MōLe-Λ: Learning the Coupled-Cluster Response State for Energies, Gradients, and Properties

Coupled-cluster (CC) theory is often considered the gold standard of quantum chemistry, but its high computational cost limits routine access to accurate energies, forces and response properties. While the right-hand $T$-amplitudes determine the correlated wavefunction, many practically important observables additionally require the left-hand $Λ$-amplitudes. We introduce MōLe-$Λ$, an extension of Molecular Orbital Learning (MōLe) that predicts the full ground-state coupled-cluster singles and doubles (CCSD) response state by jointly learning right-hand amplitudes $(T_1,T_2)$ and left-hand amplitudes $(Λ_1,Λ_2)$ from localized Hartree--Fock molecular orbitals. Architecturally, MōLe-$Λ$ extends MōLe with $Λ_1$ and $Λ_2$ readouts that mirror the symmetry constraints of the $T_1$ and $T_2$ heads, while preserving the original equivariant orbital encoder, odd sign-equivariant decoding, locality and size-extensivity. The resulting model yields accurate CC-quality energies and forces, while simultaneously recovering dipoles, quadrupoles, polarizabilities, the electron density, and 2-electron observables such as the pair density. We show that MōLe-$Λ$ further extends the speed advantage of MōLe over full CCSD while substantially expanding the accessible properties, providing a route to wavefunction-level surrogate models for correlated quantum chemistry.

Coupled Cluster con MōLe: Molecular Orbital Learning for Neural Wavefunctions

Density functional theory (DFT) is the most widely used method for calculating molecular properties; however, its accuracy is often insufficient for quantitative predictions. Coupled-cluster (CC) theory is the most successful method for achieving accuracy beyond DFT and for predicting properties that closely align with experiment. It is known as the ''gold standard'' of quantum chemistry. Unfortunately, the high computational cost of CC limits its widespread applicability. In this work, we present the Molecular Orbital Learning (MōLe) architecture, an equivariant machine learning model that directly predicts CC's core mathematical objects, the excitation amplitudes, from the mean-field Hartree-Fock molecular orbitals as inputs. We test various aspects of our model and demonstrate its remarkable data efficiency and out-of-distribution generalization to larger molecules and off-equilibrium geometries, despite being trained only on small equilibrium geometries. Finally, we also examine its ability to reduce the number of cycles required to converge CC calculations. MōLe can set the foundations for high-accuracy wavefunction-based ML architectures to accelerate molecular design and complement force-field approaches.