Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

Journal of Chemical Theory and Computation

Finite Temperature Coupled Cluster Theories for Extended Systems

Felix Hummel

Published Oct 24, 2018• 29 citations
Publisher

Abstract

At zero temperature, coupled cluster theories are widely used to predict total energies, ground state expectation values, and even excited states for molecules and extended systems. However, for systems with a small band gap, such as metals, the zero-temperature approximation does not necessarily hold. Thermal effects may even give rise to interesting chemistry on metal surfaces. Most approaches to temperature dependent electronic properties employ finite temperature perturbation theory in the Matsubara frequency formulation. Computations require a large number of Matsubara frequencies to yield sufficiently accurate results, especially at low temperatures. This work, and independently the work of White and Chan J. Chem. Theory Comput. 2018 , DOI: 10.1021/acs.jctc.8b00773 , proposes a coupled cluster implementation directly in the imaginary time domain on the compact interval [0, β], closely related to the thermal cluster cumulant approach of Sanyal et al. [ Chem. Phys. Lett. 1992 , 192 , 55 - 61 ] , Sanyal et al. [ Phys. Rev. E 1993 , 48 , 3373 - 3389 ], and Mandal et al. [ Int. J. Mod. Phys. B 2003 , 17 , 5367 - 5377 ]. Here, the arising imaginary time dependent coupled cluster amplitude integral equations are solved in the linearized direct ring doubles approximation, also referred to as Tamm-Dancoff approximation with second order (linearized) screened exchange. In this framework, the transition from finite to zero temperature is uniform and comes at no extra costs, allowing to go to temperatures as low as room temperature. In this approximation, correlation grand potentials are calculated over a wide range of temperatures for solid lithium, a metallic system, and for solid silicon, a semiconductor.

DOI
10.1021/acs.jctc.8b00793

Related papers

Finite temperature coupled cluster theories for extended systems

Felix Hummel

At zero temperature coupled cluster theory is widely used to predict total energies, ground state expectation values and even excited states for molecules and extended systems. Generalizations to finite temperature exist, however, they are in practice several orders of magnitude more demanding compared to the zero temperature case since the amplitudes must be computed and stored for many Matsubara frequencies to yield sufficiently accurate results. Instead of using Matsubara frequencies one can also work directly in the imaginary time domain on the compact interval [0, beta]. The arising imaginary time dependent coupled cluster amplitude integral equations are solved numerically on a non-uniform grid. About a dozen grid points provide sufficient accuracy if the Hamiltonian is repartitioned to include particle/hole interactions in the non-perturbative part. In this framework the transition from finite to zero temperature is uniform and comes at no extra costs, allowing, for instance, to benchmark fractional occupancy formulations of coupled cluster theories. Finite temperature direct ring coupled cluster doubles free energies are calculated for various temperatures for solid lithium, a metallic system, and for solid silicon, a semiconductor.

arXiv:1807.09687•Jul 24, 2018•0 citations

A Time-Dependent Formulation of Coupled-Cluster Theory for Many-Fermion Systems at Finite Temperature

Alec F. White, Garnet Kin‐Lic Chan

We present a time-dependent formulation of coupled cluster theory. This theory allows for direct computation of the free energy of quantum systems at finite temperature by imaginary time integration and is closely related to the thermal cluster cumulant theory of Mukherjee and co-workers [ Chem. Phys. Lett. 1992 , 192 , 55 -61 ; Phys. Rev. E 1993 , 48 , 3373 -3389 ; Chem. Phys. Lett. 2001 , 335 , 281 -288 ; Chem. Phys. Lett. 2002 , 352 , 63 -69 ; Int. J. Mod. Phys. B 2003 , 17 , 5367 -5377 ]. Our derivation of the finite-temperature theory highlights connections to perturbation theory and to zero-temperature coupled cluster theory. We show explicitly how the finite-temperature coupled cluster singles and doubles amplitude equations can be derived in analogy with the zero-temperature theory and how response properties can be efficiently computed using a variational Lagrangian. We discuss the implementation for realistic systems and showcase the potential utility of the method with calculations of the exchange correlation energy of the uniform electron gas under warm dense matter conditions.

arXiv:1807.09961•Journal of Chemical Theory and Computation•Sep 27, 2018•66 citations

Coupled Cluster Theory in Materials Science

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

Thermofield Theory for Finite-Temperature Coupled Cluster

Gaurav Harsha, Thomas M. Henderson, Gustavo E. Scuseria

We present a coupled cluster and linear response theory to compute properties of many-electron systems at nonzero temperatures. For this purpose, we make use of the thermofield dynamics, which allows for a compact wave function representation of the thermal density matrix, and extend our recently developed framework ( J. Chem. Phys. 2019, 150, 154109, DOI: 10.1063/1.5089560 ) to parametrize the so-called thermal state using an exponential ansatz with cluster operators that create thermal quasiparticle excitations on a mean-field reference. As benchmark examples, we apply this method to both model (one-dimensional Hubbard and Pairing) and ab initio (atomic Beryllium and molecular Hydrogen) systems, while comparing with exact results.

