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Hammett-style parameters can improve predictions when they are fitted to the chemical environment and target property being modelled, rather than transferred uncritically from a standard table. The key is to choose an appropriate substituent scale, estimate substituent and reaction contributions from relevant data, and test predictions on observations not used for fitting. Published studies show promise for reaction-barrier and catalyst-binding predictions, but they do not establish a universal improvement across chemistry.
What do the Hammett parameters represent?
In the conventional Hammett relationship, a substituent constant, σ, represents an electronic substituent effect, while the reaction constant, ρ, represents how sensitive a particular reaction is to that effect. A common form is log(k/k₀) = ρσ, where k and k₀ are the rates for a substituted compound and a reference compound. Related forms can describe equilibrium constants.
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Traditional σ values depend on substituent identity and ring position and are commonly based on the ionisation of substituted benzoic acids. They are not automatically the best descriptors for every scaffold or environment. When resonance interaction with a para substituent stabilises developing positive or negative charge, the σ+ or σ− scale may be more suitable than ordinary σ. The appropriate choice depends on the chemistry and the property being predicted.
What have optimisation studies demonstrated?
Published demonstrations support fitting or recalibrating parameters for a defined task. Their targets and datasets differ, so their reported results are not a common benchmark.
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| Study and domain | What was fitted or estimated | Reported result and scope |
|---|---|---|
| Royal Society of Chemistry, Chemical Science (2020), “Data enhanced Hammett-equation: reaction barriers in chemical space” | A generalised Hammett model for non-aromatic scaffolds and molecules with multiple substituents; the authors globally regressed σ and ρ for two experimental datasets and a synthetic computational activation-energy dataset. | The computational dataset contains approximately 2,400 SN2 reactions. In that dataset and setup, using the Hammett model as a baseline for delta machine learning substantially improved learning curves, with low errors reached using small training sets. This is evidence for that task, not a general guarantee. |
| Royal Society of Chemistry, Digital Discovery (2024), “Combining Hammett σ constants for Δ-machine learning and catalyst discovery” | A Hammett-inspired product model for relative ligand–metal binding energies relevant to catalyst discovery; fitted single-ligand effects were compared with published constants. | For ligand combinations in the authors’ datasets, regression-derived single-ligand values tracked experiments more closely than simply summing published Hammett values. Prediction was tested with out-of-sample folds; the result remains specific to the studied application. |
| Wiley, Journal of Physical Organic Chemistry (2023), “A G4 approach to computing the Hammett substituent constants…” | An empirically scaled G4 computational procedure for σp, σm, σ−, σ+, and σ+m. | The study evaluated 41 substituents and reports a typical mean absolute error of approximately 0.1 for its calibrated computations. Including solvation substantially improved agreement with experiment; this error is specific to the reported procedure and comparison. |
| American Chemical Society, Journal of Organic Chemistry (2023), “Machine Learning Determination of New Hammett’s Constants…” | Machine learning using quantum-chemical atomic charges to estimate constants for donor or acceptor groups. | The authors studied 90 groups and proposed 219 values, including 92 previously unavailable values. Hirshfeld charges gave the best agreement for most constant types in that study. These are proposed or calculated values, not new experimental measurements. |
| Peter Ertl, ChemRxiv (2021 preprint), “A Web Tool for Calculating Substituent Descriptors Compatible with Hammett Sigma Constants” | A charge-based method and web tool for calculating descriptors compatible with Hammett constants. | In the author-reported analysis of 200 common substituents identified from ChEMBL bioactive molecules, experimental sigma values were available for 89. This is a preprint analysis, not a peer-reviewed benchmark. |
How should parameters be optimised for a new prediction task?
- Define the target and domain. Specify whether the model predicts reaction barriers, relative rates, equilibrium constants, substituent constants, or catalyst binding energies, and identify the relevant scaffold, reaction class, and conditions. These targets are not interchangeable, so neither are their error metrics.
- Select a scale appropriate to the electronic effect. Begin with conventional σp or σm only when the target chemistry supports that choice. Consider σ+ or σ− where resonance with a para substituent and developing charge is important. State the scale and its intended use.
- Assemble relevant observations and fit the contributions. Use data representative of the target chemistry to estimate substituent effects and the reaction or environment sensitivity. Re-estimation can account for multisubstituent effects or environmental balancing that a borrowed parameter table may not capture. Report the dataset, fitting method, and any regularisation choices.
- Validate predictions out of sample. Hold out observations or use out-of-sample folds, and say exactly what was held out. A good fit to the data used to estimate parameters is not, by itself, evidence that predictions will transfer to new compounds or reactions.
- Estimate missing values cautiously. Quantum-chemical calculations and machine-learning methods can help when experimental constants are absent or inconsistent. Report the computational method, reference data, scale, solvation treatment, and uncertainty; distinguish calculated proposals from measurements.
Why can an optimised parameter set fail to transfer?
- Different target, different model. A parameterisation fitted to activation energies does not thereby predict equilibrium constants or catalyst binding energies accurately. Performance must be assessed for the intended target.
- Reaction class and conditions matter. A reaction’s sensitivity, ρ, is conditional on its chemistry and environment. Changing reaction class, solvent, or catalyst can change the relationship being modelled.
- Substituent scales encode different situations. Ordinary σ, σ+, and σ− are not interchangeable labels. Choosing a scale that does not capture the relevant resonance and charge effects can undermine an otherwise careful fit.
- Computational estimates depend on their setup. In the 2023 Wiley G4 study, the authors wrote: “However, it quickly became apparent that including a solvation correction substantially improved the correlation with experiment, and so the gas phase approach was not pursued further.” The study also identifies reactive or ionic cases as frequent outliers and notes uncertainty in some experimental reference values. Its reported typical error should therefore not be read as an accuracy guarantee for new substituents.
- Validation determines what “predictive” means. Randomly held-out observations, held-out substituents, and a new reaction class test different kinds of transfer. State the split design so readers can judge how far a reported result applies beyond its training data.
What to report with an optimised Hammett model
For a result to be interpretable and reproducible, report the target property, chemical domain and conditions; the σ scale and substituent coverage; whether inputs are experimental or calculated; the fitting procedure; and the out-of-sample validation design with target-specific error. Without those details, a claim of improved predictive power is difficult to assess or apply elsewhere.
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