Using the general anharmonic oscillator as a case study, we examine the coupled-cluster method (CCM) in some detail. Emphasis is specially placed on the accuracy of the standard ground-state energy calculation and the excitation spectrum as derived via the Emrich ansatz. We are particularly interested in problems that can arise from large differences between the exact wave function and its model counterpart that forms the starting point of the CCM. To this end we begin with a variational Hartree approximation applied to three particular anharmonic-oscillator Hamiltonians that contain, respectively, pure quartic, equally weighted cubic and quartic, and double-well perturbations. These are chosen to provide increasingly stringent tests for the CCM in the above sense. Considerable attention is paid to the variational description of the double-well case, where the various possible solutions to the Hartree equations are considered and a further calculation on the energy-level splitting is performed. The CCM is then used to improve systematically upon our chosen starting point and is shown, particularly for the ground state, both to produce extremely accurate results and to be rather resilient to even gross deficiencies in the starting wave function. The main problem is the double-well system, where the CCM shows its lack of intrinsic inbuilt tunneling mechanisms via the absence of level splitting. Even here the CCM still produces very accurate average energies for very deep wells. While clearly needing some modification in such extreme cases, the CCM is shown to be quite adaptable and robust with regard to inaccurate starting wave functions.
The coupled-cluster method (CCM) is applied in obtaining the ground-state energy and the local moments of the localized orbitals of the periodic Anderson model. Starting with the state containing a filled Fermi sea of the conduction electrons plus a set of singly occupied and antiferromagnetically ordered f orbitals, the first-order hybridization from the f orbitals to the conduction band is included in the CCM transformation operator. A simple expression for the ground-state energy has been derived, and the calculated ground-state properties are compared with those obtained using a variational scheme and the Monte Carlo method.
The coupled cluster method (CCM) is a powerful and widely applied technique of modern-day quantum many-body theory. It has been used with great success in order to understand the properties of quantum magnets at zero temperature. This is due largely to the application of computational techniques that allow the method to be applied to high orders of approximation using localised approximation schemes, e.g., such as the LSUB
m scheme. In this article, the high-order CCM formalism for the ground and excited states of quantum magnetic systems are extended to those with spin quantum number
s≥21. Solution strategies for the ket- and bra-state equations are also considered. Aspects of extrapolation of CCM expectation values are discussed and future topics regarding extrapolations are presented.
We consider a fermionic system for which there exist a single-reference configuration-interaction (CI) expansion of the ground-state wave function that converges, albeit not necessarily rapidly, with the increasing number of particle-hole excitations. For such systems, we show that, whenever the coefficients of Slater determinants (SD) with $l\ensuremath{\le}k$ excitations can be defined with a number of free parameters ${N}_{\ensuremath{\le}k}$ bounded polynomially in
k, the ground-state energy
E only depends on a small fraction of all the wave function parameters and is the solution of equations of the coupled-cluster (CC) form. This generalizes the standard CC method, for which ${N}_{\ensuremath{\le}k}$ is bounded by a constant. Based on that result and low-rank tensor decompositions (LRTD), we discuss two possible extensions of the CC approach for wave functions with general polynomial bound for ${N}_{\ensuremath{\le}k}$. The most straightforward of those extensions uses the LRTD to represent the amplitudes of the CC cluster operator
T which, unlike in the CC case, is not truncated with respect to number of excitations, and the energy and tensor parameters are given by a LRTD-adapted version of standard CC equations. The LRTD can also be used to directly parametrize the wave function coefficients, which involves different equations of the CC form. We derive those equations for the coefficients of SD's with $l\ensuremath{\le}4$ excitations, using the CC exponential wave function ansatz with a different type of excitation operator, and a representation of the Hamiltonian in terms of excited particle and hole operators. To complete the proposed computational methods, we construct compact tensor representations of coefficients, or
T amplitudes, using superpositions of tree tensor networks which take into account different possible types of entanglement between excited particles and holes. Finally, we discuss why the proposed CC extensions are theoretically applicable at larger coupling strengths than those treatable by the standard CC method.
The coupled cluster method (CCM) is a method of quantum many-body theory that may provide accurate results for the ground-state properties of lattice quantum spin systems even in the presence of strong frustration and for lattices of arbitrary spatial dimensionality. Here we present a significant extension of the method by introducing an approach that allows an efficient parallelization of computer codes that carry out high-order CCM calculations. We find that we are able to extend such CCM calculations by an order of magnitude higher than ever before utilized in a high-order CCM calculation for an antiferromagnet. Furthermore we use only a relatively modest number of processors, namely, eight. Such very high-order CCM calculations are possible only by using such a parallelized approach. An illustration of the new approach is presented for the ground-state properties of a magnetic material,
CaV4O9.
We have studied electron correlations in the doped two-dimensional (2D) Hubbard model by using the coupled-cluster method (CCM) to investigate whether or not the method can be applied to correct the independent particle approximations actually used in ab initio band calculations. The double excitation version of the CCM, implemented using the approximate coupled pair (ACP) method, accounts for most of the correlation energies of the 2D Hubbard model in the weak $(U/t\ensuremath{\simeq}1)$ and the intermediate
U/t regions $(U/t\ensuremath{\simeq}4).$ The error in this case is always less than
1 The ACP approximation gets less accurate for large
U/t $(U/t\ensuremath{\simeq}8)$ and near half-filling. Further incorporation of electron correlation effects is necessary in this region. The accuracy depends neither on the system size, nor the gap between the lowest unoccupied level and the highest occupied level in the finite size effect. Hence, the CCM may be favorably applied to ab initio band calculations on metals as well as semiconductors and insulators.