URACIL96.2075.3475.6384.0184.1092.61
120.2059.5559.0066.5165.3873.07

Table S3. Experimental isotropic 15N^{15}\mathrm{N} chemical shifts and predicted shieldings for the atomic sites in the benchmark dataset. Predicted shieldings are given for PBE and PBE0 (PET-MOLS) geometries, computed with GIPAW-DFT and with the ShiftML3 (sML3) and ShiftML4 (sML4) machine-learning models.

Detailed prediction accuracies of ShiftML4 and ShiftML3 on isotropic chemical shielding σiso\sigma_{\mathrm{iso}} DFT reference computations.

In tbl:S4 we compare prediction accuracies of isotropic chemical shieldings of ShiftML4 and ShiftML3 against their respective DFT targets on the hold-out test set, for which both GIPAW-PBE labels and molecular corrected labels are available. We evaluate both models on all environments present in the test set, rather than on the farthest-point-sampling subset used to evaluate ShiftML2. We note that our outlier protocol removed a disproportionately large number of structures containing divalent metal cations. The original ShiftML2/ShiftML3 test set contained only a small number of metal cation environments (5-7), and due to the outlier removal protocol, calcium ions are not represented in the validation and test set. For magnesium and sodium we present accuracy numbers computed on the validation set. In Figure S1 we show parity plots between ShiftML4 predictions and DFT references. The molecular corrections also slightly change the distribution of the species-wise isotropic chemical shieldings and hence, a larger absolute RMSE in tbl:S4 does not necessarily result in a larger normalized RMSE.

Parity plots of DFT reference isotropic shieldings against ShiftML4 predicted isotropic shieldings

Figure S1. Parity plots of DFT reference isotropic shieldings σisoDFT\sigma_{\mathrm{iso}}^{\mathrm{DFT}} against ShiftML4 predicted isotropic shieldings σisopred\sigma_{\mathrm{iso}}^{\mathrm{pred}}. Subplots show 1H^{1}\mathrm{H} chemical shieldings (a), 13C^{13}\mathrm{C} chemical shieldings (b), and 15N^{15}\mathrm{N} chemical shieldings (c).

NucleusNtrainN_{\mathrm{train}}NvalidationN_{\mathrm{validation}}NtestN_{\mathrm{test}}MAE [ppm] sML3MAE [ppm] sML4RMSE [ppm] sML3RMSE [ppm] sML4RMSE [%] sML3RMSE [%] sML4R2R^2 sML3R2R^2 sML4
1H^{1}\mathrm{H}487,77227,24267,9640.310.290.420.3913.212.10.980.99
13C^{13}\mathrm{C}392,54724,97862,1341.501.462.242.204.54.21.001.00
15N^{15}\mathrm{N}107,5762,7506,6564.745.0410.2011.119.59.50.990.99
17O^{17}\mathrm{O}118,4444,90211,5066.916.9510.4511.235.96.01.001.00
19F^{19}\mathrm{F}31,4154088043.873.415.554.7117.614.40.970.98
33S^{33}\mathrm{S}26,0555801,56213.2013.8721.8624.3814.915.90.980.97
31P^{31}\mathrm{P}5,7581321968.478.3118.6816.3057.348.10.670.77
35Cl^{35}\mathrm{Cl}22,90027683610.239.6214.5513.8512.713.10.980.98
23Na^{23}\mathrm{Na}900160(2.13)*(1.38)*(2.35)*(1.56)*(43.3)*(28.7)*(0.81)*(0.92)*
43Ca^{43}\mathrm{Ca}2200--------
25Mg^{25}\mathrm{Mg}3020(2.07)*(2.79)*(2.08)*(2.80)*(-)*(-)*(-)*(-)*
39K^{39}\mathrm{K}609484.133.424.463.9855.849.80.690.75

Table 4. Predicted accuracy of σiso\sigma_{\mathrm{iso}} of ShiftML3 (sML3) and ShiftML4 (sML4) on the hold-out test set. In Brackets ()* we report metrics computed on the validation set. Standardized RMSEs (RMSE[%]) are defined as the nucleus-wise RMSE divided by the nucleus-wise standard deviation of the isotropic shift. ShiftML3 is evaluated against GIPAW-PBE reference calculations and ShiftML4 against PBE0 molecular-corrected GIPAW-PBE calculations.

