We derive a renormalized many-body perturbation theory (MBPT) starting from the random phase approximation (RPA). This RPA-renormalized perturbation theory extends the scope of single-reference MBPT methods to small-gap systems without significantly increasing the computational cost. The leading correction to RPA, termed the approximate exchange kernel (AXK), substantially improves upon RPA atomization energies and ionization potentials without affecting other properties such as barrier heights where RPA is already accurate. Thus, AXK is more balanced than second-order screened exchange [A. Grüneis et al, J. Chem. Phys.131, 154115 (2009)], which tends to overcorrect RPA for systems with stronger static correlation. Similarly, AXK avoids the divergence of second-order Møller-Plesset (MP2) theory for small gap systems and delivers a much more consistent performance than MP2 across the periodic table at comparable cost. RPA+AXK thus is an accurate, non-empirical, and robust tool to assess and improve semi-local density functional theory for a wide range of systems previously inaccessible to first-principles electronic structure calculations.

Many-body perturbation theory (MBPT)1 provides an intuitive and computationally manageable approach to the quantum many-body problem. Second-order Møller-Plesset (MP2) theory2 is especially popular for large-gap systems such as closed-shell organic compounds, clusters of weakly interacting molecules, as well as insulating solids3 and their surfaces. Nowadays, the most important role of MP2 is to check and validate results from semi-local density functional theory (DFT). The key limitation of finite-order MBPT is its high sensitivity to small gaps and even modestly strong static correlation. MP2 fails catastrophically for a wide range of transition metal and lanthanide compounds with open d and f shells,4–6 but also for many radicals,7 bulk metals,8 metal clusters,9 and metallic surfaces. Including higher orders in MBPT or using a Kohn-Sham (KS) reference10,11 does not lead to an improvement.12,13 Most reactive intermediates, e.g., in transition metal catalysis, and many important materials such as nanowires or graphene sheets have small or zero gaps, leaving computational scientists with a bewildering “functional soup”14 often yielding substantially different results for these systems.

Gell-Mann and Brueckner15 showed that the random phase approximation (RPA) is the simplest remedy to the divergence of MBPT for the uniform electron gas. More recently, it has become clear that RPA can correct the deficiencies of semi-local functionals for mid- and long-range interactions for both large- and small-gap systems without empiricism.16 However, RPA performs somewhat poorly for non-isogyric processes that do not conserve the number of electron pairs, such as atomization, ionization, or spin flipping.17 These processes involve large changes in short-ranged electron-electron interactions which are known to be inaccurately described by RPA18 largely due to self-correlation of same-spin electrons.

Robust, computationally efficient, and widely applicable beyond-RPA methods have been elusive to date. The second-order screened exchange (SOSEX)19,20 method, arguably the most successful beyond-RPA method so far, does not diverge for small-gap systems and is free of one-electron self-interaction,21 but SOSEX reaction-barrier heights are less accurate than the RPA ones.22,23 Systematic corrections to RPA are accessible from coupled cluster (CC) theory;24 however, the computational cost of CC methods is substantially higher than that of low-order perturbation theory, and small-gap systems are still affected by the unphysical behavior of the underlying Hartree-Fock reference.25 Semi-local beyond-RPA corrections have some difficulty capturing non-local correlation in molecules without an empirical correction.26 Local field corrections such as the ISTLS method27 are computationally challenging and have not been widely applied to molecules to date.

Given the KS determinant Φ and the full Hamiltonian

$\hat{H}$
Ĥ⁠, the ground-state energy of a many-electron system is
$E = \langle \Phi | \hat{H} | \Phi \rangle + E^{{\rm C}}$
E=Φ|Ĥ|Φ+EC
. According to the zero-temperature fluctuation-dissipation theorem, the ground-state correlation energy within the adiabatic-connection (AC) formalism is28 

\begin{equation}E^{{\rm C}} = -\int \limits _{0}^{1}d\alpha\; \text{Im} \int \limits _{0}^{\infty }\frac{d\omega }{2\pi } \text{tr}\lbrace \mathbf {V}(\bm {\Pi }_\alpha {(\omega )} -\bm {\Pi }_0{(\omega )} ) \rbrace .\end{equation}
EC=01dαIm0dω2πtr{V(Πα(ω)Π0(ω))}.
(1)

While a correlation energy may be defined for any reference, Eq. (1) holds only in the AC formalism of density functional theory.17,

