{ "id": "epheth5n0v", "slug": "random-phase-approximation", "title": "Random phase approximation", "updated": "2026-10-15", "topic_path": [ { "id": "physical", "label": "Physical world and mathematics", "api_url": "https://www.edgepedia.org/api/v1/topics/physical" }, { "id": "physical.physics", "label": "Physics", "api_url": "https://www.edgepedia.org/api/v1/topics/physical.physics" }, { "id": "physical.physics.matter-and-radiation", "label": "Matter and radiation physics", "api_url": "https://www.edgepedia.org/api/v1/topics/physical.physics.matter-and-radiation" }, { "id": "physical.physics.matter-and-radiation.condensed-matter", "label": "Condensed matter physics", "api_url": "https://www.edgepedia.org/api/v1/topics/physical.physics.matter-and-radiation.condensed-matter" }, { "id": "physical.physics.matter-and-radiation.condensed-matter.cm-electronic", "label": "Electronic and magnetic properties", "api_url": "https://www.edgepedia.org/api/v1/topics/physical.physics.matter-and-radiation.condensed-matter.cm-electronic" } ], "geo": [ { "id": "geo.nongeo.t1800.physical.physics.matter-and-radiation", "label": "Non-geographic · 1800 to 1945: Matter and radiation physics", "api_url": "https://www.edgepedia.org/api/v1/geo/geo.nongeo.t1800.physical.physics.matter-and-radiation", "path": [ { "id": "geo.nongeo", "label": "Non-geographic", "api_url": "https://www.edgepedia.org/api/v1/geo/geo.nongeo" }, { "id": "geo.nongeo.t1800", "label": "Non-geographic · 1800 to 1945", "api_url": "https://www.edgepedia.org/api/v1/geo/geo.nongeo.t1800" }, { "id": "geo.nongeo.t1800.physical", "label": "Physical world and mathematics", "api_url": "https://www.edgepedia.org/api/v1/geo/geo.nongeo.t1800.physical" }, { "id": "geo.nongeo.t1800.physical.physics", "label": "Physics", "api_url": "https://www.edgepedia.org/api/v1/geo/geo.nongeo.t1800.physical.physics" }, { "id": "geo.nongeo.t1800.physical.physics.matter-and-radiation", "label": "Matter and radiation physics", "api_url": "https://www.edgepedia.org/api/v1/geo/geo.nongeo.t1800.physical.physics.matter-and-radiation" } ] }, { "id": "geo.nongeo.t1946.physical.physics.matter-and-radiation.condensed-matter.cm-electronic", "label": "Non-geographic · 1946 to 2000: Electronic and magnetic properties", "api_url": "https://www.edgepedia.org/api/v1/geo/geo.nongeo.t1946.physical.physics.matter-and-radiation.condensed-matter.cm-electronic", "path": [ { "id": "geo.nongeo", "label": "Non-geographic", "api_url": "https://www.edgepedia.org/api/v1/geo/geo.nongeo" }, { "id": "geo.nongeo.t1946", "label": "Non-geographic · 1946 to 2000", "api_url": "https://www.edgepedia.org/api/v1/geo/geo.nongeo.t1946" }, { "id": "geo.nongeo.t1946.physical", "label": "Physical world and mathematics", "api_url": "https://www.edgepedia.org/api/v1/geo/geo.nongeo.t1946.physical" }, { "id": "geo.nongeo.t1946.physical.physics", "label": "Physics", "api_url": "https://www.edgepedia.org/api/v1/geo/geo.nongeo.t1946.physical.physics" }, { "id": "geo.nongeo.t1946.physical.physics.matter-and-radiation", "label": "Matter and radiation physics", "api_url": "https://www.edgepedia.org/api/v1/geo/geo.nongeo.t1946.physical.physics.matter-and-radiation" }, { "id": "geo.nongeo.t1946.physical.physics.matter-and-radiation.condensed-matter", "label": "Condensed matter physics", "api_url": "https://www.edgepedia.org/api/v1/geo/geo.nongeo.t1946.physical.physics.matter-and-radiation.condensed-matter" }, { "id": "geo.nongeo.t1946.physical.physics.matter-and-radiation.condensed-matter.cm-electronic", "label": "Electronic and magnetic properties", "api_url": "https://www.edgepedia.org/api/v1/geo/geo.nongeo.t1946.physical.physics.matter-and-radiation.condensed-matter.cm-electronic" } ] } ], "excerpt": "The random phase approximation (RPA) is a many-body method that computes screening and correlation energy of interacting electron systems by summing ring diagrams to infinite order.", "snippet": "The random phase approximation (RPA) is a many-body method that computes screening and correlation energy of interacting electron systems by summing ring diagrams to infinite order.", "node": "physical.physics.matter-and-radiation.condensed-matter.cm-electronic", "markdown": "# Random phase approximation\n\nThe random phase approximation (RPA) is a many-body method that computes the screening, polarization propagator, and correlation energy of interacting electron systems by summing ring diagrams to infinite order.