Correlation energy expressions from the adiabatic-connection fluctuation-dissipation theorem approach

Motivation

• RPA is considered promising for post-Kohn-Sham correlation, but its performance for long-range interactions is poorer than semilocal density functionals at small interelectronic distances.Range separation is presented as an approach for using RPA while treating short-range interactions differently.Source: §1 · p. 2

Formulations

• The interaction-strength-averaged formulation connects to the dynamic dielectric-matrix formulation: analytical integration over interaction strength followed by numerical frequency integration yields the latter, and further analytical integration converts both intermediate forms to a common form.The text also clarifies the origin of the 1/4 prefactor in the RPAx-II plasmon formula.Source: §1 · pp. 2–3

RPA variants

• The paper studies adiabatic-connection RPA variants that differ in how exchange interactions are handled.The variants include dRPA-I, dRPA-II, RPAx-I, and RPAx-II.Source: §1 · p. 3

Contributions

• The original dRPA-II variant contracts the dRPA density matrix with antisymmetrized integrals, resembles SOSEX, and includes higher-order screened exchange effects.The paper also derives a new adiabatic-connection expression for the previously introduced RPAx-II variant.Source: §1 · p. 3

Scope

• The authors derive spin-adapted expressions and compare variants on atomic and molecular systems with and without range separation.The paper presents expressions without range separation and states that they can be generalized to range-separated interactions.Source: §1 · p. 4

Connection formulation

• Four variants arise because dRPA and RPAx correlation density matrices are not antisymmetric, so contractions with the two energy expressions are no longer equivalent.Variants I use non-antisymmetrized integrals; variants II use antisymmetrized integrals.Source: §2 · p. 5
• dRPA-I neglects exchange-correlation screening effects and its energies tend to be too strongly negative.At second order, it yields direct MP2 without the MP2 exchange term, rather than standard MP2.Source: §2 · p. 6
• dRPA-II contracts the dRPA correlation density matrix with antisymmetrized two-electron integrals, restoring the correct second-order MP2 limit; although Eq. (2) might suggest a 1/4 factor in Eq. (7), recovery requires 1/2.Source: §2 · p. 6
• RPAx-I includes exchange screening in the polarization propagator and has the correct MP2 limit, although its density matrix violates antisymmetry properties to some extent at higher orders.RPAx-I was introduced by Toulouse et al.15,33.Source: §2 · pp. 6–7
• RPAx-II contracts the RPAx density matrix with antisymmetrized integrals and properly reduces to MP2 at second order.The variant was first introduced by McLachlan and Ball.49Source: §2 · p. 7

Propagator

• The formulation uses occupied-virtual orbital products and independent one-particle excitation energies in a space sized by the occupied and virtual orbital counts.The passages define ε_ia,jb as (ε_a−ε_i)δ_ijδ_ab and identify No and Nv as the occupied and virtual orbital counts.Source: §3 · p. 7
• The fluctuation-dissipation theorem gives a supermatrix representation of the correlation density matrix from the RPA polarization propagator.The propagator is considered at interaction strength α and frequency ω, with Π0(ω) denoting the corresponding non-interacting propagator.Source: §3 · p. 9
• The dRPA and RPAx correlation density matrices have the same form in terms of eigenvector matrices X and Y, but their eigenvectors differ.The supermatrix kernel uses the Hartree kernel for dRPA and the Hartree-Fock kernel for RPAx.Source: §3 · pp. 8–9

Correlation energy expressions in spin-orbital basis

• dRPA-I and RPAx-II allow analytic integration over the adiabatic connection, yielding plasmon formulae.The paper reviews dRPA-I and provides expressions for RPAx-II, which is less documented.Source: §4 · p. 10
• dRPA-II and RPAx-I mix non-antisymmetrized V and antisymmetrized W integrals, preventing analytic integration over the adiabatic connection.The paper provides original expressions for dRPA-II and RPAx-II, and reviews RPAx-I in the range-separation context.Source: §4 · p. 10
• The review of dRPA-I provides notation and comparisons with other variants, while the RPAx-I variant had been discussed in detail for range separation.The dRPA-I variant was already well documented in the literature.Source: §4 · p. 10

4.1.1 Adiabatic-connection formula

• The dRPA-I adiabatic-connection expression can be formed directly from matrices in the RPA response equation, without explicitly calculating the eigenvector matrices.The trace is applied to block matrices that are half the size of the supermatrices.Source: §4.1.1 · p. 11

