Yale University Proposes MBGF-Net to Significantly Reduce Quantum Chemistry Computation Costs and Predict Molecular Ground and Excited State Properties}

Editor | Baicai Ye

Many-body methods provide systematic approaches for calculating electronic properties of molecules and materials, but their high computational costs limit large-scale applications. Due to the complexity of multi-electron wavefunctions, machine learning models capable of capturing fundamental many-body physics remain limited.

Here, researchers at Yale University have developed a deep learning framework centered on many-body Green's functions (MBGF), unifying the prediction of ground and excited electronic states while providing physical insights into electron correlation effects.

This approach demonstrates competitive performance in predicting single- and double-particle excitations and properties derivable from single-particle density matrices.

Moreover, the method exhibits high data efficiency and strong transferability across different chemical species, system sizes, molecular conformations, and bond dissociation strengths. This opens new possibilities for using machine learning to solve many-electron problems.

The study, titled "Unified deep learning framework for many-body quantum chemistry via Green’s functions", was published on June 4, 2025, in Nature Computational Science.

Predicting electronic properties of molecules and materials in ground and excited states is central to quantum chemistry and materials science. Density Functional Theory (DFT) is the main tool due to its balance of accuracy and efficiency, but systematic errors and uncertainties from approximate exchange-correlation functionals limit its predictive power.

Ab initio many-body electronic structure methods, such as Coupled Cluster (CC) theory and Many-Body Perturbation Theory (GW), offer more robust quantum mechanical simulations. These methods are especially effective for catalysis, strongly correlated systems, bond-breaking, excitation phenomena, and transition metal compounds. However, their high computational costs restrict their use in large systems or multi-molecule screening.

MBGF as a core physical quantity

To address these challenges, Yale researchers proposed using many-body Green's functions (MBGF) as the core physical quantity, constructing a deep learning framework that seamlessly connects ground and excited state predictions at the many-body level. The Green's function G(ω) describes electron/hole propagation in many-electron systems, with its size scaling quadratically with system size, making it more compact than wavefunctions.

Green's function theory, via Hedin equations and Bethe-Salpeter Equation (BSE), models single-particle (charged) and two-particle (neutral) excitations, providing a rigorous route to solving the Schrödinger equation. MBGF also contains most of the ground-state information: its static limit yields the single-particle density matrix, and integration along the imaginary frequency axis gives the ground-state energy.

The new model MBGF-Net

Recent advances include ab initio MBGF methods based on GW4, CC, second-order perturbation theory, algebraic diagram construction, density matrix renormalization group, and quantum Monte Carlo, achieving great success in simulating correlated molecules and materials.

MBGF is also a key quantity in quantum embedding methods, including dynamical mean-field theory and self-energy embedding theory. ML methods based on MBGF can unify predictions of various electronic properties and provide fundamental insights into electron correlation effects in large molecules and materials.

However, developing such ML methods faces a major challenge: how to represent the frequency-dependent MBGF matrix in a compact, equivariant form that captures local and non-local electron correlations encoded in MBGF.

Yale researchers addressed this by developing a graph neural network (GNN) that directly learns the many-body dynamical correlation potential (self-energy) on a compact imaginary frequency grid, using symmetry- and polarization-adaptive orbitals. The resulting model is called MBGF-Net.

Diagram: MBGF-Net workflow and architecture overview. (Source: Paper)

Performance evaluation

On a series of molecular and nanomaterial benchmarks, MBGF-Net accurately predicts ground and excited state properties, including photoluminescence, spectra, quasiparticle (QP) energies, and renormalization, matching GW and CCSD levels.

Additionally, MBGF-Net exhibits high data efficiency, predicting quantum mechanical energies for QM7 and QM9 molecules with only 2,000 training molecules, achieving mean absolute errors (MAE) below 0.02 eV.

Diagram: MBGF-Net predictions of electronic properties for QM7 and QM9 molecules. (Source: Paper)

Moreover, MBGF-Net demonstrates excellent transferability across different chemical species, conformations, system sizes, and electron correlation strengths. For example, a model trained on small silicon clusters can predict excitation spectra of larger clusters with minimal accuracy loss.

This method is also data-efficient, requiring only hundreds of training molecules in most benchmarks, greatly reducing the cost of generating many-body quantum chemistry data.

Although current research is limited to molecules, MBGF-Net can be extended to solids, such as deriving symmetry-adaptive basis functions with periodic boundary conditions. It introduces advanced many-body capabilities into Hamiltonian ML toolkits. When combined with deep learning methods for DFT Hamiltonians, it can be used to study electron correlation effects in larger systems.

Furthermore, MBGF-Net can be seamlessly integrated into widely used ab initio MBGF frameworks, such as data-driven impurity solvers in Green's function embedding methods for simulating correlated materials.

In summary, this research establishes a unified ML framework for many-body quantum chemistry and demonstrates the potential of ML-accelerated many-electron calculations for quantitative studies.

Paper link: https://www.nature.com/articles/s43588-025-00810-z