In this study, we demonstrate that the linear combination of atomic orbitals (LCAO), an approximation introduced by Pauling and Lennard-Jones in the 1920s, corresponds to graph convolutional networks (GCNs) for molecules. However, GCNs involve unnecessary nonlinearity and deep architecture. We also verify that molecular GCNs are based on a poor basis function set compared with the standard one used in theoretical calculations or quantum chemical simulations. From these observations, we describe the quantum deep field (QDF), a machine learning (ML) model based on an underlying quantum physics, in particular the density functional theory (DFT). We believe that the QDF model can be easily understood because it can be regarded as a single linear layer GCN. Moreover, it uses two vanilla feedforward neural networks to learn an energy functional and a Hohenberg--Kohn map that have nonlinearities inherent in quantum physics and the DFT. For molecular energy prediction tasks, we demonstrated the viability of an ``extrapolation,'' in which we trained a QDF model with small molecules, tested it with large molecules, and achieved high extrapolation performance. We believe that we should move away from the competition of interpolation accuracy within benchmark datasets and evaluate ML models based on physics using an extrapolation setting; this will lead to reliable and practical applications, such as fast, large-scale molecular screening for discovering effective materials.

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.

The paper compares different deep learning approaches to modeling the quantum mechanic properties of molecules, and presents a model that incorporates multiple ideas from physics. Some reviewers appreciated multiple aspects of the paper, including: - A novel approach, offering an interesting contrast to GCN approaches - Compelling numerical results in comparison to a standard GCN approach (even if this not extrapolation, it still shows the benefit of the proposed approach over GCNs). The proposed approach does not even use spherical harmonics in the LCAO representation. The fact that this "still works well" is bothering as it necessarily leads to a bad approximation of the molecular orbitals, to a point where the added benefit of chemically informed modeling might be almost gone. Overall, the impression is positive and our recommendation is to accept the paper.

In this study, we demonstrate that the linear combination of atomic orbitals (LCAO), an approximation introduced by Pauling and Lennard-Jones in the 1920s, corresponds to graph convolutional networks (GCNs) for molecules. However, GCNs involve unnecessary nonlinearity and deep architecture. We also verify that molecular GCNs are based on a poor basis function set compared with the standard one used in theoretical calculations or quantum chemical simulations. From these observations, we describe the quantum deep field (QDF), a machine learning (ML) model based on an underlying quantum physics, in particular the density functional theory (DFT). We believe that the QDF model can be easily understood because it can be regarded as a single linear layer GCN.