Fast and Accurate Modeling of Molecular Atomization Energies with Machine Learning

We introduce a machine learning model to predict atomization energies of a diverse set of organic molecules, based on nuclear charges and atomic positions only. The problem of solving the molecular Schrödinger equation is mapped onto a nonlinear statistical regression problem of reduced complexity. Regression models are trained on and compared to atomization energies computed with hybrid density-functional theory. Cross validation over more than seven thousand organic molecules yields a mean absolute error of $\sim 10\mathrm{kcal}/\mathrm{mol}$. Applicability is demonstrated for the prediction of molecular atomization potential energy curves.

Figure 1

Top: Distribution of distances, $d(\mathbf{M},{\mathbf{M}}^{\prime })$, for all molecular pairs occurring in the first 7165 small organic molecules from the GDB [15]. The inset exemplifies two distances, pyrrol/ethanol and pyrrol/thiophene (N: blue, O: red, S: yellow, C: black, and H: white). Bottom: Absolute differences in atomization energies between $\mathbf{M}$ and ${\mathbf{M}}^{\prime }$ as a function of $d(\mathbf{M},{\mathbf{M}}^{\prime })$.

Figure 2

Top: Cross-validated ML errors as a function of the number of molecules in the training set, $N$. Bottom: For $N=1000$, correlation of DFT-PBE0 [19, 20] results ($E^{\mathrm{ref}}$) with ML (cross-validated) based estimates ($E^{\mathrm{est}}$) of atomization energies. Correlations for bond counting [32] and semiempirical quantum chemistry (PM6 [33]) are also shown. Corresponding root mean square error (RMSE)/mean absolute error (MAE) for bond counting, PM6, and ML are $75.0/71.0$, $75.1/73.1$, and $30.1/14.9$ $\mathrm{kcal}/\mathrm{mol}$, respectively.

Figure 3

Energy of atomization curves of four molecules containing single bonds and branching only (${\mathrm{C}}_7{\mathrm{H}}_{16}$), a double bond (${\mathrm{C}}_6{\mathrm{H}}_{12}$), triple bonds including nitrogen (${\mathrm{C}}_6{\mathrm{NH}}_5$), and a sulfur-containing cycle with a hydroxy group (${\mathrm{C}}_4{\mathrm{SH}}_3\mathrm{OH}$). (From bottom to top in insets: black: Carbon; blue: Nitrogen; yellow: Sulfur; red: Oxygen; white: Hydrogen) (DFT-PBE0 and ML model 1k).