4.1.2 Plasmon formula

• The plasmon formula follows from an equivalent trace expression that permits analytical integration over the coupling parameter.The passages also describe the sum of direct Tamm-Dancoff excitation energies as a plasmon-formula component.Source: §4.1.2 · pp. 11–12

4.1.3 Alternative plasmon formula

• The alternative plasmon form decomposes dRPA-I correlation energy into kinetic and potential contributions.The potential contribution is the integrand of the adiabatic-connection expression at α = 1; the kinetic contribution follows by subtraction.Source: §4.1.3 · pp. 13–14
• For degenerate orbitals with ε = 0, the kinetic correlation energy vanishes, consistent with dRPA-I describing left-right static correlation in bond dissociations.The passage identifies this limit with systems having static correlation only.Source: §4.1.3 · p. 14

4.2.1 Adiabatic-connection formula

• When Pα is small, approximating Qα^-1 as 2I − Qα yields an RPAx-II energy expression analogous to dRPA-I, differing in its matrix terms.The passages describe Qα = I + Pα and state that the approximation applies when Pα is small.Source: §4.2.1 · p. 15
• The RPAx-II adiabatic-connection expression contracts Qα, obtained from the RPAx response equation, with antisymmetrized two-electron integrals B; its prefactor changes from 1/2.It is derived analogously to dRPA-I.Source: §4.2 · p. 14, §4.2.1 · p. 15

4.2.2 Plasmon formula

• The RPAx-II plasmon formula uses a factor of 1/4 because it uses antisymmetrized two-electron integrals.The passage contrasts this factor with the factor of 1/2 in the dRPA-I plasmon formula and notes that the factor had been debated.Source: §4.2.2 · p. 16

4.2.3 Alternative plasmon formula

• The alternative plasmon expression separates potential and kinetic correlation energies, and the kinetic contribution vanishes when ε = 0.The vanishing behavior is also stated for dRPA-I.Source: §4.2.3 · p. 17

• The dRPA-II correlation energy is expressed in terms of eigenvectors of the dRPA polarization response.Its expression is described as similar to the earlier RPAx-II correlation-energy expression.Source: §4.3 · pp. 17–18
• dRPA-II is closely analogous to, but usually not equal to, SOSEX in the ring-CCD formulation.The passage names this variant “adiabatic-connection SOSEX” and says its relationship to original SOSEX is discussed in Ref. 34.Source: §4.3 · p. 18

• The RPAx-I correlation energy has the same form as the dRPA-I expression, with the RPAx matrix replacing the dRPA matrix.Source: §4.4 · p. 18
• The RPAx-I formulation has been applied in range-separated density-functional theory.Source: §4.4 · p. 18

closed-shell systems

• Spin adaptation gives the RPA matrices a block-diagonal form with one singlet excitation block and three triplet blocks.The transformation is applied to the matrices in the RPA equations.Source: §5 · p. 19
• The spin-adapted dRPA-I and dRPA-II correlation energies involve only singlet excitations.For dRPA-I, the corresponding plasmon formula also contains only singlet excitation energies.Source: §5 · pp. 20, 22, 24
• RPAx spin adaptation includes a triplet B block with an opposite sign, making the adaptation less trivial.The passage identifies a minus sign in the last triplet block of the B matrix.Source: §5 · p. 21
• RPAx-II correlation energy receives contributions from both singlet and triplet excitations.Its spin-adapted plasmon formula decomposes into sums over singlet and triplet excitation energies.Source: §5 · pp. 22–23
• RPAx-I has only singlet contributions because the triplet blocks of its K matrix are zero.Source: §5 · p. 24

6.1 Atomic correlation energies

• For atoms, dRPA-II and especially dRPA-IIa (AC-SOSEX) reproduce reference correlation energies very well, whereas full-range RPAx-II usually overestimates them by more than 150%.The reference values are FCI-quality correlation energies; the figure compares ratios across several RPA variants and orbital choices.Source: Figure 1 · p. 26, §6.1 · p. 27

6.2 Bond lengths and harmonic vibrational frequencies

• For H2, HF, and N2, full-range dRPA-I and dRPA-II improve on HF and KS PBE, while dRPA-IIa is less accurate than dRPA-II; with range separation, RPAx-II appears best.Full-range RPAx response instabilities limit calculations to full-range dRPA values, whereas no instabilities are found for range-separated RPAx calculations.Source: Figure 2 · p. 29, §6.2 · p. 30

