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arXiv:2306.01573 · AIP Advances

Equation generator for equation-of-motion coupled cluster assisted by computer algebra system

Raúl Quintero-Monsebaiz, Pierre‐François Loos

Published Aug 1, 2023• 9 citations
arXivHTMLPDF

Abstract

We present an equation generator algorithm that utilizes second-quantized operators in normal order with respect to a correlated or non-correlated reference and the corresponding Wick theorem. The algorithm proposed here, written with Mathematica, enables the generation of non-redundant strings of second-quantized operators that, after classification, are directly assigned to many-body term quantities used to construct the many-body Hamiltonian. We demonstrate the capabilities of the algorithm by computing the coupled-cluster amplitude equations and various blocks of the equation-of-motion many-body Hamiltonian. A comprehensive description of this four-step algorithm is provided alongside concrete examples.

DOI
10.1063/5.0163846

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We present an equation generator algorithm that utilizes second-quantized operators in normal order with respect to a correlated or non-correlated reference and the corresponding Wick theorem. The algorithm proposed here, written with Mathematica, enables the generation of non-redundant strings of second-quantized operators that, after classification, are directly assigned to many-body term quantities used to construct the many-body Hamiltonian. We demonstrate the capabilities of the algorithm by computing the coupled-cluster amplitude equations and various blocks of the equation-of-motion many-body Hamiltonian. A comprehensive description of this four-step algorithm is provided alongside concrete examples.
This paper describes Wick&d, an implementation of the algebra of second-quantized operators normal ordered with respect to general correlated references and the corresponding Wick theorem [D. Mukherjee, Chem. Phys. Lett. 274, 561 (1997) and W. Kutzelnigg and D. Mukherjee, J. Chem. Phys. 107, 432 (1997)]. Wick&d employs a compact representation of operators and a backtracking algorithm to efficiently evaluate Wick contractions. Since Wick&d can handle both fully and partially contracted terms, it can be applied to both projective and Fock-space many-body formalisms. To demonstrate the usefulness of Wick&d, we use it to evaluate the single-reference coupled cluster equations up to octuple excitations and report an automated derivation and implementation of the second-order driven similarity renormalization group multireference perturbation theory.
A new approach to the diagrammatic formulation of many-body perturbation theory for open-shell systems is presented. The formalism is based on a generalized form of the Bloch equation that also generates the Rayleigh--Schroedinger perturbation expansion for a system with several open shells. Second quantization in the particle-hole formulation is used together with Wick's theorem in order to derive graphical rules in the usual way. The linked-diagram property of the wave operator and of the effective interaction is shown by expanding the wave operator in terms of normal products of connected diagram clusters, in analogy with the exp(S) formalism of Coester and Kuemmel for closed-shell systems. Self-consistent coupled-cluster equations are derived for the open-shell case from the generalized Bloch equation by a straightforward extension of the procedure of Cizek and Paldus. The application of such equations for investigating different properties of open-shell atoms is discussed. 56 references.
A new approach to the diagrammatic formulation of many-body perturbation theory for open-shell systems is presented. The formalism is based on a generalized form of the Bloch equation that also generates the Rayleigh-Schrödinger perturbation expansion for a system with several open shells. Second quantization in the particle-hole formulation is used together with Wick's theorem in order to derive graphical rules in the usual way. The linked-diagram property of the wave operator and of the effective interaction is shown by expanding the wave operator in terms of normal products of connected diagram clusters, in analogy with the exp(S) formalism of Coester and Kümmel for closed-shell systems. Self-consistent “coupled-cluster” equations are derived for the open-shell case from the generalized Bloch equation by a straightforward extension of the procedure of Čížek and Paldus. The application of such equations for investigating different properties of open-shell atoms is discussed.
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.
We present a symbolic manipulation algorithm for the efficient automated implementation of rigorously spin-free coupled cluster (CC) theories based on a unitary group parametrization. Due to the lack of antisymmetry of the unitary group generators under index permutations, all quantities involved in the equations are expressed in terms of non-antisymmetric tensors. Given two tensors, all possible contractions are first generated by applying Wick's theorem. Each term is then put down in the form of a non-antisymmetric Goldstone diagram by assigning its contraction topology. The subsequent simplification of the equations by summing up equivalent terms and their factorization by identifying common intermediates is performed via comparison of these contraction topologies. The definition of the contraction topology is completely general for non-antisymmetric Goldstone diagrams, which enables our algorithm to deal with noncommuting excitations in the cluster operator that arises in the unitary group based CC formulation for open-shell systems. The resulting equations are implemented in a new code, in which tensor contractions are performed by successive application of matrix-matrix multiplications. Implementation of the unitary group adapted CC equations for closed-shell systems and for the simplest open-shell case, i.e., doublets, is discussed, and representative calculations are presented in order to assess the efficiency of the generated codes.
Automated Quantum Chemistry Code Generation with the p†^\dagger†q Package
This article summarizes recent updates to the p†^\dagger†q package, which is a C++ accelerated Python library for generating equations and computer code corresponding to singly-reference many-body quantum chemistry methods such as coupled-cluster (CC) and equation-of-motion (EOM) CC theory. Since 2021, the functionality in \pq has expanded to include boson operators, coupled fermion-boson operators, unitary cluster operators, non-particle-conserving EOM operators, spin tracing, multiple single-particle subspaces, and more. Additional developments allow for the generation of C++ and Python code that minimizes floating-point operations via contraction order optimization, sub-expression elimination, and the fusion of similar terms.