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Quantum Mechanical Studies of Infinite Matter by the use of Coupled-Cluster Calculations, with an Emphasis on Nuclear Matter

Sean B. S. Miller

Published Jan 1, 2017• 0 citations
OpenAlex

Abstract

We perform a first principles investigation of full coupled-cluster doubles and triples (CCDT) calculations on infinite matter consisting of identical fermions using the Minnesota and chiral NNLOopt potentials with a closed shell model and periodic wave functions (Bloch wave functions). We do so with an emphasis on the system energy density as a function of particle number, state space size, and particle density at low to medium densities. Additionally, a CCDT study of the homogeneous electron gas (HEG) is performed. A CCDT programme has been developed with to cope with both memory and cpu consumption, and which can easily be extended to any infinite system, with or without identical fermions. A review of many-body theory and chiral perturbation theory, and also a brief introduction to nuclear forces, is given. We benchmark our data with previous CCD and CCDT-1 results for the HEG, and CCD data for the Minnesota potential. We comment on finite-size effects and wave function boundary conditions with respect to our results, and suggest possible further studies to improve upon our findings.

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Igor Ying Zhang, Andreas Grüneis

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.

arXiv:2004.06424•Frontiers in Materials•Jun 5, 2019•138 citations

Generalized Reaction Matrix Approach to the Theory of the Infinite Medium of Fermions

R. E. Prange, Abraham Klein

A closed set of coupled, nonlinear integral equations is obtained which completely describe a system of Fermi particles. The equations define the one and two-particle Green's functions which are completely analogous to those used in the field theory of fundamental particles. Special boundary conditions appropriate to the infinite medium of fermions are employed. Approximations to these equations are made which lead to a generalized Brueckner reaction-matrix formalism in the case of nuclear matter and which lead to an expression for the correlation energy in the case of the electron gas. The approximations for the system of nuclear matter are of three types: Ones which neglect the simultaneous interaction of three or more nucleons, the effective interactions of the vacant states below the Fermi sea, and the self-energy interactions with excited nucleons. The relationship between the reaction matrix and the expectation value of the actual potential energy is discussed. It is found that the energy levels given by the Brueckner formalism are nearly those given by a consideration of the actual energy, thus indicating that the shell-model energy levels for closed-shell-plus-one nuclei should be almost correct. A formula for the momentum distribution of nucleons in nuclear matter is obtained.

Physical Review•Nov 1, 1958•25 citations

Coupled-cluster singles and doubles for extended systems

So Hirata, Rafał Podeszwa, Motoi Tobita +1 more

Coupled-cluster theory with connected single and double excitation operators (CCSD) and related approximations, such as linearized CCSD, quadratic configuration interaction with single and double excitation operators, coupled-cluster with connected double excitation operator (CCD), linearized CCD, approximate CCD, and second- and third-order many-body perturbation theories, are formulated and implemented for infinitely extended one-dimensional systems (polymers), on the basis of the periodic boundary conditions and distance-based screening of integrals, density matrix elements, and excitation amplitudes. The variation of correlation energies with the truncation radii of short- and long-range lattice sums and with the number of wave vector sampling points in the first Brillouin zone is examined for polyethylene, polyacetylene, and polyyne, and is shown to be a function of the degree of pi-electron conjugation or the fundamental band gaps. The t2 and t1 amplitudes in the atomic orbital (AO) basis are obtained by first computing the t amplitudes in the Bloch-orbital basis and subsequently back-transforming them into the AO basis. The plot of these AO-based t amplitudes as a function of unit cells also indicates that the t2 amplitudes of polyacetylene and polyyne exhibit appreciably slower decay than those of polyethylene, although the asymptotic decay behavior is invariably 1/r3. The AO-based t1 amplitudes appear to correlate strongly with the electronic structure, and they decay seemingly exponentially for polyethylene whereas they stay at a constant magnitude across the seventh nearest neighbors of polyacetylene and polyyne, which attests to far reaching effects of nondynamical electron correlation mediated by orbital rotation. Nonetheless, the unit cell contributions to the correlation energies taper below 10(-6) hartree after 15 A for all three polymers. The basis set dependence of the decay behavior of t2 amplitudes is also examined for linear hydrogen fluoride polymer (HF)infinity and linear beryllium polymer (Be)infinity employing the STO-3G, 6-31G, and 6-31G* basis sets, and proves to be rather small.

