Assessment of the Second-Order Statically Screened Exchange Correction to the Random Phase Approximation for Correlation Energies

With increasing interelectronic distance, the screening of the electron–electron interaction by the presence of other electrons becomes the dominant source of electron correlation. This effect is described by the random phase approximation (RPA) which is therefore a promising method for the calculation of weak interactions. The success of the RPA relies on the cancellation of errors, which can be traced back to the violation of the crossing symmetry of the 4-point vertex, leading to strongly overestimated total correlation energies. By the addition of second-order screened exchange (SOSEX) to the correlation energy, this issue is substantially reduced. In the adiabatic connection (AC) SOSEX formalism, one of the two electron–electron interaction lines in the second-order exchange term is dynamically screened (SOSEX(W, vc)). A related SOSEX expression in which both electron–electron interaction lines are statically screened (SOSEX(W(0), W(0))) is obtained from the G3W2 contribution to the electronic self-energy. In contrast to SOSEX(W, vc), the evaluation of this correlation energy expression does not require an expensive numerical frequency integration and is therefore advantageous from a computational perspective. We compare the accuracy of the statically screened variant to RPA and RPA+SOSEX(W, vc) for a wide range of chemical reactions. While both methods fail for barrier heights, SOSEX(W(0), W(0)) agrees very well with SOSEX(W, vc) for charged excitations and noncovalent interactions where they lead to major improvements over RPA.


Introduction
The random phase approximation (RPA) 1,2 has found widespread use in quantum chemistry for the calculation of covalent and non-covalent interaction energies. [3][4][5][6][7][8][9][10] The direct (particlehole) RPA can be derived in the framework of the adiabatic connection (AC) fluctuationdissipation theorem (ACFD) [11][12][13] or as a subset of terms in the coupled cluster (CC) [14][15][16][17][18] singles and doubles (CCD) expansion. 19,20 Within Many-body perturbation theory (MBPT), [21][22][23][24] the RPA is obtained by evaluating the Klein,25 or alternatively, the Luttinger-Ward 26 functional with the self-energy in the GW approximation (GWA) using a (non-interacting) Kohn-Sham (KS) 27 Density functional theory (DFT) 28 Green's function. 29,30 In the GWA, 31 the self-energy is approximated as the first term of its expansion in terms of a screened electron-electron interaction where screening is usually calculated within a pair bubble approximation 32 24 Not only for solids but also for larger molecules it becomes decisive to consider screening which is the main reason for the popularity of the GW method in solid-state calculations. 24 The RPA is generally believed to describe long-range electron correlation very accurately Since charge screening is the dominant source of electron correlation in this limit. 12,24 CC and MBPT based methods describe screening by resummation of certain classes of self-energy diagrams to infinite order. 22,33,34 The RPA is the simplest first principle method which accounts for these effects and is implemented with O (N 4 ) scaling with system size using global density fitting (DF). 35 Modern RPA (and GW ) implementations typically use local density-fitting approaches to calculate the non-interacting polarizability, [36][37][38][39][40][41] leading to quadratic or cubic scaling in practice, and even effectively linearly scaling implementations (for sufficiently sparse and large systems) have been reported. [42][43][44][45] For these reasons, the RPA is considered a promising method to study weakly correlated large molecules. 4,10,[46][47][48] At short electron-electron distances, however, charge screening becomes less important for the description of electron correlation and taking into account higher-order contributions to the self-energy via the 4-point vertex function becomes decisive. 49 The absence of these terms in the RPA leads to Pauli exclusion principle violating contributions to the electron correlation energy. 50 As a consequence, total correlation energies are much too high compared to exact reference values. 