Supplementary Material: On the equivalence of molecular graph convolution and molecular wave function with poor basis set

The QM9 dataset [ 1 ] contains approximately 130,000 molecules made up of H, C, N, O, and F atoms along with 13 quantum chemical properties (e.g., atomization energy, HOMO, and LUMO) for each molecule. These molecular properties were calculated using a hybrid quantum simulation (Gaussian 09) at the B3L YP/6-31G(2df,p) level of theory. In this study, we created a subset of the QM9 dataset with a limited number of atoms, M 14, per molecule, which we refer to as the "QM9under14atoms" dataset in the main text. As the learning/predicting targets, we selected three kinds of energy properties: atomization energy at 0 K, zero point vibrational energy, and enthalpy at 298.15 K. The number of data samples in the QM9under14atoms dataset is approximately 15,000 molecules and we randomly shuffled and split this dataset into training, development (or validation), and test sets with a ratio of 8:1:1, in which the development set was used to tune the model and optimization hyperparameters.