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Recent developments in quantum embedding have offered an attractive approach to describing electron correlation in molecules. However, previous methods such as density matrix embedding theory (DMET) require rigid partitioning of the system into fragments, which creates significant ambiguity for molecules. Bootstrap embedding (BE) is more flexible because it allows overlapping fragments, but when done on an orbital-by-orbital basis, BE introduces ambiguity in defining the connectivity of the orbitals. In this Letter, we present an atom-based fragment definition that significantly augments BE’s performance in molecules. The resulting method, which we term atom-based BE, is very effective at recovering valence electron correlation in moderate-sized bases and delivers near-chemical-accuracy results using extrapolation. We anticipate atom-based BE may lead to a low-scaling and highly accurate approach to electron correlation in large molecules., NSF (Grant CHE-1464804)

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., National Science Foundation (U.S.) (Grant CHE-1464804)

The mean-field solns. of electronic excited states are much less accessible than ground state (e.g., Hartree-Fock) solns. Energy-based optimization methods for excited states, like Δ-SCF, tend to fall into the lowest soln. consistent with a given symmetry -- a problem known as "variational collapse". In this work, we combine the ideas of direct energy-targeting and variance-based optimization in order to describe excited states at the mean-field level. The resulting method, σ-SCF, has several advantages. First, it allows one to target any desired excited state by specifying a single parameter: a guess of the energy of that state. It can therefore, in principle, find all excited states. Second, it avoids variational collapse by using a variance-based, unconstrained local minimization. As a consequence, all states -- ground or excited -- are treated on an equal footing. Third, it provides an alternate approach to locate Δ-SCF solns. that are otherwise hardly accessible by the usual non-aufbau configuration initial guess. Numerical results on small atoms and mols. show that σ-SCF is very effective at locating excited states, including individual, high energy excitations within a dense manifold of excited states. Like all single determinant methods, σ-SCF shows prominent spin-symmetry breaking for open shell states and we show how the results can be imrpoved by a spin-projection procedure.

The idea of using fragment embedding to circumvent the high computational scaling of accurate electronic structure methods while retaining high accuracy has been a long-standing problem for quantum chemists. Traditional fragment embedding methods mainly focus on systems composed of weakly correlated parts and are insufficient when division across chem. bonds is unavoidable. Recently, d. matrix embedding theory (DMET) and other methods based on the Schmidt decompn. have emerged as a fresh approach to this problem. Despite their success on model systems, these methods can prove difficult for realistic systems because they rely on either a rigid, non-overlapping partition of the system or a specification of some special sites (i.e. "edge" and "center" sites), neither of which is well-defined in general for real mols. In this work, we present a new Schmidt decompn.-based embedding scheme that allows the combination of arbitrary overlapping fragments without the knowledge of edge sites. This method forms a convergent hierarchy in the sense that higher accuracy can be obtained by using fragments involving more sites. The computational scaling for the first few levels is lower than that of most correlated wave function methods. We present results for several small mols. in atom-centered Gaussian basis sets and demonstrate that Incremental Embedding converges quickly with fragment size and recovers most static correlation in small basis sets even when truncated at the second lowest level.

Fragment embedding approaches offer the possibility of accurate description of strongly correlated systems with low-scaling computational expense. In particular, wave function embedding approaches have demonstrated the ability to subdivide systems across highly entangled regions, promising wide applicability for a number of challenging systems. In this paper, we focus on the wave function embedding method Bootstrap Embedding, extending it to the Pariser–Parr–Pople and 2D Hubbard models in order to evaluate the behaviour of the method in systems that are less amenable to local fragment embedding. We find that Bootstrap Embedding remains accurate for these systems, and we investigate how fragment size, shape, and choice of matching conditions affect the results. We also evaluate the properties of Bootstrap Embedding that lead to the method's favourable convergence properties. Keywords: Embedding; correlation; Bootstrap; DMET, National Science Foundation (U.S.) (Grant CHE-1464804)