51,52 In contrast to RPA, the approximations to the correlation energy of Møller-Plesset (MP) perturbation theory are free of Pauli principle violating terms. Especially MP2 is relatively inexpensive and can be applied routinely to systems with more than 100 atoms even close to the complete basis set limit. However, screening effects are entirely absent in MP perturbation theory and electron correlation is described by HF quasiparticles (QP) interacting via the bare Coulomb interaction instead, neglecting the fact that the interactions between the HF QPs are generally much weaker than the ones between the undressed electrons. This issue is also present in orbital optimized MP2 in which the HF QPs are replaced by MP2 QPs. [53][54][55] Therefore, MP2 is a suitable method only for (typically small) systems in which screening effects are negligible. The divergence of MP perturbation theory for the uniform electron gas (see for instance chapter 10 in ref. 22 for a thorough discussion) is known at least since early work by Macke 1 and has been demonstrated later on for metals 56 and recently also for large, non-covalently bound organic complexes. 48 The divergence of the MP series for small-gap systems is directly related to this issue since the magnitude of the screening is proportional to the width of the fundamental gap. 57,58 There have been various approaches to regularize MP2 by an approximate treatment of higher-order screening effects, either using empirical regularizers [59][60][61][62][63][64][65][66][67][68][69] or diagrammatically motivated modifications 34,[70][71][72] or attacking the problem from a DFT perspective. 73,74 Starting from the opposite direction, there have been many attempts to correct the RPA correlation energy expression by adding additional terms to improve the description of shortrange correlation. This includes range-separation based approaches, [75][76][77][78][79][80][81][82][83][84] or augmentations by singles contributions. [85][86][87] Via MBPT, the RPA can generally be improved upon inclusion of the 4-point vertex in the electronic self-energy, either directly, or indirectly through the kernel of the Bethe-Salpeter equation (BSE) for the generalized susceptibility. Following the latter approach, approximations often start from the ACFD and go beyond the Coulomb kernel in the BSE by adding additional terms, for instance exact exchange (exx) (often denoted as exx-RPA) [88][89][90][91][92][93] and higher order contributions, [94][95][96][97] or the statically screened GW kernel, 98-100 but also empirically tuned functions of the eigenvalues of the KS density-density response. 101,102 Notice, that the BSE for the generalized susceptibility reduces to a Dyson equation for the density-density response function which makes local kernels very attractive from a computational perspective.
Instead of relying on the ACFD theorem, beyond-RPA energy expressions can also be introduced directly from approximations to the self-energy beyond the GWA. For instance, in RPAx 103-106 a local 4-point vertex obtained from the functional derivative of the local exact exchange potential calculated within the optimized effective potential method 107-109 is used in the self-energy. In Freeman's second-order screened exchange (SOSEX) correction, 110 the HF vertex (i.e. the functional derivative of the non-local HF self-energy with respect to the single-particle Green's function) is included in the self-energy directly but not in the screened interaction. 6,50,86,87,[111][112][113] Another expression for SOSEX can be obtained by including the static GW kernel in the self-energy but not in the density-density response. This possibility has not been explored until recently 114 and is the main topic of this work.
In our recent work, we have assessed the accuracy of the statically screened G3W 2 correction to the GW self-energy for charged excitations. 114 This correction has first been applied by Grüneis at al. 115 to calculate the electronic structure of solids and is obtained by calculating the self-energy to second-order in the screened Coulomb interaction (equivalent to including the full GW vertex) and then taking the static limit for both terms. The resulting energy expression fulfills the crossing symmetry of the vertex to first order in the electronelectron interaction. Preliminary results for the correlation energies of atoms have been promising. 114 This realization of SOSEX is computationally more efficient than AC-SOSEX since no expensive numerical frequency integration is required. Here, we assess the accuracy of this method for bond dissociation, atomization energies, barrier heights, charged excitations and non-covalent interactions. Our results show that the statically screened SOSEX variant is comparable in accuracy to AC-SOSEX but we observe important differences in the dissociation of diatomic molecules and charged dimers.
The remainder of this work is organized as follows. In section 2 we give a detailed derivation of the different SOSEX energy expressions. After an outline of our computational approach and implementation in section 3, we present and analyze our numerical results in section 4. Finally, section 5 summarizes and concludes this work.

Theory
The central object of MBPT is the one-particle irreducible (1PI) electronic self-energy Σ.
