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arXiv:2311.03229

Non-interacting particle systems in the bi-orthogonal and unitary coupled-cluster description

J. Schirmer

Published Nov 6, 2023• 0 citations
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Abstract

Adapting a veritable many-body method to a system of non-interacting particles (NIP), while being trivial from a physical point of view, can be of interest with regard to methodological aspects. In this article we study the NIP versions of the bi-orthogonal (BCC) and unitary (UCC) coupled-cluster methods both for the ground state and generalized excitations. The essential simplification here is that the cluster operator is confined to single (or $p$-$h$) excitations. For the ground state, specifically the CC amplitude equations, the NIP versions illustrate the enormous complexity of the UCC approach as compared to the BCC concept. In the treatment of excitations, on the other hand, the UCC outperforms the BCC method, as the UCC secular matrix is block-diagonal with respect to the excitation classes, e.g., $1h$ (one-hole), $2h$-$1p$ (two-hole-one particle), $\dots$ excitations of $N\!-\!1$ particles. Moreover, the UCC basis states are essentially equivalent with the states used in the ADC/ISR (algebraic-diagrammatic construction/intermediate state representation) approach, whose NIP version has been discussed in a recent paper [J. Schirmer, arXiv:2309.15721].

DOI
10.48550/arxiv.2311.03229

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Adapting a veritable many-body method to a system of non-interacting particles (NIP), while being trivial from a physical point of view, can be of interest with regard to methodological aspects. In this article we study the NIP versions of the bi-orthogonal (BCC) and unitary (UCC) coupled-cluster methods both for the ground state and generalized excitations. The essential simplification here is that the cluster operator is confined to single (or ppp-hhh) excitations. For the ground state, specifically the CC amplitude equations, the NIP versions illustrate the enormous complexity of the UCC approach as compared to the BCC concept. In the treatment of excitations, on the other hand, the UCC outperforms the BCC method, as the UCC secular matrix is block-diagonal with respect to the excitation classes, e.g., 1h1h1h (one-hole), 2h2h2h-1p1p1p (two-hole-one particle), …\dots… excitations of N ⁣− ⁣1N\!-\!1N−1 particles. Moreover, the UCC basis states are essentially equivalent with the states used in the ADC/ISR (algebraic-diagrammatic construction/intermediate state representation) approach, whose NIP version has been discussed in a recent paper [J. Schirmer, arXiv:2309.15721].
This article presents an alternative formulation of quasi-degenerate perturbation theory (QDPT). The development results by simplifying established many-body (MB) techniques to systems of non-interacting particles (NIP). While the physical information of a NIP system is fully equivalent with that of the underlying one-particle system, NIP adaptions of MB schemes can acquire useful new roles in the one-particle context. In particular, the algebraic-diagrammatic construction (ADC) approach to the particle-detachment part of the electron propagator, turns into a new formalism of QDPT for the original one-particle system, establishing a diagrammatic perturbation theory for the matrix elements of the effective hamiltonian and the associated effective eigenvector coefficients (EEC). Using the NIP-ADC procedure explicit QDPT expansions are derived through third order. Alternatively, the intermediate state representation (ISR), underlying the ADC approach, allows one to construct the effective secular quantities directly from successive orders in the PT expansions of the NIP ground state and ground-state energy. Moreover, exact closed-form expressions can be derived for the effective hamiltonian and the EEC matrix in terms of 'target state' eigenvectors and eigenvalues of the one-particle system. These results are fully consistent with corresponding expressions derived within the QDPT context. The NIP approach to QDPT is not confined to actual particles but applies to arbitrary quantum systems. Any quantum system can be seen as formally constituting a "hyper-particle" giving rise to a system of non-interacting hyper-particles. The NIP adaptions of the biorthogonal coupled-cluster (BCC) and unitary coupled-cluster (UCC) methods are briefly discussed as well.
Unitary cluster expansions of the electronic wavefunction have recently gained much interest because of their use in conjunction with quantum algorithms. In this contribution, we investigate some aspects of an ansatz, using generalized two-body excitation operators, which have been considered in some recent studies on quantum algorithms for quantum chemistry. Our numerical results show that, in particular, two-body operators with effective particle-hole excitation level of one in connection with the usual particle-hole double excitation operators lead to a very accurate, yet compact representation of the wavefunction. Generalized two-body operators with effective excitation rank zero have a considerably less pronounced effect. We compare with standard and unitary coupled-cluster expansions and show that the above mentioned approach matches or even surpasses the accuracy of expansions with three-body particle-hole excitations, in particular at the onset of strong correlation. A downside of the approach is that it is rather difficult to rigorously converge it to its variational minimum.
The Coupled Cluster (CC) method is used to compute the electronic correlation energy in atoms and molecules and often leads to highly accurate results. However, due to its single-reference nature, standard CC in its projected form fails to describe quantum states characterized by strong electronic correlations and multi-reference projective methods become necessary. On the other hand, quantum algorithms for the solution of many-electron problems have also emerged recently. The quantum unitary variant of CC (UCC) with singles and doubles (q-UCCSD) is a popular wavefunction Ansatz for the variational quantum eigensolver algorithm. The variational nature of this approach can lead to significant advantages compared to its classical equivalent in the projected form, in particular, for the description of strong electronic correlation. However, due to the large number of gate operations required in q-UCCSD, approximations need to be introduced in order to make this approach implementable in a state-of-the-art quantum computer. In this work, we evaluate several variants of the standard q-UCCSD Ansatz in which only a subset of excitations is included. In particular, we investigate the singlet and pair q-UCCD approaches combined with orbital optimization. We show that these approaches can capture the dissociation/distortion profiles of challenging systems, such as H4, H2O, and N2 molecules, as well as the one-dimensional periodic Fermi–Hubbard chain. These results promote the future use of q-UCC methods for the solution of challenging electronic structure problems in quantum chemistry.
