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

The Journal of Chemical Physics

On the equivalence of ring-coupled cluster and adiabatic connection fluctuation-dissipation theorem random phase approximation correlation energy expressions

Georg Jansen, Ru-Fen Liu, János G. Ángyán

Published Oct 18, 2010• 89 citations
Publisher

Abstract

The correlation energy in the direct random phase approximation (dRPA) can be written, among other possibilities, either in terms of the interaction strength averaged correlation density matrix, or in terms of the coupled cluster doubles amplitudes obtained in the direct ring approximation (drCCD). Although the corresponding dRPA correlation density matrix on the one hand, and the drCCD amplitude matrix on the other hand, differ significantly, they yield identical energies. Similarly, the analogous RPA and rCCD correlation energies calculated from antisymmetrized two-electron integrals are identical to each other despite very different underlying working equations. In the present communication, a direct correspondence between amplitudes and densities is established and investigated with perturbation theory arguments. Our analysis also sheds some light on the properties of recently proposed RPA/rCCD variants which use antisymmetrized integrals in part of the equations and nonantisymmetrized integrals in others.

DOI
10.1063/1.3481575

Related papers

Communication: Random-phase approximation excitation energies from approximate equation-of-motion coupled-cluster doubles

Timothy C. Berkelbach

The ground-state correlation energy calculated in the random-phase approximation (RPA) is known to be identical to that calculated using a subset of terms appearing in coupled-cluster theory with double excitations (CCD). In particular, for particle-hole (ph) RPA this equivalence requires keeping only those terms that generate time-independent ring diagrams, and for particle-particle (pp) RPA it requires keeping only those terms that generate ladder diagrams. Here I show that these identities extend to excitation energies, for which those calculated in each RPA are identical to those calculated using approximations to equation-of-motion coupled-cluster theory with double excitations (EOM-CCD). The equivalence requires three approximations to EOM-CCD: first, the ground-state CCD amplitudes are obtained from the ring-CCD or ladder-CCD equations (the same as for the correlation energy); second, the EOM eigenvalue problem is truncated to the minimal subspace, which is one particle + one hole for ph-RPA and two particles or two holes for pp-RPA; third, the similarity transformation of the Fock operator must be neglected, as it corresponds to a Brueckner-like dressing of the single-particle propagator, which is not present in the conventional RPA.

arXiv:1804.01037•The Journal of Chemical Physics•Jul 28, 2018•32 citations

Correlation energy from random phase approximations: A reduced density matrices perspective

Katarzyna Pernal

Abstract Random phase approximation (RPA) electron correlation methods have gained in popularity in the recent years. A number of RPA correlation energy variants emerged in the Kohn–Sham DFT framework and in the theory of strongly orthogonal geminals. Foundations of most RPA approaches trace back to an exact expression for two‐electron reduced density matrix written in terms of one‐electron density matrix and dynamic one‐electron response functions, originally presented in the seminal paper of McLachlan and Ball (Rev. Mod. Phys. 1964, 36, 844). The aim of this article is to give a pedagogical introduction to possible approaches for describing electron correlation energy based on the McLachlan and Ball formula. The focus of the presentation is to formulate electron‐interaction energy expressions as functions of reduced density matrices. On one hand, it provides a common umbrella for RPA approximations proposed for uncorrelated (Hartree–Fock or Kohn–Sham) as well as partially correlated (strongly orthogonal geminals) references. On the other hand, such presentation may stimulate new developments in density matrix functional theory.