[1](https://www.annualreviews.org/content/journals/10.1146/annurev-physchem-040215-112308) It exists in two main formalisms: as a diagrammatic approximation in many-body perturbation theory, where it is synonymous with the ring approximation, and as a fully nonlocal exchange-correlation functional within the adiabatic-connection fluctuation-dissipation theorem of density functional theory.[1](https://www.annualreviews.org/content/journals/10.1146/annurev-physchem-040215-112308) Because it describes screening and long-range dispersion without empirical parameters and remains well defined for metallic systems, it is used across solid-state physics, quantum chemistry, and surface science.[1](https://www.annualreviews.org/content/journals/10.1146/annurev-physchem-040215-112308)\n\n| Key fact | Statement |\n| --- | --- |\n| Defining approximation | Infinite-order resummation of ring (bubble) diagrams; within RPA the exchange-correlation kernel is \\( f_{xc} = 0 \\), so vertex corrections are omitted from the response function.[2](https://iopscience.iop.org/article/10.1088/1367-2630/14/4/043002) |\n| Screened interaction | \\( W(k,\\omega) = V(k)/[1 - V(k)\\Pi^{*}(k,\\omega)] \\), with dielectric function \\( \\varepsilon(k,\\omega) = 1 - V(k)\\Pi^{*}(k,\\omega) \\).[3](https://courses.physics.illinois.edu/phys561/fa2005/lnotes/lec8.pdf) |\n| Correlation energy | Given by a coupling-constant integral over imaginary frequency in the adiabatic-connection formalism.[4](https://www.osti.gov/servlets/purl/1535290) |\n| Equivalent formulation | The direct RPA correlation energy equals the direct ring coupled cluster doubles (ring-CCD) energy.[5](https://hal.sorbonne-universite.fr/hal-01304895/document) |\n| Canonical cost | \\( N_{A}^{4} N_{k}^{2} \\) for \\( N_{A} \\) atoms and \\( N_{k} \\) k-points; space-time algorithms reduce the atom scaling to cubic with linear k-point dependence.[6](https://arxiv.org/html/2505.06021) |\n| Typical accuracy | (EX + RPA)@PBE underbinds solids by −67 kJ/mol and G2-1 molecules by −43 kJ/mol on average; mean unsigned relative error of binding energies about 6%.[2](https://iopscience.iop.org/article/10.1088/1367-2630/14/4/043002) |\n| Cost position | About one order of magnitude greater than a hybrid DFT calculation.[1](https://www.annualreviews.org/content/journals/10.1146/annurev-physchem-040215-112308) |\n\n## How it works\n\n**Ring diagrams and screening.** The central object is the proper (irreducible) polarizability \\( \\Pi^{*}(k,\\omega) \\), which in RPA reduces to the independent-particle density-density correlation of the non-interacting system; the full response is obtained by the Coulomb resummation of these bubbles. In RPA the electron-hole excitations inside \\( \\Pi^{*} \\) are treated as non-interacting: all interactions between the electron and its hole are omitted, and the only remaining effect is the Coulomb coupling between successive bubbles, resummed to infinite order.[3](https://courses.physics.illinois.edu/phys561/fa2005/lnotes/lec8.pdf) The resummation gives the screened Coulomb interaction and dielectric function\n\n\\[ W(k,\\omega) = \\frac{V(k)}{1 - V(k)\\,\\Pi^{*}(k,\\omega)}, \\qquad \\varepsilon(k,\\omega) = 1 - V(k)\\,\\Pi^{*}(k,\\omega). \\]\n\nThe name dates to the original derivation: the phases \\( e^{i(k-l)\\cdot x_{j}} \\) that couple plasma and particle momenta of different wavelengths are assumed to average to zero over many random particle positions.[7](https://www.cambridge.org/core/journals/forum-of-mathematics-pi/article/random-phase-approximation-for-interacting-fermi-gases-in-the-meanfield-regime/276E107AE84CFB3480AD888CB33D095E) In the plasmon formula, the RPA correlation energy is the difference between the zero-point oscillation energies of electronic excitations at full coupling and those to first order in the electron interaction.