Detailed prediction accuracy of ShiftML4 and ShiftML3 on chemical shielding tensor principal components σPAS\sigma_{\mathrm{PAS}} DFT reference computations.

In Table S5 we compare prediction accuracies of chemical shielding tensor principal component σPAS\sigma_{\mathrm{PAS}} predictions of ShiftML4 against the DFT reference values. In Figure S2 we show parity plots between DFT reference chemical shielding tensor principal components σPAS\sigma_{\mathrm{PAS}} and ShiftML4 predicted values.

NucleusMAE [ppm]RMSE [ppm]RMSE [%]R²
sML3sML4sML3sML4sML3sML4sML3sML4
1H{}^{1}\mathrm{H}0.640.610.850.8215.214.10.980.98
13C{}^{13}\mathrm{C}2.542.443.773.684.34.11.001.00
15N{}^{15}\mathrm{N}6.937.0515.6716.419.08.80.990.99
17O{}^{17}\mathrm{O}9.619.4415.6316.255.65.61.001.00
19F{}^{19}\mathrm{F}5.204.667.546.838.77.90.990.99
33S{}^{33}\mathrm{S}20.2020.0634.3836.3115.415.90.980.97
31P{}^{31}\mathrm{P}12.6412.1623.9821.6228.726.90.920.93
35Cl{}^{35}\mathrm{Cl}13.6512.9719.9218.838.17.90.990.99
23Na{}^{23}\mathrm{Na}(2.95)*(2.79)*(3.62)*(3.37)*(29.7)*(27.7)*(0.91)*(0.92)*
43Ca{}^{43}\mathrm{Ca}--------
25Mg{}^{25}\mathrm{Mg}(5.52)*(2.79)*(6.91)*(2.92)*(25.9)*(10.9)*(0.93)*(0.99)*
39K{}^{39}\mathrm{K}4.333.845.074.9537.736.70.860.87

Table S5. Predicted accuracy of σPAS\sigma_{\mathrm{PAS}} of ShiftML3 (sML3) and ShiftML4 (sML4) on the hold-out test set. In Brackets ()* we report metrics computed on the validation set. Standardized RMSEs (RMSE[%]) are defined as the nucleus-wise RMSE divided by the nucleus-wise standard deviation of the principal components σPAS\sigma_{\mathrm{PAS}}. ShiftML3 is evaluated against GIPAW-PBE reference calculations and ShiftML4 against PBE0 molecular-corrected GIPAW-PBE calculations.

Parity plots of DFT reference chemical shielding tensor principal components $\sigma_{\mathrm{PAS}}$ against ShiftML4 predicted values

Figure S2. Parity plots of DFT reference chemical shielding tensor principal components σPAS\sigma_{\mathrm{PAS}}, against ShiftML4 predicted values. Subplots show 1H{}^{1}\mathrm{H} principal components (a), 13C{}^{13}\mathrm{C} principal components (b), and 15N{}^{15}\mathrm{N} principal components (c).

Detailed prediction accuracy of ShiftML4 and ShiftML3 on chemical shielding tensor components σij\sigma_{ij} DFT reference computations.

In Table S6 we report prediction accuracies of ShiftML4 on shielding tensor components σij\sigma_{ij} against DFT reference values. In Figure S3 we show parity plots of DFT reference shielding tensor components σij\sigma_{ij} against ShiftML4 predicted values.

NucleusMAE [ppm]RMSE [ppm]RMSE [%]R²
sML3sML4sML3sML4sML3sML4sML3sML4
1H{}^{1}\mathrm{H}0.610.590.810.786.46.11.001.00
13C{}^{13}\mathrm{C}2.302.223.433.315.95.51.001.00
15N{}^{15}\mathrm{N}5.715.6911.5311.8311.410.90.990.99
17O{}^{17}\mathrm{O}8.007.7112.7912.617.97.50.990.99
19F{}^{19}\mathrm{F}4.123.806.085.554.94.21.001.00
33S{}^{33}\mathrm{S}17.3016.7728.3728.6916.816.20.970.97
31P{}^{31}\mathrm{P}10.289.8619.7019.4214.613.70.980.98
35Cl{}^{35}\mathrm{Cl}10.8510.1216.3215.015.14.51.001.00
23Na{}^{23}\mathrm{Na}(1.95)*(1.92)*(2.59)*(2.55)*(1.0)*(1.0)*(1.00)*(1.00)*
43Ca{}^{43}\mathrm{Ca}--------
25Mg{}^{25}\mathrm{Mg}(2.78)*(3.23)*(4.25)*(3.81)*(1.5)*(1.4)*(1.00)*(1.00)*
39K{}^{39}\mathrm{K}3.233.294.154.270.70.71.001.00