6.3 London dispersion interactions

• The study also examines interaction-energy curves for the rare-gas dimers He2, Ne2, and Ar2.Source: §6.3 · p. 30
• LDA orbitals produce instabilities for Ne2 and Ar2 across a wide range of distances, whereas PBE orbitals produce none, with or without range separation.The RPA calculations use PBE orbitals for full-range calculations and a short-range PBE density functional for range-separated calculations.Source: §6.3 · p. 30
• In full-range calculations, RPAx-II overestimates binding by a factor of 3 or more, while dRPA-I largely underestimates interaction energies and predicts no He2 minimum.The absence of a He2 minimum is mostly due to the poor single-determinant energy.Source: §6.3 · p. 32
• Full-range dRPA-II gives more binding than dRPA-I but still tends to underestimate interaction energies, while dRPA-IIa stays close to dRPA-I.The methods differ by exponentially decaying exchange integrals that quickly become negligible for van der Waals complexes.Source: §6.3 · p. 32
• Full-range RPAx-I agrees quite well with reference curves for Ne2 and Ar2, but He2 without range separation has the poorest interaction curve because its EXX energy is too repulsive.For He2, the RSH energy is almost in perfect agreement with the reference, a prerequisite for the good range-separated results.Source: §6.3 · p. 32
• With range separation, the RPA variants give closer interaction-energy curves while retaining their trends; RPAx-I is globally most accurate, whereas dRPA variants underestimate and RPAx-II significantly overbinds Ar2.The curves shown are for He2, Ne2, and Ar2.Source: Figure 3 · p. 31, §6.3 · p. 32

7 Conclusions

• The variants differ in their treatment of exchange in the density matrix and integral contraction; dRPA-II is introduced here and resembles SOSEX, especially its dRPA-IIa form.dRPA-I neglects exchange, whereas dRPA-II uses a density matrix without exchange contracted with antisymmetrized integrals.Source: §7 · p. 33
• The work derives new RPAx-II expressions and connects dRPA-I and RPAx-II to plasmon formulations, explaining their different prefactors of 1/2 and 1/4, respectively.The RPA method itself is not new; the derived expressions and clarified connections are contributions of this work.Source: §7 · p. 33
• All variants except dRPA-I recover the MP2 correlation energy in the second-order interaction limit, and the paper derives spin-adapted forms for closed-shell systems.The forms were implemented and tested with and without range separation of the electron-electron interaction.Source: §7 · p. 33
• Without range separation, RPAx variants frequently suffer response-equation instabilities that can prevent meaningful correlation energies; range separation avoids the reported instabilities.The authors report that the range-separated RPAx variants can therefore be safely applied.Source: §7 · p. 33
• No variant is uniformly best: without range separation, dRPA-II performs well for atomic correlation and equilibrium molecular properties, while RPAx-I is more accurate for dispersion energies.dRPA-II significantly underestimates London dispersion interaction energies.Source: §7 · p. 33
• With range separation, all RPA variants tend to be more accurate and more similar; RPAx-II appears best for equilibrium molecular properties, while RPAx-I is best for dispersion energies.These tests do not establish a single best variant across properties.Source: §7 · pp. 33–34

A Second-order approximations to the RPA correlation energy

• dRPA-I has a direct-MP2 second-order limit, without exchange.The section identifies this as a MP2-like correlation energy rather than the normal MP2 expression.Source: A Second-order approximations to the… · p. 38
• RPAx-II and its approximation have the usual MP2 expression as their second-order limit.The limit is exactly MP2, subject to possible replacement of Hartree-Fock orbitals and energies with Kohn-Sham quantities.Source: A Second-order approximations to the… · p. 38
• The dRPA-II and RPAx-I correlation-energy variants also have the usual MP2 expression as their second-order limit.For dRPA-II, antisymmetry of B and its prefactor of 1/2 give another form of the usual MP2 expression.Source: A Second-order approximations to the… · pp. 38–39
• Closed-shell orbital degeneracies invalidate the non-degenerate expansion, so the response equation is spin-adapted before perturbation expansion.The resulting singlet and triplet expansions are formally identical if no further orbital-energy-difference degeneracies occur.Source: A Second-order approximations to the… · p. 39
• At second order, the spin-adapted variants reduce to MP2 except dRPA-I, which reduces to direct MP2.The closed-shell derivation checks that the spin-adapted correlation expressions recover these limits.Source: A Second-order approximations to the… · p. 39