A full coupled-cluster expansion suitable for sparse algebraic operations is developed by expanding the commutators of the Baker-Campbell-Hausdorff series explicitly for cluster operators in binary representations. A full coupled-cluster reduction that is capable of providing very accurate solutions of the many-body Schrödinger equation is then initiated employing screenings to the projection manifold and commutator operations. The projection manifold is iteratively updated through the single commutators ⟨κ|[H[over ^],T[over ^]]|0⟩ comprised of the primary clusters T[over ^]_{λ} with a substantial contribution to the connectivity. The operation of the commutators is further reduced by introducing a correction, taking into account the so-called exclusion-principle-violating terms that provides a fast and near-variational convergence in many cases.
A full coupled-cluster expansion suitable for sparse algebraic operations is developed by expanding the commutators of the Baker-Campbell-Hausdorff series explicitly for cluster operators in binary representations. A full coupled-cluster reduction that is capable of providing very accurate solutions of the many-body Schrödinger equation is then initiated employing screenings to the projection manifold and commutator operations. The projection manifold is iteratively updated through the single commutators ⟨κ|[H[over ^],T[over ^]]|0⟩ comprised of the primary clusters T[over ^]_{λ} with a substantial contribution to the connectivity. The operation of the commutators is further reduced by introducing a correction, taking into account the so-called exclusion-principle-violating terms that provides a fast and near-variational convergence in many cases.
We introduce a Lagrangian implementation of the full coupled-cluster reduction [Xu et al., Phys. Rev. Lett. 121, 113001 (2018)], that is, a selected coupled-cluster (CC) based on an arbitrary-order full CC expansion using direct commutator expansions. In this method, the screening for the products of cluster amplitudes plays a central role to reduce the computational cost for the nonlinear commutator operations, while the convergence of the total energy in the standard energy expression is not rapid with tightening the threshold. The new implementation using Lagrangian is robust, containing error only quadratic to those of amplitudes, allowing a much larger screening threshold. We demonstrate the performance of the new implementation by investigating the calculations of N2 and C6H6. The accuracy and applicability are also demonstrated for the potential energy curve of H2O in comparison with conventional quantum chemical methods.
We propose a modified coupled cluster Monte Carlo algorithm that stochastically samples connected terms within the truncated Baker-Campbell-Hausdorff expansion of the similarity-transformed Hamiltonian by construction of coupled cluster diagrams on the fly. Our new approach-diagCCMC-allows propagation to be performed using only the connected components of the similarity-transformed Hamiltonian, greatly reducing the memory cost associated with the stochastic solution of the coupled cluster equations. We show that for perfectly local, noninteracting systems diagCCMC is able to represent the coupled cluster wavefunction with a memory cost that scales linearly with system size. The favorable memory cost is observed with the only assumption of fixed stochastic granularity and is valid for arbitrary levels of coupled cluster theory. Significant reduction in memory cost is also shown to smoothly appear with dissociation of a finite chain of helium atoms. This approach is also shown not to break down in the presence of strong correlation through the example of a stretched nitrogen molecule. Our novel methodology moves the theoretical basis of coupled cluster Monte Carlo closer to deterministic approaches.
Standard approaches to approximating solutions to the Schroedinger equation rely upon the assumption that there is a single dominant electronic configuration with a small number of low-lying excited states. As such, they are referred to as single reference approaches and are accurate only when the band gap is small in comparison with the inter-electronic repulsion of the valence electrons. Systems for which this is the case are weakly-correlated. If the inter-electronic repulsion larger than the band gap, then a single electronic reference is not an appropriate ansatz for the physical wavefunction. As the strength of the interaction is increased many Slater determinants become near-degenerate, each providing non-negligible contributions. Such systems are strongly-correlated. The goal of the thesis was to employ wavefunctions which are: non-trivial, tractable, size-consistent and extensive, and not given in an active space. We have outlined approaches based on wavefunction forms more suited to describe strong correlation. Built on the Lie algebra su(2), our first trial wavefunction was the Bethe ansatz solution to the reduced BCS Hamiltonian derived by Richardson. This wavefunction is expressed as a product of quasi-particles each parametrized in terms of quasi-momenta. Consistency among the quasi-momenta is ensured by a set of non-linear equations to be solved numerically. This approach is feasible in both variational and projected flavours. We next considered a wavefunction of the same form without requiring it to be an eigenvector of the reduced BCS Hamiltonian. A variational approach in this case is only feasible provided there are two equivalent descriptions in terms of particles and holes. In any case, a bivariational principle and a projected approach are both feasible. Also based on su(2), there is AP1roG (coupled cluster pair doubles). Open-shell singlets and triplets were constructed with the Lie algebras sp(N) and so(4N). The resulting formulae appear to be infeasible unless it is possible to efficiently compute generalizations of determinants to three-dimensional objects. Treating each spatial orbital individually, all four possible occupations close the Lie superalgebra gl(2|2).
