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A 2023 study shows how machine learning, fed by quantum-chemical atomic charges, can estimate missing Hammett substituent constants for defined meta- and para-substituted benzoic-acid systems. It is a computational extension of an 86-year-old relationship—not a replacement for measurements and not a general-purpose reaction-prediction engine.
What is the Hammett equation?
The Hammett equation is a physical-organic chemistry relationship that connects a substituent’s electronic influence with a measured reaction equilibrium or rate. For a reaction series containing substituted and unsubstituted compounds, it is commonly written as log(K/K₀) = ρσ for equilibria or log(k/k₀) = ρσ for rates.
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- σ (sigma) is the substituent constant. It describes whether a group withdraws electron density from, or donates electron density to, an aromatic ring relative to hydrogen.
- ρ (rho) is the reaction constant. It measures how sensitive that particular reaction series is to electronic substitution.
- K/K₀ or k/k₀ compares the substituted compound with the reference compound under the same experimental conditions.
A positive σ generally indicates an electron-withdrawing group, while a negative σ generally indicates an electron-donating group, although the sign and magnitude must be interpreted with the reaction series and the σ definition used. The same substituent can have different constants for different reaction contexts, including meta versus para attachment and different reaction types.
Louis Hammett’s original 1937 work connected hydrolysis rates of substituted ethyl esters with equilibrium positions for the corresponding acid ionisation. That pairing established the practical idea that electronic effects could be transferred from one measured series to another, provided the compounds and mechanism were sufficiently comparable. The historical explanation is summarized by Chemistry World.
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Why missing σ constants are a practical problem
Hammett analysis depends on a consistent set of constants. Experimental tables contain gaps because some substituents are difficult to synthesize, unstable under the measurement conditions, poorly soluble, or simply never tested in a particular positional and reaction series. Values can also be inconsistent when they come from different solvents, temperatures, reference reactions, or definitions of σ.
A missing value prevents a chemist from making a clean linear free-energy comparison. An estimate can therefore be useful for selecting compounds, checking whether a proposed trend is plausible, or deciding which measurement would be most informative. The estimate still needs to be labeled as computational and tied to the chemical series for which it was produced.
What the 2023 machine-learning study actually did
Gabriel Monteiro-de-Castro, Julio Cesar Duarte, and Itamar Borges Jr. reported the work in The Journal of Organic Chemistry, published online 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 charge calculations with several machine-learning algorithms.
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The calculations concern substituent groups bonded to benzene and meta- and para-substituted benzoic-acid derivatives. The models target several Hammett-constant types for 90 donor or acceptor groups. This scope matters: the paper does not establish a model for every aromatic scaffold, every substitution position, or every reaction mechanism.
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2. Calculate atomic charges
Quantum-chemical calculations provide numerical descriptors for the molecules. The study tested four charge schemes:
| Charge scheme | Role in the study | Reported agreement |
|---|---|---|
| Mulliken | Atomic-charge descriptors used as machine-learning inputs | Compared with the other schemes; not the best overall according to the abstract |
| Löwdin | Alternative population-analysis descriptors | Compared with the other schemes; not the best overall according to the abstract |
| Hirshfeld | Atomic-charge descriptors used as machine-learning inputs | Best agreement for most σ-value types among the four methods |
| ChelpG | Electrostatic-potential-derived charge descriptors | Compared with the other schemes; not the best overall according to the abstract |
The charges are not Hammett constants by themselves. They are molecular features from which a statistical model can learn a relationship to known σ values.
3. Fit machine-learning relationships
The authors trained different algorithms on available charge and σ-value data. They also developed linear expressions based on carbon charges. In practical terms, the models learn how the electronic environment around relevant carbon atoms tracks experimentally established substituent constants, then use that relationship to estimate constants for groups lacking a reported value.
4. Compare estimates with known values
The predicted constants were compared with original experimental values. The abstract describes the estimates as close to the experimental values, with the most accurate reported results for the meta- and para-substituted benzoic-acid derivatives in the study’s data set.
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What the study added
| Finding | What it means |
|---|---|
| 219 proposed new σ values, including 92 previously unknown — Monteiro-de-Castro, Duarte, and Borges, 2023. | The work expands the set of constants available for the specified donor and acceptor groups and benzoic-acid substitution patterns. |
| 90 donor or acceptor groups examined | The number describes the group set used in the computational study, not all substituents in organic chemistry. |
| Hirshfeld charges had the best agreement for most constant types | Feature choice affects model quality; a different charge partitioning scheme could change an estimate. |
| Linear expressions based on carbon charges were developed | Some estimates can be represented with relatively simple charge-based relationships rather than an opaque prediction alone. |
Chemistry World reports three notable examples—–CCl₃, –NHCHO, and –NHCONH₂—for which the team computationally obtained values that had previously been available only experimentally; the reported estimates corresponded to literature experimental values. Those examples illustrate gap filling, not a claim that every substituent can be predicted equally well.
Can machine learning predict Hammett constants?
