where ℓ = 1,2,,L is the number of hidden layers (ψ(1)(ri) = ψ(ri) and L is the final layer), ReLU is the nonlinear activation function, W (ℓ) E RN N is the weight matrix in layer ℓ,and b

These molecular properties were calculated using a hybrid quantum simulation (Gaussian 09) at the B3LYP/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. E RN is the bias vector in layer ℓ. The LCAO considers the normalization for the coefficients in Eq. (6) in the main text. Additionally, the normalization term in Eq. (7) in the main text is calculated as follows: Z(qn,ζn)=