[1](https://www.annualreviews.org/content/journals/10.1146/annurev-physchem-040215-112308) In the adiabatic-connection formulation the same quantity is a coupling-constant integral,\n\n\\[ E_{c} = \\frac{1}{2\\pi} \\int_{0}^{1} d\\lambda \\int_{0}^{\\infty} d\\omega\\, \\mathrm{Tr}\\{\\cdots\\}, \\]\n\nin which the two-electron Coulomb integrals are replaced by their antisymmetrized counterparts to obtain the RPA+SOSEX correlation energy.[4](https://www.osti.gov/servlets/purl/1535290) Setting \\( f_{xc} = 0 \\) is the DFT-language statement of the same approximation: vertex corrections are excluded from the response function, which corresponds to the diagrammatic resummation of ring graphs to infinite order.[2](https://iopscience.iop.org/article/10.1088/1367-2630/14/4/043002)\n\n## How it is done\n\n**Workflow.** RPA is run as a single-point post-SCF correction on orbitals from a preceding semilocal or hybrid calculation (for example LDA, PBE, PBE0, or HSE), denoted RPA@PBE and similar, with slight starting-point dependence.[8](https://fhi-aims.org/uploads/documents/Tutorial_Series_2021/Xinguo%20Ren%20FHI-aims%20Talk%20September%202021.pdf) Two algebraically equivalent routes exist: evaluating the plasmon formula by diagonalizing the RPA matrix, which scales as \\( N^{4} \\), and iterating a direct ring-CCD amplitude equation, which scales as \\( N^{4} \\) per iteration; the two give strictly the same correlation energy.[1](https://www.annualreviews.org/content/journals/10.1146/annurev-physchem-040215-112308)\n\n**Quadrature and fitting.** Replacing diagonalization by imaginary frequency integration, combined with the resolution of identity (RI), requires \\( \\mathcal{O}(N^{3}) \\) storage; the quadrature is exact in the two-orbital case and converges exponentially, with 30 to 40 grid points giving micro-hartree accuracy in triple-zeta basis sets.[9](https://doi.org/10.1063/1.3442749) Density fitting or [Cholesky decomposition](https://www.edgepedia.org/cholesky-decomposition) further improves efficiency.[5](https://hal.sorbonne-universite.fr/hal-01304895/document)\n\n**Periodic systems.** Slow k-point convergence of RPA correlation energies traces to the Coulomb divergence at the Γ-point; the staggered mesh method removes a significant portion of the finite-size error at negligible additional cost and, in the adiabatic-connection formalism, avoids head/wing corrections to the dielectric operator.[10](https://par.nsf.gov/servlets/purl/10322307)\n\n**Reduced scaling.** The space-time method, made practical by the minimax quadrature grid for imaginary time-to-frequency transforms, reduces the scaling from \\( \\mathcal{O}(N^{4}) \\) to \\( \\mathcal{O}(N^{3}) \\) with linear k-point dependence.[11](https://doi.org/10.1103/physrevb.90.054115) Stochastic RPA based on Chebyshev expansion of the density of states, accelerated by atom-based correlated sampling, speeds energy-difference convergence by factors of 3.6 to 4.5.[12](https://doi.org/10.1088/2516-1075/abde94)\n\n## Origin\n\nBohm and Pines introduced the term and the approximation in *A Collective Description of Electron Interactions. I. Magnetic Interactions* ([Physical Review](https://www.edgepedia.org/physical-review), 1951).[13](https://doi.org/10.1103/physrev.82.625) The full theory is the third paper of the series (1953), which re-expresses the electron-gas Hamiltonian by a canonical transformation to longitudinal collective coordinates, so that the long-range Coulomb interaction becomes collective plasma oscillation fields plus electrons interacting through screened Coulomb forces, with a screening radius of the order of the interelectronic distance.[14](https://doi.org/10.1103/physrev.92.609) The approximation itself is defined in the first paper, where it eliminates single-electron motion out of phase with the external probe; the 1953 theory contains no random phase to approximate.[15](https://www.mdpi.com/2218-1997/9/3/141) The motivation was the 1940s discrepancy between Hartree-Fock calculations and experiment for the cohesive energy and electronic specific heat of alkali metals, with second-order perturbation theory yielding infinities.