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  • Open Access

Diagrammatic theory of the irreducible coupled-cluster self-energy

Christopher J. N. Coveney1,* and David P. Tew2,†

  • *Contact author: christopher.coveney@physics.ox.ac.uk
  • †Contact author: david.tew@chem.ox.ac.uk
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Phys. Rev. B 112, 045104 – Published 1 July, 2025

DOI: https://doi.org/10.1103/p41w-bl6p

Abstract

Coupled-cluster and Green's function theories are highly successful in treating many-body electron correlation, and there has been significant interest in identifying and leveraging connections between them. Here we present a diagrammatic definition of the irreducible coupled-cluster self-energy that directly embeds coupled-cluster (CC) theory within the framework of many-body field theory. The equation-of-motion coupled-cluster (EOM-CC) treatment emerges naturally from our definition via the Dyson equation and the Bethe-Salpeter equation (BSE), providing a unified description of the random phase approximation (RPA), GW-BSE, and CC theory for ground state and excitation energies. This clarifies the origin of previously established connections between RPA, GW-BSE, and CC theory, and it exposes the relationship between vertex corrections and the coupled-cluster amplitude equations.

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References (58)

  1. A. Fetter and J. D. Walecka, Quantum Theory of Many Particle Systems (McGraw-Hill, New York City, New York, 1971).
  2. G. D. Mahan, Many-Particle Physics (Springer, Berlin, 2000).
  3. G. Stefanucci and R. Van Leeuwen, Nonequilibrium Many-Body Theory of Quantum Systems: A Modern Introduction (Cambridge University Press, Cambridge, 2013).
  4. F. J. Dyson, The radiation theories of Tomonaga, Schwinger and Feynman, Phys. Rev. 75, 486 (1949).
  5. L. Hedin, New method for calculating the one-particle Green's function with application to the electron-gas problem, Phys. Rev. 139, A796 (1965).
  6. P. Pokhilko, S. Iskakov, C.-N. Yeh, and D. Zgid, Evaluation of two-particle properties within finite-temperature self-consistent one-particle Green's function methods: Theory and application to GW and GF2, J. Chem. Phys. 155, 024119 (2021).
  7. R. M. Martin, L. Reining, and D. M. Ceperley, Interacting Electrons (Cambridge University Press, Cambridge, 2016).
  8. C. Verdozzi, R. W. Godby, and S. Holloway, Evaluation of GW approximations for the self-energy of a Hubbard cluster, Phys. Rev. Lett. 74, 2327 (1995).
  9. M. Puig von Friesen, C. Verdozzi, and C.-O. Almbladh, Successes and failures of Kadanoff-Baym dynamics in Hubbard nanoclusters, Phys. Rev. Lett. 103, 176404 (2009).
  10. P. Romaniello, S. Guyot, and L. Reining, The self-energy beyond GW: Local and nonlocal vertex corrections, J. Chem. Phys. 131, 154111 (2009).
  11. M. Puig von Friesen, C. Verdozzi, and C.-O. Almbladh, Kadanoff-Baym dynamics of Hubbard clusters: Performance of many-body schemes, correlation-induced damping and multiple steady and quasi-steady states, Phys. Rev. B 82, 155108 (2010).
  12. P. Romaniello, F. Bechstedt, and L. Reining, Beyond the GW approximation: Combining correlation channels, Phys. Rev. B 85, 155131 (2012).
  13. W. Kohn and J. M. Luttinger, Ground-state energy of a many-fermion system, Phys. Rev. 118, 41 (1960).
  14. J. M. Luttinger and J. C. Ward, Ground-state energy of a many-fermion system. II, Phys. Rev. 118, 1417 (1960).
  15. J. M. Luttinger, Analytic properties of single-particle propagators for many-fermion systems, Phys. Rev. 121, 942 (1961).
