Equivariant analytical mapping of first principles Hamiltonians to accurate and transferable materials models

www.nature.com/npjcompumats ARTICLE OPEN Equivariant analytical mapping of first principles Hamiltonians to accurate and transferable materials models Liwei Zhang 1, Berk Onat2, Geneviève Dusson3, Adam McSloy2, G. Anand James R. Kermode 2 ✉ 4 , Reinhard J. Maurer5, Christoph Ortner1 and We propose a scheme to construct predictive models for Hamiltonian matrices in atomic orbital representation from ab initio data as a function of atomic and bond environments. The scheme goes beyond conventional tight binding descriptions as it represents the ab initio model to full order, rather than in two-centre or three-centre approximations. We achieve this by introducing an extension to the atomic cluster expansion (ACE) descriptor that represents Hamiltonian matrix blocks that transform equivariantly with respect to the full rotation group. The approach produces analytical linear models for the Hamiltonian and overlap matrices. Through an application to aluminium, we demonstrate that it is possible to train models from a handful of structures computed with density functional theory, and apply them to produce accurate predictions for the electronic structure. The model generalises well and is able to predict defects accurately from only bulk training data. 1234567890():,; npj Computational Materials (2022)8:158 ; https://doi.org/10.1038/s41524-022-00843-2 INTRODUCTION The availability of accurate and highly efficient interatomic potentials is crucial for the atomistic simulation of materials phenomena with intrinsic length and time scales inaccessible to first principles electronic structure theory. Examples in materials science include failure processes such as crack propagation1 and chemical dynamics at reactive surfaces2. The advent of machinelearning-based interatomic potentials (MLIPs) has meant that highfidelity interatomic potentials based on Kohn–Sham density functional theory (KS-DFT) and beyond have become much more widely available3–5. Yet, the effort to generate MLIPs that are both transferable and accurate is still significant and heavily depends on the configurational space spanned by the underlying training data set6. Very few MLIPs have been reported that are able to capture different materials phases, surface terminations, and the effects of complex defects on the stability and structure of the material5,7,8. More importantly, MLIPs and conventional interatomic potentials fundamentally neglect explicit electronic degrees of freedom of molecules and materials thereby removing access to the simulation of observables beyond structure and stability, such as electric conductivity and optical response, which depend on the electronic subsystem and electron–phonon coupling. While the ability to predict optical and electronic properties is desirable, the inclusion of electronic degrees of freedom will likely also benefit the transferability of MLIPs. For decades, semi-empirical and tight-binding (TB) models of electronic structure have sought to combine the efficiency of interatomic potentials with the explicit description of electrons. A plethora of approaches based on two-centre and three-centre integral approximations have led to established method frameworks such as the AM1 and PM3 methods9,10, the density functional tight-binding (DFTB) method11,12, the Sankey–Niklewski approach as implemented in the FIREBALL code13,14, and the xTB approach15. Unfortunately, the rigid mathematical form of the integral tabulations in most approaches means that TB parametrizations are limited in accuracy and often do not transfer beyond the materials classes for which they were originally intended. As ML methods make inroads across a diverse range of molecular simulation workflows16, approaches beyond MLIPs are being pursued that incorporate electronic properties. For molecules, Li et al. have proposed a neural-network-based parametrization pipeline for DFTB17, while Stoehr et al. have proposed deep tensor neural networks (DTNNs) to construct beyond-pairwise repulsion potentials18. Qiao et al. have shown that the use of symmetryadapted atomic-orbital features can significantly improve transferability and prediction accuracy of molecular stability19. In the realm of condensed phase materials, the automated construction of tight-binding models from ab initio data has been a topic of great interest as it can benefit high-throughput materials screening studies20. Most commonly, electronic structure simulations of materials are performed in non-atom-centred basis representations such as the pseudopotential plane wave framework, which is not easily amenable to the construction of TB models. TB Hamiltonians are typically constructed via transformation into a maximally localised Wannier function representation21, which provides a compact atom-centred basis representation with local support22. It is also possible to fit Slater–Koster parameters directly to DFT calculations in a data-driven fashion23,24. Materials simulations in atom-centred orbital representations as provided by, for example, the FHI-aims code25 are becoming more common, where Wannierization is not necessary and the basis representation provided by the code is directly amenable to machine learning approaches based on local representations of atomic neighbourhoods6. Examples of such representations include Behler–Parinello symmetry functions3,26, the SOAP descriptor27 or the atomic cluster expansion28,29. First efforts of direct machine learning prediction of electronic structure have been reported in literature. For example, SchNOrb30 is a DTNN representation of molecular mean-field electronic structure Hamiltonians, which 1 Department of Mathematics, University of British Columbia, 1984 Mathematics Road, Vancouver, BC V6T 1Z2, Canada. 2Warwick Centre for Predictive Modelling, School of Engineering, University of Warwick, Coventry CV4 7AL, UK. 3Laboratoire de Mathématiques, UMR CNRS 6623, Université Bourgogne Franche-Comté, 16 route de Gray, 25030 Besançon, France. 4Department of Metallurgy and Materials Engineering, Indian Institute of Engineering Science and Technology-Shibpur, Howrah, WB, India. 5Department of Chemistry, University of Warwick, Coventry CV4 7AL, UK. ✉email: Published in partnership with the Shanghai Institute of Ceramics of the Chinese Academy of Sciences L. Zhang et al. 1234567890():,; 2 has been used to predict Hamiltonians in local atomic orbital and optimised effective minimal basis representations for organic molecules including up to 13 heavy atoms30,31. Hedge and Bowen32 employed Kernel ridge regression with a bispectrum representation33 for an analytical representation of a minimal basis DFT Hamiltonian for bulk copper and diamond. Equivariant parameterisations for molecular systems along similar lines to what we describe here have been reported, learning either from the Hamiltonian34 or from wavefunctions and electronic densities35. These works apply linear or nonlinear equivariant models, respec (...truncated)