Alchemical harmonic approximation based potential for iso-electronic diatomics: Foundational baseline for $\Delta$-machine learning

We introduce the alchemical harmonic approximation (AHA) of the absolute electronic energy for charge-neutral iso-electronic diatomics at fixed interatomic distance $d_0$. To account for variations in distance, we combine AHA with this Ansatz for the electronic binding potential, $E(d)=(E_{u}-E_s) \left(\frac{E_c-E_s}{E_u-E_s}\right)^{\sqrt{d/d_0}}+E_s$, where $E_u,E_c,E_s$ correspond to the energies of united atom, calibration at $d_0$, and sum of infinitely separated atoms, respectively. Our model covers the entire two-dimensional electronic potential energy surface spanned by distance and difference in nuclear charge from which only one single point (with elements of nuclear charge $Z_1,Z_2$ and distance $d_0$) is drawn to calibrate $E_c$. Using reference data from pbe0/cc-pVDZ, we present numerical evidence for the electronic ground-state of all neutral diatomics with 8, 10, 12, 14 electrons. We assess the validity of our model by comparison to legacy interatomic potentials (Harmonic oscillator, Lennard-Jones, and Morse) within the most relevant range of binding (0.7 - 2.5 A), and find comparable accuracy if restricted to single diatomics, and significantly better predictive power when extrapolating to the entire iso-electronic series. We also investigated $\Delta$-learning of the electronic absolute energy using our model as baseline. This baseline model results in a systematic improvement, effectively reducing training data needs for reaching chemical accuracy by up to an order of magnitude from $\sim$1000 to $\sim$100. By contrast, using AHA+Morse as a baseline hardly leads to any improvement, and sometimes even deteriorates the predictive power. Inferring the energy of unseen CO converges to a prediction error of $\sim$0.1 Ha in direct learning, and $\sim$0.04 Ha with our baseline.