Many computational methods in ab initio quantum chemistry are formulated in terms of high-order tensor contractions, whose cost determines the size of system that can be studied. We introduce stochastic tensor contraction to perform such operations with greatly reduced cost, and present its application to the gold-standard quantum chemistry method, coupled cluster theory with up to perturbative triples. For total energy errors more stringent than chemical accuracy, we reduce the computational scaling to that of mean-field theory, while starting to approach the mean-field absolute cost, thereby challenging the existing cost-to-accuracy landscape. Benchmarks against state-of-the-art local correlation approximations further show that we achieve an order-of-magnitude improvement in both total computation time and error, with significantly reduced sensitivity to system dimensionality and electron delocalization. We conclude that stochastic tensor contraction is a powerful computational primitive to accelerate a wide range of quantum chemistry.
Many computational methods in ab initio quantum chemistry are formulated in terms of high-order tensor contractions, whose cost determines the size of system that can be studied. We introduce stochastic tensor contraction to perform such operations with greatly reduced cost, and present its application to the gold-standard quantum chemistry method, coupled cluster theory with up to perturbative triples. For total energy errors more stringent than chemical accuracy, we reduce the computational scaling to that of mean-field theory, while starting to approach the mean-field absolute cost, thereby challenging the existing cost-to-accuracy landscape. Benchmarks against state-of-the-art local correlation approximations further show that we achieve an order-of-magnitude improvement in both total computation time and error, with significantly reduced sensitivity to system dimensionality and electron delocalization. We conclude that stochastic tensor contraction is a powerful computational primitive to accelerate a wide range of quantum chemistry.
A constant goal of quantum chemistry is devising accurate and computationally effective methods for molecular simulations. In this work an application of tensor decompositions in the context of highly accurate coupled cluster theory, which is often considered a "gold standard", is investigated. The scheme we develop is aimed to mitigate the steep growth of the computational cost with the system size, and hence to overcome the "curse of dimensionality" common in many potent methods in electronic structure. We show how to reduce the computational effort of the restricted coupled cluster with singles and doubles (RCCSD) by two orders of magnitude by introducing alternative parameterizations of the method using "canonical polyad decomposition" (CPD) and "tensor hypercontraction" (THC) formats. After describing CPD and THC formats in detail, we demonstrate how to cast regular index based tensors into a decomposed form. The number of parameters and the accuracy of these representations depend on the expansion length (rank) of the approximation. We investigate the dependence of rank upon the size of the system and a target accuracy and show it to be low for typical tensors in electronic structure. We then provide a generic procedure to reformulate any coupled cluster method using tensor decompositions. Two specific approximate methods, THC-RCCSD and CPD-RCCSD, are derived. We demonstrate the accuracy of these new approaches by calculating energies of a large set of organic molecules, as well as by simulations of Hubbard models. Finally, it is shown how the restriction of the number of parameters in approximate coupled cluster can improve the accuracy in the challenging strong correlation regime. We conclude by discussing a connection of our findings to other new developments in the coupled cluster theory and propose possible extensions of our approach.
Calculation of high-rank tensor contractions plays a central role in computational physics, quantum chemistry, and computer science. The ability to perform a tensor contraction within a given computational budget poses a challenge to feasibility and restricts the types of systems that can be investigated computationally. We present the development and implementation of the Stratified Stochastic Tensor Contraction (SSTC) method as an alternative to sequential evaluation of tensor contraction. The SSTC approach partitions the tensor indices into stratified segments and employs composite index mappings to guide efficient sampling. By leveraging intrinsic structure within tensors, SSTC achieves rapid convergence to exact results with controllable sampling errors. We demonstrate the method on representative cases, two-point Coulomb integrals, four-point kernels, and correlation energy calculations using 2nd order Møller–Plesset perturbation theory, all of which are relevant to explicitly correlated electronic structure theory. In all examples, the SSTC method provides an accurate estimate of the tensor contraction with systematically reducible errors. We present the theory of exact tensor contractions as a foundation for the stochastic approach, followed by the mathematical derivation of the SSTC framework. Key components include the construction of a composite index to enable loop compression, and the application of functional minimization techniques for variance reduction. A discussion comparing sequential and stochastic contraction strategies, analysis of the distribution of sampling points through cumulative distribution functions and correlations between sampling assignments and kernel values are presented. A description of the SSTC method’s impact on computational clock times and dependence on segment number is included. Additional considerations include sampling with vs without replacement and the computational advantages of index mapping. These results highlight the effectiveness of the SSTC method as a versatile and scalable alternative to the conventional sequential tensor contraction approach, with broad applicability to quantum chemistry, many-body physics, stochastic quantum mechanics, and tensor-based machine learning.
