M=oLe-Lambda: 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 M=oLe-Lambda, an extension of Molecular Orbital Learning (M=oLe) that predicts the full ground-state coupled-cluster singles and doubles (CCSD) response state by jointly learning right-hand amplitudes (T1,T2) and left-hand amplitudes (Lambda1,Lambda2) from localized Hartree--Fock molecular orbitals. Architecturally, M=oLe-Lambda extends M=oLe with Lambda1 and Lambda2 readouts that mirror the symmetry constraints of the T1 and T2 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 M=oLe-Lambda further extends the speed advantage of M=oLe over full CCSD while substantially expanding the accessible properties, providing a route to wavefunction-level surrogate models for correlated quantum chemistry.