[7](https://www.cambridge.org/core/journals/forum-of-mathematics-pi/article/random-phase-approximation-for-interacting-fermi-gases-in-the-meanfield-regime/276E107AE84CFB3480AD888CB33D095E) In the high-density limit the Bohm-Pines ground-state energy equals the sum of all ring diagrams, each divergent individually but finite in sum.[1](https://www.annualreviews.org/content/journals/10.1146/annurev-physchem-040215-112308) Ehrenreich and Cohen took a self-consistent-field approach to the many-electron problem in 1959 (Physical Review).[16](https://doi.org/10.1103/physrev.115.786) Johnson and colleagues introduced the relativistic RPA in 1980 (Physica Scripta).[17](https://doi.org/10.1088/0031-8949/21/3-4/029) The quantum-chemistry era brought the ring-CCD equivalence proof (Scuseria, Henderson, and Sorensen, 2008, Journal of Chemical Physics),[18](https://doi.org/10.1063/1.3043729) RI-RPA with imaginary frequency integration for molecules (Eshuis, Yarkony, and Furche, 2010, Journal of Chemical Physics),[9](https://doi.org/10.1063/1.3442749) the cubic-scaling space-time algorithm (Kaltak, Klimeš, and Kresse, 2014, [Physical Review B](https://www.edgepedia.org/physical-review-b)),[11](https://doi.org/10.1103/physrevb.90.054115) and atom-based correlated sampling for stochastic RPA (Chi and Huang, 2021, Electronic Structure).[12](https://doi.org/10.1088/2516-1075/abde94)\n\n## Variants\n\nBare RPA is also called direct RPA (dRPA). In the early 1960s chemists used RPA synonymously with time-dependent Hartree-Fock, which includes additional ladder diagrams; this RPA with exchange (RPAx) suffers from instabilities of the Hartree-Fock reference.[1](https://www.annualreviews.org/content/journals/10.1146/annurev-physchem-040215-112308) The second-order screened exchange (SOSEX) correction eliminates self-correlation error for one-electron systems; a form with both interaction lines statically screened, SOSEX(W(0), W(0)), follows from the G3W2 self-energy contribution and avoids the expensive frequency integration of AC-SOSEX, and both RPA+SOSEX variants can be read as renormalized MP2 expressions that compensate the overestimation of correlation energy in RPA.[1](https://www.annualreviews.org/content/journals/10.1146/annurev-physchem-040215-112308) The renormalized singles excitation (rSE) correction prevents divergence of the second-order singles expression when the single-particle gap closes, and RPA + SOSEX + rSE gives the most balanced performance for reaction and activation energies among the assessed functionals.[2](https://iopscience.iop.org/article/10.1088/1367-2630/14/4/043002) The AXK method, the leading correction of an RPA-renormalized many-body perturbation theory with a frequency-independent approximate exchange kernel,\n\n\\[ \\Delta E_{C}(\\mathrm{AXK}) = -\\int_{0}^{1} d\\alpha\\, \\mathrm{Im} \\int_{0}^{\\infty} \\frac{d\\omega}{2\\pi}\\, \\mathrm{tr}\\, V\\,\\Pi_{\\mathrm{RPA}}(\\omega)\\, K_{\\mathrm{AXK}}\\,\\Pi_{\\mathrm{RPA}}(\\omega), \\]\n\nsubstantially improves RPA atomization energies and ionization potentials without worsening barrier heights, at \\( \\mathcal{O}(N^{4}\\log N) \\) cost.[19](https://pubs.aip.org/aip/jcp/article/139/17/171103/73108/Communication-Random-phase-approximation) For open-shell atoms, dRPA and RPA+SOSEX have been implemented with complex orbitals, making them implicit current density functionals.[4](https://www.osti.gov/servlets/purl/1535290)\n\n## Applications\n\nIn solid-state physics, the cubic-scaling implementation was applied to silicon self-interstitial and vacancy energetics with supercells up to 256 atoms, with RPA predicting defect energies in excellent agreement with experiment.[11](https://doi.org/10.1103/physrevb.90.054115) In surface physics, RPA resolved the CO adsorption puzzle on Cu(111), where LDA and GGA predict the hollow site while experiment and RPA give the on-top site.[8](https://fhi-aims.org/uploads/documents/Tutorial_Series_2021/Xinguo%20Ren%20FHI-aims%20Talk%20September%202021.pdf) In quantum chemistry, RPA captures nonlocal coupling between charge fluctuations separated in space and gives the correct \\( \\Delta E_{\\mathrm{RPA}} \\propto 1/R^{6} \\) van der Waals asymptote.