The ultimate goal of electronic structure theory is solving the electronic Schr¨odinger Equation. However, even accurate approximations of solving Schr¨odinger Equation, such as high order coupled cluster theories, require computational efforts that are too demanding to be applied on large chemical systems. This thesis tackles the problem of curse of dimensionality: how to reduce the time complexity of high-accuracy coupled cluster methods in order to accelerate computations of molecular energy. On one hand, we believe that low-rank approximation (i.e. Tensor HyperContraction) of high-order tensors appearing in coupled cluster theory is a promising way to achieve rank-reduced coupled cluster theory. On the other hand, we think that the work of deriving, optimizing and implementing low-rank approximated coupled cluster theory is both tedious and repetitious, and shall be performed by an automatic code engine other than chemists. In this thesis, we introduce our chemistry equation compiler, Autom: a code engine that is designed for automatically deriving, optimizing and implementing low-rank approximated coupled cluster theories. With Autom, the development of electronic theories is greatly simplified. Autom can easily generate runnable codes in minutes for tensor-algebra based chemistry equations with customized low-rank approximation rules; conventionally, such procedure may take an expert weeks or even months to manually program. This thesis roughly consists of two parts. In the first part, we aim to cover the lowrank approximation techniques for high-accuracy coupled cluster theories, emphasizing on two new factorization techniques we developed especially for several coupled cluster theories including triple contributions. In the second part, we demonstrate the algorithms of our automatic code engine. We also summarize recent advances of domain specific compilers in Machine Learning community and discuss future directions for compute graph optimization. We believe that this new software infrastructure holds great promise for future chemistry package designs.
The ultimate goal of electronic structure theory is solving the electronic Schr¨odinger Equation. However, even accurate approximations of solving Schr¨odinger Equation, such as high order coupled cluster theories, require computational efforts that are too demanding to be applied on large chemical systems. This thesis tackles the problem of curse of dimensionality: how to reduce the time complexity of high-accuracy coupled cluster methods in order to accelerate computations of molecular energy. On one hand, we believe that low-rank approximation (i.e. Tensor HyperContraction) of high-order tensors appearing in coupled cluster theory is a promising way to achieve rank-reduced coupled cluster theory. On the other hand, we think that the work of deriving, optimizing and implementing low-rank approximated coupled cluster theory is both tedious and repetitious, and shall be performed by an automatic code engine other than chemists. In this thesis, we introduce our chemistry equation compiler, Autom: a code engine that is designed for automatically deriving, optimizing and implementing low-rank approximated coupled cluster theories. With Autom, the development of electronic theories is greatly simplified. Autom can easily generate runnable codes in minutes for tensor-algebra based chemistry equations with customized low-rank approximation rules; conventionally, such procedure may take an expert weeks or even months to manually program. This thesis roughly consists of two parts. In the first part, we aim to cover the lowrank approximation techniques for high-accuracy coupled cluster theories, emphasizing on two new factorization techniques we developed especially for several coupled cluster theories including triple contributions. In the second part, we demonstrate the algorithms of our automatic code engine. We also summarize recent advances of domain specific compilers in Machine Learning community and discuss future directions for compute graph optimization. We believe that this new software infrastructure holds great promise for future chemistry package designs.
The continuous demand for higher accuracy in computational chemistry necessitates the development of advanced many-body electronic structure methods. However, the derivation and efficient implementation of these theories constitute a significant bottleneck. As the rank of the associated tensors increases, the governing equations explode in complexity, rendering manual implementation labor-intensive, error-prone, and difficult to optimize for modern hardware. To address this challenge, this dissertation presents SeQuant, a comprehensive framework for the automated derivation and parallel implementation of many-body quantum chemistry methods. Built upon a robust symbolic algebra engine, SeQuant allows for the expression of theories in the natural language of second quantization. It automates the transformation of high-level theoretical ansatzes into explicit tensor contraction expressions and subsequently generates optimized, high-performance C++ code. A central innovation of this work is the extension of automated implementation to reduced-scaling methods, which exploit the sparsity inherent in electronic correlation. We introduce a novel "tensor-of-tensors" data structure designed to manage the irregular sparsity patterns of Pair Natural Orbital (PNO) formulations. This development enables the first fully automated implementation of PNO-Coupled Cluster (PNO-CC) methods, bridging the gap between symbolic abstraction and the runtime requirements of sparse tensor algebra. The results demonstrate that SeQuant not only reproduces established dense methods (such as CCSD and CCSDT) with high fidelity but also effectively handles the complexity of sparse, local correlation approaches. By decoupling the complexity of the physics from the details of the implementation, this framework establishes a new paradigm for method development, dramatically accelerating the translation of theoretical insights into computational reality.
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.
As both electronic structure methods and the computers on which they are run become increasingly complex, the task of producing robust, reliable, high-performance implementations of methods at a rapid pace becomes increasingly daunting. In this paper we present an overview of the Tensor Contraction Engine (TCE), a unique effort to address issues of both productivity and performance through automatic code generation. The TCE is designed to take equations for many-body methods in a convenient high-level form and acts like an optimizing compiler, producing an implementation tuned to the target computer system and even to the specific chemical problem of interest. We provide examples to illustrate the TCE approach, including the ability to target different parallel programming models, and the effects of particular optimizations.
