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The Journal of Physical Chemistry A

Projector-Based Quantum Embedding for Molecular Systems: An Investigation of Three Partitioning Approaches

Jonathan M. Waldrop, Theresa L. Windus, Niranjan Govind

Published Jul 14, 2021• 23 citations
Publisher

Abstract

Projector-based embedding is a relatively recent addition to the collection of methods that seek to utilize chemical locality to provide improved computational efficiency. This work considers the interactions between the different proposed procedures for this method and their effects on the accuracy of the results. The interplay between the embedded background, projector type, partitioning scheme, and level of atomic orbital (AO) truncation are investigated on a selection of reactions from the literature. The Huzinaga projection approach proves to be more reliable than the level-shift projection when paired with other procedural options. Active subsystem partitioning from the subsystem projected AO decomposition (SPADE) procedure proves slightly better than the combination of Pipek-Mezey localization and Mulliken population screening (PMM). Along with these two options, a new partitioning criteria is proposed based on subsystem von Neumann entropy and the related subsystem orbital occupancy. This new method overlaps with the previous PMM method, but the screening process is computationally simpler. Finally, AO truncation proves to be a robust option for the tested systems when paired with the Huzinaga projection, with satisfactory results being acquired at even the most severe truncation level.

DOI
10.1021/acs.jpca.1c03821

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Projector-based embedding is a relatively recent addition to the collection of methods that seek to utilize chemical locality to provide improved computational efficiency. This work considers the interactions between the different proposed procedures for this method and their effects on the accuracy of the results. The interplay between the embedded background, projector type, partitioning scheme, and level of atomic orbital (AO) truncation are investigated on a selection of reactions from the literature. The Huzinaga projection approach proves to be more reliable than the level-shift projection when paired with other procedural options. Active subsystem partitioning from the subsystem projected AO decomposition (SPADE) procedure proves slightly better than the combination of Pipek-Mezey localization and Mulliken population screening (PMM). Along with these two options, a new partitioning criteria is proposed based on subsystem von Neumann entropy and the related subsystem orbital occupancy. This new method overlaps with the previous PMM method, but the screening process is computationally simpler. Finally, AO truncation proves to be a robust option for the tested systems when paired with the Huzinaga projection, with satisfactory results being acquired at even the most severe truncation level.
Abstract Using wave function (WF) in density functional theory (DFT) embedding methods provides a framework for performing localized, high-accuracy WF calculations on a system, while not incurring the full computational cost of the WF calculation on the full system. To effectively partition a system into localized WF and DFT subsystems, we utilize the Huzinaga level-shift projection operator within an absolutely localized basis. In this work, we study the ability of the absolutely localized Huzinaga level-shift projection operator method to study complex WF and DFT partitions, including partitions between multiple covalent bonds, a double bond, and transition-metal–ligand bonds. We find that our methodology can accurately describe all of these complex partitions. Additionally, we study the robustness of this method with respect to the WF method, specifically where the embedded systems were described using a multiconfigurational WF method. We found that the method is systematically improvable with respect to both the number of atoms in the WF region and the size of the basis set used, with energy errors less than 1 kcal/mol. Additionally, we calculated the adsorption energy of H2 to a model of an iron metal–organic framework (Fe-MOF-74) to within 1 kcal/mol compared to CASPT2 calculations performed on the full model while incurring only a small fraction of the full computational cost. This work demonstrates that the absolutely localized Huzinaga level-shift projection operator method is applicable to very complex systems with difficult electronic structures.
A widely used strategy to reduce the computational cost of quantum-chemical calculations is to partition the system into an active subsystem, which is the focus of the computational efforts, and an environment that is treated at a lower computational level. The system partitioning is mostly based on localized molecular orbitals. When reaction paths or energy differences are to be calculated, it is crucial to keep the orbital space consistent for all structures. Inconsistencies in orbital space can lead to unpredictable errors on the potential energy surface. While successful strategies to ensure this consistency have been established for organic and even metal-organic systems, these methods often fail for metal clusters or nanoparticles with a high density of near-degenerate and delocalized molecular orbitals. However, such systems are highly relevant for catalysis. Accurate yet feasible quantum-mechanical ab initio calculations are therefore highly desired. In this work, we present an approach based on the subsystem projected atomic orbital decomposition algorithm that allows us to ensure automated and consistent partitioning even for systems with delocalized and near-degenerate molecular orbitals and demonstrate the validity of this method for the binding energies of small molecules on transition-metal clusters.
