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International Journal of Modern Physics B

A BIOGRAPHY OF THE COUPLED CLUSTER METHOD

Hermann G. Kümmel

Published Nov 10, 2003• 58 citations
Publisher

Abstract

The origins of the coupled cluster method are described. Special attention is paid to the arguments put forward for the exponential structure of the wave functions. Various approximation schemes invented during the last 40 years are presented. The problems arising from these approximations necessarily truncating or destroying the exponential form are discussed and ways to deal with them are described.

DOI
10.1142/s0217979203020442

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A BIOGRAPHY OF THE COUPLED CLUSTER METHOD

Hermann G. Kümmel

Recent Progress in Many-Body Theories•Dec 1, 2002•17 citations
The origins of the coupled cluster method are described. Special attention is paid to the arguments put forward for the exponential structure of the wave functions. Various approximation schemes invented during the last 40 years are presented. The problems arising from these approximations necessarily truncating or destroying the exponential form are discussed and ways to deal with them are described.
The nonperturbative Hamiltonian eigenvalue problem for bound states of a quantum field theory is formulated in terms of Dirac's light-front coordinates and then approximated by the exponential operator technique of the many-body coupled-cluster method. This approximation eliminates any need for the usual approximation of Fock-space truncation. Instead, the exponentiated operator is truncated, and the terms retained are determined by a set of nonlinear integral equations. These equations are solved simultaneously with an effective eigenvalue problem in the valence sector, where the number of constituents is small. Matrix elements can be calculated, with extensions of techniques from many-body coupled-cluster theory, to obtain form factors and other observables.
The nonperturbative Hamiltonian eigenvalue problem for bound states of a quantum field theory is formulated in terms of Dirac's light-front coordinates and then approximated by the exponential-operator technique of the many-body coupled-cluster method. This approximation eliminates any need for the usual approximation of Fock-space truncation. Instead, the exponentiated operator is truncated, and the terms retained are determined by a set of nonlinear integral equations. These equations are solved simultaneously with an effective eigenvalue problem in the valence sector, where the number of constituents is small. Matrix elements can be calculated, with extensions of techniques from standard coupled-cluster theory, to obtain form factors and other observables.
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.