# M=oLe-Lambda: Learning the Coupled-Cluster Response State for Energies, Gradients, and Properties

arXiv: https://arxiv.org/abs/2605.29622 · HTML: https://mlsanity.com/paper/2605.29622/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- Builds on 11 of its 44 references that are in the map.
- First posted on arXiv 28 May 2026.

## Builds on (most cited first, 11 of 11)

- [The Devil in Linear Transformer](https://mlsanity.com/paper/2210.10340/) (2022)
- [By-passing the Kohn-Sham equations with machine learning](https://mlsanity.com/paper/1609.02815/) (2016)
- [Efficient and Equivariant Graph Networks for Predicting Quantum Hamiltonian](https://mlsanity.com/paper/2306.04922/) (2023)
- [Generalizing Neural Wave Functions](https://mlsanity.com/paper/2302.04168/) (2023)
- [Equivariant graph neural networks for fast electron density estimation of molecules, liquids, and solids](https://mlsanity.com/paper/2112.00652/) (2021)
- [Enhancing the Scalability and Applicability of Kohn-Sham Hamiltonians for Molecular Systems](https://mlsanity.com/paper/2502.19227/) (2025)
- [HIP: Hessian Interatomic Potentials without derivatives](https://mlsanity.com/paper/2509.21624/) (2025)
- [Efficient and Scalable Density Functional Theory Hamiltonian Prediction through Adaptive Sparsity](https://mlsanity.com/paper/2502.01171/) (2025)
- [High-order Equivariant Flow Matching for Density Functional Theory Hamiltonian Prediction](https://mlsanity.com/paper/2505.18817/) (2025)
- [ELECTRA: A Cartesian Network for 3D Charge Density Prediction with Floating Orbitals](https://mlsanity.com/paper/2503.08305/) (2025)
- [Surface Stability Modeling with Universal Machine Learning Interatomic Potentials: A Comprehensive Cleavage Energy Benchmarking Study](https://mlsanity.com/paper/2508.21663/) (2025)

## Built on by (most cited first, 1 of 1)

- [Grand Canonical Generators](https://mlsanity.com/paper/2610.00683/) (2026)

## Abstract (arXiv)

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.