The random phase approximation (RPA) for the Kohn-Sham correlation energy, evaluated on exact-exchange orbitals, systematically overestimates correlation energies of atoms, and its simple remedies — the RPA+ short-range local correction and the RSOX partially resummed second-order exchange — were validated only for closed- and half-filled-shell spherical atoms (Jiang and Engel, 2007). The open problem is twofold: (i) establish whether RPA+-type short-range and RSOX/SOSEX-type screened-exchange corrections can be made uniformly reliable for degenerate or near-degenerate systems, statically correlated molecules, and extended inhomogeneous systems; and (ii) construct a computationally feasible, systematically improvable approximation to the fully screened second-order exchange that is size-consistent and free of one-electron self-correlation, at a cost comparable to RPA itself. Later literature (2009-2025) supplies extensive benchmarks showing the existing corrections are accurate in some regimes and degrade in others, and provides progressively better screened-exchange schemes (SOSEX, AXK, short-range SOSEX, sigma-functionals) that remain approximate — so the combined objective, a uniform and feasible beyond-RPA correlation functional, is still open.

Background

In Kohn-Sham density functional theory (DFT) the exchange-correlation energy is usually approximated by the local density approximation (LDA) or generalized gradient approximations (GGA), which cannot describe dispersion (van der Waals) binding. The adiabatic-connection fluctuation-dissipation (ACFD) theorem expresses the exact correlation energy as an integral over the density response of the interacting system at coupling strengths between zero and one; the random phase approximation (RPA) truncates the response at first order in the Coulomb interaction, equivalent to summing the infinite series of direct 'ring' diagrams built from Kohn-Sham orbitals and eigenvalues. RPA is non-empirical, size consistent (the energy of infinitely separated fragments is additive), and captures long-range dispersion, but it includes unphysical 'self-correlation' — a spurious correlation energy even for one-electron systems — and substantially overestimates atomic correlation energies, much like the second-order perturbation functional Ec(2)E_c^{(2)} obtained by expanding in the Kohn-Sham Green's function (the Kohn-Sham variant of second-order Møller-Plesset perturbation theory, KS-MP2). Ec(2)E_c^{(2)} is even variationally unstable for systems with a small gap between the highest occupied and lowest unoccupied molecular orbital (HOMO-LUMO gap) and fails for the nitrogen dimer, the prototype of static correlation (near-degeneracy of several electronic configurations, e.g. at stretched multiple bonds). The second-order exchange (SOX) diagram — the exchange counterpart of the second-order direct term — overcorrects when inserted unscreened. Jiang and Engel (J. Chem. Phys. 127, 184108 (2007)) therefore benchmarked three extensions for spherical closed- and half-filled-shell atoms, evaluated perturbatively on exact-exchange (EXX) orbitals: (a) RPA+, adding a short-range LDA correction for the non-RPA correlation (introduced for isoelectronic energy changes by Yan, Perdew and Kurth, Phys. Rev. B 61, 16430 (2000), following Kurth and Perdew's jellium-surface work); (b) RPA+SOX with the unscreened second-order exchange; and (c) RPA+RSOX, a partial resummation of the perturbation series including second-order exchange (a 'renormalized' SOX with a modified denominator). Extensions (a) and (c) clearly improved total energies and ionization potentials, while the fully screened SOX contribution — obtained by replacing the bare Coulomb interaction in the second-order exchange diagram with the same dynamically screened interaction as in the direct (ring) term, following the line of thought used by Hu and Langreth (Phys. Rev. B 33, 943 (1986)) in deriving GGAs, and conceptually the right object, since screening is what resums the ring diagrams in the direct channel — was computationally too demanding for their atomic real-space scheme. Their conclusions also note that ladder-resummed Epstein-Nesbet-type functionals (second-order terms whose denominators use exact orbital-energy differences rather than MP2-style occupied-virtual sums, partially resummed as particle-hole ladders) 'face fundamental problems in the case of degenerate or near-degenerate systems', that a gradient correction to the RPA+ short-range part 'suggests itself', that the RPA energy turned out insensitive to the input orbitals for the atoms studied, and that their multi-shell frequency-integration partitioning should help larger systems. A related, slightly different object is SOSEX (second-order screened exchange, Grüneis, Marsman, Harl, Schimka and Kresse, J. Chem. Phys. 131, 154115 (2009)): a screened second-order exchange correction that is one-electron self-correlation free and computable at O(N5)O(N^5) cost (fifth power of system size, as for canonical MP2).