Table S6. Predicted accuracy of σij\sigma_{ij} of ShiftML3 (sML3) and ShiftML4 (sML4) on the hold-out test set. In Brackets ()* we report metrics computed on the validation set. Standardized RMSEs (RMSE[%]) are defined as the nucleus-wise RMSE divided by the nucleus-wise standard deviation of the tensor components σij\sigma_{ij}. ShiftML3 is evaluated against GIPAW-PBE reference calculations and ShiftML4 against PBE0 molecular-corrected GIPAW-PBE calculations.

Parity plots of DFT reference chemical shielding tensor components against ShiftML4 predicted values

Figure S3. Parity plots of DFT reference chemical shielding tensor components σij\sigma_{ij} against ShiftML4 predicted values. Subplots show 1H{}^{1}\mathrm{H} shielding tensor components (a), 13C{}^{13}\mathrm{C} shielding tensor components (b) and 15N{}^{15}\mathrm{N} shielding tensor components (c).

Evaluating ShiftML4 and ShiftML3 on 15^{15}N shielding tensor components on an experimental benchmark compared to DFT-PBE0 shielding predictions

Parity plots comparing sML4, sML3 and DFT-PBE0 predictions of experimental $^{15}$N principal shielding tensor components

Figure S4. Parity plots between ShiftML4 (a), ShiftML3 (b), (c) DFT-PBE0 fragment computation 15^{15}N predicted principal components σPAS\sigma_{\mathrm{PAS}} of the shielding tensor and experimental reference values taken from Reference 105. Subplot (c) shows DFT-PBE0 fragment reference calculations taken directly from Ref. 105. The black line shows the linear relation, converting computational principal components into experimental components. The slope and intercept for each method are indicated in the respective subplot. In blue and red, PBE and PBE0 (PET-MOLS) geometries were taken as starting points for shielding calculations.

Evaluating differences of ShiftML4 and ShiftML3 predictions on 1^{1}H, 13^{13}C and 15^{15}N experimental benchmarks

Evaluating ShiftML4 Uncertainty Estimates

ShiftML4 is based on a committee of nanoPET models. The mean y‾(Ai)\overline{y}(A_i) of the individual model predictions yk(Ai)y^k(A_i) for a given environment AiA_i, is the ShiftML4 prediction. We can interpret the spread of the committee predictions σpred2(Ai)\sigma_{\mathrm{pred}}^2(A_i) as a measure of the uncertainty of the ShiftML4 predictions:

σpred2(Ai)=1Nens−1∑k=1Nens[yk(Ai)−y‾(Ai)]2(2)\sigma_{\mathrm{pred}}^2(A_i) = \frac{1}{N_{\mathrm{ens}}-1} \sum_{k=1}^{N_{\mathrm{ens}}} \left[y^k(A_i)-\overline{y}(A_i)\right]^2 \tag*{(2)}

Committee based uncertainty estimators tend to be globally miscalibrated, meaning that their uncertainty estimates are globally over- or underconfident. [106, 107] This miscalibration can be corrected by computing an input-independent rescaling factor α\alpha on a hold-out calibration set, as in (3). [108, 109] Here ziz_i are the prediction errors ∣y‾(Ai)−yref∣|\overline{y}(A_i)-y_{\mathrm{ref}}| of the ensemble predictions against DFT reference labels, and σi,pred2\sigma_{i,\mathrm{pred}}^2 the committee uncertainty. The scaled, or calibrated uncertainty estimates are then obtained by scaling the committee spread σscaled=ασpred\sigma_{\mathrm{scaled}}=\alpha\sigma_{\mathrm{pred}}. We use one calibration factor per chemical species in the validation database. We omit calcium, given that there are no calcium environments in the validation dataset. In Table S7 we list the species wise α\alpha calibration factors according to (3).

