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Random-phase approximation and its applications in computational chemistry and materials science
Xinguo Ren, Patrick Rinke, Christian Joas, Matthias Scheffler
TL;DR
RPA addresses the need for an electronic-structure method that improves beyond traditional DFT while treating correlation and realistic systems. The review synthesizes its history, theoretical formulations, applications, limitations, corrections, and computational cost. It presents RPA as promising for quantum chemistry and materials science, while identifying computational expense and known accuracy limitations.
Problem
Traditional local and semi-local DFT approximations face accuracy and reliability limits, motivating improved treatments of electronic correlation for realistic systems.
Method
The paper reviews RPA through its historical development, adiabatic-connection, DFT, and many-body perturbation formulations, applications, corrections, and computational implementations.
Results
RPA includes long-range interactions automatically, applies beyond the homogeneous electron gas, and has shown promise across computational chemistry, physics, and materials science.
Takeaways & Limitations
RPA and its generalizations are expected to play an increasingly important role in computational materials science and related fields.
Abstract
from arXiv · showhide
The random-phase approximation (RPA) as an approach for computing the electronic correlation energy is reviewed. After a brief account of its basic concept and historical development, the paper is devoted to the theoretical formulations of RPA, and its applications to realistic systems. With several illustrating applications, we discuss the implications of RPA for computational chemistry and materials science. The computational cost of RPA is also addressed which is critical for its widespread use in future applications. In addition, current correction schemes going beyond RPA and directions of further development will be discussed.
I. INTRODUCTION
RPA emerged as a first-principles approach addressing deficiencies of local and semi-local DFT while retaining access to non-local correlation and long-range interactions. This review traces its development, applications, limitations, corrections, and computational challenges.
- Motivation: Exact exchange removes Hartree self-interaction exactly, while RPA correlation is fully non-local and includes long-range van der Waals interactions automatically.RPA correlation nevertheless contains self-correlation and is non-zero for one-electron systems.
- Motivation: RPA is applicable to small-gap or metallic systems where finite-order many-body perturbation theories break down.
- Practical formulations: Standard RPA is commonly performed non-self-consistently using a preceding LDA or GGA reference calculation.
- Limitations and corrections: Standard RPA systematically underestimates binding energies and fails to describe stretched radicals, motivating local, non-local, and range-separated corrections.Range-separated approaches can also avoid slow basis-function convergence associated with the short-range part of RPA.
- Computational challenges: RPA calculations remain computationally costly, and input-orbital dependence makes self-consistency desirable even though OEP-RPA is numerically demanding.Practical calculations therefore are expected to remain mainly post-processing calculations in the near future.
II. THEORY AND CONCEPTS
RPA can be derived through the adiabatic-connection framework, which connects a reference Hamiltonian to the fully interacting system. The resulting energy construction depends on the chosen coupling path and mean-field reference.
- Adiabatic connection: The adiabatic connection introduces coupling-strength-dependent Hamiltonians linking a reference Hamiltonian at λ = 0 to the physical interacting Hamiltonian at λ = 1.
- Reference system: The reference Hamiltonian is mean-field and consists of a sum of single-particle Hamiltonians.
- Reference system: The mean-field potential can be the Hartree-Fock potential or the Hartree plus exchange-correlation potential in DFT.
- Energy construction: The interacting ground-state energy is obtained from the coupling-dependent wave function and the Hellmann-Feynman theorem under a normalized-state condition.
- Choice of path: DFT chooses an adiabatic-connection path that keeps the electron density fixed, whereas many-body perturbation theory often uses a linear connection path.
B. RPA derived from ACFD
The ACFD formulation expresses exchange-correlation energy through density-response functions along an adiabatic-connection path. RPA approximates these response functions simply using the Kohn-Sham reference response.
- DFT foundation: In Kohn-Sham DFT, the total energy separates into kinetic, external, Hartree, and exchange-correlation terms, with many-body complexity concentrated in the exchange-correlation energy.
- Density correlations: The exchange-correlation hole describes density depletion around an electron and is related to the density-density correlation function.
- Response formulation: The fluctuation-dissipation theorem links density fluctuations to the system’s response properties.