Background: Ab initio many-body methods have been developed over the past 10 yr to address closed-shell nuclei up to mass $A\ensuremath{\approx}130$ on the basis of realistic two- and three-nucleon interactions. A current frontier relates to the extension of those many-body methods to the description of open-shell nuclei. Several routes to address open-shell nuclei are currently under investigation, including ideas that exploit spontaneous symmetry breaking.Purpose: Singly open-shell nuclei can be efficiently described via the sole breaking of U(1) gauge symmetry associated with particle-number conservation as a way to account for their superfluid character. While this route was recently followed within the framework of self-consistent Green's function theory, the goal of the present work is to formulate a similar extension within the framework of coupled cluster theory.Methods: We formulate and apply Bogoliubov coupled cluster (BCC) theory, which consists of representing the exact ground-state wave function of the system as the exponential of a quasiparticle excitation cluster operator acting on a Bogoliubov reference state. Equations for the ground-state energy and the cluster amplitudes are derived at the singles and doubles level (BCCSD) both algebraically and diagrammatically. The formalism includes three-nucleon forces at the normal-ordered two-body level. The first BCC code is implemented in mmm scheme, which will permit the treatment of doubly open-shell nuclei via the further breaking of SU(2) symmetry associated with angular momentum conservation.Results: Proof-of-principle calculations in an Nmax=6{N}_{\text{max}}=6Nmax​=6 spherical harmonic oscillator basis for 16,18O^{16,18}\mathrm{O}16,18O and 18Ne^{18}\mathrm{Ne}18Ne in the BCCD approximation are in good agreement with standard coupled cluster results with the same chiral two-nucleon interaction, while 20O^{20}\mathrm{O}20O and 20Mg^{20}\mathrm{Mg}20Mg display underbinding relative to experiment. The breaking of U(1) symmetry, monitored by computing the variance associated with the particle-number operator, is relatively constant for all five nuclei, in both the Hartree-Fock-Bogoliubov and BCCD approximations.Conclusions: The newly developed many-body formalism increases the potential span of ab initio calculations based on single-reference coupled cluster techniques tremendously, i.e., potentially to reach several hundred additional midmass nuclei. The new formalism offers a wealth of potential applications and further extensions dedicated to the description of ground and excited states of open-shell nuclei. Short-term goals include the implementation of three-nucleon forces at the normal-ordered two-body level. Midterm extensions include the approximate treatment of triples corrections and the development of the equation-of-motion methodology to treat both excited states and odd nuclei. Long-term extensions include exact restoration of U(1) and SU(2) symmetries.
The traditional state universal multi-reference coupled cluster (SUMRCC) theory uses the Jeziorski-Monkhorst (JM) based Ansatz of the wave operator: Ω = Σ(μ)Ω(μ)|φ(μ)><φ(μ)|, where Ω(μ) = exp(T(μ)) is the cluster representation of the component of Ω inducing virtual excitations from the model function φ(μ). In the first formulations, φ(μ)s were chosen to be single determinants and T(μ)s were defined in terms of spinorbitals. This leads to spin-contamination for the non-singlet cases. In this paper, we propose and implement an explicitly spin-free realization of the SUMRCC theory. This method uses spin-free unitary generators in defining the cluster operators, {T(μ)}, which even at singles-doubles truncation, generates non-commuting cluster operators. We propose the use of normal-ordered exponential parameterization for Ω:Σ(μ){exp(T(μ))}|φ(μ)><φ(μ)|, where {} denotes the normal ordering with respect to a common closed shell vacuum which makes the "direct term" of the SUMRCC equations terminate at the quartic power. We choose our model functions {φ(μ)} as unitary group adapted (UGA) Gel'fand states which is why we call our theory UGA-SUMRCC. In the spirit of the original SUMRCC, we choose exactly the right number of linearly independent cluster operators in {T(μ)} such that no redundancies in the virtual functions {χ(μ)(l)} are involved. Using example applications for electron detached/attached and h-p excited states relative to a closed shell ground state we discuss how to choose the most compact and non-redundant cluster operators. Although there exists a more elaborate spin-adapted JM-like ansatz of Datta and Mukherjee (known as combinatoric open-shell CC (COS-CC), its working equations are more complex. Results are compared with those from COS-CC, equation of motion coupled cluster methods, restricted open-shell Hartree-Fock coupled cluster, and full configuration interaction. We observe that our results are more accurate with respect to most other theories as a result of the use of the cluster expansion structure for our wave operator. Our results are comparable to those from the more involved COS-CC, indicating that our theory captures the most important aspects of physics with a considerably simpler scheme.
Along with the surge in interest in quantum computing, interest in the unitary coupled cluster (UCC) Ansatz has reemerged. Although extensively studied within electronic structure theory, the UCC Ansatz remains relatively unexplored for the problem of molecular vibrations. In this contribution, working equations for the unitary vibrational coupled cluster (UVCC) Ansatz are derived, implemented, and benchmarked. Accuracy and convergence of state-specific excitation energies toward the full vibrational configuration interaction (FVCI) limit are observed to be comparable to vibrational coupled cluster theory. In addition, the overlap of a truncated UVCC state with the FVCI state is shown to exhibit some interesting properties from the perspective of fault-tolerant quantum computing.