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. 2026 Jan 28;11(5):8594–8599. doi: 10.1021/acsomega.5c11679

Estimating the Hydrogen Bond Strength by Machine Learning Approaches

Nahera Samangani 1, Stefan Zahn 1,*
PMCID: PMC12902869  PMID: 41696234

Abstract

The capabilities of machine learning approaches were investigated to predict the hydrogen bond energy based on partial charges, bond orders, bond distances, and element types. Support vector regression in combination with gradient boosting resulted in a mean absolute percentage error of 3%, which is a significant improvement compared to previous models. The best models include Löwdin partial charges and bond orders from BLYP or B3LYP with the def2-SVP double-ζ basis set. Furthermore, the semiempirical GFN2-xTB approach can be employed to calculate Mulliken partial charges and Wiberg bond orders as input features of a regression model, resulting in a mean absolute percentage error of 4%. All models were fitted on coupled cluster energies with singles, doubles, and perturbative triples extrapolated to the complete basis set limit.

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Introduction

Hydrogen bonds (HB) are noncovalent interactions. They are well-known to people outside the scientific community due to their central role for ice and liquid water as well as the structure of proteins. Linus Pauling described HB as electrostatic interactions about 100 years ago where a positively charged hydrogen atom links two negatively charged ions. The charged atom closer to hydrogen with a bond order close to one is called hydrogen bond donor atom (HBD) while the one with a bond order to the hydrogen atom close to zero is called hydrogen bond acceptor atom (HBA). The strong electrostatic interactions were confirmed by symmetry-adapted perturbation theory (SAPT) calculations in the 90s. , Nonetheless, covalent interactions contribute to HB as well. Weinhold suggested based on the natural bond orbital (NBO) analysis that the nonbonded natural orbital of the HBA pushes electron density into the antibonding σ-orbital between the hydrogen atom and HBD. This results in covalent interactions between the HBA and hydrogen atom while the covalent interactions between hydrogen atom and HBD are weakened. Nonetheless, neither the SAPT nor the NBO approach allow directly to quantify intramolecular hydrogen bond strengths.

Recently, machine learning force fields (MLFF) have become quite popular. − Foundation models can be applied to a broad range of systems to get the total energy of the system at the accuracy of quantum chemistry methods. Commonly, the total energy is based on a sum of atomic energies where each atom considers all atoms within a cutoff radius. Therefore, foundation models have the same drawback as density functional theory: It is not possible to determine the energy of a single hydrogen bond in systems with multiple noncovalent interactions or intramolecular hydrogen bonds. Therefore, it is reasonable to train machine learning models which employ local properties as input features to predict the hydrogen bond energy.

Geometric criteria are well established to detect hydrogen bonds in molecular dynamics simulations or protein structures. , Furthermore, the distance between HBA and hydrogen atom (d AH) can be linked to the energy of the hydrogen bond (E HB). , Reiher et al. suggested to use a linear fit of the two-center shared electron number (2c-SEN) between HBA and hydrogen atom. Since the hydrogen bond criteria is based on the electron density, it can be also applied to detect hydrogen bonds where geometric criteria are not established so far. Thus, a linear fit based on 2c-SEN provides an overall good estimate for the hydrogen bond strength. ,, A linear fit of frequency shifts are an alternative to the 2c-SEN approach. , This approach requests a frequency calculation where the hydrogen atom interacts solely with the HBD. Thus, the 2c-SEN ansatz can be better applied to a broad range of systems. This is also the case for a linear fit based on polarization charge densities proposed by Klamt et al. , Zou et al. investigated diverse linear fits including the electron density at bond critical points as well as second-order perturbation stabilization energies by the natural orbital at the hydrogen bond acceptor atom as input. Nonetheless, all previous mentioned linear fits are not reliable for systems where cooperative effects might play a central role. E HB will increase with decreasing d AH. This results in large errors for small d AH where the HB geometry is in a repulsive arrangement. Thus, it seems reasonable to employ machine learning approaches which can consider multiple input features and can fit complex functions.

