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Optimising Hammett parameters means fitting substituent effects (σ) and reaction sensitivity (ρ) to data from the chemical system you want to predict, rather than assuming a published scale will transfer unchanged. Studies of reaction barriers and catalyst binding show that this can improve prediction in particular domains—but the result depends on the target property, substituent scale, chemical environment and validation design.
What is being optimised?
The Hammett relationship separates two contributions: σ describes the electronic effect assigned to a substituent, while ρ describes how sensitive a particular reaction is to that effect. In its traditional form, the relationship connects these terms to a relative reaction rate or equilibrium constant. A substituent’s σ value depends on its identity and position; ρ belongs to the reaction and its conditions.
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Optimisation estimates or recalibrates these parameters against observations relevant to a defined chemical domain. It does not make σ or ρ universal constants for every reaction, solvent or measured property. Reaction barriers, rates, equilibrium constants and catalyst binding energies are different targets, so their fitted parameters and prediction errors are not interchangeable.
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What published demonstrations show
| Study and target | Approach and scope | Reported result | What the result supports |
|---|---|---|---|
| Royal Society of Chemistry, Chemical Science (2020), “Data enhanced Hammett-equation: reaction barriers in chemical space” | Generalised a Hammett-style model to non-aromatic scaffolds and molecules with multiple substituents; 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, as described by the authors. | The authors report that using the Hammett model as a baseline for delta machine learning substantially improved learning curves, with low errors reached using small training sets. A specific error value is not stated here. | Fitting a chemically informed baseline can help a particular reaction-barrier learning task. This is not a universal benchmark of Hammett models. |
| Royal Society of Chemistry, Digital Discovery (2024), “Combining Hammett σ constants for Δ-machine learning and catalyst discovery” | Extended a Hammett-inspired product model to relative ligand–metal binding energies relevant to catalyst discovery; compared fitted ligand effects with published constants and evaluated predictions using out-of-sample folds. | For combinations of ligands in the authors’ datasets, regression-derived single-ligand values tracked experiments more closely than simply summing published Hammett values. A common numerical error metric is not stated here. | Environment-specific fitting can be useful for this catalyst-binding application. The finding is specific to the studied datasets and validation setup. |
These demonstrations address different targets and datasets. They show why recalibration can be useful, not that one optimised parameter set will improve every chemical prediction.
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Choose the scale and domain before fitting
Define the prediction target
State what the model predicts—such as an activation barrier, relative rate, equilibrium constant or ligand–metal binding energy—and identify the reaction or catalyst family. A model fitted to one of these quantities should not be treated as validated for another.
Match the σ scale to the electronic situation
Ordinary σp and σm values are established from the ionisation of substituted benzoic acids. If a developing positive or negative charge can interact by resonance with a para substituent, σ+ or σ− may better represent the electronic effect. Scale choice is part of the model specification, not a cosmetic change.
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Decide whether inherited values are adequate
Published constants provide a useful starting point, but multisubstituted molecules and different reaction or catalyst environments may include interactions or balancing effects that an inherited set does not capture. When there are enough relevant observations, fit parameters to the target environment and retain the domain definition alongside the fitted values.
Ways to estimate or improve parameters
Regression against relevant observations
Fit σ and ρ, or the corresponding substituent contributions in an extended model, to experimental or computed data from the intended domain. The 2020 reaction-barrier and 2024 catalyst-binding studies illustrate this strategy for distinct targets. The fitted values describe those model setups; they should not be presented as replacements for established constants in every context.
Quantum-chemical calibration
A 2023 Journal of Physical Organic Chemistry study by Yett and coauthors describes an empirically scaled G4 approach for σp, σm, σ−, σ+ and σ+m, reporting values for 41 substituents. The authors report a typical mean absolute error of approximately 0.1 for their calibrated computations and comparison data. That figure belongs to this procedure and dataset; it is not an accuracy guarantee for a new substituent or reaction.
Solvation mattered in that work. The authors write: “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.” They also identify reactive or ionic cases as common outliers and note that some experimental reference values may themselves be uncertain.
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Machine-learning estimates for missing substituents
A 2023 Journal of Organic Chemistry study used machine learning with quantum-chemical atomic charges for 90 donor or acceptor groups and proposed 219 constants, including 92 values that had not previously been available. The authors report that Hirshfeld charges gave the best agreement for most of the constant types studied. These are proposed, calculated values from a specific method—not new experimental measurements—and should be labelled accordingly when used.
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Peter Ertl’s 2021 ChemRxiv preprint describes a charge-based method and web tool for calculating descriptors compatible with Hammett constants. In an author-reported analysis of 200 common substituents identified from ChEMBL bioactive molecules, experimental σ values were available for 89. Because this work is a preprint, and web-tool availability can change, distinguish its reported analysis from peer-reviewed validation and confirm tool availability before relying on it.
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How to test whether optimisation improved prediction
- Specify the domain: name the target property, reaction or catalyst family, substituent positions, scale and relevant conditions, including solvent where it is part of the dataset.
- Record the source of each input: distinguish experimental constants and outcomes from quantum-derived estimates, machine-learning proposals and computed reaction data. Give the method and calibration context for calculated values.
- Fit only on the training data: estimate parameters using the designated training observations. Keep a held-out set or use out-of-sample folds to test predictions on observations not used in fitting.
- Compare like with like: evaluate the optimised model against an appropriate baseline on the same target and held-out observations. Report the error metric and validation design; an in-sample fit alone does not establish predictive power.
- Describe the boundary of the result: state what was held out and which substituents, scaffolds, reaction conditions or catalyst combinations the test represents. Do not claim transfer to untested chemistry without evidence.
For any reported accuracy, keep the target, dataset, scale, fitting method and validation design attached to the number. A mean absolute error for calculated substituent constants is not directly comparable with a reaction-barrier learning curve or a catalyst-binding prediction.
Quick Recap
Common reasons a fitted model may not transfer
- A different reaction or property: ρ reflects reaction sensitivity; a fit for barriers or binding energies does not automatically predict rates or equilibria.
- An unsuitable substituent scale: ordinary σ values may not capture resonance interaction with a developing charge where σ+ or σ− is more appropriate.
- A changed chemical environment: solvent effects, multisubstituent interactions and catalyst context can alter the relationship learned in the original domain.
- Incomplete or uncertain inputs: calculated constants depend on their computational method and calibration, while some experimental reference values may also be uncertain.
- Validation that does not match the claim: fitting and testing on the same observations measures fit, not out-of-sample prediction; a narrow held-out test supports only a correspondingly narrow conclusion.
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