International Journal of Quantum Chemistry•Aug 28, 2017•32 citations

Correlation energy within exact-exchange adiabatic connection fluctuation-dissipation theory: Systematic development and simple approximations

Nicola Colonna, Maria Hellgren, Stefano de Gironcoli

We have calculated the correlation energy of the homogeneous electron gas (HEG) and the dissociation energy curves of molecules with covalent bonds from an efficient implementation of the adiabatic connection fluctuation dissipation expression including the exact-exchange (EXX) kernel. The EXX kernel is defined from first-order perturbation theory and used in the Dyson equation of time-dependent density-functional theory. Within this approximation (RPAx), the correlation energies of the HEG are significantly improved with respect to the random phase approximation (RPA) up to densities of the order of ${r}_{s}\ensuremath{\approx}10$. However, beyond this value, the RPAx response function exhibits an unphysical divergence and the approximation breaks down. Total energies of molecules at equilibrium are also highly accurate, but we find a similar instability at stretched geometries. Staying within an exact first-order approximation to the response function, we use an alternative resummation of the higher-order terms. This slight redefinition of RPAx fixes the instability in total energy calculations without compromising the overall accuracy of the approach.

arXiv:1409.0354•Physical Review B•Sep 26, 2014•47 citations

Particle-particle and quasiparticle random phase approximations: Connections to coupled cluster theory

Gustavo E. Scuseria, Thomas M. Henderson, Ireneusz W. Bulik

We establish a formal connection between the particle-particle (pp) random phase approximation (RPA) and the ladder channel of the coupled cluster doubles (CCD) equations. The relationship between RPA and CCD is best understood within a Bogoliubov quasiparticle (qp) RPA formalism. This work is a follow-up to our previous formal proof on the connection between particle-hole (ph) RPA and ring-CCD. Whereas RPA is a quasibosonic approximation, CC theory is a "correct bosonization" in the sense that the wavefunction and Hilbert space are exactly fermionic, yet the amplitude equations can be interpreted as adding different quasibosonic RPA channels together. Coupled cluster theory achieves this goal by interacting the ph (ring) and pp (ladder) diagrams via a third channel that we here call "crossed-ring" whose presence allows for full fermionic antisymmetry. Additionally, coupled cluster incorporates what we call "mosaic" terms which can be absorbed into defining a new effective one-body Hamiltonian. The inclusion of these mosaic terms seems to be quite important. The pp-RPA and qp-RPA equations are textbook material in nuclear structure physics but are largely unknown in quantum chemistry, where particle number fluctuations and Bogoliubov determinants are rarely used. We believe that the ideas and connections discussed in this paper may help design improved ways of incorporating RPA correlation into density functionals based on a CC perspective.

arXiv:1306.6360•The Journal of Chemical Physics•Sep 12, 2013•110 citations

Tension between predicting accurate ground state correlation energies and excitation energies from adiabatic approximations in TDDFT

Lucas M. Everhart, Julio A. Derteano, Jefferson E. Bates

The connection between the adiabatic excitation energy of time-dependent density functional theory and the ground state correlation energy from the adiabatic connection fluctuation-dissipation theorem (ACFDT) is explored in the limiting case of one excited state. An exact expression is derived for any adiabatic Hartree-exchange-correlation kernel that connects the excitation energy and the potential contribution to correlation. The resulting formula is applied to the asymmetric Hubbard dimer, a system where this limit is exact. Results from a hierarchy of approximations to the kernel, including the random phase approximation (RPA) with and without exchange and the adiabatically exact (AE) approximation, are compared to the exact ones. At full coupling, the numerical results indicate a tension between predicting an accurate excitation energy and an accurate potential contribution to correlation. The AE approximation is capable of making accurate predictions of both quantities, but only in parts of the parameter space that classify as weakly correlated, while RPA tends to be unable to accurately predict these properties simultaneously anywhere. For a strongly correlated dimer, the AE approximation greatly overestimates the excitation energy yet continues to yield an accurate ground state correlation energy due to its accurate prediction of the adiabatic connection integrand. If similar trends hold for real systems, the development of correlation kernels will be important for applications of the ACFDT in systems with large potential contributions to correlation.