Ruggiu et al. fitted a support vector regression (SVR) model on 537 compounds and obtained determination coefficient of 0.91 on a test set of 451 compounds for the hydrogen bond basicity scale. The hydrogen bond basicity scale should be correlated to the hydrogen bond strength for the same hydrogen bond donor. Recently, Melo et al. investigated several machine learning models to predict the hydrogen bond basicity scale with orbital stabilization energies from a NBO analysis as input features and obtained a determination coefficient of 0.92 for the best model. Also a model based on Gaussian process regression (GRP) proposed by Göller and co-workers is limited to the hydrogen bond basicity scale and cannot predict the hydrogen bond energy itself. Glavatskikh et al. reported the first fit by SVR on 3373 examples of various types of HB donor and acceptor atoms which can be applied to determine the hydrogen bond strength. A determination coefficient of 0.74 and root-mean-square error of 3.20 kJ/mol was obtained for the test set of 641 complexes. Input features to predict the experimental Gibbs energy of HB complexes were fragment descriptors provided by the ISIDA software package.

Within this work, we will use local properties obtained by density functional theory or low cost semiempirical extended tight-binding calculations to predict the hydrogen bond energy trained on data at the coupled cluster theory level with singles, doubles and perturbative triples extrapolated to the complete basis set limit. Investigated machine learning approaches will cover linear regression, polynomial regression, random forests (RF), support vector regression (SVR), nearest neighbor regression, Gaussian process regression (GPR) and neural networks. Additionally, gradient boosting and AdaBoost are investigated to improve the best fit obtained by SVR.

Computational Details

The machine learning model was trained and validated on a data set with N, O, F, S, Cl as HBA and N, O, F, S as HBD. Covered functional groups include hydroxyl groups, ethers, aldehydes, carbonic acids, carboxylates, thioles, thioethers, amines, amides, ammonium, imidazole, imidazolium, chloride and fluoride. Thus, the data set includes cations such as ammonium or imidazolium as well as anions such as chloride or carboxylates. Systems where an anion interacts with a cation were not included since the strong Coulomb forces between the counterions will strongly affect the hydrogen bond energy determined by following reaction:

HBA+HBD→HBA···HBD 1

Please note, the dimers were optimized with constraints to obtain reference energies also in the attractive and repulsive region which might play a crucial role if cooperative effects affect the structure. The structure of reactants was optimized without any constraints. Therefore, the training data set include also some structures with repulsive interactions. Very strong interactions were observed for ions like chloride or ammonium forming a hydrogen bond to an uncharged compound. Each data set had 1638 instances.

The geometry of the compounds was obtained by structure optimization with the B3LYP functional in combination with def2-TZVP basis set. Nine different distances between HBA and HBD were constrained during the structure optimization of the hydrogen bonded complex to obtain structures outside the equilibrium distance. Furthermore, the bond angle between the HBA, hydrogen atom and HBD was constrained to generate a data set with solely a single hydrogen bond as noncovalent interaction. The third version of the dispersion correction proposed by Grimme et al. was employed in combination with a Becke-Johnson damping scheme to include an empirical dispersion correction in all Kohn–Sham density functional theory (DFT) calculations. , Single point calculations were carried out with domain-based local pair natural orbital coupled cluster with singles, doubles, and perturbative triples (DLPNO–CCSD­(T)) and an extrapolation scheme to the complete basis set limit − with the cc-pVDZ , and cc-pVTZ , basis set as implemented in ORCA to obtain the final electronic energy of all compounds. The convergence criterion of the self-consistent field cycle was set to 10–8 Hartree. The energy gap on this level of theory for the reaction in eq was used as reference value to train and validate the machine learning approach. Löwdin bond orders and partial charges were obtained with single point calculations with B3LYP in combination with a def2-SVP basis set as well as BLYP in combination with def2-SVP basis set. We have chosen a double-ζ basis set to provide a cheap approach feasible for large systems. Additionally, single point calculations were carried out with the semiempirical GFN2-xTB (xTB2) approach for which Wiberg bond orders and Mulliken partial charges were determined by the xTB program. All static quantum chemistry calculations were accelerated by the resolution of identity (RI) approximation for the Coulomb integrals and by the chain of sphere approach (COSX) for the exchange contributions. ORCA 5.0.3 was used throughout this work except for the xTB2 calculations. Our data set is available at the Zenodo repository by following DOI: 10.5281/zenodo.17483320 It includes also the Jupyter notebooks to train and validate the models.

