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Journal of Chemical Theory and Computation

Machine Learning-Assisted Selection of Active Spaces for Strongly Correlated Transition Metal Systems

Pavlo Golub, Andrej Antalík, Libor Veis, Jiří Brabec

Published Sep 27, 2021• 37 citations
Publisher

Abstract

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 nontrivial task. In this article, we present a neural network-based approach for automatic selection of active spaces, focused on transition metal systems. The training set has been formed from artificial systems composed of one transition metal and various ligands, on which we have performed the density matrix renormalization group and calculated the 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 machine learning models could predict the active space orbitals with reasonable accuracy. We also tested the transferability on out-of-the-model systems, including bimetallic complexes and complexes with ligands, which were not involved in the training set. Also, we tested the correctness of the automatically selected active spaces on a Fe(II)-porphyrin model, where we studied the lowest states at the DMRG level and compared the energy difference between spin states or the energy difference between conformations of ferrocene with recent studies.

DOI
10.1021/acs.jctc.1c00235

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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 nontrivial task. In this article, we present a neural network-based approach for automatic selection of active spaces, focused on transition metal systems. The training set has been formed from artificial systems composed of one transition metal and various ligands, on which we have performed the density matrix renormalization group and calculated the 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 machine learning models could predict the active space orbitals with reasonable accuracy. We also tested the transferability on out-of-the-model systems, including bimetallic complexes and complexes with ligands, which were not involved in the training set. Also, we tested the correctness of the automatically selected active spaces on a Fe(II)-porphyrin model, where we studied the lowest states at the DMRG level and compared the energy difference between spin states or the energy difference between conformations of ferrocene with recent studies.
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.
The selection of a good set of active orbitals for modeling strongly correlated quantum states is difficult to automate as it is highly dependent on the state and molecule of interest. As such, although many approaches have been proposed with some success, no single approach has worked in all cases. Here, we propose an improved framework for automated selection in which (i) multiple wave functions based on different active-spaces are generated, and (ii) the resulting wave functions are chosen between by some means of facile evaluation. Using this framework, we propose a method in which (i) we construct different active space orbitals through diagonalization of a parameterized operator, and (ii) we choose the state with the lowest tPBE absolute energy from multiconfigurational pair density functional theory (MC-PDFT) averaged over the targeted states of interest. We test the method using density matrix renormalization group (DMRG) wave functions with 40 active orbitals and bond index of 700, with no further orbital optimization following the active space selection. We find that with only four values of the parameter (i.e., with only four trial wave functions), we can obtain a mean unsigned error of only 0.19 eV for 199 vertical excitation energies in the QUESTDB database. Furthermore, the tPBE absolute energy proves robust in selecting between active spaces of very different sizes, over and above the wave function (CASCI/DMRG-CASCI) absolute energy. We believe this new framework is promising for the application of active space methods to chemical problems in a high-throughput fashion.
Accurate electronic structure calculations are essential in modern materials science, but strongly correlated systems pose a significant challenge due to their computational cost. Traditional methods, such as complete active space self-consistent field (CASSCF), scale exponentially with system size, while alternative methods like the density matrix renormalization group (DMRG) scale more favorably, yet remain limited for large systems. In this work, we demonstrate how a simple machine learning model can enhance quantum chemical DMRG calculations, improving their accuracy to chemical precision, even for systems that would otherwise require considerably higher computational resources. The systems under study are polycyclic aromatic hydrocarbons, which are typical candidates for DMRG calculations and are highly relevant for advanced technological applications.
The quantum chemical version of the density matrix renormalization group (DMRG) method has established itself as one of the methods of choice for calculations of strongly correlated molecular systems. Despite its great ability to capture strong electronic correlation in large active spaces, it is not suitable for computations of dynamical electron correlation. In this work, we present a new approach to the electronic structure problem of strongly correlated molecules, in which DMRG is responsible for a proper description of the strong correlation, whereas dynamical correlation is computed via the recently developed adiabatic connection (AC) technique which requires only up to two-body active space reduced density matrices. We report the encouraging results of this approach on typical candidates for DMRG computations, namely, n -acenes ( n = 2 → 7), Fe(II)–porphyrin, and the Fe 3 S 4 cluster.
An estimation of optoelectronic properties is a crucial necessity and an ongoing challenge in modern functional material science. This involves accurate accessing of ground and excited states in systems with complex electronic structure that are often too computationally expensive for accurate many-body methods. Building on our previous work on ground states [J. Phys. Chem. Lett. 2025, 16, 3295-3301], we present an efficient and cost-effective approach for evaluating electronic excitations in functional materials, combining the density matrix renormalization group method as a complete active space solver with machine learning techniques. We demonstrate its performance on π-electron-correlated systems, namely polycyclic aromatic compounds. The transferability and effectiveness of the derived machine learning model are demonstrated on a number of challenging examples with up to 34 π-electrons.
The quality of multi-scale modelling techniques in molecular electronic structure calculations, such as embedding and subspace methods, relies upon the chosen active space. The automation of active space selection is vital for ensuring the accuracy, reproducibility, and scalability in such calculations. In this work, we introduce Tensor Network Active Space Selection using the Entanglement Feature. Through the isolation of strongly correlated electrons, this method provides a scalable foundation for embedding methods in multi-scale modelling. By representing the purities of all possible orbital partitions as a Matrix Product State, our method isolates regions of strong electron correlation without requiring manual preselection of target atoms or the calculation of expensive high-order density matrices. The results demonstrate that this approach leads to lower ground state energies and more accurate dipole moments than other fully automated selection schemes such as those based solely on single-orbital entropy or the selection of spatial orbitals around the HOMO/LUMO gap.