- ACFD formulation: The ACFD expression transforms exchange-correlation-energy evaluation into calculating response functions for fictitious systems along the adiabatic-connection path.
- RPA approximation: RPA uses a particularly simple approximation to the response function, with χ0 given by the independent-particle response of the Kohn-Sham reference system.
- Energy components: The RPA exchange-correlation energy separates into exact exchange and an RPA correlation term.
C. RPA derived from MBPT
The MBPT formulation builds correlation energy from perturbative and diagrammatic expansions, then connects RPA to infinite-order ring-diagram summations and GW theory. It also clarifies reference dependence and distinctions between perturbative RPA and self-consistent GW.
- Perturbation theory: A linear adiabatic-connection path permits a Taylor expansion of the ground-state wave function and an order-by-order expansion of the interacting energy.
- Perturbation theory: The sum of zeroth- and first-order terms gives Hartree-Fock energy, while higher-order terms constitute correlation energy.
- Goldstone diagrams: Second-order perturbation theory contains single- and double-excitation contributions, with the double-excitation term splitting into direct and exchange MP2 energies.
- Green-function formulation: Green-function and self-energy formulations become preferable when arbitrarily high-order selective diagram summations are required.
- RPA and GW: RPA correlation energy arises from applying the GW approximation to ring-structured self-energy diagrams and integrating over the interaction strength.
- RPA diagrammatics: The leading RPA term corresponds to the second-order direct MP2 contribution.
- Reference dependence: Perturbative RPA can use Hartree-Fock or local/semi-local Kohn-Sham references, whereas self-consistent GW includes improper self-energy diagrams and yields a different total energy.
D. Link to coupled cluster theory
RPA is closely connected to coupled-cluster theory through ring diagrams and the ring-CCD formulation. Solving a nonlinear Riccati equation yields amplitudes from which the RPA correlation energy can be expressed.
- RPA has an intimate relationship with coupled-cluster theory, motivating its study in quantum chemistry.
- Coupled-cluster theory uses an exponential wave-function ansatz built from excitation operators of different orders acting on a non-interacting reference state.
- Projecting the many-body Schrödinger equation onto excited configurations produces coupled nonlinear equations for the coupled-cluster amplitudes, solved self-consistently.
- The CCD approximation retains only double excitations, while its diagrams include ring, ladder, and mixed diagram classes.
- The ring-CCD amplitudes satisfy a matrix Riccati equation involving occupied and virtual states and two-electron Coulomb integrals.
- Ring-CCD is analytically equivalent to plasmonic RPA, but the nonlinear equation has multiple solutions requiring judicious amplitude selection.
A. RPA implementations and scaling
RPA implementations use local orbitals, plane waves, or augmented-plane-wave bases, with computational scaling strongly dependent on the representation and factorization strategy. RI-based local-orbital implementations reduce the dominant scaling to O(N^4).
- RPA implementations can use local orbitals, plane waves, or linearized augmented plane waves, with scaling and efficiency central because RPA is computationally expensive.
- Furche’s molecular particle-hole implementation scales as O(N^6), reducible to O(N^5) with a plasmon-pole formulation or O(N^4) with resolution of identity.
- RI-RPA expands occupied-virtual orbital-pair products in auxiliary basis functions, reducing the response-matrix rank from Nocc·Nvir to Naux.
- The resulting RPA correlation-energy evaluation is a low-rank matrix calculation, while building the response matrix χ0 dominates and scales as O(N^4).
- Plane-wave implementations also scale as O(N^4), with plane waves serving the role of auxiliary basis functions.
B. Speed-up of RPA with iterative methods
Iterative RPA approaches exploit the fact that only a small fraction of dielectric-function eigenvalues differs substantially from one. Dominant-eigenvalue calculations retain formal O(N^4) scaling while substantially reducing the prefactor.
- The RPA correlation energy can be rewritten using the eigenvalues of the dielectric function represented in the auxiliary basis.
- Eigenvalues close to one contribute vanishingly to the correlation energy, so the full dielectric spectrum is unnecessary for accurate calculations.
- Only a small fraction of dielectric eigenvalues differs significantly from one across a set of different materials.