Results and Discussion

Regression Models Trained on the B3LYP Data Set

Initially, diverse regression models were investigated for the data set where partial charges and bond orders were obtained from DFT calculations with the B3LYP functional and the def2-SVP basis set. All data sets had 1638 data points and were split in a test and training data set with the size of 328 and 1310 data points, respectively. Subsequently, a grid search in combination with a cross-validation was employed to identify the best hyperparameters of each investigated model. Input features of the model optimized for the B3LYP data set were the bond order between the hydrogen atom and HBA (BOAH), the bond order between the hydrogen atom and HBD (BODH), the Löwdin partial charge of the hydrogen atom (q H) in a.u., the Löwdin partial charge of HBA (q A) in a.u., the Löwdin partial charge of the HBD (q D) in a.u., the distance between the hydrogen atom and HBA (d AH) in Å, the distance between hydrogen atom and HBD (d DH) in Å as well as the element type of HBA and HBD. A one hot encoder was employed to transform the element types into a one hot numeric array for regression. Target value of the regression was the hydrogen bond energy (E HB) obtained by the electronic energy for the reaction from the relaxed geometries of the noninteracting compounds to the interacting one at DLPNO–CCSD­(T) extrapolated to the complete basis set limit. E HB in the data set is between −200.6 and 45.6 kJ/mol with 50% of the data points between −48.2 kJ/mol and −15.0 kJ/mol. Please note, our approach does not consider nuclear quantum effects.

The quality of each optimized model was validated by a 5-fold cross-validation where the average root-mean-square error (aRMSE) of the five trained models is given in Table . The linear regression model showed an overall poor performance with an aRMSE of 9.5 kJ/mol. The poor performance was expected since the data set is composed of geometries with constrained distances between HBA and HBD. It highlights the disadvantage of linear regression models to fit the hydrogen bond energy in the attractive and repulsive region, which have been mainly used in the literature. A polynomial regression of degree 3 with a bias term reduced the aRMSE to 3.5 kJ/mol, showing the advantage of more complex regression models. Higher polynomials showed significantly poorer performance. This can be attributed to overfitting visible by a large RMSE gap between training and validation data set of the learning curve, see Figure . Therefore, we have investigated Lasso, Ridge and ElasticNet as regularized regression models to fit a polynomial of fourth or fifth degree. Solely Ridge regression showed a significant improvement where an aRMSE of 2.1 kJ/mol was obtained in combination with a fifth degree polynomial.

1. Comparison of Average Root-Mean-Square Error (aRMSE) of a 5-Fold Cross-Validation on the Training Data Set with Partial Charges and Bond Orders from the B3LYP Data Set.

model aRMSE in kJ/mol
linear regression 9.5
polynomial regression (degree: 3) 3.5
ElasticNet regression of fifth degree polynomial 3.7
lasso regression of sixth degree polynomial 3.3
ridge regression of fifth degree polynomial 2.1
random forests 4.5
nearest neighbors regression 5.0
GPR (RBF kernel) 3.1
SVR (RBF kernel) 2.1
multilayer perceptron regressor (size hidden layers: 10/10/10) 3.3
ridge regression of fifth degree polynomial with AdaBoost 2.7
SVR (RBF kernel) with AdaBoost 1.7
ridge regression of fifth degree polynomial with gradient boosting 1.7
SVR (RBF kernel) with gradient boosting 1.3
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1.

1

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Learning curves of a polynomial regression with degrees 3 and 4 for the B3LYP data set.