Projection-based embedding offers a robust framework for modeling chemical processes in large molecular systems. It partitions computations into low-level treatments for the environmental subsystems and high-level treatments for the active subsystems. To address the computational bottleneck of applying high-accuracy wavefunction theory to the active regions, we introduce frozen natural orbitals (FNOs) approach. This technique efficiently compresses the virtual orbital space to generate a compact set of natural orbitals, which significantly accelerates the convergence of correlation energy recovery. We evaluated the FNO-based embedding approach in comparison with two alternative virtual space truncation approaches for a diverse range of molecular systems. Our results demonstrate that the FNO-embedded method exhibits superior performance by delivering accurate correlation energies with a substantially fewer number of virtual orbitals. Its potential for quantum computing is also discussed, as this reduction in orbitals can be directly translated to lower qubit requirements and thereby facilitate the quantum simulations on near-term quantum hardware.
Projection-based quantum embedding methodologies provide a framework for performing wave function-in-density functional theory (WF-in-DFT) calculations. The total WF-in-DFT energy is dependent on the partitioning of the total system and requires similar partitioning in each system for accurate energy differences. To achieve this, we enforce an absolute localization of the WF orbitals to basis functions only associated with the WF subsystem. This absolute localization, followed by iterative optimization of the subsystems' orbitals, provides improved energy differences for WF-in-DFT while simultaneously improving the computational efficiency.
Fragment embedding is one way to circumvent the high computational scaling of accurate electron correlation methods. The challenge of applying fragment embedding to molecular systems primarily lies in the strong entanglement and correlation that prevent accurate fragmentation across chemical bonds. Recently, Schmidt decomposition has been shown effective for embedding fragments that are strongly coupled to a bath in several model systems. In this work, we extend a recently developed quantum embedding scheme, bootstrap embedding (BE), to molecular systems. The resulting method utilizes the matching conditions naturally arising from using overlapping fragments to optimize the embedding. Numerical simulation suggests that the accuracy of the embedding improves rapidly with fragment size for small molecules, whereas larger fragments that include orbitals from different atoms may be needed for larger molecules. BE scales linearly with system size (apart from an integral transform) and hence can potentially be useful for large-scale calculations.
We discuss the partitioning of a quantum system through subsystem separation by unitary block-diagonalization (SSUB) applied to a Fock operator. For a one-particle Hilbert space, this separation can be formulated in a very general way. Therefore, it can be applied to very different partitionings ranging from those driven by features in the molecular structure (such as a solute surrounded by solvent molecules or an active site in an enzyme) to those that aim at an orbital separation (such as core-valence separation). Our framework embraces recent developments of Manby and Miller as well as the older ones of Huzinaga and Cantu. Projector-based embedding is simplified and accelerated by SSUB. Moreover, it directly relates to decoupling approaches for relativistic four-component many-electron theory. For a Fock operator based on the Dirac one-electron Hamiltonian, one would like to separate the so-called positronic (negative-energy) states from the electronic bound and continuum states. The exact two-component (X2C) approach developed for this purpose becomes a special case of the general SSUB framework and may therefore be viewed as a system-environment decoupling approach. Moreover, for SSUB, there exists no restriction with respect to the number of subsystems that are generated-in the limit, decoupling of all single-particle states is recovered, which represents exact diagonalization of the problem. The fact that a Fock operator depends on its eigenvectors poses challenges to all system-environment decoupling approaches and is discussed in terms of the SSUB framework. Apart from improved conceptual understanding, these relations bring about technical advances as developments in different fields can immediately cross-fertilize one another. As an important example, we discuss the atomic decomposition of the unitary block-diagonalization matrix in X2C-type approaches that can inspire approaches for the efficient partitioning of large total systems based on SSUB.