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A 2023 study combines quantum-chemical atomic charges with machine-learning models to estimate missing Hammett substituent constants. It proposes 219 new σ values—including 92 reported as previously unknown—for donor and acceptor groups attached to benzene, while keeping experimental measurements as the reference standard.
What the Hammett equation measures
The Hammett equation is a way to connect a substituent’s electronic influence with a measured chemical response. In its standard form, the relationship is written as log(K/K0) = ρσ for equilibria, or with an analogous rate ratio when reaction rates are compared. K (or a rate constant) is measured for a substituted compound, K0 is the value for the unsubstituted reference, σ is the substituent constant, and ρ describes how sensitive the particular reaction series is to electronic effects.
A positive σ generally indicates that a group withdraws electron density relative to hydrogen; a negative value indicates donation. The sign and size are not universal properties of a group in every circumstance. They depend on the reference series, the position of substitution and the definition of the constant being used. Meta and para positions can therefore have different constants, and specialized series are used when resonance changes the response.
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Why calculate new Hammett constants?
Hammett constants are most reliable when they come from a carefully defined experimental series. In practice, however, a chemist may encounter a donor or acceptor group for which no value has been measured, or find values that differ between compilations because the reaction, solvent, temperature or analysis differed. Running a complete experimental series for every new group is expensive and can be impractical when compounds are unstable or difficult to prepare.
The 2023 work treats machine learning as a computational extension for those gaps. Instead of claiming that a model can replace the Hammett experiment, it uses quantum-chemical descriptors that can be generated for a candidate substituent and learns how those descriptors relate to known σ values. The resulting estimate can guide reaction analysis or help decide which measurements are most useful.
What the 2023 study actually did
Gabriel Monteiro-de-Castro, Julio Cesar Duarte and Itamar Borges Jr. reported the method in The Journal of Organic Chemistry on July 11, 2023. Their paper, “Machine Learning Determination of New Hammett’s Constants for meta- and para-Substituted Benzoic Acid Derivatives Employing Quantum Chemical Atomic Charge Methods,” combines quantum-chemical atomic charges with several machine-learning algorithms.
| Aspect | What the paper reports |
|---|---|
| Substituent set | 90 donor or acceptor groups |
| Molecular scope | Groups bonded to benzene, evaluated in meta- and para-substituted benzoic-acid derivatives |
| Predicted output | Several types of Hammett σ constants |
| New values proposed | 219 σ values, including 92 described as previously unknown |
| Descriptor families tested | Mulliken, Löwdin, Hirshfeld and ChelpG atomic charges |
| Best agreement reported | Hirshfeld charges for most of the constant types examined |
| Additional model form | Linear expressions based on carbon atomic charges |
The authors describe their estimates as close to original experimental values, with the most accurate reported results for the meta- and para-substituted benzoic-acid derivatives in the study. Those qualifiers matter: the paper does not establish equivalent accuracy for every aromatic framework, substitution pattern, solvent or reaction class.
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The primary article is available from the American Chemical Society at https://pubs.acs.org/doi/abs/10.1021/acs.joc.3c00410.
How the machine-learning workflow works
1. Define the chemical series
The calculation starts with a specific structural and electronic problem: substituent groups attached to a benzene ring in the meta or para position of benzoic-acid derivatives. Fixing that framework prevents the model from mixing unrelated reaction environments.
2. Calculate quantum-chemical charges
Quantum-chemical calculations assign partial charges to atoms. These are not directly observable charges; they are model-dependent ways of partitioning the molecule’s electron density. The study generated charge descriptors using four named schemes: Mulliken, Löwdin, Hirshfeld and ChelpG.
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1Clear out junk files and repair common Windows errors2Scan for outdated or missing drivers - takes under a minute3Repair Windows errors before they cause bigger problems3. Pair descriptors with known constants
For substituents with established experimental Hammett constants, the calculated charge features become inputs and the measured σ values become targets. The algorithm can then learn the relationship between a charge on a relevant carbon atom and the corresponding substituent effect.
4. Fit and compare machine-learning models
Different machine-learning algorithms were tested rather than assuming one model would suit every constant type. The paper also reports linear expressions built from carbon charges, which provides a simpler, more interpretable alternative to a more complex algorithm.
5. Estimate values for gaps
Once a model is fitted, a new donor or acceptor group can be processed through the same quantum-chemical charge calculation. The model returns an estimated σ value, which can be compared with any later experimental result and used provisionally where no measurement exists.
Why the choice of charge scheme matters
Atomic charges depend on the mathematical partitioning scheme, so changing the scheme changes the descriptors supplied to the model. The study compared four approaches rather than treating “the atomic charge” as a single unambiguous quantity.