Projected Hartree-Fock theory provides an accurate description of many kinds of strong correlations but does not properly describe weakly correlated systems. On the other hand, single-reference methods, such as configuration interaction or coupled cluster theory, can handle weakly correlated problems but cannot properly account for strong correlations. Ideally, we would like to combine these techniques in a symmetry-projected coupled cluster approach, but this is far from straightforward. In this work, we provide an alternative formulation to identify the so-called disentangled cluster operators, which arise when we combine these two methodological strands. Our formulation shows promising results for model systems and small molecules.
Abstract. Coupled cluster theory produced arguably the most widely used high-accuracy computational quantum chemistry methods. Despite the approach’s overall great computational success, its mathematical understanding is so far limited to results within the realm of functional analysis. The coupled cluster amplitudes, which are the targeted objects in coupled cluster theory, correspond to solutions to the coupled cluster equations, which is a system of polynomial equations of at most degree four. The high dimensionality of the electronic Schrödinger equation and the nonlinearity of the coupled cluster ansatz have so far stalled a formal analysis of this polynomial system. In this article, we present algebraic investigations that shed light on the coupled cluster equations and the root structure of this ansatz. This is of importance for the a posteriori evaluation of coupled cluster calculations. To that end, we investigate the root structure by means of Newton polytopes. We derive a general v-description, which is subsequently turned into an h-description for explicit examples. This perspective reveals an apparent connection between Pauli’s exclusion principle and the geometrical structure of the Newton polytopes. We also propose an alternative characterization of the coupled cluster equations projected onto singles and doubles as cubic polynomials on an algebraic variety with certain sparsity patterns. Moreover, we provide numerical simulations of two computationally tractable systems, namely, the two electrons in four spin-orbitals system and the three electrons in six spin-orbitals system. These simulations provide novel insight into the root structure of the coupled cluster solutions when the coupled cluster ansatz is truncated.
In this paper, we propose a novel many-body approach for determining the amplitudes of generalized operators in a projection-based formalism. To implicitly account for the effects of higher-order excitations, we begin with the well-established double-exponential coupled-cluster (CC) ansatz, parametrized by both one- and two-body excitation operators, complemented by a set of vacuum-annihilating two-body generalized operators with effective excitation rank of one. A systematic formalism is developed that effectively bypasses the constraints due to the vacuum-annihilation property of the generalized operators toward a set of closed-form residual equations for their optimization. Such a strategy requires the removal of the underlying redundancy in high-rank excited determinants, generated due to the presence of the generalized operators in the ansatz, by projecting them onto an internally contracted lower-dimensional manifold. This many-body formalism is integrated with the near-term projective quantum eigensolver (PQE) framework that leverages the conventional CC-like residual minimization to iteratively decouple the excited manifold from the reference. With the application of several molecular systems within PQE architecture, we demonstrate that the developed methodology enables us to achieve similar accuracy to the disentangled unitary coupled cluster with singles, doubles, and triples ansatz while utilizing an order of magnitude fewer quantum resources. Furthermore, when simulated under stochastic Gaussian noise or depolarizing hardware noise, our method shows significantly improved noise resilience compared to the other members of PQE family and the state-of-the-art variational quantum eigensolver.