$\bm {\Pi }_\alpha$
Πα is the causal polarization propagator29 at coupling strength α and V denotes the bare Coulomb interaction.
$\bm {\Pi }_\alpha$
Πα
and V are 2 × 2 supermatrices30 of dimension 2 × Nph, where Nph is the dimension of the particle-hole (ph) space. Denoting the ph Coulomb interaction by
$\mathbf {B}^{{\rm H}}_{iajb} = \langle ij \vert ab \rangle$
BiajbH=ij|ab
, where ⟨ij|ab⟩ is an electron repulsion integral in Dirac notation,
$\mathbf {V} = \left(\begin{array}{cc}{\scriptstyle\mathbf {B}^{{\rm H}}} & {\scriptstyle\mathbf {B}^{{\rm H}}}\\{\scriptstyle\mathbf {B}^{{\rm H}}} & {\scriptstyle\mathbf {B}^{{\rm H}}}\end{array}\right)$
V=BHBHBHBH
. Here, indices i, j, …  denote occupied, and a, b, …  denote unoccupied KS orbitals. In the non-interacting (α = 0) limit,
$\bm {\Pi }_{\alpha }(\omega )$
Πα(ω)
reduces to the uncoupled KS polarization propagator,

\begin{equation}\bm {\Pi }_{0}(\omega ) = -\left( \begin{array}{c@{\quad}c}\bm {\Delta } - \omega \mathbf {1} & 0 \\0 & \bm {\Delta } + \omega \mathbf {1} \end{array} \right)^{-1} ,\end{equation}
Π0(ω)=Δω100Δ+ω11,
(2)

where

$\bm {\Delta }_{iajb} = (\epsilon _{a}-\epsilon _{i})\delta _{ij}\delta _{ab}$
Δiajb=(εaεi)δijδab contains the bare canonical KS orbital energy differences.

$\bm {\Pi }_{\alpha }$
Πα and
$\bm {\Pi }_{0}$
Π0
may be related through the Bethe-Salpeter equation31 (BSE)

\begin{equation}\bm {\Pi }_{\alpha }(\omega ) = \big[\frac{}{} \bm {\Pi }_{0}^{-1}(\omega ) -\alpha \mathbf {V} - \mathbf {K}_{\alpha }(\omega ) \big]^{-1} .\end{equation}
Πα(ω)=Π01(ω)αVKα(ω)1.
(3)

The frequency-dependent BSE kernel Kα(ω) is a four-point kernel corresponding to spatially non-local interactions and must be distinguished from the frequency-dependent exchange-correlation kernel of time-dependent DFT corresponding to a local interaction.32 Starting from Eq. (3), geometric series expansion of

$\bm {\Pi }_{\alpha }$
Πα in powers of
$\bm {\Pi }_{0}$
Π0
generates an unscreened perturbation theory that reduces to Görling-Levy10 MBPT if Kα is expanded in powers of α. Since
$\bm {\Pi }_{0}$
Π0
is a non-interacting quantity containing bare orbital energy denominators, see Eq. (2), this series diverges for small-gap systems. The central idea of this paper is to expand
$\bm {\Pi }_{\alpha }$
Πα
in powers of

\begin{equation}\bm {\Pi }_{\alpha }^{{\rm RPA}}(\omega ) = \big[\bm {\Pi }_{0}^{-1}(\omega ) - \alpha \mathbf {V} \big]^{-1} .\end{equation}
Πα RPA (ω)=Π01(ω)αV1.
(4)

In contrast to

$\bm {\Pi }_{0}$
Π0⁠,
$\bm {\Pi }_{\alpha }^{{\rm RPA}}$
Πα RPA
includes screening through the Hartree kernel αV. Since no exchange terms are included, this approach is also known as direct RPA (dRPA).21,33 Thus, the RPA-renormalized series

\begin{eqnarray}\bm {\Pi }_\alpha {(\omega )} &=& \big[\big(\bm {\Pi }_{\alpha }^{{\rm RPA}}(\omega )\big)^{-1} - \mathbf {K}_{\alpha }(\omega ) \big]^{-1} \nonumber\\&=& \bm {\Pi }_\alpha ^{{\rm RPA}}{(\omega )} + \bm {\Pi }_\alpha ^{{\rm RPA}}{(\omega )} \mathbf {K}_\alpha {(\omega )} \bm {\Pi }_\alpha ^{{\rm RPA}}{(\omega )} + \ldots\qquad\end{eqnarray}
Πα(ω)=Πα RPA (ω)1Kα(ω)1=Πα RPA (ω)+Πα RPA (ω)Kα(ω)Πα RPA (ω)+...
(5)

produces finite correlation energies, even for zero-gap systems such as metals.

