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International Journal of Quantum Chemistry

Coupled cluster response theory in parameter subspaces

Linus Boman, Henrik Koch A

Published Oct 16, 2008• 1 citation
Publisher

Abstract

Abstract We introduce a response function formalism that enables smaller number of parameters than that defined in standard coupled cluster response theory. This is essential in the development of reduced scaling methods. The formalism is general and it applies to all parameterizations at all levels of the coupled cluster hierarchy. We show that to achieve physically reasonable results the parameterization must fulfill certain criteria. The linear response functions are derived and discussed in the context of optimized virtual orbitals and Cholesky decomposition of the cluster amplitudes. © 2008 Wiley Periodicals, Inc. Int J Quantum Chem, 2009

DOI
10.1002/qua.21898

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Abstract We introduce a response function formalism that enables smaller number of parameters than that defined in standard coupled cluster response theory. This is essential in the development of reduced scaling methods. The formalism is general and it applies to all parameterizations at all levels of the coupled cluster hierarchy. We show that to achieve physically reasonable results the parameterization must fulfill certain criteria. The linear response functions are derived and discussed in the context of optimized virtual orbitals and Cholesky decomposition of the cluster amplitudes. © 2008 Wiley Periodicals, Inc. Int J Quantum Chem, 2009
Abstract The advantages and possibilities of a unitary coupled‐cluster (CC) theory are examined. It is shown that using a unitary parameterization of the wave function guarantees agreement between a sum‐over‐states polarization propagator and response theory calculation of properties of arbitrary order, as opposed to the case in conventional CC theory. Then, using the Zassenhaus expansion for noncommuting exponential operators, explicit diagrams for an extensive and variational method based on unitary CC theory are derived. Possible extensions to the approximations developed are discussed as well. © 2006 Wiley Periodicals, Inc. Int J Quantum Chem, 2006
We introduce a time-dependent coupled cluster based Lagrangian that includes orbital rotation. This Lagrangian is shown to give gauge invariant response properties for one-electron operators in the limit of a complete one-electron basis. The pole structure of the linear response function is compatible with that of the exact response function and the notorious problem of unphysical second-order poles in the Brueckner coupled cluster response theory is not present in this model. The total energy of the model is identical to the coupled cluster model using optimized orbitals recently revived by Sherrill et al. [J. Chem. Phys. 109, 4171 (1998)]. The model provides a straightforward approach for calculating magnetic response properties in a gauge invariant manner using coupled cluster type wave functions.
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.
The Coupled-Cluster (CC) method is one of the most popular and efficient correlation methods in quantum chemistry. \nEspecially the CCSD(T) approximation, which includes single and double excitations by means of the application of the cluster operator \nto a reference determinant and triple excitations via a perturbative treatment, has become a standard tool \nin quantum chemical applications. However, the method is restricted to relatively small system sizes due to its unfavourable scaling (N^7, where N is the number of basis functions applied). \nFor the full treatment of triple excitations the scaling advances to N^8 and every further excitation level increases the exponent by two. \n \nThe goal of this work is to reduce the calculation time for closed shell systems at least by a factor growing with the excitation level for arbitrary truncation levels. \nThis is done by restricting the spatial parts of the spin orbitals and thus treat pairs of spin orbitals on the same footing. The restrictions can be easily constructed \nfor the CCSD model by spin integration. \nThe derivation of the restrictions arising in higher excited case will be done employing \nthe spatial orbital excitation operators E. \n \nIn the first part of this work an algorithm is derived that is capable of the derivation of the energy and amplitude equations for arbitrary excitation levels. In the second part \nan implementation of this algorithm is presented.
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.
The coupled cluster method (CCM) is one of the most powerful and most successful fully microscopic, ab initio formulations available for quantum N-body theory, with N finite or infinite. It has probably been applied to more systems in quantum chemistry, nuclear, condensed matter and other areas of physics, and quantum field theory than any other competing method. In nearly all such cases the numerical results are either the best or among the best available. The CCM can deal with ground- and excited-state energies of closed- and open-shell systems, density matrices and hence other properties, sum rules, and the sub-sum-rules that follow from imbedding linear response theory within it. Extensions exist to deal with systems at nonzero temperature and out of equilibrium. At the formal level it provides an exact mapping of the quantum-mechanical problem onto a classical Hamiltonian phase space where the multiconfigurational canonical classical coordinates have specific cluster and locality properties. In this way it can provide exact hierarchical generalizations of mean-field theory and the random phase approximation. We discuss here both the formalism itself and a selection of its applications.