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.
Accurate correlated electronic structure methods involve a significant amount of computations and can be only employed to small molecular systems. For example, the coupled-cluster singles, doubles, and perturbative triples model (CCSD(T)), which is known as the ``gold standard" of quantum chemistry for its accuracy, usually can treat molecules with 20-30 atoms. To extend the reach of accurate correlated electronic structure methods to larger molecular systems, we work towards two directions: parallel computing and reduced-cost/scaling approaches. Parallel computing can utilize more computational resources to handle systems that demand more substantial computational efforts. Reduced-cost/scaling approaches, which introduce approximations to the existing electronic structure methods, can significantly reduce the amount of computation and storage requirements. In this work, we introduce a new distributed-memory massively parallel implementation of standard and explicitly correlated (F12) coupled-cluster singles and doubles (CCSD) with canonical bigO{N^6} computational complexity ( C. Peng, J. A. Calvin, F. Pavov{s}evi'c, J. Zhang, and E. F. Valeev, textit{J. Phys. Chem. A} 2016, textbf{120}, 10231.), based on the TiledArray tensor framework. Excellent strong scaling is demonstrated on a multi-core shared-memory computer, a commodity distributed-memory computer, and a national-scale supercomputer. We also present a distributed-memory implementation of the density-fitting (DF) based CCSD(T) method. (C. Peng, J. A. Calvin, and E. F. Valeev, textit{in preparation for submission}) An improved parallel DF-CCSD is presented utilizing lazy evaluation for tensors with more than two unoccupied indices, which makes the DF-CCSD storage requirements always smaller than those of the non-iterative triples correction (T). Excellent strong scaling is observed on both shared-memory and distributed-memory computers equipped with conventional Intel Xeon processors and the Intel Xeon Phi (Knights Landing) processors. With the new implementation, the CCSD(T) energies can be evaluated for systems containing 200 electrons and 1000 basis functions in a few days using a small size commodity cluster, with even more massive computations possible on leadership-class computing resources. The inclusion of F12 correction to the CCSD(T) method makes it converge to basis set limit much more rapidly. The large-scale parallel explicitly correlated coupled-cluster program makes the accurate estimation of the coupled-cluster basis set limit for molecules with 20 or more atoms a routine. Thus, it can be used rigorously to test the emerging reduced-scaling coupled-cluster approaches. Moreover, we extend the pair natural orbital (PNO) approach to excited states through the equation-of-motion coupled cluster singles and doubles (EOM-CCSD) method. (C. Peng, M. C. Clement, and E. F. Valeev, textit{submitted}) We simulate the PNO-EOM-CCSD method using an existing massively parallel canonical EOM-CCSD program. We propose the use of state-averaged PNOs, which are generated from the average of the pair density of excited states, to span the PNO space of all the excited states. The doubles amplitudes in the CIS(D) method are used to compute the state-averaged pair density of excited states. The issue of incorrect states in the state-averaged pair density, caused by an energy reordering of excited states between the CIS(D) and EOM-CCSD, is resolved by simply computing more states than desired. We find that with a truncation threshold of
10−7, the truncation error for the excitation energy is already below 0.02 eV for the systems tested, while the average number of PNOs is reduced to 50-70 per pair. The accuracy of the PNO-EOM-CCSD method on local, Rydberg and charge transfer states is also investigated.
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.
Strong correlation poses a difficult problem for electronic structure theory, with computational cost scaling quickly with system size. Fragment embedding is an attractive approach to this problem. By dividing a large complicated system into smaller manageable fragments "embedded" in an approximate description of the rest of the system, we can hope to ameliorate the steep cost of correlated calculations. While appealing, these methods often converge slowly with fragment size because of small errors at the boundary between fragment and bath. We describe a new electronic embedding method, dubbed "Bootstrap Embedding," a self-consistent wavefunction-in-wavefunction embedding theory that uses overlapping fragments to improve the description of fragment edges. We apply this method to the one dimensional Hubbard model and a translationally asymmetric variant, and find that it performs very well for energies and populations. We find Bootstrap Embedding converges rapidly with embedded fragment size, overcoming the surface-area-to-volume-ratio error typical of many embedding methods. We anticipate that this method may lead to a low-scaling, high accuracy treatment of electron correlation in large molecular systems.
We review the current capabilities of local electron correlation methods up to the coupled cluster model with single, double, and perturbative triple excitations [CCSD(T)], which is a gold standard in quantum chemistry. The main computational aspects of the local method types are assessed from the perspective of applications, but the focus is kept on how to achieve chemical accuracy (i.e., <1 kcal/mol uncertainty), as well as on the broad scope of chemical problems made accessible. The performance of state-of-the-art methods are also compared, including the most employed DLPNO and, in particular, our local natural orbital (LNO) CCSD(T) approach. The high accuracy and efficiency of the LNO method makes chemically accurate CCSD(T) computations accessible for molecules of hundreds of atoms with resources affordable to a broad computational community (days on a single CPU and 10–100 GB of memory). Recent developments in LNO-CCSD(T) enable systematic convergence and robust error estimates even for systems of complicated electronic structure or larger size (up to 1000 atoms). The predictive power of current local CCSD(T) methods, usually at about 1–2 order of magnitude higher cost than hybrid density functional theory (DFT), has become outstanding on the palette of computational chemistry applicable for molecules of practical interest. We also review more than 50 LNO-based and other advanced local-CCSD(T) applications for realistic, large systems across molecular interactions as well as main group, transition metal, bio-, and surface chemistry. The examples show that properly executed local-CCSD(T) can contribute to binding, reaction equilibrium, rate constants, etc. which are able to match measurements within the error estimates. These applications demonstrate that modern, open-access, and broadly affordable local methods, such as LNO-CCSD(T), already enable predictive computations and atomistic insight for complicated, real-life molecular processes in realistic environments.
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.
We present a novel multiscale embedding scheme that links conventional QM/MM embedding and bootstrap embedding (BE) to allow simulations of large chemical systems on limited quantum devices. We also propose a mixed-basis BE scheme that facilitates BE calculations on extended systems using classical computers with limited memory resources. Benchmark data suggest the combination of these two strategies as a robust path in attaining the correlation energies of large realistic systems, combining the proven accuracy of BE with chemical and biological systems of interest in a lower computational cost method. Due to the flexible tunability of the resource requirements and systematic fragment construction, future developments in the realization of quantum computers naturally offer improved accuracy for multiscale BE calculations.