Advances in Bootstrap Embedding Towards Large Molecular Systems

Abstract

In the pursuit of a quantitative understanding of molecular energetics, theoretical chemistry methods must carefully balance accuracy with computational cost. This challenge is particularly acute in quantum chemical approaches, which aim to describe electronic structure and interactions with high fidelity but whose exact solutions scale exponentially with system size. Quantum embedding techniques address this challenge by exploiting the locality of electron correlation to reduce computational expense. Among these, bootstrap embedding (BE), developed in the Van Voorhis group, offers a compelling compromise between the steep scaling of correlated wavefunction methods and their ability to accurately describe chemical systems. BE achieves this by partitioning molecules or solids into fragments, treating each fragment at a high level of theory, and recombining the fragment results to recover correlated-level descriptions of extended systems.

In this thesis, we present several developments that extend this linear-scaling, wavefunction-in-wavefunction embedding framework to larger molecular systems and dense basis sets, often beyond the reach of competing approaches. We demonstrate that localizing orbitals using intrinsic atomic orbitals (IAOs) in large basis sets yields highly accurate results that systematically converge toward full-system energies as fragment sizes increase. This localization also accelerates convergence and enables more efficient integral transformations. Using fragments that extend two coordination shells beyond a central atom, we consistently recover more than 99.7% of the total correlation energy across all tested systems and basis sets.

Building on this localization strategy, we introduce an alternative fragment construction scheme designed specifically for large basis sets. This approach employs cluster natural orbitals (CNOs) to capture interactions between each fragment and its environment, augmenting the fragment spaces in a manner inspired by local correlation methods. The inclusion of CNOs significantly improves the accuracy of BE calculations, and when combined with extrapolation techniques, enables the recovery of more than 99% of the total correlation energy in double-zeta basis sets, often outperforming the BE(3) result where calculable.

We further introduce a mixed-basis-set framework that enables BE calculations at the CCSD level for systems containing hundreds of atoms in triple-zeta basis sets. We combine a multiscale basis-set description with the ensemble of overlapping fragments, leveraging the locality of chemically-important interactions, to recover over 99% of the target BE energy for systems well beyond traditional size limits. When coupled with QM/MM techniques, this approach is demonstrated on extended biological systems containing approximately 40,000 atoms.

In addition, we explore the application of BE in a different computational regime by employing selected configuration interaction (SCI) as the fragment solver. We show that incorporating SCI within the BE framework reliably extends its applicability to systems containing tens of hydrogen atoms and suggests a viable path toward treating carbon-based molecules. This represents a substantial expansion beyond the conventional capabilities of SCI and points toward the possibility of achieving near–full configuration interaction accuracy with a linearly scaling approach.

Finally, we describe the implementation of these methods and others in the open-source software package QuEmb, which enables the broad use and further development of BE. Through this platform, we aim to facilitate the application of BE to new classes of systems and encourage collaborative efforts to continue advancing the method.