Problem Statement

Starting from the atom benchmarks of Jiang and Engel — whose own outlook calls for 'a more suitable partial resummation scheme ... to establish a universally applicable, implicit XC functional, the RPA and its extensions being the prime candidates' — the open problem is to complete the RPA-based correlation-functional program in two linked respects. (1) Universal reliability: given that every individual scheme tested to date has been shown to fail this uniformity, each existing correction improving some observables while degrading others (see the benchmark literature, refs. 3-20), establish whether a correction of the RPA+/RSOX type — a short-range semilocal addition and/or a partially resummed second-order exchange — can be constructed that is uniformly reliable beyond spherical closed- and half-filled-shell atoms, specifically for (i) degenerate or near-degenerate systems, including open-shell atoms and small-HOMO-LUMO-gap systems where Ec(2)E_c^{(2)} is variationally unstable, (ii) molecules with substantial static correlation such as stretched multiple bonds (the regime where Ec(2)E_c^{(2)} fails for the nitrogen dimer), and (iii) extended and strongly inhomogeneous systems such as solids and surfaces, with uniformly improved total correlation energies and energy differences (ionization potentials, atomization and reaction energies) relative to uncorrected RPA. (2) Feasible screened exchange: devise an approximation to the fully screened second-order exchange contribution defined in the Background (of which unscreened SOX, RSOX, and SOSEX are partial realizations) that is size consistent, free of one-electron self-correlation, computationally feasible at a cost comparable to RPA or canonical MP2 (i.e. O(N5)O(N^5) or better including stable frequency integration for many-shell and extended systems), and systematically improvable in a controlled way — order by order in the Kohn-Sham perturbation expansion underlying the ACFD diagrammatics, each higher-order correction defined unambiguously rather than fitted — and demonstrate its reliability over the same three regimes. The two parts are linked to one objective — a generally applicable, non-empirical RPA-based correlation functional: part (2) is the constructive realization of the reliability question in part (1), so an approximation satisfying (2) settles (1) affirmatively, while a demonstration that no uniformly reliable correction of this type exists settles (1) negatively — either outcome resolves the question.