Random forests regression as well as nearest neighbor regression showed a poorer performance compared to the polynomial fit, see Table . SVR as well as GPR with a radial basis function (RBF) kernel showed already a better performance with the employed data set size than the polynomial fit without any regularization. Furthermore, the aRMSE of 2.1 kJ/mol of the best SVR model cannot be surpassed by a deep learning model based on a fully connected multilayer perceptron with three hidden layers and ten perceptrons in each layer. To improve the SVR model and the model based on the Ridge regression further, we investigated two boosting techniques. Adaboost decreased the aRMSE of the SVR model to 1.7 kJ/mol while it shows no improvement for the Ridge regression. Gradient boosting for the SVR model led to the best model with an aRMSE of 1.3 kJ/mol, see Table . Ridge regression can be improved by gradient boosting as well but resulted with an aRMSE of 1.7 kJ/mol in poorer performance than the SVR model in combination with gradient boosting. Therefore, we selected the SVR model with gradient boosting and trained it on the full training data set in combination with standardization of the input features, a one hot encoder to transform the element types, a RBF kernel with an epsilon of 0.1, a regularization parameter C of 5000 and 10–3 kJ/mol as tolerance of the stopping criterion for the SVR, a learning rate of 0.1, 70 boosting stages and a maximum depth of five for the individual regression estimators as hyperparameters of the gradient boosting. All other hyperparameters were the default values of scikit-learn. The obtained model showed a mean absolute error of 0.5 kJ/mol and a mean absolute percentage error of 3% on the test data set. The coefficient of determination is 0.999. Thus, it is more accurate than previous models and can also determine the hydrogen bond strength where the geometry is affected by cooperative effects, a limit of previous linear regression models.

Finally, we analyzed the Gini feature importance by a random forest model trained on the whole training data set. Please note, that this model shows a determination coefficient of 0.983 for the test data set trained on the full training data. The most important input feature is q A with 38.4% followed by BODH with 25,3% and q D with 18,3%, see Table . Thus, the inclusion of the partial charges contributes most to the accuracy of the model.

2. Gini Feature Importance in % of the Löwdin Partial Charges q A, q D, and q H, Löwdin Bond Orders BOAH and BODH, Bond Distances d AH and d DH, and Element Types of HBA (ETA) and HBD (ETD).

q A q D q H BOAH BODH d AH d DH ETA ETD
38.4 18.3 7.4 2.1 25.3 2.5 2.4 2.6 1.1
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Regression Model Trained on the BLYP Data Set

A second data set was investigated where Löwdin partial charges and Löwdin bond orders were obtained from the electron density calculated by the general gradient approximation (GGA) functional BLYP instead of the hybrid functional B3LYP as in the first data set. Since the GGA functional does not include an exact Hartree–Fock exchange term, it can be employed to larger systems. Since the input features are very similar to the B3LYP data set, only four model types were investigated. The polynomial regression of degree 3 showed an aRMSE of 3.4 kJ/mol, which is slightly better than the same regression model for B3LYP. A fifth degree polynomial with ridge regression reduced the aRMSE to 2.1 kJ/mol. The best SVR model showed an aRMSE of 2.1 kJ/mol which can be decreased to 1.3 kJ/mol with gradient bosting, similar as for the B3LYP data set. We trained the last mentioned model on the full training data set in combination with standardization of the input features, a one hot encoder to transform the element types, a RBF kernel with an epsilon of 0.1, a regularization parameter C of 5000 and 10–3 kJ/mol as tolerance of the stopping criterion for the SVR, a learning rate of 0.1, 70 boosting stages and a maximum depth of six for the individual regression estimators as hyperparameters of the gradient boosting. The coefficient of determination for this regression model is 0.999, the mean absolute error is 0.5 kJ/mol and the mean absolute percentage error is 3% for the test data set. Thus, there is no benefit to employ a more expansive hybrid functional instead of a GGA to calculate the Löwdin bond orders and Löwdin partial charges from the electron density for the input features. This is not unexpected since the data set includes solely systems where the self-interaction error should play a negligible role.

