Can an Alchemical Harmonic Approximation Baseline Cut Machine Learning Training Data?

The baseline is a one-point alchemical interpolation

The paper introduces the alchemical harmonic approximation (AHA) of the absolute electronic energy for charge-neutral iso-electronic diatomics at a fixed interatomic distance d0. To handle changes in distance, AHA is combined with an electronic binding potential of the form E(d) = (Eu − Es)((Ec − Es)/(Eu − Es))^(√(d/d0)) + Es, where Eu is the united-atom energy, Ec the calibration energy at d0, and Es the sum of infinitely separated atoms.

The resulting model covers the whole two-dimensional electronic potential energy surface spanned by distance and difference in nuclear charge, yet only one single point — a diatomic with nuclear charges Z1 and Z2 at distance d0 — is drawn to calibrate Ec. That single-point calibration is what makes the model usable as a cheap baseline rather than as another fitted potential.

One calibration point spans an iso-electronic series

Using reference data from pbe0/cc-pVDZ, the authors present numerical evidence for the electronic ground state of all neutral diatomics with 8, 10, 12, and 14 electrons. The model is assessed against legacy interatomic potentials — harmonic oscillator, Lennard-Jones, and Morse — within the binding range 0.7 to 2.5 Å.

Within that range, accuracy is comparable when the comparison is restricted to a single diatomic, but the AHA-based potential has significantly better predictive power when extrapolating across the entire iso-electronic series. The paper attributes this to the model's correct and systematic behavior at d = 0 and d → ∞, and to its dependence on only one calibration point, almost regardless of the choice of d and λ.

Morse fits the calibration point but not the series

The Morse potential describes the calibration calculation (BF) better than the authors' potential does. That local advantage does not survive generalization: when Morse is placed inside the AHA and used as a Δ-learning baseline, it hardly leads to any improvement and sometimes even deteriorates predictive power.

The paper traces this to how conventional potentials handle repulsion. Morse, Lennard-Jones, and the harmonic potential all model distance-dependent nuclear repulsion and electronic attraction together, even though the Coulombic part could have been subtracted. At short distances the Coulombic and empirical repulsion terms do not cancel exactly — for instance, the Lennard-Jones repulsive term proportional to r^−12 is not a physical repulsion — so fitting its parameters to the electronic energy produces a cancellation of questionable quality.

Δ-learning with AHA cuts training data tenfold

The authors test Δ-learning of the electronic absolute energy with their model as baseline. The difference between the baseline and pbe0/cc-pVDZ reference data is learned with kernel ridge regression using a Laplacian kernel with the Manhattan norm, 8-fold cross-validation, training set sizes N_train = 8, 16, 32, 64, …, 4096, and a test set of N_test = 3072.

With the AHA-based baseline, the model produces a systematic improvement that effectively reduces the training data needed to reach chemical accuracy by up to an order of magnitude, from roughly 1000 to roughly 100 points. The comparison with direct learning shows almost one order of magnitude gain in accuracy across all learning curves. By contrast, the AHA+Morse baseline hardly improves anything and sometimes worsens predictions.

A four-number global representation beats large local ones

The study compares global and local representations for the diatomics. The small global CM(n) representation contains only four numbers per diatomic, while cMBDF has 40 per atom, FCHL19 has 3952 per atom, and SOAP has 44296 per atom; SOAP is especially costly because every element species of the 14-electron diatomics, H through Si, is included.

CM(n) clearly works best, though only marginally. The difference between global and local kernels diminishes as the AHA parabola becomes smaller along the series N2 to C2 to B2 to Be2. The paper expects this, since diatomics have no three- or higher-order many-body terms and can be adequately described using interatomic distance information alone.

Held-out CO exposes the limits of the baseline

To test out-of-sample prediction, the authors systematically remove one specific diatomic, such as CO, from the training set and use it as the test set. Training data are restricted to λ ≠ 1 and test data to λ = 1, and only the CM(n) representations are considered because local representations no longer work in this setting.

All three models approach an average energy prediction early and gain no accuracy as the training set grows further. The AHA-based baseline still gives a significant edge over direct learning and over Δ-learning with AHA+Morse. The paper reports that inferring the energy of unseen CO converges to a prediction error of about 0.1 Ha in direct learning and about 3 Ha with the baseline — a reminder that the baseline's advantage is about learning efficiency, not about making held-out extrapolation accurate.

The advantage shrinks as the AHA parabola narrows

The procedure was repeated for different calibration systems (N2, CO, BeNe, LiNa, HeMg, HAl) and for the iso-electronic series of 8 (Be2, LiB, HeC, HN), 10 (B2, BeC, LiN, HeO, HF), and 12 (C2, BN, BeO, LiF, HeNe, HNa) electrons, with similar results. Odd electron counts are possible but lead to half-integer λm, λ0, and λu for physical diatomics; the authors did not study those because open-shell systems would require more sophisticated treatment.

In the 8-electron series, Δ-learning with an AHA+Morse baseline proves harmful, with mean absolute error increasing beyond direct learning. The paper's own baseline applies to all iso-electronic diatomics considered, independent of training set composition, but the size of its benefit tracks how wide the AHA parabola is — so the practical priority is to check the series and calibration system before assuming the tenfold data reduction carries over.

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