A simple approximation to Kα(ω)28,34 is the static approximate exchange kernel (AXK)

\begin{align}\mathbf {K}_\alpha ^{{\rm AXK}} = \alpha \left(\begin{array}{c@{\quad}c}\mathbf {B}^{{\rm X}} & \mathbf {B}^{{\rm X}} \\\mathbf {B}^{{\rm X}} & \mathbf {B}^{{\rm X}} \end{array} \right) ,\end{align}
Kα AXK =αBXBXBXBX,
(6)

where

$\mathbf {B}^{{\rm X}}_{iajb} =- \langle ij \vert ba\rangle$
BiajbX=ij|ba is a ph exchange integral. By Eqs. (1) and (5), the AXK second-order beyond-RPA correlation energy is

\begin{eqnarray}&&\Delta E^{{\rm C}}(\text{AXK}) \nonumber\\&&\quad = - \int \limits _0^1 d\alpha\, \text{Im} \int \limits _0^\infty \frac{d\omega }{2\pi } \text{tr}\big\lbrace \mathbf {V} \bm {\Pi }_\alpha ^{{\rm RPA}}{(\omega )} \mathbf {K}_\alpha ^{{\rm AXK}} \bm {\Pi }_\alpha ^{{\rm RPA}}{(\omega )} \big\rbrace ,\qquad\end{eqnarray}
ΔEC(AXK)=01dαIm0dω2πtrVΠα RPA (ω)Kα AXK Πα RPA (ω),
(7)

with the total correlation energy EC(AXK) = EC(RPA) + ΔEC(AXK). EC(RPA) is computed by replacing

$\bm {\Pi }_{\alpha }$
Πα with
$\bm {\Pi }^{{\rm RPA}}_{\alpha }$
Πα RPA
in Eq. (1).
$\mathbf {K}^{{\rm AXK}}_{\alpha }$
Kα AXK
is a low-order approximation to the full Kα(ω) designed to produce correlation energies correct to second order in α. Higher-order corrections must balance the truncation of the geometric series (5) and of the kernel Kα(ω). For example, using the exchange-only Kα to infinite order34 in Eq. (5), also known as RPA with exchange (RPAX),17 leads to instabilities in Πα precluding applications to small-gap systems.28,35

To further analyze the AXK method, we use the spectral representation36 of

$\bm{\Pi }_{\alpha }^{{\rm RPA}}(\omega )$
Πα RPA (ω)29 to carry out the frequency integration in Eq. (7) analytically, yielding

\begin{eqnarray}&&\Delta E^{{\rm C}}(\text{AXK}) \nonumber\\&&\quad = - \int \limits _0^1 d\alpha \sum _{n,m} \alpha \nonumber\\&&\qquad \times\,\frac{ \big[ (\mathbf {X}\!+\!\mathbf {Y})_m^{\alpha ,T} \mathbf {B}^{{\rm H}} (\mathbf {X}\!+\!\mathbf {Y})_n^{\alpha } \big]\big[ (\mathbf {X}\!+\!\mathbf {Y})_n^{\alpha ,T} \mathbf {B}^{{\rm X}} (\mathbf {X}\!+\!\mathbf {Y})_m^{\alpha } \big] }{ \Omega _n^\alpha \!+\! \Omega _m^\alpha },\nonumber\\\end{eqnarray}
ΔEC(AXK)=01dαn,mα×(X+Y)mα,TBH(X+Y)nα(X+Y)nα,TBX(X+Y)mαΩnα+Ωmα,
(8)

where

$\left\lbrace \Omega _n^\alpha ,\mathbf {X}_{n}^{\alpha },\mathbf {Y}_{n}^{\alpha }\right\rbrace$
Ωnα,Xnα,Ynα are the interacting excitation energies and transition vectors within RPA.37 

If both

$\bm {\Pi }_\alpha ^{{\rm RPA}}{(\omega )}$
Πα RPA (ω) are replaced in Eq. (7) by their zeroth-order approximation,
$\bm {\Pi }_0{(\omega )}$
Π0(ω)
, ΔEC(AXK) reduces to second-order exchange (SOX), see Fig. 1(a). In this case, the excitation energies
$\Omega _n^\alpha$
Ωnα
become bare KS orbital energy differences, and the
$(\mathbf {X}+\mathbf {Y})_n^\alpha$
(X+Y)nα
vectors reduce to unit vectors corresponding to single-particle excitations. Bare SOX is not viable beyond-RPA correction38 because it inherits the problems of unscreened MBPT for small-gap systems. In ACSOSEX, which is identical to SOSEX for practical purposes,39 one of the two
$\bm {\Pi }_\alpha ^{{\rm RPA}}$
Πα RPA
in Eq. (7) is replaced by
$\bm {\Pi }_0$
Π0
, see Fig. 1(b). ACSOSEX removes the one-electron self-interaction error, but difficulties for small-gap systems and strong static correlation remain, as illustrated below. AXK, on the other hand, is fully screened and consistent through second order in
$\bm {\Pi }_{\alpha }^{{\rm RPA}}$
Πα RPA
, see Fig. 1(c). The coupling strength integration in Eq. (7) may be carried out analytically,29 and the resulting expression may be evaluated with at most
$\mathcal {O}(N^4 \log {(N)})$
O(N4log(N))
operations, where N denotes the system size.