Regression Model Trained on the xTB2 Data Set

The cheap semiempirical xTB2 approach was employed to calculate Mulliken partial charges and Wiberg bond orders as input features of the last set of regression models. We did not include BOAH as input feature since the xTB program is limited to Wiberg bond orders larger than 0.1. Our aim of investigating this data set was to provide an approach applicable to large systems on a laptop by a broad community. As can be seen in Table , BOAH is significantly less important than BODH. To determine the influence of BOAH on the quality of the DFT models, we trained a SVR model with gradient boosting on the B3LYP data set where BOAH was omitted. The mean absolute error on the test data set is slightly increased to 0.6 kJ/mol. Nonetheless, the mean absolute percentage error is still 3%. Thus, there should be no significant shortcoming in accuracy if solely BOAH is omitted as input feature.

The polynomial fit with the xTB2 data set provided an aRMSE of 5.4 kJ/mol which is larger than for the DFT based polynomial fit (3.5 kJ/mol). Also Ridge regression in combination with a fifth degree polynomial results in a poor model with an aRMSE of 3.9 kJ/mol compared to the DFT data set. The best SVR model for the xTB2 data set had an aRMSE of 2.2 kJ/mol and thus, is comparable to the DFT data set based SVR models (2.1 kJ/mol). However, subsequent gradient boosting could reduce the aRMSE solely to 1.8 kJ/mol which is larger than for the DFT model including all features (1.3 kJ/mol) or the B3LYP data set without BOAH (1.5 kJ/mol). The mean absolute error of the xTB2-model on the test data set is 0.7 kJ/mol while the mean absolute percentage error is 4%. The determination coefficient is 0.998. Thus, the best regression model on the xTB2 data set is still sufficient accurate for a fast investigation of the hydrogen bond strength. It can be only slightly improved by the best models of the two previous subsections. We trained this model on the full training data set in combination with standardization of the input features, a one hot encoder to transform the element types, a RBF kernel with an epsilon of 0.1, a regularization parameter C of 5000 and 10–3 kJ/mol as tolerance of the stopping criterion for the SVR, a learning rate of 0.1, 70 boosting stages and a maximum depth of six for the individual regression estimators as hyperparameters of the gradient boosting.

Conclusions

Initially, a data set was generated to train a regression model for hydrogen bond energy at CCSD­(T) extrapolated to the complete basis set limit. The DFT models employed a double-ζ basis set with BLYP or B3LYP to calculate the Löwdin partial charges and bond orders as input features in addition to bond distances and element types. The polynomial fit of degree 3 provided already a significant improvement compared to the linear regression models and can estimate the hydrogen bond energy where cooperative effects play a central role. The best fit was obtained by a SVR model with gradient boosting which had a determination coefficient of 0.999 and a mean absolute percentage error of 3%. Mulliken partial charges and Wiberg bond orders calculated by the xTB2 approach provide a slightly less accurate model with a determination coefficient of 0.998 and a mean absolute percentage error of 4%. Thus, all three SVR models with gradient boosting provide an efficient approach to determine intra- and intermolecular hydrogen bonds affected by cooperative effects. A Gini feature importance analysis revealed that the partial charge of the acceptor atom, the partial charge of the donor atom and the bond order between the hydrogen bond donor atom and hydrogen atom strongly affect the accuracy of the regression models.

Our approach can be employed, for example, in drug design to screen for weak hydrogen bonds, in catalysis to determine the role of hydrogen bonds in stabilizing intermediates and transition states or in identifying the role of hydrogen bonds for bonding of micropollutants on a functionalized polymer membrane surface. It might be interesting to study how temperature dependent nuclear quantum effects or electric fields can be included in a regression model to predict the hydrogen bond strength. Furthermore, foundation models might be extended to predict partial charges and bond orders, which can be employed subsequently to determine local properties such as hydrogen bond energies.

Acknowledgments

Computational time from the University Computing Center of Leipzig University is gratefully acknowledged.

N.S. investigated initially the capability of ML approaches to determine the hydrogen bond energy under guidance of S.Z. S.Z. generated the data set based on quantum chemistry calculations, wrote the scripts to extract the data from the calculations, investigated all ML models outlined in this manuscript and wrote the draft of the publication.

The authors declare no competing financial interest.

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