References

  1. Hong Jiang and Eberhard Engel, 'Random-phase-approximation-based correlation energy functionals: Benchmark results for atoms', J. Chem. Phys. 127, 184108 (2007), DOI 10.1063/1.2795707; arXiv:0707.1615, https://doi.org/10.1063/1.2795707
  2. Zidan Yan, John P. Perdew, and Stefan Kurth, 'Density functional for short-range correlation: Accuracy of the random phase approximation for isoelectronic energy changes', Phys. Rev. B 61, 16430-16439 (2000), DOI 10.1103/physrevb.61.16430, https://doi.org/10.1103/PhysRevB.61.16430
  3. Andreas Grüneis, Martijn Marsman, Judith Harl, Laurids Schimka, and Georg Kresse, 'Making the random phase approximation to electronic correlation accurate', J. Chem. Phys. 131, 154115 (2009), DOI 10.1063/1.3250347, https://doi.org/10.1063/1.3250347
  4. Judith Harl, Laurids Schimka, and Georg Kresse, 'Assessing the quality of the random phase approximation for lattice constants and atomization energies of solids', Phys. Rev. B 81, 115126 (2010), DOI 10.1103/physrevb.81.115126, https://doi.org/10.1103/PhysRevB.81.115126
  5. Julien Toulouse, Iann C. Gerber, Georg Jansen, Andreas Savin, and János G. Ángyán, 'Adiabatic-Connection Fluctuation-Dissipation Density-Functional Theory Based on Range Separation', Phys. Rev. Lett. 102, 096404 (2009), DOI 10.1103/physrevlett.102.096404, https://doi.org/10.1103/PhysRevLett.102.096404
  6. János G. Ángyán, Ru-Fen Liu, Julien Toulouse, and Georg Jansen, 'Correlation Energy Expressions from the Adiabatic-Connection Fluctuation-Dissipation Theorem Approach', J. Chem. Theory Comput. 7, 3116-3130 (2011), DOI 10.1021/ct200501r, https://doi.org/10.1021/ct200501r
  7. Robert M. Irelan, Thomas M. Henderson, and Gustavo E. Scuseria, 'Long-range-corrected hybrids using a range-separated Perdew-Burke-Ernzerhof functional and random phase approximation correlation', J. Chem. Phys. 135, 094105 (2011), DOI 10.1063/1.3630951, https://doi.org/10.1063/1.3630951
  8. Xinguo Ren, Alexandre Tkatchenko, Patrick Rinke, and Matthias Scheffler, 'Beyond the Random-Phase Approximation for the Electron Correlation Energy: The Importance of Single Excitations', Phys. Rev. Lett. 106, 153003 (2011), DOI 10.1103/physrevlett.106.153003, https://doi.org/10.1103/PhysRevLett.106.153003
  9. Henk Eshuis, Julian Yarkony, and Filipp Furche, 'Fast computation of molecular random phase approximation correlation energies using resolution of the identity and imaginary frequency integration', J. Chem. Phys. 132, 234114 (2010), DOI 10.1063/1.3442749, https://doi.org/10.1063/1.3442749
  10. Henk Eshuis, Jefferson E. Bates, and Filipp Furche, 'Electron correlation methods based on the random phase approximation', Theor. Chem. Acc. 131, 1084 (2012), DOI 10.1007/s00214-011-1084-8, https://doi.org/10.1007/s00214-011-1084-8
  11. Joachim Paier, Xinguo Ren, Patrick Rinke, Gustavo E. Scuseria, Andreas Grüneis, Georg Kresse, and Matthias Scheffler, 'Assessment of correlation energies based on the random-phase approximation', New J. Phys. 14, 043002 (2012), DOI 10.1088/1367-2630/14/4/043002, https://doi.org/10.1088/1367-2630/14/4/043002
  12. Xinguo Ren, Patrick Rinke, Christian Joas, and Matthias Scheffler, 'Random-phase approximation and its applications in computational chemistry and materials science', J. Mater. Sci. 47, 7447-7471 (2012), DOI 10.1007/s10853-012-6570-4, https://doi.org/10.1007/s10853-012-6570-4
  13. Adrienn Ruzsinszky, John P. Perdew, and Gábor I. Csonka, 'The RPA Atomization Energy Puzzle', J. Chem. Theory Comput. 6, 127-134 (2010), DOI 10.1021/ct900518k, https://doi.org/10.1021/ct900518k
  14. Maria Hellgren, Daniel R. Rohr, and E. K. U. Gross, 'Correlation potentials for molecular bond dissociation within the self-consistent random phase approximation', J. Chem. Phys. 136, 034106 (2012), DOI 10.1063/1.3676174, https://doi.org/10.1063/1.3676174
  15. Jefferson E. Bates and Filipp Furche, 'Communication: Random phase approximation renormalized many-body perturbation theory', J. Chem. Phys. 139, 171103 (2013), DOI 10.1063/1.4827254, https://doi.org/10.1063/1.4827254
  16. Matthias Beuerle and Christian Ochsenfeld, 'Short-range second order screened exchange correction to RPA correlation energies', J. Chem. Phys. 147, 204107 (2017), DOI 10.1063/1.4998647, https://doi.org/10.1063/1.4998647
  17. Yannick Lemke and Christian Ochsenfeld, 'Highly accurate sigma- and tau-functionals for beyond-RPA methods with approximate exchange kernels', J. Chem. Phys. 159, 194104 (2023), DOI 10.1063/5.0173042, https://doi.org/10.1063/5.0173042
  18. Mario Benites, Angel Rosado, and Efstratios Manousakis, 'Accurate electron correlation energy functional: Expansion in the interaction renormalized by the random-phase approximation', Phys. Rev. B 110, 195151 (2024), DOI 10.1103/physrevb.110.195151, https://doi.org/10.1103/PhysRevB.110.195151
  19. Rong Shi, Peize Lin, Min-Ye Zhang, Lixin He, and Xinguo Ren, 'Subquadratic-scaling real-space random phase approximation correlation energy calculations for periodic systems with numerical atomic orbitals', Phys. Rev. B 109, 035103 (2024), DOI 10.1103/physrevb.109.035103, https://doi.org/10.1103/PhysRevB.109.035103
  20. Thomas Pitts, Damian Contant, and Maria Hellgren, 'Self-consistent random phase approximation and optimized hybrid functionals for solids', Phys. Rev. B 112, 085137 (2025), DOI 10.1103/1ymj-3ycc, https://doi.org/10.1103/1ymj-3ycc