FIG. 1.

Goldstone diagrams for the correlation energy coupling strength integrand for (a) ΔEC(SOX), (b) ΔEC(ACSOSEX), and (c) ΔEC(AXK). Single black lines correspond to bare KS p or h excitations, and the interactions to BH and BX as in Eq. (8). For diagrams (b) and (c) double blue lines correspond to dressing a pair of KS ph excitations into interacting RPA excitations

$(\mathbf {X}+\mathbf {Y})_{n}^{\alpha }(\mathbf {X}+\mathbf {Y})_{n}^{\alpha ,T}$
(X+Y)nα(X+Y)nα,T and the corresponding KS orbital eigenvalue difference in the denominator to
$\Omega _{n}^{\alpha }$
Ωnα
.

FIG. 1.

Goldstone diagrams for the correlation energy coupling strength integrand for (a) ΔEC(SOX), (b) ΔEC(ACSOSEX), and (c) ΔEC(AXK). Single black lines correspond to bare KS p or h excitations, and the interactions to BH and BX as in Eq. (8). For diagrams (b) and (c) double blue lines correspond to dressing a pair of KS ph excitations into interacting RPA excitations

$(\mathbf {X}+\mathbf {Y})_{n}^{\alpha }(\mathbf {X}+\mathbf {Y})_{n}^{\alpha ,T}$
(X+Y)nα(X+Y)nα,T and the corresponding KS orbital eigenvalue difference in the denominator to
$\Omega _{n}^{\alpha }$
Ωnα
.

Close modal

The performance of AXK was assessed for atomization energies, ionization potentials, barrier heights, and weakly bound complexes using the HEAT,40 G2-1 atomization energy41,42 (G21AE), G21IP, BH76, BH76RC,43,44 and A2445 test sets. Correlation energies were extrapolated to the complete basis set (CBS) limit.46 The KS reference Φ was generated using the TPSS meta-generalized gradient approximation.47 The resulting mean absolute errors (MAEs) are summarized in Table I. TPSS orbitals were used because the beyond-RPA results were less sensitive compared to PBE orbitals, see supplementary material.29 The equilibrium geometry, dissociation energy, and harmonic vibrational frequency of eight transition metal monoxides were also computed as these systems are known to exhibit strong static correlation. Computational details, complete results, and additional statistics are provided as supplementary material.29 

Table I.

Mean absolute errors in energy differences (kcal/mol) for the six test sets described in the text (upper panel), and mean absolute errors for the binding energies (De, kcal/mol), equilibrium bond lengths (re, pm), and harmonic vibrational frequencies (ωe, cm−1) of monoxides MO for M = Ca−Fe, Cu (lower panel). MR indicates the mean reference value. Computational details are given in the supplementary material.29 

Test setMRMP2TPSSRPAACSOSEXAXK
HEAT −212.5 6.26 3.72 9.07 9.38 5.72 
G21AE −220.9 6.48 4.58 9.08 6.24 4.28 
G21IP 257.6 3.44 3.96 7.86 2.60 1.71 
BH76 18.12 4.36 8.59 2.06 4.17 1.63 
BH76RC −20.89 3.65 3.76 1.55 2.10 1.19 
A24 −1.73 0.13 0.77 0.33 0.21 0.28 
Parameter             
re 166.2 5.3 0.4 1.3 3.5 0.9 
De 118.6 20.1 14.3 3.3 14.4 7.2 
ωe 872 290 24 37 92 25 
Test setMRMP2TPSSRPAACSOSEXAXK
HEAT −212.5 6.26 3.72 9.07 9.38 5.72 
G21AE −220.9 6.48 4.58 9.08 6.24 4.28 
G21IP 257.6 3.44 3.96 7.86 2.60 1.71 
BH76 18.12 4.36 8.59 2.06 4.17 1.63 
BH76RC −20.89 3.65 3.76 1.55 2.10 1.19 
A24 −1.73 0.13 0.77 0.33 0.21 0.28 
Parameter             
re 166.2 5.3 0.4 1.3 3.5 0.9 
De 118.6 20.1 14.3 3.3 14.4 7.2 
ωe 872 290 24 37 92 25 