It is the sum of all 1PI skeleton diagrams (diagrams which do not contain any self-energy insertions) of nth order in the electron-electron interaction v c . It maps the interacting singleparticle Green's function G to its non-interacting counterpart G (0) by means of Dyson's equation, 116 Space, spin, and imaginary time indices are collected as 1 = (r 1 , σ 1 , iτ 1 ). One can always switch between imaginary time and imaginary frequency using the Laplace transforms 117 and In (1), G = G 1 is defined by Here, Ψ is the ground state of an N -electron system, T is the time-ordering operator and ψ is the field operator. Σ is given by where the second term on the r.h.s. can be written as For a detailed deviation we refer to the supporting information. We note, that Maggio and Kresse 118 and Martin et al. 24 used a similar expression. Equation (6) combines several quantities. These are the particle-hole irreducible 4-point vertex (i.e. the sum of all diagrams which cannot be cut into parts by removing a particle and a hole line), 119 the non-interacting generalized susceptibility, and the screened (bare) Coulomb interaction W (W (0) ). These quantities are related by the Dyson equation with given in terms of the bare coulomb interaction v c and the reducible polarizability with χ is related to its non-interacting counterpart χ which reduces to a Dyson equation for the polarizability P when the xc-contribution to the 4-point vertex is set to zero. One can then also introduce the irreducible polarizability P (0) as Using this quantity, (9) can also be written as Note, that the equations above are completely equivalent to Hedin's equations. 31 Their form given here has the advantages that the BSE appears explicitly and that only 2-point or 4-point quantities occur. Therefore, the resulting equations are invariant under unitary transformations of the basis, as has for instance been pointed out by Starke and Kresse. 121 or in ref. 122 The xc-contribution to the self-energy defined in (6) can also be obtained as the functional Φ is a universal functional of the interacting G and is defined by 24,26,123 As for instance discussed in refs., 30,123 If this expression is evaluated with a non-interacting Green's function one directly obtains the exchange-correlation energy from it. A suitable non-interacting Green's function G s can be obtained from G (0) by where v s (1, 2) = v H (1, 2)δ(1, 2) + v xc (r 1 , r 2 )δ(τ 12 ) and with v xc being a KS xc-potential mixed with a fraction of HF exchange and τ 12 = τ 1 −τ 2 .
The correlation energy is then given by 30 The Hx contribution to the electron-electron interaction energy is obtained as In case G s is the Hartree-Fock (HF) Green's function, (22) is the HF expression for the Hartree and exchange energy.
In the GWA, the self-energy (6) is approximated as Σ ≈ Σ H + iGW . W is typically calculated within the RPA which consists in approximating Γ (0) H in the BSE (13).
Making both approximations and using eqs. (9) and (14), the RPA exchange-correlation energy is obtained. 123 Isolating the exchange contribution to the Hartree-exchange energy, we obtain the RPA correlation energy and using (2) as well as the symmetry of the polarizability on the imaginary frequency axis, its well-known representation due to Langreth and Perdew 12 is obtained, In this work, we are interested in approximations to the self-energy beyond the GWA. It follows from the antisymmetry of Fermionic Fock space that G 2 needs to change sign when the two creation or annihilation operators in (4) are interchanged. This property is known as crossing symmetry. 124 In the RPA, the crossing symmetry is violated which leads to the wellknown overestimation of absolute correlation energies. However, when the 4-point vertex is approximated by the functional derivative of the Hartree-exchange self-energy the crossing symmetry is fulfilled. We show this explicitly in the supporting information.
Approximations to the self-energy in Hedin's equations always violate the crossing symmetry. 125,126 However, with each iteration of Hedin's pentagon, the crossing symmetry is fulfilled up to an increasingly higher order in v c . We can then expect to obtain improvements over the RPA energies expressions by choosing a self-energy which fulfills the crossing symmetry to first order in v c . The easiest approximation to the self-energy of this type is obtained from the HF vertex, Using this expression in (6) with (8) yields the AC-SOSEX contribution to the self-energy. 118,127 We first notice that within the pair bubble approximation for W , (6) becomes where we have indicated the screening of the electron-electron interaction in the SOSEX expression in the superscript on the l.h.s. of (28). Here we have used the identity W χ (0) = Using the GW self-energy in (7), to first order in W (0) (ignoring the vriation of W with respect to G) the screened exchange kernel is obtained, The resulting self-energy is the complete second-order term in the expansion of the self-energy in terms of the screened electron-electron interaction, 31 and contains the AC-SOSEX self-energy. The G3W 2 self-energy can be decomposed into The double and single wiggly lines are screened and bare electron-electron interactions, respectively a) Greater and lesser contributions to the full G3W 2 the self-energy term. b) Greater and lesser components of the SOSEX self-energy c) Greater and lesser components of the MP2 self-energy. The static approximation to the G3W 2 self-energy is the same, with the bare electron-electron interaction lines replaced by the statically screened ones. The black part of the diagrams are the contributions to the self-energy only which, combined with the blue lines, yield the corresponding single-particle propagator.