The Journal of Chemical Physics•Feb 4, 2004•162 citations

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Alessandro Roggero, Abhishek Mukherjee, Francesco Pederiva

We introduce a many-body method that combines two powerful many-body techniques, viz., quantum Monte Carlo and coupled cluster theory. Coupled cluster wave functions are introduced as importance functions in a Monte Carlo method designed for the configuration interaction framework to provide rigorous upper bounds to the ground-state energy. We benchmark our method on the homogeneous electron gas in momentum space. The importance function used is the coupled cluster doubles wave function. We show that the computational resources required in our method scale polynomially with system size. Our energy upper bounds are in very good agreement with previous calculations of similar accuracy, and they can be systematically improved by including higher order excitations in the coupled cluster wave function.

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From Interacting Nucleons to Hamiltonian Lattice Gauge Theories: A Unified Many-Body Treatment

R. F. Bishop

What has since become known as the coupled cluster method (CCM) was invented some 35 years ago to calculate ground-state energies of closed-shell atomic nuclei.The method has since been extended in many different ways to the point where it is nowadays recognized as providing one of the most universally applicable, most powerful, and numerically most accurate of all ab initio methods in the microscopic quantum many-body theory of finite or infinite systems.We review here the applications within nuclear physics, and thereby show how the CCM provides a unifying approach of almost unique stature.The early calculations on both closed-and open-shell nuclei and on infinite nuclear matter are thus related to the recent extensions which provide both extremely accurate calculations for light nuclei and a framework for handling various problems in nuclear field theory.The latter include pion-nucleon field theory and such Hamiltonian lattice gauge field theories as the Abelian U(l) model and the non-Abelian SU(2) model.In all cases we stress the underlying concepts and the high quality of the results obtained.

•Jan 1, 1994•0 citations

Ab initioBogoliubov coupled cluster theory for open-shell nuclei

A. Signoracci, T. Duguet, G. Hagen +1 more

Background: Ab initio many-body methods have been developed over the past 10 yr to address closed-shell nuclei up to mass $A\ensuremath{\approx}130$ on the basis of realistic two- and three-nucleon interactions. A current frontier relates to the extension of those many-body methods to the description of open-shell nuclei. Several routes to address open-shell nuclei are currently under investigation, including ideas that exploit spontaneous symmetry breaking.Purpose: Singly open-shell nuclei can be efficiently described via the sole breaking of U(1) gauge symmetry associated with particle-number conservation as a way to account for their superfluid character. While this route was recently followed within the framework of self-consistent Green's function theory, the goal of the present work is to formulate a similar extension within the framework of coupled cluster theory.Methods: We formulate and apply Bogoliubov coupled cluster (BCC) theory, which consists of representing the exact ground-state wave function of the system as the exponential of a quasiparticle excitation cluster operator acting on a Bogoliubov reference state. Equations for the ground-state energy and the cluster amplitudes are derived at the singles and doubles level (BCCSD) both algebraically and diagrammatically. The formalism includes three-nucleon forces at the normal-ordered two-body level. The first BCC code is implemented in $m$ scheme, which will permit the treatment of doubly open-shell nuclei via the further breaking of SU(2) symmetry associated with angular momentum conservation.Results: Proof-of-principle calculations in an ${N}_{\text{max}}=6$ spherical harmonic oscillator basis for $^{16,18}\mathrm{O}$ and $^{18}\mathrm{Ne}$ in the BCCD approximation are in good agreement with standard coupled cluster results with the same chiral two-nucleon interaction, while $^{20}\mathrm{O}$ and $^{20}\mathrm{Mg}$ display underbinding relative to experiment. The breaking of U(1) symmetry, monitored by computing the variance associated with the particle-number operator, is relatively constant for all five nuclei, in both the Hartree-Fock-Bogoliubov and BCCD approximations.Conclusions: The newly developed many-body formalism increases the potential span of ab initio calculations based on single-reference coupled cluster techniques tremendously, i.e., potentially to reach several hundred additional midmass nuclei. The new formalism offers a wealth of potential applications and further extensions dedicated to the description of ground and excited states of open-shell nuclei. Short-term goals include the implementation of three-nucleon forces at the normal-ordered two-body level. Midterm extensions include the approximate treatment of triples corrections and the development of the equation-of-motion methodology to treat both excited states and odd nuclei. Long-term extensions include exact restoration of U(1) and SU(2) symmetries.