The Journal of Chemical Physics•Feb 25, 2022•0 citations

Exchange-correlation energy from pairing matrix fluctuation and the particle-particle random-phase approximation

Helen van Aggelen, Yang Yang, Weitao Yang

We formulate an adiabatic connection for the exchange-correlation energy in terms of pairing matrix fluctuation. This connection opens new channels for density functional approximations based on pairing interactions. Even the simplest approximation to the pairing matrix fluctuation, the particle-particle random phase approximation (pp-RPA), has some highly desirable properties. It has no delocalization error with a nearly linear energy behavior for systems with fractional charges, describes van der Waals interactions similarly and thermodynamic properties significantly better than particle-hole RPA, and eliminates static correlation error for single-bond systems. Most significantly, the pp-RPA is the first known functional that has an explicit and closed-form dependence on the occupied and unoccupied orbitals and captures the energy derivative discontinuity in strongly correlated systems. These findings illustrate the potential of including pairing interactions within a density functional framework.

arXiv:1306.4957•Physical Review A•Sep 24, 2013•105 citations
The correlation energy in the direct random phase approximation (dRPA) can be written, among other possibilities, either in terms of the interaction strength averaged correlation density matrix, or in terms of the coupled cluster doubles amplitudes obtained in the direct ring approximation (drCCD). Although the corresponding dRPA correlation density matrix on the one hand, and the drCCD amplitude matrix on the other hand, differ significantly, they yield identical energies. Similarly, the analogous RPA and rCCD correlation energies calculated from antisymmetrized two-electron integrals are identical to each other despite very different underlying working equations. In the present communication, a direct correspondence between amplitudes and densities is established and investigated with perturbation theory arguments. Our analysis also sheds some light on the properties of recently proposed RPA/rCCD variants which use antisymmetrized integrals in part of the equations and nonantisymmetrized integrals in others.
The ground-state correlation energy calculated in the random-phase approximation (RPA) is known to be identical to that calculated using a subset of terms appearing in coupled-cluster theory with double excitations (CCD). In particular, for particle-hole (ph) RPA this equivalence requires keeping only those terms that generate time-independent ring diagrams, and for particle-particle (pp) RPA it requires keeping only those terms that generate ladder diagrams. Here I show that these identities extend to excitation energies, for which those calculated in each RPA are identical to those calculated using approximations to equation-of-motion coupled-cluster theory with double excitations (EOM-CCD). The equivalence requires three approximations to EOM-CCD: first, the ground-state CCD amplitudes are obtained from the ring-CCD or ladder-CCD equations (the same as for the correlation energy); second, the EOM eigenvalue problem is truncated to the minimal subspace, which is one particle + one hole for ph-RPA and two particles or two holes for pp-RPA; third, the similarity transformation of the Fock operator must be neglected, as it corresponds to a Brueckner-like dressing of the single-particle propagator, which is not present in the conventional RPA.
Abstract Random phase approximation (RPA) electron correlation methods have gained in popularity in the recent years. A number of RPA correlation energy variants emerged in the Kohn–Sham DFT framework and in the theory of strongly orthogonal geminals. Foundations of most RPA approaches trace back to an exact expression for two‐electron reduced density matrix written in terms of one‐electron density matrix and dynamic one‐electron response functions, originally presented in the seminal paper of McLachlan and Ball (Rev. Mod. Phys. 1964, 36, 844). The aim of this article is to give a pedagogical introduction to possible approaches for describing electron correlation energy based on the McLachlan and Ball formula. The focus of the presentation is to formulate electron‐interaction energy expressions as functions of reduced density matrices. On one hand, it provides a common umbrella for RPA approximations proposed for uncorrelated (Hartree–Fock or Kohn–Sham) as well as partially correlated (strongly orthogonal geminals) references. On the other hand, such presentation may stimulate new developments in density matrix functional theory.
