Equivariant neural network for Green's functions of molecules and materials (2312.14680v2)

Abstract: The many-body Green's function provides access to electronic properties beyond density functional theory level in ab inito calculations. In this manuscript, we propose a deep learning framework for predicting the finite-temperature Green's function in atomic orbital space, aiming to achieve a balance between accuracy and efficiency. By predicting the self-energy matrices in Lehmann representation using an equivariant message passing neural network, our method respects its analytical property and the $E(3)$ equivariance. The Green's function is obtained from the predicted self-energy through Dyson equation with target total number of electrons. We present proof-of-concept benchmark results for both molecules and simple periodic systems, showing that our method is able to provide accurate estimate of physical observables such as energy and density of states based on the predicted Green's function.

• The paper presents a novel equivariant neural network framework for accurately predicting finite-temperature Green's functions in atomic orbital space.
• The methodology utilizes message passing on molecular graphs while preserving E(3) symmetry, reducing the computational cost of many-body quantum calculations.
• Tests on molecular and periodic systems show predictions meeting chemical accuracy thresholds, offering scalable alternatives to traditional DFT methods.

Equivariant Neural Network for Green's Functions of Molecules and Materials

Introduction

The paper introduces an equivariant neural network framework to predict finite-temperature Green's functions in atomic orbital space. This approach has the potential to enhance the efficiency and accuracy of calculations in quantum field theory for condensed matter systems. By leveraging the analytical properties of Green's functions and $E(3)$ equivariance, the proposed method aims to replace computationally heavy many-body calculations with a scalable and efficient deep learning approach.

Methodology

Green's Function Formalism

Within the Born-Oppenheimer approximation, the paper outlines the Hamiltonian for electron systems and the formulation of single-particle Green's functions. These functions, which provide insights into electronic properties, are expressed through matrices that capture the relationships between atoms represented by creation and annihilation operators. The central task is solving the Dyson equation to recover the full Green's function from the non-interacting counterpart and self-energy terms.

Equivariant Message Passing Neural Networks (EMPNNs)

The methodology centers around using an equivariant message passing neural network (EMPNN) to predict the Green's functions. These neural networks operate on graphs where nodes correspond to atoms, and edges describe interactions based on geometric configurations. The EMPNN respects rotational, translational, and reflectional symmetries, making it well-suited for predicting matrix-valued functions that exhibit such symmetry properties.

Predicting Green's Functions and Self-Energies

The neural network predicts the self-energy matrices in Lehmann representation based on the structure of atomic configurations. These predictions are constrained by physical properties to ensure causality and the conservation of physical observables. The approach uses the discrete Lehmann representation to efficiently handle frequency space representations of the Green's function.

Workflow

The overall workflow (Figure 1) involves generating input atomic structures, predicting self-energies and the Fock matrix using the neural network, computing the Green's function from these predictions, and finally determining physical observables such as total energy and density of states. The training involves minimizing a loss function composed of differences in predicted and labeled self-energies.

Figure 1: Flow chart of predicting the Green's function and downstream physical observables from atomic charges $\{Z_i\}$ and positions $\{r_i\}$.

Results

Proof-of-Concept Benchmarking

The paper provides results for both molecular systems, like water and small organics, and periodic systems like diamond and silicon. The results indicate that the neural network is capable of achieving high accuracy in predicting energies and band gaps, demonstrating its potential for use in real-world configurations.

Figure 2: Learning curve for single water molecule with sto-3g basis in terms of the MAE of total energy.

Testing configurations reveal that significant accuracy improvements occur as the training set size increases. Specifically, models repeatedly achieve mean absolute errors for total energy predictions below the chemical accuracy threshold for various molecules and crystals (Tables presented in the paper summary).

Analysis and Implications

The experimental results suggest that EMPNNs can circumvent traditional computational bottlenecks associated with finite-temperature Green's functions by rapidly synthetizing multi-body effects. This work provides the foundation for deploying machine learning models in quantum physics to access properties that are beyond the reach of conventional Density Functional Theory (DFT) approaches, while still remaining a computationally scalable option.

Conclusion

The proposed equivariant neural network effectively lowers the computational cost of predicting electronic properties by bypassing extensive many-body diagrammatic approaches. With appropriate training, such models can offer substantial gains in speed and efficiency while maintaining the predictive accuracy required for reliable quantitative studies. Future enhancements could involve the inclusion of more sophisticated equivariant mechanisms and the integration of additional observables in the predictive model's loss function. This demonstrates promising directions for adopting machine learning insights in the broader domain of quantum chemistry and material science.