arXiv:1412.2696•Physical Review C•Jun 29, 2015•96 citations
We perform a first principles investigation of full coupled-cluster doubles and triples (CCDT) calculations on infinite matter consisting of identical fermions using the Minnesota and chiral NNLOopt potentials with a closed shell model and periodic wave functions (Bloch wave functions). We do so with an emphasis on the system energy density as a function of particle number, state space size, and particle density at low to medium densities. Additionally, a CCDT study of the homogeneous electron gas (HEG) is performed. A CCDT programme has been developed with to cope with both memory and cpu consumption, and which can easily be extended to any infinite system, with or without identical fermions. A review of many-body theory and chiral perturbation theory, and also a brief introduction to nuclear forces, is given. We benchmark our data with previous CCD and CCDT-1 results for the HEG, and CCD data for the Minnesota potential. We comment on finite-size effects and wave function boundary conditions with respect to our results, and suggest possible further studies to improve upon our findings.
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 closed set of coupled, nonlinear integral equations is obtained which completely describe a system of Fermi particles. The equations define the one and two-particle Green's functions which are completely analogous to those used in the field theory of fundamental particles. Special boundary conditions appropriate to the infinite medium of fermions are employed. Approximations to these equations are made which lead to a generalized Brueckner reaction-matrix formalism in the case of nuclear matter and which lead to an expression for the correlation energy in the case of the electron gas. The approximations for the system of nuclear matter are of three types: Ones which neglect the simultaneous interaction of three or more nucleons, the effective interactions of the vacant states below the Fermi sea, and the self-energy interactions with excited nucleons. The relationship between the reaction matrix and the expectation value of the actual potential energy is discussed. It is found that the energy levels given by the Brueckner formalism are nearly those given by a consideration of the actual energy, thus indicating that the shell-model energy levels for closed-shell-plus-one nuclei should be almost correct. A formula for the momentum distribution of nucleons in nuclear matter is obtained.
Coupled-cluster theory with connected single and double excitation operators (CCSD) and related approximations, such as linearized CCSD, quadratic configuration interaction with single and double excitation operators, coupled-cluster with connected double excitation operator (CCD), linearized CCD, approximate CCD, and second- and third-order many-body perturbation theories, are formulated and implemented for infinitely extended one-dimensional systems (polymers), on the basis of the periodic boundary conditions and distance-based screening of integrals, density matrix elements, and excitation amplitudes. The variation of correlation energies with the truncation radii of short- and long-range lattice sums and with the number of wave vector sampling points in the first Brillouin zone is examined for polyethylene, polyacetylene, and polyyne, and is shown to be a function of the degree of pi-electron conjugation or the fundamental band gaps. The t2 and t1 amplitudes in the atomic orbital (AO) basis are obtained by first computing the t amplitudes in the Bloch-orbital basis and subsequently back-transforming them into the AO basis. The plot of these AO-based t amplitudes as a function of unit cells also indicates that the t2 amplitudes of polyacetylene and polyyne exhibit appreciably slower decay than those of polyethylene, although the asymptotic decay behavior is invariably 1/r3. The AO-based t1 amplitudes appear to correlate strongly with the electronic structure, and they decay seemingly exponentially for polyethylene whereas they stay at a constant magnitude across the seventh nearest neighbors of polyacetylene and polyyne, which attests to far reaching effects of nondynamical electron correlation mediated by orbital rotation. Nonetheless, the unit cell contributions to the correlation energies taper below 10(-6) hartree after 15 A for all three polymers. The basis set dependence of the decay behavior of t2 amplitudes is also examined for linear hydrogen fluoride polymer (HF)infinity and linear beryllium polymer (Be)infinity employing the STO-3G, 6-31G, and 6-31G* basis sets, and proves to be rather small.
