# On the equivalence of molecular graph convolution and molecular wave function with poor basis set

arXiv: https://arxiv.org/abs/2011.07929 · HTML: https://mlsanity.com/paper/2011.07929/

## Key facts

- Cited by 7 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 16 Nov 2020.

## Built on by (most cited first, 7 of 7)

- [SizeShiftReg: a Regularization Method for Improving Size-Generalization in Graph Neural Networks](https://mlsanity.com/paper/2207.07888/) (2022)
- [Graph Neural Network with Local Frame for Molecular Potential Energy Surface](https://mlsanity.com/paper/2208.00716/) (2022)
- [Equivariant Neural Operator Learning with Graphon Convolution](https://mlsanity.com/paper/2311.10908/) (2023)
- [Formulation Graphs for Mapping Structure-Composition of Battery Electrolytes to Device Performance](https://mlsanity.com/paper/2307.03811/) (2023)
- [Edge-featured Graph Neural Architecture Search](https://mlsanity.com/paper/2109.01356/) (2021)
- [On the Problem of Consistent Anomalies in Zero-Shot Industrial Anomaly Detection](https://mlsanity.com/paper/2510.10456/) (2025)
- [Detect the Interactions that Matter in Matter: Geometric Attention for Many-Body Systems](https://mlsanity.com/paper/2106.02549/) (2021)

## Abstract (arXiv)

In this study, we demonstrate that the linear combination of atomic orbitals (LCAO), an approximation of quantum physics 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. This will lead to reliable and practical applications for discovering effective materials. The implementation is available at https://github.com/masashitsubaki/QuantumDeepField_molecule.