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| Charge scheme | Role in the study | Reported outcome |
|---|---|---|
| Hirshfeld | Quantum-chemical atomic-charge descriptors | Best agreement for most σ-constant types tested |
| Mulliken | Alternative charge descriptors | Compared with the other schemes; no overall best result reported |
| Löwdin | Alternative charge descriptors | Compared with the other schemes; no overall best result reported |
| ChelpG | Alternative charge descriptors | Compared with the other schemes; no overall best result reported |
“Best agreement” is a result for the tested constant types and molecular scope, not a general declaration that Hirshfeld charges are always superior in every computational-chemistry application.
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What was new in the reported results
219 proposed values, 92 previously unknown
The headline numerical result is 219 proposed new σ values, including 92 previously unknown — Monteiro-de-Castro, Duarte, and Borges, 2023. The count covers the several Hammett-constant types modeled for the 90 donor and acceptor groups, not 219 independent substituent molecules in a different chemical family.
Computational recovery of three literature-only values
Chemistry World reports that the team computationally obtained values for –CCl3, –NHCHO and –NHCONH2, groups for which values had previously been available only experimentally. The calculated results corresponded to the literature experimental values, according to that report (Chemistry World). These examples illustrate the intended use: provide a check or an initial estimate when a direct measurement is missing, rather than erase the distinction between calculation and experiment.
Computed estimates versus experimental constants
| Question | Experimental value | Machine-learning estimate in this study |
|---|---|---|
| How is it obtained? | Measured reaction equilibrium or rate in a defined series | Predicted from quantum-chemical charge features and trained models |
| What does it describe? | A substituent effect under the tested reaction conditions | An estimate for the modeled benzoic-acid derivative framework |
| Best use | Reference for quantitative interpretation and validation | Filling gaps, prioritizing measurements and screening candidates |
| Main uncertainty | Experimental error and differences in conditions or compilations | Model error, descriptor dependence and the limits of available training data |
A predicted number should therefore be labeled as computed and tied to its model and molecular context. It should not be copied into a database as though it were a newly measured universal constant.
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- It does not show that the model works equally well for all aromatic systems, heteroaromatics or aliphatic substituents.
- It does not demonstrate a universal reaction-prediction engine. The Hammett equation still requires a defined reaction series and a reaction-dependent ρ value.
- It does not remove the need to check solvent, temperature, mechanism and substitution position when comparing σ values.
- It does not prove that one charge scheme will be optimal outside the tested calculations.
The distinction between a substituent constant and a full reaction prediction is especially important. A σ estimate captures an electronic descriptor for a defined reference framework; predicting an actual yield, rate or selectivity also requires a mechanistic model and conditions that the Hammett calibration may not cover.
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Why limited DFT data can limit accuracy
Kristaps Ermanis of the University of Nottingham described the approach as something that “can fill in values where the data hasn’t been previously found,” but cautioned that it “relies on limited amounts of DFT data, which limits the number of parameters in the machine learning method, and therefore potentially also limits its accuracy.” Chemistry World published those comments in its coverage (Chemistry World, July 31, 2023).
In practical terms, a small or narrowly distributed training set makes it harder to support a highly parameterized model. A model can fit known examples yet behave unpredictably for a substituent unlike those examples. Charge calculations also carry choices about molecular geometry, electronic-structure method and charge partitioning. Those choices create additional reasons to report the computational setup alongside every predicted constant.
Matthew Grayson’s group at the University of Bath called the work a “valuable idea that allows experimentalists to access previously unknown Hammett constants using simple and readily available atomic charge features.” That assessment captures the trade-off: the descriptors are comparatively accessible, but accessibility does not turn an estimate into a measurement.
How to use a predicted σ value responsibly
- Check the framework. Confirm that the molecule is a benzene derivative and that the substituent position and constant type match the study’s meta or para benzoic-acid scope.
- Record provenance. Keep the charge scheme, quantum-chemical settings, model type and date with the predicted number.
- Compare like with like. Do not combine a computed value from one constant type with an experimental value from another or with a different reaction series.
- Use it for prioritization. Apply the estimate to rank candidates, design a substrate set or decide which missing value deserves experimental measurement.
- Validate when decisions depend on precision. If the value controls a mechanistic claim, a publication-quality correlation or a safety-critical process, measure the relevant reaction series or obtain an independent experimental check.
Why this is an update rather than a replacement
The original Hammett framework remains useful because it separates a substituent-dependent term, σ, from a reaction-dependent sensitivity, ρ. Machine learning adds a way to infer σ when the experimental table is incomplete. Quantum-chemical charges provide a physically motivated input, while the learned relationship supplies a fast estimate for groups that have not been characterized in the same way.
That combination can expand the practical reach of an 86-year-old relationship without changing what the relationship means. Experimental constants continue to anchor the scale; computed values extend it provisionally within a stated chemical domain. The strongest interpretation of the 2023 paper is therefore methodological: it demonstrates a route to missing or inconsistent Hammett data, with Hirshfeld-based descriptors performing best for most of the tested constant types, while leaving the limits of extrapolation visible.
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