  16. M. Potthoff, Making use of self-energy functionals: The variational cluster approximation, arXiv:1407.4065.
  17. M. J. van Setten, F. Weigend, and F. Evers, The GW-method for quantum chemistry applications: Theory and implementation, J. Chem. Theor. Comput. 9, 232 (2013).
  18. M. Nooijen and J. G. Snijders, Coupled-cluster approach to the single-particle Green's function, Int. J. Quantum Chem. 44, 55 (1992).
  19. M. Nooijen and J. G. Snijders, Coupled-cluster Green's function method: Working equations and applications, Int. J. Quantum Chem. 48, 15 (1993).
  20. B. Peng and K. Kowalski, Green's function coupled-cluster approach: Simulating photoelectron spectra for realistic molecular systems, J. Chem. Theor. Comput. 14, 4335 (2018).
  21. A. Shee and D. Zgid, Coupled-cluster as an impurity solver for Green's function embedding methods, J. Chem. Theor. Comput. 15, 6010 (2019).
  22. O. J. Backhouse and G. H. Booth, Constructing “full-frequency” spectra via moment constraints for coupled-cluster Green's functions, J. Chem. Theor. Comput. 18, 6622 (2022).
  23. S. J. Bintrim and T. C. Berkelbach, Full-frequency GW without frequency, J. Chem. Phys. 154, 041101 (2021).
  24. S. J. Bintrim and T. C. Berkelbach, Full-frequency dynamical Bethe–Salpeter equation without frequency and a study of double excitations, J. Chem. Phys. 156, 044114 (2022).
  25. C. J. C. Scott, O. J. Backhouse, and G. H. Booth, A moment-conserving reformulation of GW theory, J. Chem. Phys. 158, 124102 (2023).
  26. R. Quintero-Monsebaiz, E. Monino, A. Marie, and P.-F. Loos, Connections between many-body perturbation and coupled-cluster theories, J. Chem. Phys. 157, 231102 (2022).
  27. E. Monino and P.-F. Loos, Connections and performances of Green's function methods for charged and neutral excitations, J. Chem. Phys. 159, 034105 (2023).
  28. J. Tölle and G. K.-L. Chan, Exact relationships between the GW approximation and equation-of-motion coupled-cluster theories through the quasi-boson formalism, J. Chem. Phys. 158, 124123 (2023).
  29. G. E. Scuseria, T. M. Henderson, and D. C. Sorensen, The ground state correlation energy of the random phase approximation from a ring coupled cluster doubles approach, J. Chem. Phys. 129, 231101 (2008).
  30. T. C. Berkelbach, Communication: Random-phase approximation excitation energies from approximate equation-of-motion coupled-cluster doubles, J. Chem. Phys. 149, 041103 (2018).
  31. V. Rishi, A. Perera, and R. J. Bartlett, A route to improving RPA excitation energies through its connection to equation-of-motion coupled-cluster theory, J. Chem. Phys. 153, 234101 (2020).
  32. M. F. Lange and T. C. Berkelbach, On the relation between equation-of-motion coupled-cluster theory and the GW approximation, J. Chem. Theor. Comput. 14, 4224 (2018).
  33. V. M. Galitskii and A. B. Migdal, Application of quantum field theory methods to the many-body problem, Sov. Phys. JETP 7, 18 (1958).
  34. M. S. Hybertsen and S. G. Louie, Electron correlation in semiconductors and insulators: Band gaps and quasiparticle energies, Phys. Rev. B 34, 5390 (1986).
  35. J. Schirmer, L. S. Cederbaum, and O. Walter, New approach to the one-particle Green's function for finite Fermi systems, Phys. Rev. A 28, 1237 (1983).
  36. F. Caruso, P. Rinke, X. Ren, A. Rubio, and M. Scheffler, Self-consistent GW: All-electron implementation with localized basis functions, Phys. Rev. B 88, 075105 (2013).
  37. F. Raimondi and C. Barbieri, Algebraic diagrammatic construction formalism with three-body interactions, Phys. Rev. C 97, 054308 (2018).
  38. J. Schirmer, Many-Body Methods for Atoms, Molecules and Clusters (Springer, Berlin, 2018), Vol. 94.
