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arXiv:2310.09911 · Journal of Chemical Theory and Computation

Machine Learning Many-Body Green’s Functions for Molecular Excitation Spectra

Christian Venturella, Christopher Hillenbrand, Jiachen Li, Tianyu Zhu

Published Dec 27, 2023• 17 citations
arXivHTMLPDF

Abstract

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.

DOI
10.1021/acs.jctc.3c01146

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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.
Abstract We demonstrate that a machine learning framework based on kernel ridge regression can encode and predict the self-energy of one-dimensional Hubbard models using only mean-field features such as static and dynamic Hartree–Fock quantities and first-order GW calculations. This approach is applicable across a wide range of on-site Coulomb interaction strengths U / t , ranging from weakly interacting systems ( U / t ≪ 1 ) to strong correlations ( U / t > 8 ). The predicted self-energy is transformed via Dyson’s equation and analytic continuation to obtain the real-frequency Green’s function, which allows access to the spectral function and density of states. This method can be used for nearest-neighbor interactions t and long-range hopping terms t ′, t ′′, and t ′ ′ ′ .
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.
The Anderson Impurity Model (AIM) is a canonical model of quantum many-body physics. Here we investigate whether machine learning models, both neural networks (NN) and kernel ridge regression (KRR), can accurately predict the AIM spectral function in all of its regimes, from empty orbital, to mixed valence, to Kondo. To tackle this question, we construct two large spectral databases containing approximately 410 000 and 600 000 spectral functions of the single-channel impurity problem. We show that the NN models can accurately predict the AIM spectral function in all of its regimes, with pointwise mean absolute errors down to 0.003 in normalized units. We find that the trained NN models outperform models based on KRR and enjoy a speedup on the order of 105{10}^{5}105 over traditional AIM solvers. The required size of the training set of our model can be significantly reduced using farthest point sampling in the AIM parameter space, which is important for generalizing our method to more complicated multichannel impurity problems of relevance to predicting the properties of real materials.
Predicting molecular properties is a cornerstone of drug discovery, materials science, and quantum chemistry, where accuracy and speed are critical. Classical machine learning mod- els offer rapid predictions but often fail to capture the intricate quantum interactions governing molecular behavior. Our team investigated quantum kernel methods to predict properties of the QM9 dataset, a widely recognized benchmark containing134,000\mathbf{1 3 4, 0 0 0}134,000organic molecules with diverse chemical structures. By encoding molecular features into quantum circuits, we mapped classical data into highdimensional Hilbert spaces to enhance prediction accuracy. Our pipeline encompasses data preprocessing, quantum circuit design, model training with support vector regression, and evaluation of ten key molecular properties. The quantum kernel approach delivered promising results, achieving mean absolute errors of 0.13 Debye for dipole moment and 0.08 eV for HOMOLUMO gap, even when executed on noisy quantum devices. This work provides a comprehensive, reproducible pipeline, benchmarks quantum and classical models, and discusses practical challenges, contributing significantly to the field of quantumenhanced computational chemistry.
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We present results for many-body perturbation theory for the one-body Green's function at finite temperatures using the Matsubara formalism. Our method relies on the accurate representation of the single-particle states in standard Gaussian basis sets, allowing to efficiently compute, among other observables, quasiparticle energies and Dyson orbitals of atoms and molecules. In particular, we challenge the second-order treatment of the Coulomb interaction by benchmarking its accuracy for a well-established test set of small molecules, which includes also systems where the usual Hartree-Fock treatment encounters difficulties. We discuss different schemes how to extract quasiparticle properties and assess their range of applicability. With an accurate solution and compact representation, our method is an ideal starting point to study electron dynamics in time-resolved experiments by the propagation of the Kadanoff-Baym equations.