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arXiv:1307.1803 · Physical Review C

Self-consistent random-phase approximation from a coupled-cluster wave function perspective

Mohsen Jemaï, Doru Sabin Delion, Peter Schuck

Published Oct 25, 2013• 24 citations
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Abstract

Self-consistent random-phase approximation is rederived in a consistent way with the help of the coupled cluster ground state wave function truncated at the two-body level. An exact killing operator for this wave function is introduced allowing for a detailed discussion of the approximation scheme. Several exactly solvable models are reanalyzed under this new perspective giving rise to a quantitative evaluation of the performances of this many-body method.

DOI
10.1103/physrevc.88.044004

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Self-consistent random-phase approximation is rederived in a consistent way with the help of the coupled cluster ground state wave function truncated at the two-body level. An exact killing operator for this wave function is introduced allowing for a detailed discussion of the approximation scheme. Several exactly solvable models are reanalyzed under this new perspective giving rise to a quantitative evaluation of the performances of this many-body method.
The correlated-basis-functions method of Feenberg and the coupled-cluster formalism of Coester and K\"ummel are joined to form a new ground-state many-body method combining the advantages of both older methods and avoiding their disadvantages. From the point of view of the correlated-basis-functions method, coupled-cluster theory is used to sum the perturbation series partially to arbitrary order. From the point of view of the coupled-cluster method, correlated basis functions are used to take out the repulsive core of the two-body interaction in order to allow more efficient truncation schemes. It is found that powerful renormalizations are possible. Explicit equations are given for the two-body subsystems embodying generalized Bethe-Goldstone and random-phase equations summing, in the correlated basis, ladder and ring diagrams to arbitrary order.
The random phase approximation (RPA), the renormalized RPA (RRPA), and the self-consistent RPA (SCRPA) methods are applied to calculate the wave functions of the ground and excited states of an exactly solvable model. The approximated wave functions are expanded in the basis of the exact solutions. It is found that, when the RPA collapses, the RPA wave functions are orthogonal to the exact solutions while the RRPA and SCRPA ones have small but finite overlaps with the exact results. In spite of the apparently good agreement between the results of the RRPA, the SCRPA, and the exact solution, for the energy of the first excited state beyond the point of collapse, it is found that these approximations do not correctly describe the exact wave functions.
Coupled equations for even and odd particle number correlation functions are set up via the equation of motion method. For the even particle number case this leads to self-consistent random-phase approximation equations already known from the literature. From the equations of the odd particle number case the single-particle occupation probabilities are obtained in a self-consistent way. This is the essential new procedure of this work. Both even and odd particle number cases are based on the same correlated vacuum and, thus, are coupled equations. Applications to the Lipkin model and to the one-dimensional Hubbard model give very good results.
The workhorse method of computational materials science is undeniably density functional theory in the Kohn-Sham framework of approximate exchange and correlation energy functionals. However, the need for highly accurate predictions of ground and excited state properties in materials science motivates the further development and exploration of alternative as well as complementary techniques. Among these alternative approaches, quantum chemical wavefunction based theories and in particular coupled cluster theory hold the promise to fill a gap in the toolbox of computational materials scientists. Coupled cluster (CC) theory provides a compelling framework of approximate infinite-order perturbation theory in the form of an exponential of cluster operators describing the true quantum many-body effects of the electronic wave function at a computational cost that, despite being significantly more expensive than DFT, scales polynomially with system size. The hierarchy of size-extensive approximate methods established in the framework of CC theory achieves systematic improvability for many materials properties. This is in contrast to currently available density functionals that often suffer from uncontrolled approximations that limit the accuracy in the prediction of materials properties. In this tutorial-style review we will introduce basic concepts of coupled cluster theory and recent developments that increase its computational efficiency for calculations of molecules, solids and materials in general. We will touch upon the connection between coupled cluster theory and the random-phase approximation to bridge the gap between traditional quantum chemistry and many-body Green’s function theories that are widely-used in the field of solid state physics. We will discuss various approaches to improve the computational performance without compromising accuracy. These approaches include large-scale parallel design as well as techniques that reduce the prefactor of the computational complexity. A central part of this article discusses the convergence of calculated properties to the thermodynamic limit which is of significant importance for reliable predictions of materials properties and constitutes an additional challenge compared to calculations of large molecules. Furthermore we mention technical aspects of computer code implementations of periodic coupled cluster theories in different numerical frameworks of the one-electron orbital basis; the projector-augmented-wave formalism using a plane wave basis set and the numeric atom-centered-orbital with resolution-of-identity.
A coupled-cluster approach for systems of NNN bosons in external traps is developed. In the coupled-cluster approach the exact many-body wave function is obtained by applying an exponential operator expT\mathrm{exp}{T}expT to the ground configuration $\ensuremath{\mid}{\ensuremath{\phi}}_{0}⟩$. The natural ground configuration for bosons is, of course, when all reside in a single orbital. Because of this simple structure of $\ensuremath{\mid}{\ensuremath{\phi}}_{0}⟩$, the appearance of excitation operators $T={\ensuremath{\sum}}_{n=1}^{N}{T}_{n}$ for bosons is much simpler than for fermions. We can treat very large numbers of bosons with coupled-cluster expansions. In a substantial part of this work, we address the issue of size consistency for bosons and enquire whether truncated coupled-cluster expansions are size consistent. We show that, in contrast to the familiar situation for fermions for which coupled-cluster expansions are size consistent, for bosons the answer to this question depends on the choice of ground configuration. Utilizing the natural ground configuration, working equations for the truncated coupled-cluster with T=T1+T2T={T}_{1}+{T}_{2}T=T1​+T2​, i.e., coupled-cluster singles doubles are explicitly derived. Finally, an illustrative numerical example for a condensate with up to N=10000N=10\phantom{\rule{0.2em}{0ex}}000N=10000 bosons in an harmonic trap is provided and analyzed. The results are highly promising.