eight skeleton diagrams on the Keldysh contour, 128 but the AC-SOSEX self-energy only into four. 129 Diagrammatically, this is shown in figure 1a) and 1b), respectively. In practice, the evaluation of the resulting energy expression requires to perform a double frequency integration while the evaluation of the AC-SOSEX energy only requires a single frequency integration. Since the computation of the AC-SOSEX term is already quite cumbersome, the complete G3W 2 energy expression is therefore not a good candidate for an efficient beyond-RPA correction. Instead, we take the static limit in both W in (30) to arrive at a self-energy expression similar to AC-SOSEX, whose diagrammatic form is shown in figure 1c). Due to the presence of the two δ-functions, only two out of the eight diagrams of the G3W 2 term remain. This expression is similar to the MP2 self-energy, with the only difference that the bare electron-electron interaction is replaced by the statically screened one. However, the resulting expression for the correlation energy will be different due to the factors 1 n in (17). Using (9), eq. (31) can be written as with the first term being the second-order exchange (SOX) term in MP2 and with the remainder accounting for the screening of the electron-electron interaction. Defining it can be written as In the same way one can see, that the statically screened GW vertex contains the HF vertex. The same is obviously true for all other flavors of SOSEX, and therefore all of them fulfill the crossing symmetry of the full 4-point vertex to first order in the electron-electron interaction. Therefore, all of these approximations compensate the overestimation of the electron correlation energy in the RPA.
In contrast to the RPA which is efficiently evaluated in a localized basis, beyond-RPA energies are most easily formulated in the molecular spin-orbital basis {φ i (r, σ)} in which the time-ordered KS Green's function is diagonal, The k denote KS eigenvalues which are understood to be measured relative to the chemical potential µ and f ( k ) denotes the occupation number of the kth orbital. One can now obtain energy expressions analogous to (26). For example, inserting the AC-SOSEX self-energy (28) into (21), we obtain In contrast to the RPA energy expression, the terms in this equation cannot be summed exactly due to the presence of the 1/n-terms. However, defining we can rewrite (21) as an integral over a coupling constant λ, Therefore, (37) becomes where W (λ) is defined as in (15), with W (0) replaced by λW (0) . Defining and the correlation energy becomes The integral in (40) and when the statically screened G3W 2 self-energy (31)  gives the MP2 correlation energy (evaluated with G s ). 30 Using (42), simple expressions for the AC-SOSEX energy in the basis of KS orbitals is obtained. With eqs. (28), (35) and (42) we have In going from the second equations, we have used (2) to transform W to the imaginary frequency axis. The integral over τ 3 can be evaluated by splitting it at τ 1 and using the definition of the KS Green's function (35), The remaining integral over τ 12 is so that the correlation energy becomes Each of the nominators can only give a non-vanishing contribution if one of the two occupation numbers are zero. If the difference of the occupation numbers is −1, we simply flip sign in the denominator. Without loss of generality we can then decide that the indices r and p belong to occupied and the indices s and q to virtual single-particle states. Equation (47) then becomes For a closed-shell system we can also sum over spins which gives us an additional factor of 2. The resulting expression is then equivalent to the one of ref. 87 In the spin-orbital basis, the SOSEX(W (0), W (0)) Correlation energy is obtained from (30) and (35) as This is the expression we have introduced in ref. 114. It is completely equivalent to the exchange term in MP2 with the bare electron-electron interaction replaced by the statically screened, coupling constant averaged one. Both RPA+SOSEX variants can be understood as renormalized MP2 expressions and allow for a clear diagrammatic interpretation. In the next section, we briefly outline our implementation of these expressions, before we proceed by assessing their accuracy for correlation energies in sec. 4.

Technical and Computational Details
All expressions presented herein have been implemented in a locally modified development version of the Amsterdam density functional (ADF) engine of the Amsterdam modelling suite 2022 (AMS2022). 133 The non-interacting polarizability needed to evaluate (26) and (15) where F and F denote even and odd parts of F , respectively. The transformation from imaginary frequency to imaginary time only requires the (pseudo)inversion of Ω (c) and Ω (s) , respectively. Our procedure to calculate Ω (c) and Ω (s) as well as T and W follows Kresse and coworkers. [138][139][140] The technical specifications of our implementation have been described in the appendix of ref. 134.
