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International Journal of Quantum Chemistry

Biorthogonal method of moments of coupled‐cluster equations: Alternative derivation, further considerations, and application to a model magnetic system

Piotr Piecuch B, Jeffrey R. Gour, Marta Włoch

Published Jan 1, 2008• 16 citations
Publisher

Abstract

Abstract The energy expansion defining the biorthogonal method of moments of coupled‐cluster equations (MMCC) [Piecuch and Włoch, J Chem Phys, 2005, 123, 224105 and Piecuch et al., Chem Phys Lett 2006, 418, 467], which leads to the size extensive completely renormalized (CR) coupled‐cluster (CC) approach with singles, doubles, and noniterative triples employing the left eigenstates of the similarity‐transformed Hamiltonian, termed CR‐CC(2,3), is overviewed and rederived. The rederivation of the biorthogonal MMCC expansion presented in this work is based on a direct resummation and subsequent elimination of the many‐body components of the exponential wave operator of CC theory that appear at individual moment contributions in the original MMCC energy expansion [Kowalski and Piecuch, J Chem Phys, 2000, 113, 18; Kowalski and Piecuch, J Chem Phys 2001, 115, 2966], enabling one to understand why the CR‐CC(2,3) method using the biorthogonal MMCC theory is more accurate than the earlier CR‐CCSD(T) approach. The superiority of the CR‐CC(2,3) method over the CR‐CCSD(T) and other previously developed single‐reference CC methods with a noniterative treatment of triply excited clusters, including the widely used CCSD(T) approach and the triples corrections defining the CCSD(2) schemes, is illustrated by examining the singlet–triplet gap of the (HFH)− magnetic system in which two paramagnetic centers are linked via a polarizable diamagnetic bridge. © 2008 Wiley Periodicals, Inc. Int J Quantum Chem, 2008