arXiv:1907.11286•Journal of Chemical Theory and Computation•Sep 19, 2019•46 citations

On the Chemical Potential of Many-Body Perturbation Theory in Extended Systems

Felix Hummel

Finite-temperature many-body perturbation theory in the grand-canonical ensemble is fundamental to numerous methods for computing electronic properties at nonzero temperature, such as finite-temperature coupled-cluster. In most applications it is the average number of electrons that is known rather than the chemical potential. Expensive correlation calculations must be repeated iteratively in search for the interacting chemical potential that yields the desired average number of electrons. In extended systems with mobile charges the situation is particular, however. Long-ranged electrostatic forces drive the charges such that the average ratio of negative and positive charges is one for any finite chemical potential. All properties per electron are expected to be virtually independent of the chemical potential, as they are in an electric wire at different voltage potentials. This work shows that per electron, the exchange-correlation free energy and the exchange-correlation grand potential indeed agree in the infinite-size limit. Thus, only one expensive correlation calculation suffices for each system size, sparing the search for the interacting chemical potential. This work also demonstrates the importance of regularizing the Coulomb interaction such that each electron on average interacts only with as many electrons as there are electrons in the simulation, avoiding interactions with periodic images. Numerical calculations of the warm uniform electron gas have been conducted with the Spencer-Alavi regularization employing the finite-temperature Hartree approximation for the self-consistent field and linearized finite-temperature direct-ring coupled-cluster doubles for treating correlation.

arXiv:2210.05548•Journal of Chemical Theory and Computation•Feb 15, 2023•0 citations

Conservation laws in coupled cluster dynamics at finite temperature

Ruojing Peng, Alec F. White, Huanchen Zhai +1 more

We extend the finite-temperature Keldysh non-equilibrium coupled cluster theory (Keldysh-CC) [A. F. White and G. K.-L. Chan, J. Chem. Theory Comput. 15, 6137–6253 (2019)] to include a time-dependent orbital basis. When chosen to minimize the action, such a basis restores local and global conservation laws (Ehrenfest’s theorem) for all one-particle properties while remaining energy conserving for time-independent Hamiltonians. We present the time-dependent Keldysh orbital-optimized coupled cluster doubles method in analogy with the formalism for zero-temperature dynamics, extended to finite temperatures through the time-dependent action on the Keldysh contour. To demonstrate the conservation property and understand the numerical performance of the method, we apply it to several problems of non-equilibrium finite-temperature dynamics: a 1D Hubbard model with a time-dependent Peierls phase, laser driving of molecular H2, driven dynamics in warm-dense silicon, and transport in the single impurity Anderson model.

