Recent developments in quantum embedding theories have provided attractive approaches to correlated calculations for large systems. In this work, we extend our previous work [J. Chem. Theory Comput. 2019, 15, 4497–4506; J. Phys. Chem. Lett. 2019, 10, 6368–6374] on bootstrap embedding (BE) to enable correlated ab initio calculations at the coupled cluster with singles and doubles (CCSD) level for large molecules. We introduce several new algorithmic developments that significantly reduce the computational cost of BE, while maintaining its accuracy. The resulting implementation scales as O(N 3) for the integral transform and O(N) for the CCSD calculation. Numerical results on a series of conjugated molecules suggest that BE with reasonably sized fragments can recover more than 99.5% of the total correlation energy of a full CCSD calculation, while the required computational resources (time and storage) compare favorably to one popular local correlation scheme: domain localized pair natural orbital (DLPNO). The largest BE calculation in this work involves ∼2900 basis functions and can be performed on a single node with 16 CPU cores and 64 GB of memory in a few days. We anticipate that these developments represent an important step toward the application of BE to solve practical problems.

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.

A new framework based on density matrix embedding theory (DMET) capable of directly targeting excited electronic states is proposed and implemented. DMET has previously been shown to be an effective method of calculating the ground state energies of systems exhibiting strong static correlation, but has never been applied to calculate excited state energies. In this work, the Schmidt decomposition is applied directly on excited states, approximated by higher lying SCF solutions. The DMET prescription is applied following this Schmidt decomposition allowing for a direct embedding of excited states. Initial results are obtained for a system of multiple hydrogen dimers and the lithium hydride dissociation. We analyze the nature of each part of the excited state DMET calculation and identify challenges. These challenges to the implementation of excited state DMET are discussed and potential suggestions moving forward are recommended.

A suite of quantum embedding methods have recently been developed where the Schmidt decomposition is applied to the full system wavefunction to derive basis states that preserve the entanglement between the fragment and the bath. The quality of these methods can depend heavily on the quality of the initial full system wavefunction. Most of these methods, including bootstrap embedding (BE) [M. Welborn et al; J. Chem. Phys. 145, 074102 (2016)], start from a spin-restricted mean-field wavefunction [call this restricted BE (RBE)]. Given that spin-unrestricted wavefunctions can capture a significant amount of strong correlation at the mean-field level, we suspect that starting from a spin-unrestricted mean-field wavefunction will improve these embedding methods for strongly correlated systems. In this work, BE is generalized to an unrestricted Hartree–Fock bath [call this unrestricted BE (UBE)], and UBE is applied to model hydrogen ring systems. UBE’s improved versatility over RBE is utilized to calculate high spin symmetry states that were previously unattainable with RBE. Ionization potentials, electron affinities, and spin-splittings are computed using UBE with accuracy on par with spin-unrestricted coupled cluster singles and doubles. Even for cases where RBE is viable, UBE converges more reliably. We discuss the limitations or weaknesses of each calculation and how improvements to RBE and density matrix embedding theory these past few years can also improve UBE.

The limitations associated with the common practice of fitting a quadratic Hamiltonian to vibronic levels of a Jahn-Teller system have been explored quantitatively. Satisfactory results for the prototypicalX 2 E state of Li 3 are obtained from fits to both experimental spectral data and to an "artificial" spectrum calculated by a quartic Hamiltonian which accurately reproduces the adiabatic potential obtained from state-of-the-art quantum chemistry calculations. However the values of the Jahn-Teller parameters, stabilization energy, and pseudo-rotation barrier obtained from the quadratic fit differ markedly from those associated with the ab initio potential. Nonetheless the RMS deviations of the fits are not strikingly different. Guidelines are suggested for comparing parameters obtained from fits to experiment to those obtained by direct calculation, but a principal conclusion of this work is that such comparisons must be done with a high degree of caution.