The HEAT benchmark is composed of small molecule atomization energies of first and second row compounds, while the G21AE set also contains third row elements. For the G21AE benchmark, the RPA, ACSOSEX, and AXK MAEs are 8.85 kcal/mol, 6.27 kcal/mol, and 4.17 kcal/mol, respectively, see Table I. On the other hand, ACSOSEX worsens the RPA result for the HEAT set, while AXK systematically improves it by several kcal/mol. Since RPA is inaccurate for non-isogyric processes,17 RPA ionization potentials exhibit a large MAE of 7.86 kcal/mol for the G21IP set. The addition of screened exchange dramatically decreases the error of RPA, with ACSOSEX and AXK yielding MAEs of 2.60 and 1.71 kcal/mol, respectively. As opposed to IPs, RPA barrier heights are quite accurate without any corrections.17 The addition of ACSOSEX increases the mean absolute and maximum errors for the BH76 and BH76RC tests.22 AXK systematically improves upon RPA for both sets resulting in MAEs of less than 2 kcal/mol. For the weakly bound complexes contained in the A24 set, both ACSOSEX and AXK reduce the statistical errors compared to RPA, see Table I. Similar improvements upon semi-local DFT with empirical dispersion corrections and other beyond-RPA approaches have been reported for rare-gas dimers,48 the S22 set,49 and the S66 set.22 

Transition metal monoxides are minimal model systems for heterogeneous catalysis; their considerable multi-reference character and effective one- and three-electron bonding are susceptible to self-interaction error and pose a serious challenge.50 The MP2 results for binding energies and vibrational frequencies are erratic, see Table I, with the largest errors occurring for the later transition metals with significant d occupation. The TPSS MGGA performs well for structures and frequencies, but exhibits considerable overbinding.51 This is corrected by RPA, yielding an MAE of 3.3 kcal/mol for the binding energies.29 Addition of ACSOSEX is counterproductive, leading to a roughly three-fold increase of the MAE in dissociation energies, bond lengths, and vibrational frequencies. AXK decreases the RPA error in bond lengths and frequencies while the error in binding energies increases.

The

${\rm H}_{2}^{+}$
H2+ and closed-shell H2 potential energy curves, Fig. 2, summarize why AXK is more balanced than ACSOSEX. ACSOSEX is exact for
${\rm H}_{2}^{+}$
H2+
, where it cancels the spurious RPA self-correlation, but it reintroduces the well-known symmetry dilemma for dissociating H2, confounding the qualitatively correct behavior of RPA.21,52 On the other hand, AXK captures the strong static correlation in stretched H2 and significantly improves upon RPA and TPSS close to the equilibrium bond distance of
${\rm H}_{2}^{+}$
H2+
, but breaks down for stretched
${\rm H}_{2}^{+}$
H2+
due to self-interaction error.

FIG. 2.

Potential energy curves for

${\rm H}_{2}^{+}$
H2+ (top) and H2 (bottom) computed using TPSS input orbitals and aug-cc-pV5Z basis sets. ACSOSEX is exact for
${\rm H}_{2}^{+}$
H2+
, whereas CCSD is exact for H2.

FIG. 2.

Potential energy curves for

${\rm H}_{2}^{+}$
H2+ (top) and H2 (bottom) computed using TPSS input orbitals and aug-cc-pV5Z basis sets. ACSOSEX is exact for
${\rm H}_{2}^{+}$
H2+
, whereas CCSD is exact for H2.

Close modal

In conclusion, RPA-renormalized MBPT removes the most debilitating deficiencies of conventional single-reference MBPT without significantly increasing the computational cost. Thus, it delivers on the promise that low-order perturbation theory could be a viable and cost-effective treatment of electron correlation for diverse chemical systems independent of the gap. The second-order AXK variant systematically improves RPA for non-isogyric processes such as atomization or ionization and does not worsen the results where RPA is already accurate such as barrier heights. While AXK removes most of the self-correlation contained in RPA, the marked improvement of AXK over ACSOSEX appears to result from a tradeoff between some self-interaction error and an improved description of static correlation. While more tests are desirable, the present results suggest that AXK may be the long sought efficient and broadly applicable first-principles method that can serve as quality control for semi-local DFT.

The authors would like to thank Kieron Burke and Henk Eshuis for useful discussions, as well as Joachim Paier and Thomas Henderson for providing SOSEX reference values. This material is based upon work supported by the National Science Foundation under CHE-1213382.

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