- Selecting the dominant eigenvalues preserves formal O(N^4) scaling but reportedly reduces the computational prefactor by 100–1000.
IV. COMPUTIONAL SCHEMES BEYOND RPA
Standard RPA describes long-range interactions well but misses important short-range and multi-center correlation effects. Corrections therefore range from semi-local RPA+ schemes to fully non-local approaches, with trade-offs between properties.
- Improving standard RPA focuses on achieving better accuracy through correction schemes beyond the original approximation.
- RPA describes long-range interactions well, but its short-range correlation is inadequate and can make the HEG pair-correlation function spuriously negative at small separations.
- RPA+ adds a semi-local correction for inhomogeneous systems, but it does not significantly improve energy differences such as small-molecule atomization energies.
- The limited improvement of RPA+ has been attributed to inaccurate treatment of multi-center non-locality in the correlation hole, which semi-local corrections cannot correct.
- A fully non-local correction introduces a functional of the difference between GGA and exact-exchange energy densities, with an empirical parameter α.
- Choosing F(f)=f fits atomization energies but destroys the correct H2 dissociation limit obtained with standard RPA, motivating a more complex functional form.
B. Screened second-order exchange (SOSEX)
SOSEX modifies the ring-CCD formulation of RPA by replacing unsymmetrized Coulomb integrals with antisymmetrized ones. The review also situates SOSEX alongside renormalized single excitations and their combined r2PT scheme, highlighting improved energies alongside a dissociation trade-off.
- SOSEX formulation: SOSEX is obtained in ring-CCD by inserting antisymmetrized Coulomb integrals instead of unsymmetrized integrals.The replacement yields the RPA+SOSEX correlation-energy expression.
- SOSEX effects: SOSEX removes one-electron self-correlation and substantially reduces RPA’s short-range over-correlation, producing significantly better total energies.The correction has been examined for both solids and molecular properties.
- SOSEX effects: SOSEX substantially reduces RPA’s underestimation of atomization energies but worsens the dissociation of covalent diatomic molecules.RPA describes covalent dissociation well, whereas SOSEX does not preserve that behavior.
- SOSEX formulation: RPA and SOSEX correlation energies are usually evaluated with orbitals from a preceding KS or generalized KS calculation.Within many-body perturbation theory, they represent infinite-order summations of selected diagram classes.
- Single excitation correction: Renormalized single excitations sum higher-order single-excitation diagrams to address the divergence of the second-order SE term for systems with zero direct gap.The summation is illustrated with Goldstone diagrams whose crossed dashed lines represent Δv_pq matrix elements.
- Combination with SOSEX: RPA, SOSEX, and rSE form distinct infinite diagrammatic series whose leading terms match second-order RSPT, motivating their combination as r2PT.The resulting RPA+SOSEX+rSE scheme is described as a renormalization of normal second-order perturbation theory.
D. Other “beyond-RPA” activities
Beyond-RPA developments add exchange, response-kernel, range-separation, and perturbative corrections, but their performance remains system-dependent. Benchmarks show improvements for selected molecular properties, alongside trade-offs and a lack of uniformly accurate binding curves.
- Correction strategies: Beyond-RPA schemes improve RPA by adding exchange effects, exchange-correlation kernels, range separation, or higher-order perturbative corrections.RPAx includes exchange-type contributions, kernel approaches modify the density response, range separation replaces short-range RPA, and some corrections target third-order exactness.
- Dissociation of diatomic molecules: Standard RPA and RPA+ correctly reach covalent molecules’ atomic limits at large separations, but exhibit a positive intermediate-distance bump.This behavior is reported for H2 and N2, while MP2 diverges in the dissociation limit for both molecules.
- Dissociation of diatomic molecules: SOSEX reduces RPA’s atomization-energy underestimation but worsens covalent-dimer dissociation, whereas rSE slightly improves equilibrium binding and can shift N2 below zero asymptotically.The SOSEX-induced overestimation at large separations persists in r2PT because rSE does not remove it.
- Dissociation of diatomic molecules: For Ar2, all RPA-based methods reproduce C6/R6 asymptotics, RPA underestimates C6 by ∼9%, and rSE substantially improves equilibrium binding toward the Tang-Toennies reference.MP2 overestimates C6 by ∼18%, while SOSEX has little effect for Ar2.