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.
Slater determinants have underpinned quantum chemistry for nearly a century, yet their full potential has remained challenging to exploit. In this work, we show that a variational wavefunction composed of a few hundred optimized non-orthogonal determinants can achieve energy accuracies comparable to the state of the art. This is obtained by introducing an optimization method that leverages the quadratic dependence of the variational energy on the orbitals of each determinant, enabling an exact iterative optimization, and uses an efficient tensor-contraction algorithm to evaluate the effective Hamiltonian with a computational cost that scales as the fourth power of the number of basis functions. We benchmark the accuracy of the proposed method with exact full-configuration interaction results where available, and we achieve lower variational energies than coupled cluster (CCSD(T)) for several molecules in the double-zeta basis. Accurate description of electronic wave functions typically requires large numbers of Slater determinants. Here, the authors develop an optimization method that achieves state-of-the-art accuracy for small molecules with only few hundred determinants.
For many decades, quantum chemical method development has been dominated by algorithms which involve increasingly complex series of tensor contractions over one-electron orbital spaces. Procedures for their derivation and implementation have evolved to require the minimum amount of logic and rely heavily on computationally efficient library-based matrix algebra and optimised paging schemes. In this regard, the recent development of exact stochastic quantum chemical algorithms to reduce computational scaling and memory overhead requires a contrasting algorithmic philosophy, but one which when implemented efficiently can achieve higher accuracy/cost ratios with small random errors. Additionally, they can exploit the continuing trend for massive parallelisation which hinders the progress of deterministic high-level quantum chemical algorithms. In the Quantum Monte Carlo community, stochastic algorithms are ubiquitous but the discrete Fock space of quantum chemical methods is often unfamiliar, and the methods introduce new concepts required for algorithmic efficiency. In this paper, we explore these concepts and detail an algorithm used for Full Configuration Interaction Quantum Monte Carlo (FCIQMC), which is implemented and available in MOLPRO and as a standalone code, and is designed for high-level parallelism and linear-scaling with walker number. Many of the algorithms are also in use in, or can be transferred to, other stochastic quantum chemical methods and implementations. We apply these algorithms to the strongly correlated chromium dimer to demonstrate their efficiency and parallelism.
Wavefunction-based quantum methods are some of the most accurate tools for predicting and analyzing the electronic structure of molecules, in particular for accounting for dynamical electron correlation. However, most methods of including dynamical correlation beyond the simple second-order Møller-Plesset perturbation theory (MP2) level are too computationally expensive to apply to large molecules. Approximations which reduce scaling with system size are a potential remedy, such as the tensor hyper-contraction (THC) technique of Hohenstein et al., but also result in additional sources of error. In this work, we correct errors in THC-approximated methods using machine learning. Specifically, we apply THC to third-order Møller-Plesset theory (MP3) as a simplified model for coupled cluster with single and double excitations (CCSD), and train several regression models on observed THC errors from the Main Group Chemistry Database (MGCDB84). We compare performance of multiple linear regression models and nonlinear Kernel Ridge regression models. We also investigate correlation procedures using absolute and relative corrections and evaluate the corrections for both molecule and reaction energies. We discuss the potential for using regression techniques to correct THC-MP3 errors by comparing it to the "canonical" MP3 reference values and find the optimum technique based on accuracy. We find that nonlinear regression models reduced root mean squared errors between THC- and canonical MP3 by a factor of 6-9× for total molecular energies and 2-3× for reaction energies.