We have calculated the correlation energy of the homogeneous electron gas (HEG) and the dissociation energy curves of molecules with covalent bonds from an efficient implementation of the adiabatic connection fluctuation dissipation expression including the exact-exchange (EXX) kernel. The EXX kernel is defined from first-order perturbation theory and used in the Dyson equation of time-dependent density-functional theory. Within this approximation (RPAx), the correlation energies of the HEG are significantly improved with respect to the random phase approximation (RPA) up to densities of the order of ${r}_{s}\ensuremath{\approx}10$. However, beyond this value, the RPAx response function exhibits an unphysical divergence and the approximation breaks down. Total energies of molecules at equilibrium are also highly accurate, but we find a similar instability at stretched geometries. Staying within an exact first-order approximation to the response function, we use an alternative resummation of the higher-order terms. This slight redefinition of RPAx fixes the instability in total energy calculations without compromising the overall accuracy of the approach.
We establish a formal connection between the particle-particle (pp) random phase approximation (RPA) and the ladder channel of the coupled cluster doubles (CCD) equations. The relationship between RPA and CCD is best understood within a Bogoliubov quasiparticle (qp) RPA formalism. This work is a follow-up to our previous formal proof on the connection between particle-hole (ph) RPA and ring-CCD. Whereas RPA is a quasibosonic approximation, CC theory is a "correct bosonization" in the sense that the wavefunction and Hilbert space are exactly fermionic, yet the amplitude equations can be interpreted as adding different quasibosonic RPA channels together. Coupled cluster theory achieves this goal by interacting the ph (ring) and pp (ladder) diagrams via a third channel that we here call "crossed-ring" whose presence allows for full fermionic antisymmetry. Additionally, coupled cluster incorporates what we call "mosaic" terms which can be absorbed into defining a new effective one-body Hamiltonian. The inclusion of these mosaic terms seems to be quite important. The pp-RPA and qp-RPA equations are textbook material in nuclear structure physics but are largely unknown in quantum chemistry, where particle number fluctuations and Bogoliubov determinants are rarely used. We believe that the ideas and connections discussed in this paper may help design improved ways of incorporating RPA correlation into density functionals based on a CC perspective.
The connection between the adiabatic excitation energy of time-dependent density functional theory and the ground state correlation energy from the adiabatic connection fluctuation-dissipation theorem (ACFDT) is explored in the limiting case of one excited state. An exact expression is derived for any adiabatic Hartree-exchange-correlation kernel that connects the excitation energy and the potential contribution to correlation. The resulting formula is applied to the asymmetric Hubbard dimer, a system where this limit is exact. Results from a hierarchy of approximations to the kernel, including the random phase approximation (RPA) with and without exchange and the adiabatically exact (AE) approximation, are compared to the exact ones. At full coupling, the numerical results indicate a tension between predicting an accurate excitation energy and an accurate potential contribution to correlation. The AE approximation is capable of making accurate predictions of both quantities, but only in parts of the parameter space that classify as weakly correlated, while RPA tends to be unable to accurately predict these properties simultaneously anywhere. For a strongly correlated dimer, the AE approximation greatly overestimates the excitation energy yet continues to yield an accurate ground state correlation energy due to its accurate prediction of the adiabatic connection integrand. If similar trends hold for real systems, the development of correlation kernels will be important for applications of the ACFDT in systems with large potential contributions to correlation.
We formulate an adiabatic connection for the exchange-correlation energy in terms of pairing matrix fluctuation. This connection opens new channels for density functional approximations based on pairing interactions. Even the simplest approximation to the pairing matrix fluctuation, the particle-particle random phase approximation (pp-RPA), has some highly desirable properties. It has no delocalization error with a nearly linear energy behavior for systems with fractional charges, describes van der Waals interactions similarly and thermodynamic properties significantly better than particle-hole RPA, and eliminates static correlation error for single-bond systems. Most significantly, the pp-RPA is the first known functional that has an explicit and closed-form dependence on the occupied and unoccupied orbitals and captures the energy derivative discontinuity in strongly correlated systems. These findings illustrate the potential of including pairing interactions within a density functional framework.