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 $\Lambda$-amplitudes. We introduce MōLe-$\Lambda$, 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 $(\Lambda_1,\Lambda_2)$ from localized Hartree--Fock molecular orbitals. Architecturally, MōLe-$\Lambda$ extends MōLe with $\Lambda_1$ and $\Lambda_2$ 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-$\Lambda$ further extends the speed advantage of M\ōLe over full CCSD while substantially expanding the accessible properties, providing a route to wavefunction-level surrogate models for correlated quantum chemistry.