In this thesis the application of novel tensor decomposition and tensor representation techniques in highly accurate post Hartree-Fock methods is evaluated. These representation techniques can help to overcome the steep scaling behaviour of high level ab-initio calculations with increasing system size and therefore break the "curse of dimensionality". After a comparison of various tensor formats the application of the "canonical polyadic" format (CP) is described in detail. There, especially the casting of a normal, index based tensor into the CP format (tensor decomposition) and a method for a low rank approximation (rank reduction) of the two-electron integrals in the AO basis are investigated. The decisive quantity for the applicability of the CP format is the scaling of the rank with increasing system and basis set size. The memory requirements and the computational effort for tensor manipulations in the CP format are only linear in the number of dimensions but still depend on the expansion length (rank) of the approximation. Furthermore, the AO-MO transformation and a MP2 algorithm with decomposed tensors in the CP format is evaluated and the scaling with increasing system and basis set size is investigated. Finally, a Coupled-Cluster algorithm based only on low-rank CP representation of the MO integrals is developed. There, especially the successive tensor contraction during the iterative solution of the amplitude equations and the error propagation upon multiple application of the reduction procedure are discussed. In conclusion the overall complexity of a Coupled-Cluster procedure with tensors in CP format is evaluated and some possibilities for improvements of the rank reduction procedure tailored to the needs in electronic structure calculations are shown.
We have developed a symbolic manipulation program and program generator (tensor contraction engine or TCE) that abstracts and automates the time-consuming, error-prone processes of deriving the working equations of a well-defined model of second-quantized many-electron theories and synthesizing efficient parallel computer programs on the basis of these equations. Provided an ansatz of a many-electron theory model, TCE performs valid contractions of creation and annihilation operators according to Wick's theorem, consolidates identical terms, and reduces the expressions into the form of multiple tensor contractions acted upon by permutation operators. It subsequently determines the binary contraction order for each multiple tensor contraction with the minimal operation and memory cost, factorizes common binary contractions (defines intermediate tensors), and identifies reusable intermediates. The resulting ordered list of binary tensor contractions, additions, and index permutations is translated into an optimized program that is combined with the NWCHEM and UTCHEM computational chemistry software packages. The programs synthesized by TCE take advantage of spin symmetry (within the spin−orbital formalisms), real Abelian point-group symmetry, and index permutation symmetry at every stage of the calculations to minimize the number of arithmetic operations and storage requirement, adjust the peak local memory usage by index-range tiling, and support parallel I/O interfaces and dynamic load balancing for parallel executions. We demonstrate the utility of TCE through automatic derivation and implementation of parallel programs for a range of predictive computational methods configuration-interaction theory (CISD, CISDT, CISDTQ), generalized many-body perturbation theory [MBPT(2), MBPT(3), MBPT(4)], and coupled-cluster theory (LCCD, CCD, LCCSD, CCSD, QCISD, CCSDT, and CCSDTQ), some for the first time and discuss the performance of the implemented programs.
This thesis deals with selected aspects of a new implementation of many-body methods which can be formulated in the framework of second quantization, in particular the coupled-cluster (CC) method with arbitrary excitation level. Coupled-cluster is one of the most successful and widely used quantum chemical methods for accurate calculations on small to medium-sized molecules. Since it employs a nonlinear parametrization of the wave function, its implementation is rather difficult, in particular if higher (i.e. more than double) excitations are to be included. The latter is necessary to obtain highly accurate results and also as a prerequisite for the generalization to multi-reference cases. The implementation described here has a twofold focus. One is on generality and flexibility regarding the method to be implemented, the other is on efficiency. To achieve flexibility, it is useful to have a machinery which automatically derives working equations for a given method. We realize this by applying techniques of second quantization. This work treats in particular the last step of this procedure, namely the simplification of the resulting equations by the identification of equivalent terms. The algorithm used here is based on the interpretation of algebraic terms as graphs. The derived CC equations then have to be solved iteratively. The efficiency of the program is mainly determined by the evaluation of the occurring expressions, which has to be done in each iteration step. The evaluation is split up in a sequence of tensor contractions. Their generic implementation is complicated by the particular structure of the involved tensors. We reduce each contraction to a sequence of matrix multiplications, which requires a previous data rearranging. But since matrix multiplication is the most efficient operation on modern computers, this additional effort pays off. Preliminary tests show that our program is at least as fast as the most efficient general coupled-cluster implementation so far, and the relation is expected to improve for calculations with larger basis sets where the matrix multiplication becomes the time-determining step. Finally, we give an outlook to possible further developments. In particular, the preparation of the equations before the actual evaluation (factorization) offers much potential for optimizations which we do not exploit at the moment, in contrast to the program with which we compare, which employs at least a partial optimization at this point.