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arXiv:2312.14680 · Physical review. B./Physical review. B

Equivariant neural network for Green's functions of molecules and materials

Xinyang Dong, Emanuel Gull, Lei Wang

Published Feb 6, 2024• 11 citations
arXivHTMLPDF

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 the 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 an accurate estimate of physical observables such as energy and density of states based on the predicted Green's function.

DOI
10.1103/physrevb.109.075112

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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)E(3)E(3) equivariance. The Green's function is obtained from the predicted self-energy through the 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 an accurate estimate of physical observables such as energy and density of states based on the predicted Green's function.
We present a machine learning (ML) framework for predicting Green’s functions of molecular systems, from which photoemission spectra and quasiparticle energies at quantum many-body level can be obtained. Kernel ridge regression is adopted to predict self-energy matrix elements on compact imaginary frequency grids from static and dynamical mean-field electronic features, which gives direct access to real-frequency many-body Green’s functions through analytic continuation and Dyson’s equation. Feature and self-energy matrices are represented in a symmetry-adapted intrinsic atomic orbital plus projected atomic orbital basis to enforce rotational invariance. We demonstrate good transferability and high data efficiency of the proposed ML method across molecular sizes and chemical species by showing accurate predictions of density of states (DOS) and quasiparticle energies at the level of many-body perturbation theory ( GW ) or full configuration interaction. For the ML model trained on 48 out of 1995 molecules randomly sampled from the QM7 and QM9 data sets, we report the mean absolute errors of ML-predicted highest occupied and lowest unoccupied molecular orbital energies to be 0.13 and 0.10 eV, respectively, compared to GW @PBE0. We further showcase the capability of this method by applying the same ML model to predict DOS for significantly larger organic molecules with up to 44 heavy atoms.
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Given access to accurate solutions of the many-electron Schrödinger equation, nearly all chemistry could be derived from first principles.Exact wave functions of interesting chemical systems are out of reach because they are NP-hard to compute in general, but approximations can be found using polynomially scaling algorithms.The key challenge for many of these algorithms is the choice of wave function approximation, or Ansatz, which must trade off between efficiency and accuracy.Neural networks have shown impressive power as accurate practical function approximators and promise as a compact wave-function Ansatz for spin systems, but problems in electronic structure require wave functions that obey Fermi-Dirac statistics.Here we introduce a novel deep learning architecture, the Fermionic neural network, as a powerful wave-function Ansatz for many-electron systems.The Fermionic neural network is able to achieve accuracy beyond other variational quantum Monte Carlo Ansatz on a variety of atoms and small molecules.Using no data other than atomic positions and charges, we predict the dissociation curves of the nitrogen molecule and hydrogen chain, two challenging strongly correlated systems, to significantly higher accuracy than the coupled cluster method, widely considered the most accurate scalable method for quantum chemistry at equilibrium geometry.This demonstrates that deep neural networks can improve the accuracy of variational quantum Monte Carlo to the point where it outperforms other ab initio quantum chemistry methods, opening the possibility of accurate direct optimization of wave functions for previously intractable many-electron systems.
We present a novel approach that combines machine learning with direct variational energy optimization via the density matrix to solve the Kohn-Sham equation in density functional theory. Instead of relying on the conventional self-consistent field method, our approach directly optimizes the ground state by predicting the density matrix using a neural network, thereby bypassing Hamiltonian matrix diagonalization. Our model employs equivariant neural networks to generate a physically constrained density matrix, enabling stable and efficient energy minimization. This method integrates the construction of training sets directly into the model training process and achieves high accuracy in predicting ground-state properties across various molecular and extended systems, thereby establishing a powerful machine learning paradigm for electronic structure optimization and paving the way for large-scale quantum simulations.
Strongly correlated materials exhibit complex electronic phenomena that are challenging to capture with traditional theoretical methods, yet understanding these systems is crucial for discovering new quantum materials. Addressing the computational bottlenecks in studying such systems, we present a proof-of-concept machine learning-based approach to accelerate Dynamical Mean Field Theory (DMFT) calculations. Our method predicts interacting Green's functions on arbitrary two-dimensional lattices using a two-step ML framework. First, an autoencoder-based network learns and generates physically plausible band structures of materials, providing diverse training data. Next, a dense neural network predicts interacting Green's functions of these physically-possible band structures, expressed in the basis of Legendre polynomials. We demonstrate that this architecture can serve as a substitute for the computationally demanding quantum impurity solver in DMFT, significantly reducing computational cost while maintaining accuracy. This approach offers a scalable pathway to accelerate simulations of strongly correlated systems and lays the groundwork for future extensions to multi-band systems.
Recent neural network-based wave functions have achieved state-of-the-art accuracies in modeling ab-initio ground-state potential energy surface. However, these networks can only solve different spatial arrangements of the same set of atoms. To overcome this limitation, we present Graph-learned orbital embeddings (Globe), a neural network-based reparametrization method that can adapt neural wave functions to different molecules. Globe learns representations of local electronic structures that generalize across molecules via spatial message passing by connecting molecular orbitals to covalent bonds. Further, we propose a size-consistent wave function Ansatz, the Molecular orbital network (Moon), tailored to jointly solve Schrödinger equations of different molecules. In our experiments, we find Moon converging in 4.5 times fewer steps to similar accuracy as previous methods or to lower energies given the same time. Further, our analysis shows that Moon's energy estimate scales additively with increased system sizes, unlike previous work where we observe divergence. In both computational chemistry and machine learning, we are the first to demonstrate that a single wave function can solve the Schrödinger equation of molecules with different atoms jointly.