We introduce a many-body method that combines two powerful many-body techniques, viz., quantum Monte Carlo and coupled cluster theory. Coupled cluster wave functions are introduced as importance functions in a Monte Carlo method designed for the configuration interaction framework to provide rigorous upper bounds to the ground-state energy. We benchmark our method on the homogeneous electron gas in momentum space. The importance function used is the coupled cluster doubles wave function. We show that the computational resources required in our method scale polynomially with system size. Our energy upper bounds are in very good agreement with previous calculations of similar accuracy, and they can be systematically improved by including higher order excitations in the coupled cluster wave function.
What has since become known as the coupled cluster method (CCM) was invented some 35 years ago to calculate ground-state energies of closed-shell atomic nuclei.The method has since been extended in many different ways to the point where it is nowadays recognized as providing one of the most universally applicable, most powerful, and numerically most accurate of all ab initio methods in the microscopic quantum many-body theory of finite or infinite systems.We review here the applications within nuclear physics, and thereby show how the CCM provides a unifying approach of almost unique stature.The early calculations on both closed-and open-shell nuclei and on infinite nuclear matter are thus related to the recent extensions which provide both extremely accurate calculations for light nuclei and a framework for handling various problems in nuclear field theory.The latter include pion-nucleon field theory and such Hamiltonian lattice gauge field theories as the Abelian U(l) model and the non-Abelian SU(2) model.In all cases we stress the underlying concepts and the high quality of the results obtained.
Background: Ab initio many-body methods have been developed over the past 10 yr to address closed-shell nuclei up to mass $A\ensuremath{\approx}130$ on the basis of realistic two- and three-nucleon interactions. A current frontier relates to the extension of those many-body methods to the description of open-shell nuclei. Several routes to address open-shell nuclei are currently under investigation, including ideas that exploit spontaneous symmetry breaking.Purpose: Singly open-shell nuclei can be efficiently described via the sole breaking of U(1) gauge symmetry associated with particle-number conservation as a way to account for their superfluid character. While this route was recently followed within the framework of self-consistent Green's function theory, the goal of the present work is to formulate a similar extension within the framework of coupled cluster theory.Methods: We formulate and apply Bogoliubov coupled cluster (BCC) theory, which consists of representing the exact ground-state wave function of the system as the exponential of a quasiparticle excitation cluster operator acting on a Bogoliubov reference state. Equations for the ground-state energy and the cluster amplitudes are derived at the singles and doubles level (BCCSD) both algebraically and diagrammatically. The formalism includes three-nucleon forces at the normal-ordered two-body level. The first BCC code is implemented in mmm scheme, which will permit the treatment of doubly open-shell nuclei via the further breaking of SU(2) symmetry associated with angular momentum conservation.Results: Proof-of-principle calculations in an Nmax=6{N}_{\text{max}}=6Nmax​=6 spherical harmonic oscillator basis for 16,18O^{16,18}\mathrm{O}16,18O and 18Ne^{18}\mathrm{Ne}18Ne in the BCCD approximation are in good agreement with standard coupled cluster results with the same chiral two-nucleon interaction, while 20O^{20}\mathrm{O}20O and 20Mg^{20}\mathrm{Mg}20Mg display underbinding relative to experiment. The breaking of U(1) symmetry, monitored by computing the variance associated with the particle-number operator, is relatively constant for all five nuclei, in both the Hartree-Fock-Bogoliubov and BCCD approximations.Conclusions: The newly developed many-body formalism increases the potential span of ab initio calculations based on single-reference coupled cluster techniques tremendously, i.e., potentially to reach several hundred additional midmass nuclei. The new formalism offers a wealth of potential applications and further extensions dedicated to the description of ground and excited states of open-shell nuclei. Short-term goals include the implementation of three-nucleon forces at the normal-ordered two-body level. Midterm extensions include the approximate treatment of triples corrections and the development of the equation-of-motion methodology to treat both excited states and odd nuclei. Long-term extensions include exact restoration of U(1) and SU(2) symmetries.