1H^{1}\mathrm{H}13C^{13}\mathrm{C}15N^{15}\mathrm{N}17O^{17}\mathrm{O}19F^{19}\mathrm{F}33S^{33}\mathrm{S}31P^{31}\mathrm{P}35Cl^{35}\mathrm{Cl}23Na^{23}\mathrm{Na}25Mg^{25}\mathrm{Mg}39K^{39}\mathrm{K}
α\alpha0.421.291.441.541.252.121.001.520.650.820.87

Table S7. Per-nucleus uncertainty calibration factors α\alpha of ShiftML4, determined on the validation set

α=1Nsamples∑i=1Nsampleszi2σi,pred2(3)\alpha= \sqrt{\frac{1}{N_{\mathrm{samples}}}\sum_{i=1}^{N_{\mathrm{samples}}}\frac{z_i^2}{\sigma_{i,\mathrm{pred}}^2}} \tag*{(3)}

In Figure S6 we show double logarithmic scatter plots in which we plot ShiftML4 predicted uncertainties against prediction errors against DFT reference labels. Overall, the ShiftML4 committee uncertainties require only moderate calibration, with values of α\alpha ranging from 0.42 to 2.12.

Scatter plots of ShiftML4 predicted and calibrated uncertainty estimates for 1H, 13C, and 15N

Figure S6. ShiftML4 predicted and calibrated uncertainty estimates of the isotropic shieldings std⁡pred[σiso]\operatorname{std}_{\mathrm{pred}}[\sigma_{\mathrm{iso}}], and prediction errors against DFT labels z[σiso]z[\sigma_{\mathrm{iso}}]. Subplots show scatter plots for 1H^{1}\mathrm{H} predictions (a), 13C^{13}\mathrm{C} predictions(b) and 15N^{15}\mathrm{N} predictions (c). The black dashed line indicates the unit line. Grey lines indicate quantiles of the expected log-folded-normal distribution. For example, for any given predicted uncertainty estimate std⁡pred[σiso]\operatorname{std}_{\mathrm{pred}}[\sigma_{\mathrm{iso}}] 90% of the samples in the scatter plot should fall between the two outermost grey lines, that correspond to quantile lines of 5% and 95% of the expected error distribution.

Note that committee uncertainties can also be used to propagate uncertainties through arbitrarily complex workflows, such as entire NMR crystallography workflows, or evaluating uncertainties on tensor-derived quantities such as the shielding tensor principal components.

Reference ORCA computations

In this section we list the ORCA input files, required to reproduce the molecular corrections, computed in this work. First, we list the input file of the PBE reference computation (as written by the ORCA python API):

!pbe
!sp
!nmr
!rijcosx
!cc-pvtz
!autoaux
!cpcm(dichloromethane)
!defgrid3
!tightscf
%
    nprocs 12
end

%
    jsonpropfile true
    jsongbwfile true
end

Second we list the input file of the PBE0 reference computation (as written by the ORCA python API):

!pbe0
!sp
!nmr
!rijcosx
!cc-pvtz
!autoaux
!cpcm(dichloromethane)
!defgrid3
!tightscf
%
    nprocs 12
end

%
    jsonpropfile true
    jsongbwfile true
end

cc-pVTZ [92, 93] is not parametrized for potassium. To stay consistent with the original work from Ramos et al. [72] we still use cc-pVTZ throughout, and set the correction for the isolated potassium atoms produced by the cutting procedure to zero, in line with the near-zero corrections found for the other metal ions. This affects only the correction term, so ShiftML4 can still be evaluated meaningfully on potassium-containing structures.

Implementation

ShiftML4 is available via the ShiftML PyPI package. The package interfaces the ShiftML models via a convenient calculator implementation to the Atomic Simulation Environment (ASE). [83]

from ase.build import bulk
from shiftml.ase import ShiftML
import numpy as np

frame = bulk("C", "diamond", a=3.566)
model = ShiftML("ShiftML4")

Ypred = model.get_cs_tensor(frame)

# Get the committee predictions:
cs_committee_iso = model.get_cs_iso_ensemble(frame)

# Compute uncertainty estimates for the isotropic chemical shieldings
cs_iso_uncertainty = np.std(cs_committee_iso, axis=1, ddof=1)

The package can be easily installed from PyPI via:

pip install shiftml

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