  39. C. J. N. Coveney and D. P. Tew, A regularized second-order correlation method from Green's function theory, J. Chem. Theor. Comput. 19, 3915 (2023).
  40. O. J. Backhouse and G. H. Booth, Efficient excitations and spectra within a perturbative renormalization approach, J. Chem. Theor. Comput. 16, 6294 (2020).
  41. G. E. Scuseria, On the connections between Brueckner–coupled-cluster, density-dependent Hartree–Fock, and Density Functional Theory, Int. J. Quantum Chem. 55, 165 (1995).
  42. T. Helgaker, P. Jorgensen, and J. Olsen, Molecular Electronic-Structure Theory (Wiley, Hoboken, NJ, 2013).
  43. I. Shavitt and R. J. Bartlett, Many-Body Methods in Chemistry and Physics: MBPT and Coupled-cluster Theory (Cambridge University Press, Cambridge, 2009).
  44. J. Gauss and J. F. Stanton, Coupled-cluster calculations of nuclear magnetic resonance chemical shifts, J. Chem. Phys. 103, 3561 (1995).
  45. S. Hirata, I. Grabowski, J. V. Ortiz, and R. J. Bartlett, Nonconvergence of the Feynman-Dyson diagrammatic perturbation expansion of propagators, Phys. Rev. A 109, 052220 (2024).
  46. G. E. Scuseria, T. M. Henderson, and I. W. Bulik, Particle-particle and quasiparticle random phase approximations: Connections to coupled-cluster theory, J. Chem. Phys. 139, 104113 (2013).
  47. M. Rohlfing and S. G. Louie, Electron-hole excitations and optical spectra from first principles, Phys. Rev. B 62, 4927 (2000).
  48. P. Ring and P. Schuck, The Nuclear Many-body Problem (Springer Science & Business Media, New York, 2004).
  49. A. Schindlmayr and R. W. Godby, Systematic vertex corrections through iterative solution of Hedin's equations beyond the GW approximation, Phys. Rev. Lett. 80, 1702 (1998).
  50. L. G. Molinari, Hedin's equations and enumeration of Feynman diagrams, Phys. Rev. B 71, 113102 (2005).
  51. H. Ness, L. K. Dash, M. Stankovski, and R. W. Godby, GW approximations and vertex corrections on the Keldysh time-loop contour: Application for model systems at equilibrium, Phys. Rev. B 84, 195114 (2011).
  52. G. Lani, P. Romaniello, and L. Reining, Approximations for many-body Green's functions: Insights from the fundamental equations, New J. Phys. 14, 013056 (2012).
  53. A. L. Kutepov, Self-consistent solution of Hedin's equations: Semiconductors and insulators, Phys. Rev. B 95, 195120 (2017).
  54. V. Harkov, A. I. Lichtenstein, and F. Krien, Parametrizations of local vertex corrections from weak to strong coupling: Importance of the Hedin three-leg vertex, Phys. Rev. B 104, 125141 (2021).
  55. A. L. Kutepov, Full versus quasiparticle self-consistency in vertex-corrected GW approaches, Phys. Rev. B 105, 045124 (2022).
  56. C. Mejuto-Zaera and V. Vlček, Self-consistency in GWΓ formalism leading to quasiparticle-quasiparticle couplings, Phys. Rev. B 106, 165129 (2022).
  57. G. Riva, P. Romaniello, and J. A. Berger, Multichannel Dyson equation: Coupling many-body Green's functions, Phys. Rev. Lett. 131, 216401 (2023).
  58. P.-O. Löwdin, Partitioning technique, perturbation theory, and rational approximations, Int. J. Quantum Chem. 21, 69 (1982).

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Phys. Rev. B 112, 045104– Published 1 July, 2025
  • Received 19 September 2023
  • Revised 9 May 2025
  • Accepted 11 June 2025
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Published by the American Physical Society under the terms of the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to the author(s) and the published article's title, journal citation, and DOI.

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