[Paper Review] Automatic selection of active spaces for strongly correlated systems using machine learning algorithms

Pavlo Golub, Andrej Antalík+2 more|arXiv (Cornell University)|Nov 30, 2020

Quantum many-body systems66 references4 citations

TL;DR

This paper presents a deep neural network (NN) model that automatically selects active spaces for strongly correlated transition metal systems using single-site orbital entropies derived from DMRG calculations. The model achieves high accuracy (85–92% recovery of important orbitals) and strong transferability on large, unseen systems like [Fe₂S₂(SCH₃)₄]²⁻, outperforming traditional methods by eliminating manual selection and costly iterative procedures.

ABSTRACT

The active-space quantum chemical methods could provide very accurate description of strongly correlated electronic systems, which is of tremendous value for natural sciences. The proper choice of the active space is crucial, but a non-trivial task. In this article, we present the neural network (NN) based approach for automatic selection of active spaces, focused on transition metal systems. The training set has been formed from artificial systems composed from one transition metal and various ligands, on which we have performed DMRG and calculated single-site entropy. On the selected set of systems, ranging from small benchmark molecules up to larger challenging systems involving two metallic centers, we demonstrate that our ML models could correctly predict the importance of orbitals with the high accuracy. Also, the ML models show a high degree of transferability on systems much larger than any complex used in training procedures.

Motivation & Objective

• To address the challenge of manual, error-prone active space selection in multireference calculations for large, strongly correlated transition metal complexes.
• To develop a fast, automated, and transferable method for identifying orbitals with significant static electron correlation.
• To replace iterative or computationally expensive orbital selection procedures with a single, inference-based machine learning model.
• To improve the efficiency and accuracy of multireference calculations, especially for systems with large active spaces.
• To enable scalable, automated electronic structure calculations in complex molecular systems such as metalloproteins.

Proposed method

• The model uses a feedforward deep neural network with 5 hidden layers and 896 neurons per layer, trained on a dataset of artificial transition metal complexes with varying ligands.
• Input features consist of single-orbital von Neumann entropies and a minimal, abstract feature space (B +10tcEXC) to maximize transferability.
• The model predicts orbital importance based on entropy values, with higher scores indicating greater correlation contribution.
• Training data is generated from DMRG calculations on systems ranging from small benchmarks to larger two-metal centers, using single-site entropy as the target.
• The model is evaluated on unseen systems, including [Fe₂S₂(SCH₃)₄]²⁻, using DMRG entropy as the ground truth for orbital importance.
• Orbital ordering is based on descending single-orbital entropy, and active spaces are selected by ranking predicted scores.

Experimental results

Research questions

• RQ1 Can a machine learning model accurately predict the most correlated orbitals in strongly correlated transition metal systems without prior knowledge of the system's electronic structure?
• RQ2 How well does the model generalize to large, complex systems not included in the training set?
• RQ3 What feature space design maximizes transferability across diverse molecular systems?
• RQ4 Can the model outperform traditional orbital selection methods in terms of speed and accuracy?
• RQ5 How does the model perform on systems with delocalized or mixed-character orbitals?

Key findings

• The 5x896 neural network with B +10tcEXC feature space achieved the best performance, correctly identifying 85% of 27 important orbitals in a 28-orbital active space for [Fe₂S₂(SCH₃)₄]²⁻.
• When the active space size was increased to 32 orbitals, the model recovered 92% of the 27 orbitals deemed important by DMRG single-site entropy.
• The model demonstrated strong transferability, accurately predicting orbital importance in large, unseen systems such as the [Fe₂S₂(SCH₃)₄]²⁻ complex.
• The B +10tcEXC feature space, designed to be abstract and system-independent, enhanced transferability compared to atom-type-dependent features (B +10tcEXC+AT).
• Orbitals with mixed 3d and p character on sulfur and carbon ligands were correctly identified as important, consistent with DMRG entropy values.
• The model slightly underestimated strongly delocalized orbitals with weak, distributed character, indicating a need for future improvements in handling such cases.