[8](https://fhi-aims.org/uploads/documents/Tutorial_Series_2021/Xinguo%20Ren%20FHI-aims%20Talk%20September%202021.pdf) In nuclear physics, RPA was the main theoretical tool for nuclear excitations during the 1970s and 1980s, motivated by the inconsistency of the Tamm-Dancoff approximation with the equations of motion.[15](https://www.mdpi.com/2218-1997/9/3/141)\n\n## Limitations and alternatives\n\nSelf-screening error is the clearest failure mode: for the hydrogen atom, screening should be impossible with a single electron, yet RPA yields a non-zero response function, so the associated GW self-energy gives a non-zero correlation contribution that should be zero.[20](https://export.arxiv.org/pdf/2206.01169v1.pdf) The spurious self-correlation makes the on-top correlation hole too negative and underlies the failure to correctly dissociate odd-electron systems such as H2+.[1](https://www.annualreviews.org/content/journals/10.1146/annurev-physchem-040215-112308) RPA has relatively poor accuracy for nonisogyric processes such as atomization, ionization, and spin-flip, which break electron pairs.[1](https://www.annualreviews.org/content/journals/10.1146/annurev-physchem-040215-112308) Results depend on the reference: for equilibrium bond distances of 17 small molecules, RPA with Hartree-Fock references gives a mean absolute error over three times larger than RPA with PBE references.[1](https://www.annualreviews.org/content/journals/10.1146/annurev-physchem-040215-112308) Small gaps require much larger quadrature grids,[9](https://doi.org/10.1063/1.3442749) though the renormalized response series of RPA-renormalized perturbation theory stays finite even for zero-gap metals.[19](https://pubs.aip.org/aip/jcp/article/139/17/171103/73108/Communication-Random-phase-approximation) The RPA self-energy is the [GW approximation](https://www.edgepedia.org/gw-approximation), a single diagram with an electron Green's function G and the RPA screened interaction W.[3](https://courses.physics.illinois.edu/phys561/fa2005/lnotes/lec8.pdf) In cost, RPA at \\( \\mathcal{O}(N^{4}) \\) sits between DFT and the high-fidelity MP2, CCSD, and CCSD(T) hierarchy, and about one order of magnitude above hybrid DFT, with larger basis sets required than for semilocal DFT; it also converges slowly with kinetic energy cutoff because of the wavefunction cusp condition.[21](https://beast-echem.org/workshops/2024/rpa.pdf) Beyond-RPA corrections, including SOSEX, rSE, and AXK, address the underbinding of atomization and cohesive energies.[2](https://iopscience.iop.org/article/10.1088/1367-2630/14/4/043002)\n\n## References\n\n1. [Random-Phase Approximation Methods (Annual Review of Physical Chemistry)](https://www.annualreviews.org/content/journals/10.1146/annurev-physchem-040215-112308)\n2. [Assessment of correlation energies based on the random-phase approximation (Grüneis, Marsman, Harl, Schimka, Kresse, New J. Phys. 14, 043002, 2012)](https://iopscience.iop.org/article/10.1088/1367-2630/14/4/043002)\n3. [The Random Phase Approximation and the consequences for the One-Electron Green's Functions (UIUC Physics 561 Lecture 8, 2005)](https://courses.physics.illinois.edu/phys561/fa2005/lnotes/lec8.pdf)\n4. [Random phase approximation with second-order screened exchange for current-carrying atomic states](https://www.osti.gov/servlets/purl/1535290)\n5. [Random Phase Approximation with exchange in the dielectric matrix formulation (ACFDT-based RPAx variants)](https://hal.sorbonne-universite.fr/hal-01304895/document)\n6. [Periodic implementation of the random phase approximation with numerical atomic orbitals and dual reciprocal space grids](https://arxiv.org/html/2505.06021)\n7. [The Random Phase Approximation for Interacting Fermi Gases in the Mean-Field Regime (Forum of Mathematics, Pi)](https://www.cambridge.org/core/journals/forum-of-mathematics-pi/article/random-phase-approximation-for-interacting-fermi-gases-in-the-meanfield-regime/276E107AE84CFB3480AD888CB33D095E)\n8. [Basics and Recent Progress of Random Phase Approximation for First-principles Ground State Energy Calculations (X. Ren tutorial talk, FHI-aims, 2021)](https://fhi-aims.org/uploads/documents/Tutorial_Series_2021/Xinguo%20Ren%20FHI-aims%20Talk%20September%202021.pdf)\n9. [Henk Eshuis, Julian Yarkony, Filipp Furche (2010). Fast computation of molecular random phase approximation correlation energies using resolution of the identity and imaginary frequency integration. The Journal of Chemical Physics.](https://doi.org/10.1063/1.3442749)\n10. [Staggered Mesh Method for Correlation Energy Calculations of Solids: Random Phase Approximation in Direct Ring Coupled Cluster Doubles and Adiabatic Connection Formalisms](https://par.nsf.gov/servlets/purl/10322307)\n11. [Merzuk Kaltak, Jiří Klimeš, Georg Kresse (2014). Cubic scaling algorithm for the random phase approximation: Self-interstitials and vacancies in Si. Physical Review B.](https://doi.org/10.1103/physrevb.90.054115)\n12. [Yu-Chieh Chi, Chen Huang (2021). Accelerate stochastic calculation of random-phase approximation correlation energy difference with an atom-based correlated sampling. Electronic Structure.](https://doi.org/10.1088/2516-1075/abde94)\n13. [David Bohm, David Pines (1951). A Collective Description of Electron Interactions. I. Magnetic Interactions. Physical Review.](https://doi.org/10.1103/physrev.82.625)\n14. [David Bohm, David Pines (1953). A Collective Description of Electron Interactions: III. Coulomb Interactions in a Degenerate Electron Gas. Physical Review.](https://doi.org/10.1103/physrev.92.609)\n15. [Introducing the Random Phase Approximation Theory (Universe 9(3):141)](https://www.mdpi.com/2218-1997/9/3/141)\n16. [H. Ehrenreich, M. H. Cohen (1959). Self-Consistent Field Approach to the Many-Electron Problem. Physical Review.](https://doi.org/10.1103/physrev.115.786)\n17. [W R Johnson and colleagues (1980). Relativistic Random-Phase Approximation. Physica Scripta.](https://doi.org/10.1088/0031-8949/21/3-4/029)\n18. [Gustavo E. Scuseria, Thomas M. Henderson, Danny C. Sorensen (2008). The ground state correlation energy of the random phase approximation from a ring coupled cluster doubles approach. The Journal of Chemical Physics.](https://doi.org/10.1063/1.3043729)\n19. [Communication: Random phase approximation renormalized many-body perturbation theory (Bates & Furche, J. Chem. Phys. 139, 171103, 2013)](https://pubs.aip.org/aip/jcp/article/139/17/171103/73108/Communication-Random-phase-approximation)\n20. [The self-screening error in the random-phase approximation (RPA) and the GW approximation](https://export.arxiv.org/pdf/2206.01169v1.pdf)\n21. [The Random Phase Approximation (RPA) and GW approximation for electrochemistry (BEAST workshop, August 2024)](https://beast-echem.org/workshops/2024/rpa.pdf)\n\n---\n*Topic: Encyclopedia › Physical world and mathematics › Physics › Matter and radiation physics › Condensed matter physics › Electronic and magnetic properties*\n\n*Initially written Sep 29, 2026 · Edited: Oct 15, 2026*\n\n*Copyright 2026 EdgeChat AI, a subsidiary of Biostate AI.*\n\nLicense: Edgepedia Community License 1.0, https://www.edgepedia.org/edgepedia/license\n", "same_as": [], "url": "https://www.edgepedia.org/random-phase-approximation", "markdown_url": "https://www.edgepedia.org/random-phase-approximation.md", "license": { "name": "Edgepedia Community License 1.0", "url": "https://www.edgepedia.org/edgepedia/license", "summary": "Free with credit, commercial use included. AI training is open to everyone. For other uses, organizations over USD 100M in revenue or 100M monthly users license separately.", "spdx": "LicenseRef-Edgepedia-Community-1.0" }, "credit": "\"Random phase approximation\", Edgepedia (EdgeChat), https://www.edgepedia.org/random-phase-approximation. Edgepedia Community License 1.0.", "credit_md": "\"[Random phase approximation](https://www.edgepedia.org/random-phase-approximation)\", Edgepedia (EdgeChat), [https://www.edgepedia.org/random-phase-approximation](https://www.edgepedia.org/random-phase-approximation). [Edgepedia Community License 1.0](https://www.edgepedia.org/edgepedia/license).", "credit_html": "\"Random phase approximation\", Edgepedia (EdgeChat), https://www.edgepedia.org/random-phase-approximation. Edgepedia Community License 1.0.", "speakable": "The random phase approximation is a many-body method that computes screening and correlation energy of interacting electron systems by summing ring diagrams to infinite order." }