DOI
10.1002/qua.21745

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Abstract The energy expansion defining the biorthogonal method of moments of coupled‐cluster equations (MMCC) [Piecuch and Włoch, J Chem Phys, 2005, 123, 224105 and Piecuch et al., Chem Phys Lett 2006, 418, 467], which leads to the size extensive completely renormalized (CR) coupled‐cluster (CC) approach with singles, doubles, and noniterative triples employing the left eigenstates of the similarity‐transformed Hamiltonian, termed CR‐CC(2,3), is overviewed and rederived. The rederivation of the biorthogonal MMCC expansion presented in this work is based on a direct resummation and subsequent elimination of the many‐body components of the exponential wave operator of CC theory that appear at individual moment contributions in the original MMCC energy expansion [Kowalski and Piecuch, J Chem Phys, 2000, 113, 18; Kowalski and Piecuch, J Chem Phys 2001, 115, 2966], enabling one to understand why the CR‐CC(2,3) method using the biorthogonal MMCC theory is more accurate than the earlier CR‐CCSD(T) approach. The superiority of the CR‐CC(2,3) method over the CR‐CCSD(T) and other previously developed single‐reference CC methods with a noniterative treatment of triply excited clusters, including the widely used CCSD(T) approach and the triples corrections defining the CCSD(2) schemes, is illustrated by examining the singlet–triplet gap of the (HFH)− magnetic system in which two paramagnetic centers are linked via a polarizable diamagnetic bridge. © 2008 Wiley Periodicals, Inc. Int J Quantum Chem, 2008
Coupled cluster (CC) theory is the “gold standard” of molecular electronic structure theory, yet the full arc of the subject—from its variational origins in Hartree–Fock theory, through the exponential ansatz and its diagrammatic machinery, to its modern reformulation as a system of polynomial equations on an algebraic variety, its multireference generalizations, and its emerging reincarnation as a quantum circuit—is rarely presented as a single connected narrative. This review builds that narrative completely from first principles and carries it through to today’s research frontier. Starting from the electronic Schrödinger equation, we derive Hartree–Fock theory, second quantization, and Wick’s theorem; construct the coupled cluster singles-and-doubles (CCSD) equations via the similarity-transformed Hamiltonian; and show how these same equations reappear, verbatim, as a truncated eigenvalue problem on a projective variety generalizing the Grassmannian, with a computable CC degree controlling the number of solutions. We then develop the perturbative triples correction CCSD(T), the linear-response equationof-motion (EOM-CC) framework for excited, ionized, and electron-attached states, the effective-Hamiltonian formalism underlying multireference coupled cluster theory and its notorious intruder-state problem, and the unitary coupled cluster (UCC) ansatz that restores the variational principle at the cost of an infinite operator expansion—tamed, on a quantum computer, by Trotterization and the variational quantum eigensolver. We close with a synthesis of what is rigorously known about the resulting nonlinear equations, a map of directions deliberately left out of scope, and an assessment of the open question of quantum advantage in this domain. Every claim is stated formally, with a full statement and proof or proof sketch, and cited to its origin; the computational cost of every method is compared explicitly throughout.
The Coupled-Cluster (CC) method is one of the most popular and efficient correlation methods in quantum chemistry. \nEspecially the CCSD(T) approximation, which includes single and double excitations by means of the application of the cluster operator \nto a reference determinant and triple excitations via a perturbative treatment, has become a standard tool \nin quantum chemical applications. However, the method is restricted to relatively small system sizes due to its unfavourable scaling (N^7, where N is the number of basis functions applied). \nFor the full treatment of triple excitations the scaling advances to N^8 and every further excitation level increases the exponent by two. \n \nThe goal of this work is to reduce the calculation time for closed shell systems at least by a factor growing with the excitation level for arbitrary truncation levels. \nThis is done by restricting the spatial parts of the spin orbitals and thus treat pairs of spin orbitals on the same footing. The restrictions can be easily constructed \nfor the CCSD model by spin integration. \nThe derivation of the restrictions arising in higher excited case will be done employing \nthe spatial orbital excitation operators E. \n \nIn the first part of this work an algorithm is derived that is capable of the derivation of the energy and amplitude equations for arbitrary excitation levels. In the second part \nan implementation of this algorithm is presented.
Nonunitary theories are commonly seen in the classical simulations of quantum systems. Among these theories, the method of moments of coupled cluster equations (MMCCs) and the ensuing classes of the renormalized coupled cluster (CC) approaches have evolved into one of the most accurate approaches to describe correlation effects in various quantum systems. The MMCC formalism provides an effective way for correcting the energies of approximate CC formulations (parent theories) using moments, or CC equations, that are not used to determine approximate cluster amplitudes. In this paper, we propose a quantum algorithm for computing MMCC ground-state energies that provides two main advantages over classical computing or other quantum algorithms: (i) the possibility of forming superpositions of CC moments of arbitrary ranks in the entire Hilbert space and using an arbitrary form of the parent cluster operator for MMCC expansion, and (ii) significant reduction in the number of measurements in quantum simulation through a compact unitary representation for a generally nonunitary operator. We illustrate the robustness of our approach over a broad class of test cases, including $\ensuremath{\sim}40$ molecular systems with varying basis sets encoded using 4--40 qubits, and we exhibit the detailed MMCC analysis for the 8-qubit half-filled, four-site, single impurity Anderson model and the 12-qubit hydrogen fluoride molecular system from the corresponding noise-free and noisy MMCC quantum computations. We also outline the extension of the MMCC formalism to the case of a unitary CC wave-function Ansatz.
To discuss the usefulness of the entries of the complete set of solutions to approximate coupled-cluster (CC) equations for the description of the states of many-electron systems, one has to be able to establish a correspondence between the states and solutions. We have explored four methods of finding links between the states (represented for the model by the full configuration interaction wave functions and energies) and solutions to the CC equations. Numerical studies for the P4 model, which belongs to the simplest realistic many-electron model systems, show that the states-solutions correspondence pattern strongly depends on the method employed. Only for a few states all methods considered have assigned the same solutions. Among these states are those that are energetically the closest to the reference determinants. For the remaining states the assignment is not unique and the accuracy of description depends on the physical features considered. The article is concluded with a comparison of the present results and results of the most recent studies on the structure of the complete sets of solutions to the CC equations with the findings and intuitions being the content of the work by Živković and Monkhorst (J Math Phys 1978, 19, 1007). ©1999 John Wiley & Sons, Inc. Int J Quant Chem 75: 483–496, 1999
We present a general-order spin-free formulation of the single-reference closed-shell coupled-cluster method. We show that the working equations of a fully biorthogonal contravariant projection formulation of the residual equations, as near-universally used in closed-shell CCSD, can also be defined at the CCSDT and CCSDTQ levels, despite singularities in the spin projection manifolds. We describe permutation-group based techniques for obtaining and simplifying the equations encountered in general second-quantization-based methods; this includes a permutation group based approach of evaluating second-quantized matrix elements into tensor contraction networks, and the use of Portugal's double coset canonical representation technique [Int. J. Mod. Phys. C 13, 859 (2002)] for eliminating redundant terms. A computer implementation of our techniques is simple, because no operator-valued symbolic algebra is required. Explicit working equation lists for closed-shell CCSD, CCSDT, and CCSDTQ in the semi-biorthogonal formulation are provided. We also release open-source computer programs for both deriving and numerically evaluating these equations.
The standard and renormalized coupled cluster methods with singles, doubles, and noniterative triples and their generalizations to excited states, based on the equation of motion coupled cluster approach, are applied to the 4He and 16O nuclei. A comparison of coupled cluster results with the results of the exact diagonalization of the Hamiltonian in the same model space shows that the quantum chemistry inspired coupled cluster approximations provide an excellent description of ground and excited states of nuclei. The bulk of the correlation effects is obtained at the coupled cluster singles and doubles level. Triples, treated noniteratively, provide the virtually exact description.