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Physical Review A

Variational and coupled-cluster calculations of the spectra of anharmonic oscillators

R. F. Bishop, M. F. Flynn

Published Sep 1, 1988• 53 citations

Abstract

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.

DOI: 10.1103/physreva.38.2211

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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) of quantum many-body theory has been widely and very successfully applied to a broad spectrum of condensed-matter systems. At a given level of approximation in a hierarchy of truncations which are necessary to implement the method in practice, the many-body Schr\"odinger equation is decomposed into a finite set of coupled nonlinear equations for the various amplitudes that otherwise exactly describe the correlated clusters or subsystems within the interacting many-body medium. The properties of the multiple solutions to these equations and their implications for the method itself are the primary concern here. We perform a detailed investigation for the simple but illustrative one-body problem of a quartic anharmonic oscillator, which provides a very stringent test of the convergence properties of any method rooted in perturbation theory. The problem is treated as a model field theory in 0+1 dimensions in order to illuminate general properties of the CCM. The various supercoherent states generated by the CCM at other than the lowest levels of truncation are unnormalizable but yield finite and well-defined estimates for quantities of physical interest. Their possible use as multiphoton generalizations of the two-photon squeezed coherent states is pointed out.

Both single-well and multiwell one-dimensional anharmonic oscillators have provided an extremely fruitful testing ground for various modern techniques of microscopic quantum many-body theory and quantum field theory. They have served both as (0+1)-dimensional field-theory models of their important counterparts in a higher number of dimensions, and as algebraically simple yet highly nontrivial problems on which various approximation or truncation schemes could be tested to very high orders. Such techniques have included perturbation theory and its many possible resummation schemes, and the coupled-cluster method. In the present paper the spotlight is focused on variational methods, and we concentrate on their application to the particularly demanding double-well quartic anharmonic oscillator. We examine in detail both single-state and multistate variational calculations based on either orthogonal (uncorrelated) or nonorthogonal (correlated) basis functions. Particular attention is paid to the question of the sensitivity and accuracy of the results with respect to any nonlinear parameters that characterize the basis states. Practical schemes are suggested for their choice and are shown to be capable of very high accuracy. Use of this simple model also allows detailed comparisons to be made with other techniques of a perturbative or perturbative-variational hybrid character.

The widely used coupled cluster method (CCM) in quantum many-body theory has recently provided very accurate descriptions of a large number of extended systems. Although its earlier applications to closed-shell and neighboring finite nuclei were also very successful, they have been shrouded in algebraic and technical complexity. Furthermore, they are difficult to compare with more traditional calculations of generalized shell-model theory since, at least at the important level of two-body correlations, they have been largely implemented in relative-coordinate space rather than the more usual oscillator configuration space. The CCM is reviewed here in the precise context of applications to simple finite systems. Special attention is paid to formulate it in such a way that comparison may be made with generalized shell-model or configuration-interaction (CI) theories. Particular regard is paid to an exact incorporation of translational invariance, so that any spuriosity associated with the center-of-mass motion is always avoided. An important side benefit is that the number of many-body configurations in the usual oscillator basis is dramatically reduced. We are thereby able to present both CI and CCM calculations on

^{4}\mathrm{He}

up to the essentially unprecedented level of 60\ensuremath{\Elzxh}\ensuremath{\omega} in oscillator excitation energy, for two popular and quasirealistic choices of the nucleon-nucleon interaction for which exact Monte Carlo results are available for this nucleus. Although even our simplest approximations attain about 95% of the total binding energy, the convergence in the oscillator configuration space is shown to be both very slow and of a complicated nonuniform nature. Strong implications are drawn for standard implementations of generalized shell-model techniques for heavier nuclei.

We present a coupled cluster method (CCM) with optimized excitation operators. The efficiency comes from a parameterized form of excitation operators. The parameters are found by variational optimization procedure. The resultant number of excitation operators is much smaller than that of the conventional CCM theory. This property makes it possible to apply the method in systems of solid state physics. Starting from Hartree-Fock state as the reference state, i.e., the Fermi sea, we search for particle-hole excitation operators such that the wave function of configuration interaction in terms of these excitation operators spans a good approximation to the ground state. The Match-pursuit algorithm is capable of doing the search of the excitation operators. The resultant operators are our excitation operators for the CCM wave function. We test the method by two dimensional fermionic Hubbard model on a square lattice.

Recent applications of the coupled-cluster method to the (${\ensuremath{\varphi}}^{4}$

{)}_{2}

quantum field theory showed the necessity of a detailed analysis of its convergence behavior. The anharmonic oscillator has always served as a good test case for different approximation schemes. We treat the model both in the maximum-overlap condition and in the Hartree approximation. The ground-state energy is reproduced very well for all values of the coupling strength and already for a low-order truncation scheme. The expansion in correlation amplitudes can be carried out in extremely high order and shows a divergent tendency of the amplitudes. Introducing temperature dependence allows us to select the stable ground state out of a variety of solutions of the hierarchy of equations.

In this work, an unusual approach is taken to calculate the ground state and first excited state energies of the anharmonic oscillator potential. By using the variational method, analytical wave functions which have extremely accurate expectation values for the quartic or sextic oscillators are obtained. The corrections due to the other terms are then calculated perturbatively. It is shown that each approach, which starts with a different choice of zeroth‐order term, provides results that agree with those computed from numerical integration and series solution. Joining the above schemes, yields a combined variational and perturbative treatment of anharmonic oscillator problems.