arXiv:2106.02691•The Journal of Chemical Physics•Jul 26, 2021•14 citations
At zero temperature, coupled cluster theories are widely used to predict total energies, ground state expectation values, and even excited states for molecules and extended systems. However, for systems with a small band gap, such as metals, the zero-temperature approximation does not necessarily hold. Thermal effects may even give rise to interesting chemistry on metal surfaces. Most approaches to temperature dependent electronic properties employ finite temperature perturbation theory in the Matsubara frequency formulation. Computations require a large number of Matsubara frequencies to yield sufficiently accurate results, especially at low temperatures. This work, and independently the work of White and Chan J. Chem. Theory Comput. 2018 , DOI: 10.1021/acs.jctc.8b00773 , proposes a coupled cluster implementation directly in the imaginary time domain on the compact interval [0, β], closely related to the thermal cluster cumulant approach of Sanyal et al. [ Chem. Phys. Lett. 1992 , 192 , 55 - 61 ] , Sanyal et al. [ Phys. Rev. E 1993 , 48 , 3373 - 3389 ], and Mandal et al. [ Int. J. Mod. Phys. B 2003 , 17 , 5367 - 5377 ]. Here, the arising imaginary time dependent coupled cluster amplitude integral equations are solved in the linearized direct ring doubles approximation, also referred to as Tamm-Dancoff approximation with second order (linearized) screened exchange. In this framework, the transition from finite to zero temperature is uniform and comes at no extra costs, allowing to go to temperatures as low as room temperature. In this approximation, correlation grand potentials are calculated over a wide range of temperatures for solid lithium, a metallic system, and for solid silicon, a semiconductor.
At zero temperature coupled cluster theory is widely used to predict total energies, ground state expectation values and even excited states for molecules and extended systems. Generalizations to finite temperature exist, however, they are in practice several orders of magnitude more demanding compared to the zero temperature case since the amplitudes must be computed and stored for many Matsubara frequencies to yield sufficiently accurate results. Instead of using Matsubara frequencies one can also work directly in the imaginary time domain on the compact interval [0, beta]. The arising imaginary time dependent coupled cluster amplitude integral equations are solved numerically on a non-uniform grid. About a dozen grid points provide sufficient accuracy if the Hamiltonian is repartitioned to include particle/hole interactions in the non-perturbative part. In this framework the transition from finite to zero temperature is uniform and comes at no extra costs, allowing, for instance, to benchmark fractional occupancy formulations of coupled cluster theories. Finite temperature direct ring coupled cluster doubles free energies are calculated for various temperatures for solid lithium, a metallic system, and for solid silicon, a semiconductor.
We present a time-dependent formulation of coupled cluster theory. This theory allows for direct computation of the free energy of quantum systems at finite temperature by imaginary time integration and is closely related to the thermal cluster cumulant theory of Mukherjee and co-workers [ Chem. Phys. Lett. 1992 , 192 , 55 -61 ; Phys. Rev. E 1993 , 48 , 3373 -3389 ; Chem. Phys. Lett. 2001 , 335 , 281 -288 ; Chem. Phys. Lett. 2002 , 352 , 63 -69 ; Int. J. Mod. Phys. B 2003 , 17 , 5367 -5377 ]. Our derivation of the finite-temperature theory highlights connections to perturbation theory and to zero-temperature coupled cluster theory. We show explicitly how the finite-temperature coupled cluster singles and doubles amplitude equations can be derived in analogy with the zero-temperature theory and how response properties can be efficiently computed using a variational Lagrangian. We discuss the implementation for realistic systems and showcase the potential utility of the method with calculations of the exchange correlation energy of the uniform electron gas under warm dense matter conditions.
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.
We present a coupled cluster and linear response theory to compute properties of many-electron systems at nonzero temperatures. For this purpose, we make use of the thermofield dynamics, which allows for a compact wave function representation of the thermal density matrix, and extend our recently developed framework ( J. Chem. Phys. 2019, 150, 154109, DOI: 10.1063/1.5089560 ) to parametrize the so-called thermal state using an exponential ansatz with cluster operators that create thermal quasiparticle excitations on a mean-field reference. As benchmark examples, we apply this method to both model (one-dimensional Hubbard and Pairing) and ab initio (atomic Beryllium and molecular Hydrogen) systems, while comparing with exact results.
Finite-temperature many-body perturbation theory in the grand-canonical ensemble is fundamental to numerous methods for computing electronic properties at nonzero temperature, such as finite-temperature coupled-cluster. In most applications it is the average number of electrons that is known rather than the chemical potential. Expensive correlation calculations must be repeated iteratively in search for the interacting chemical potential that yields the desired average number of electrons. In extended systems with mobile charges the situation is particular, however. Long-ranged electrostatic forces drive the charges such that the average ratio of negative and positive charges is one for any finite chemical potential. All properties per electron are expected to be virtually independent of the chemical potential, as they are in an electric wire at different voltage potentials. This work shows that per electron, the exchange-correlation free energy and the exchange-correlation grand potential indeed agree in the infinite-size limit. Thus, only one expensive correlation calculation suffices for each system size, sparing the search for the interacting chemical potential. This work also demonstrates the importance of regularizing the Coulomb interaction such that each electron on average interacts only with as many electrons as there are electrons in the simulation, avoiding interactions with periodic images. Numerical calculations of the warm uniform electron gas have been conducted with the Spencer-Alavi regularization employing the finite-temperature Hartree approximation for the self-consistent field and linearized finite-temperature direct-ring coupled-cluster doubles for treating correlation.
We extend the finite-temperature Keldysh non-equilibrium coupled cluster theory (Keldysh-CC) [A. F. White and G. K.-L. Chan, J. Chem. Theory Comput. 15, 6137–6253 (2019)] to include a time-dependent orbital basis. When chosen to minimize the action, such a basis restores local and global conservation laws (Ehrenfest’s theorem) for all one-particle properties while remaining energy conserving for time-independent Hamiltonians. We present the time-dependent Keldysh orbital-optimized coupled cluster doubles method in analogy with the formalism for zero-temperature dynamics, extended to finite temperatures through the time-dependent action on the Keldysh contour. To demonstrate the conservation property and understand the numerical performance of the method, we apply it to several problems of non-equilibrium finite-temperature dynamics: a 1D Hubbard model with a time-dependent Peierls phase, laser driving of molecular H2, driven dynamics in warm-dense silicon, and transport in the single impurity Anderson model.