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Chemical Physics

Biorthogonal moment expansions in coupled-cluster theory: Review of key concepts and merging the renormalized and active-space coupled-cluster methods

Jun Shen, Piotr Piecuch B

Published Dec 7, 2011• 88 citations
Publisher
DOI
10.1016/j.chemphys.2011.11.033

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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

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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
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.
In this article, we investigate the numerical and theoretical aspects of the coupled-cluster method tailored by matrix-product states. We investigate formal properties of the used method, such as energy size consistency and the equivalence of linked and unlinked formulation. The existing mathematical analysis is here elaborated in a quantum chemical framework. In particular, we highlight the use of what we have defined as a complete active space-external space gap describing the basis splitting between the complete active space and the external part generalizing the concept of a HOMO-LUMO gap. Furthermore, the behavior of the energy error for an optimal basis splitting, i.e., an active space choice minimizing the density matrix renormalization group-tailored coupled-cluster singles doubles error, is discussed. We show numerical investigations on the robustness with respect to the bond dimensions of the single orbital entropy and the mutual information, which are quantities that are used to choose a complete active space. Moreover, the dependence of the ground-state energy error on the complete active space has been analyzed numerically in order to find an optimal split between the complete active space and external space by minimizing the density matrix renormalization group-tailored coupled-cluster error.