Yes, within a defined chemical domain and when the model is trained on comparable data. The 2023 results support using quantum-derived charge features to estimate several σ types for the studied meta- and para-benzoic-acid systems. They do not support treating machine learning as a universal σ calculator.
| Approach | What is obtained | Best use | Main caution |
|---|---|---|---|
| Direct experimental measurement | A value tied to stated solvent, temperature, reaction, position and protocol | Reference data and final validation | Can be slow, costly or impossible for an unstable or inaccessible compound |
| Literature σ value | A previously measured constant under the source’s conditions | Routine analysis when definitions and conditions match | Values from different series may not be interchangeable |
| Charge-plus-machine-learning estimate | A computed σ estimate for a model’s chemical domain | Prioritizing experiments and filling documented data gaps | Accuracy depends on training coverage, charge method and DFT data quality |
The distinction is especially important when comparing σ values from different positions or constant families. A para constant should not automatically be substituted for a meta constant, and a value learned for benzoic-acid derivatives should not be assumed valid for a mechanistically unrelated aromatic reaction.
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Repair common Windows errors and clear accumulated junk for a smoother, more stable PC - no reinstall needed.Free scan · no reinstallWhy the choice of charge method matters
Atomic charges are not directly observable quantities with one universally correct partition. They depend on the quantum-chemical method and on how electron density is divided among atoms. Mulliken and Löwdin charges use orbital-population partitions; Hirshfeld charges divide density relative to reference atoms; ChelpG fits charges to reproduce an electrostatic potential at selected points.
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Because the machine-learning model uses these charges as input features, changing the charge scheme changes the numerical representation of the same molecule. In this comparison, Hirshfeld charges produced the best agreement for most of the σ-value categories reported in the abstract. That is a result for this data set and modeling setup, not a general ranking for every chemistry problem.
How a chemist could use the approach responsibly
- Identify the exact target. Record whether the needed value is a meta or para constant, which σ definition is required, and whether the downstream equation uses a rate or an equilibrium.
- Check measured sources first. Confirm that a literature value matches the solvent, temperature, reference reaction and substituent position. Do not merge values with incompatible definitions simply because they share the σ symbol.
- Confirm chemical-domain similarity. The published model concerns groups attached to benzene and meta- and para-substituted benzoic-acid derivatives. A different aromatic framework should be treated as an extrapolation.
- Generate the same type of quantum descriptor. If reproducing the method, use a consistent quantum-chemical protocol and the charge partitioning being evaluated. Mixing charge methods between training and prediction undermines comparability.
- Apply the published relationship or model. Use the reported linear carbon-charge expressions or the selected machine-learning algorithm with the same feature definitions. Keep the output as an estimate, not a measured constant.
- Check against neighboring chemistry. Compare the estimate with known substituents of similar electronic character and inspect whether the sign and relative ordering are chemically plausible.
- Validate important decisions experimentally. If the value will determine a mechanistic conclusion, a process change or a high-consequence compound choice, measure the relevant reaction series rather than relying solely on the estimate.
What limits accuracy?
Limited DFT training data
Kristaps Ermanis of the University of Nottingham described the benefit and the limitation directly: the method “can fill in values where the data hasn’t been previously found,” but 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.” The comment was reported by Chemistry World.
With a small or narrow training set, adding many adjustable parameters can make a model fit the available examples without learning a relationship that transfers reliably to new chemistry. More data are not automatically better if they mix incompatible computational methods or experimental definitions.
Extrapolation outside the paper’s scope
- The reported systems are benzene-based, with meta and para substitution on benzoic-acid derivatives.
- The study covers several σ types, but not every Hammett constant used in physical-organic chemistry.
- The abstract does not establish performance across all aromatic rings, ortho substitution, heteroaromatics, non-benzoic-acid reaction series or unrelated mechanisms.
- “Close to experimental values” describes the reported comparison; it is not a guarantee of a stated error for an arbitrary new molecule.
Experimental definitions still control interpretation
A Hammett constant is meaningful only with its definition and reference context. Solvent effects, temperature, ionisation state, conformation and reaction mechanism can alter the apparent electronic response. A computed number may be internally consistent with the model while still being unsuitable for a reaction series that probes a different physical effect.
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How this update changes the role of the Hammett equation
The equation itself has not been replaced. Its role remains to organize measured substituent effects through σ and ρ. Machine learning adds a way to estimate missing σ entries from quantum-chemical descriptors, making the existing framework more usable when experimental tables are incomplete.
Matthew Grayson’s group at the University of Bath called the approach a “valuable idea that allows experimentalists to access previously unknown Hammett constants using simple and readily available atomic charge features,” according to Chemistry World. The practical interpretation is conservative: use the model to prioritize compounds, formulate hypotheses and identify promising measurements, then retain experiment as the authority for critical values.
Bottom line
The 2023 paper demonstrates a useful computational extension of Hammett analysis. By combining quantum-chemical atomic charges with machine-learning models, it estimates multiple σ constants for 90 donor or acceptor groups in a defined set of meta- and para-substituted benzoic-acid derivatives. Hirshfeld charges performed best for most reported constant types, and the work proposed a substantial set of previously missing values. The method is valuable precisely when treated as scoped estimation: it can fill gaps and guide experiments, but it does not remove the need for compatible measurements or justify universal reaction predictions.
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