- Dissociation of diatomic molecules: For Be2, rSE improves agreement with reference binding energies, SOSEX reduces the intermediate bump but weakens equilibrium binding, and r2PT remains too weak at equilibrium.Be2 combines static correlation with long-range vdW interactions, producing more complex correction behavior than the other dimers.
- Scope and limitations: No discussed RPA-based approach is quantitatively accurate for binding-energy curves across all bonding situations, and more benchmarks are needed for some beyond-RPA corrections.Standard RPA performs remarkably well for activation energies, while existing beyond-RPA corrections do not improve that accuracy.
3. vdW interactions: S22 set
The S22 benchmark shows that RPA substantially improves non-covalent binding energies over semilocal functionals, while corrections improve some interaction types but leave challenging cases and quantitative limitations.
- S22 benchmark: RPA improves binding energies considerably over semilocal functionals for the 22 weakly bound S22 complexes.The set contains hydrogen-bonded, dispersion-bonded, and mixed complexes.
- S22 benchmark: 2.79 kcal/mol, 0.90 kcal/mol, and 0.41 kcal/mol are reported standard-RPA MAEs for S22, with discrepancies attributed to basis-set incompleteness and BSSE.
- Corrections beyond RPA: rSE systematically reduces errors; SOSEX has little effect on dispersion interactions, while r2PT overshoots hydrogen bonding but improves the other two bonding types on average.
- Remaining challenges: π-π stacking, exemplified by the slip-parallel benzene dimer, is the most challenging S22 case for RPA-based methods.RPA has the largest relative error there; rSE provides little improvement and SOSEX slightly worsens it.
- Corrections beyond RPA: RPA+rSE gives the smallest MAE, while r2PT gives the smallest MAPE among the investigated approaches.The approaches are compared using ME, MAE, MAPE, and MaxAPE.
- Reaction barriers: Standard RPA significantly improves barrier-height performance over PBE0, but rSE and SOSEX deteriorate it, while their combination largely cancels the errors.The comparison uses HTBH38 and NHTBH38 test sets.
C. Adsorption at surfaces
RPA is presented as a promising surface-adsorption method because it balances descriptions of solids, adsorbates, and interfaces, including systems where common functionals fail.
- Applications: RPA has been successfully applied to adsorbates and surfaces including Xe, PTCDA, CO, benzene, graphene, and multiple metals.
- CO adsorption puzzle: For CO on Cu(111), conventional local and semilocal DFT favors the hollow site, whereas experiment identifies the on-top site as most stable.
- CO adsorption puzzle: RPA is reported as the only approach giving both good adsorption energies and surface energies across the examined surfaces.GGAs and hybrid functionals were reported to yield at most one of these properties well.
- Long-range behavior: RPA captures the expected -C3/(d-d0)^3 behavior at large molecule-surface separations.
- Outlook: RPA-based methods are expected to become more important in computational materials science, while further accuracy and cost improvements remain needed.The outlook identifies corrections, reduced scaling, and atomic forces as development directions.
Appendix A: RI-RPA implementation in FHI-aims
The RI-RPA implementation represents response quantities in an auxiliary basis, converting the correlation-energy calculation into matrix operations and reducing the effective rank of the problem.
- RI formulation: RI-RPA represents both the independent response function χ0(iω) and Coulomb interaction v in an auxiliary basis.
- RI formulation: The transformation matrix C reduces matrix rank from Nocc*Nvir to Naux, where Nocc, Nvir, and Naux denote occupied, virtual, and auxiliary dimensions.
- Efficiency: Choosing Naux much smaller than Nocc*Nvir considerably reduces computational effort.The auxiliary basis is constructed for accuracy and efficiency with atom-centered basis functions.
- Matrix operations: The trace operation in the RPA correlation-energy expression can be reinterpreted as a summation over auxiliary-basis indices.
- Computational cost: The most expensive RI-RPA step is constructing the independent response function, with scaling proportional to NauxNoccNvir^2.Plane-wave implementations have an analogous role for the number of response-function plane waves.