We use in all calculations grids of 24 points in imaginary time and imaginary frequency which is more than sufficient for convergence. 137 The final correlation energies are then extrapolated to the complete basis set limit using the relation , 141 where E QZ (E T Z ) denotes the total energies at the QZ6P (TZ3P) level.

Dissociation Curves
The potential energy curves of small diatomic molecules serve as an important test for electronic structure methods. We first consider molecules with different bonding types for which we were able to calculate FCI reference values: H 2 is covalently bound, LiH is an ionic molecule, and He 2 has a very weak, non-covalent bond.
The dissociation curve of H 2 calculated with RPA+SOSEX(W (0),W (0))@PBE is the red  91,112,145 Here we find that also RPA+SOSEX(W (0),W (0)) dissociates the hydrogen molecule correctly and that the potential energy curve has a similar shape than the RPA one. Henderson and Scuseria have argued that the self-correlation in the RPA mimics static correlation effects 145 whose good description is necessary to dissociate H 2 correctly. The fact that in RPA+SOSEX(W (0),W (0) the self-correlation error is eliminated to some large extent (also see table 1 in the SI) but not completely therefore explains the similarity to the RPA dissociation curve.
To rationalize this result further, we also calculated the dissociation curve within the static limit of RPA+SOSEX(W ,v c ), RPA+SOSEX(W (0),v c ) (blue curve). This shows that the screening of the second electron-electron interaction line is responsible for the qualita-tive differences between SOSEX(W ,v c ) and SOSEX(W (0),W (0)). It should also be noted that the RPA+SOSEX(W (0),W (0)) dissociation curve of H 2 very closely resembles the one calculated by Bates and Furche using the approximate exchange kernel (AXK) correction to the RPA. 94 SOSEX(W (0),W (0)) and the AXK kernel have in common that both electronelectron interaction lines are screened. For LiH, we find a similar behavior than for H 2 . For He 2 (notice that we plotted here the binding energy and not the total energy) we see that all approaches give the correct dissociation limit.

Dissociation of charged Dimers
In

Thermochemistry and Kinetics
We move on to assess the performance of RPA+SOSEX(W (0),W (0)) for reaction types which are relevant for thermochemistry and kinetics. Total atomization energies, ionization potentials and electron affinities as well as barrier heights of different reactions serve hereby as important testing grounds. For this work, we calculated the atomization energies (defined as the total energy of the molecule minus the sum of the energies of the atomic fragments) of the 144 small and medium molecules in the W4-11 dataset. 151 The reference values have been calculated using the highly accurate W4 protocol. 152 For barrier heights, we use the BH76 database which is a compilation of the HTBH38 153 and NHTBH38 154 databases for barrier heights by Truhlar and coworkers, which are typically used in benchmarks of (beyond-)RPA methods. 5,6,86,87 The reference values have been calculated with the W2-F12 protocol. 149,155 To benchmark the performance for ionization potentials and electron affinities we employ the G21IP and G21EA databases by Pople and coworkers and use the original experimental reference values. 156 To start with, we assess the effect of the Green's function G s used to calculate the correlation energies. RPA calculations can in principle be performed self-consistently using a variety of approaches. 88   . This has been confirmed later by Ren at al. 6 and Paier et al. 86 for the 55 covalently bound molecules in the G2-I set. 156 The same holds for RPA+SOSEX(W ,v c ), but compared to RPA the magnitude of the error is reduced on average. 6, 86 We observe here that unlike SOSEX(W, v c ), the addition of SOSEX(W (0), W (0) substantially overcorrects the RPA atomization energies which are now much too high in magnitude. 171 Adding bare SOX to RPA leads to underestimated correlation energies. 52 This effect is expected to be more pronounced for the molecule than for the individual atoms since more electrons are correlated in the former. Therefore, RPA+SOX will substantially overestimate atomization energies and due to underestimated screening of the SOX term in SOSEX(W (0), W (0), RPA+SOSEX(W (0), W (0) inherits this problem.

Non-covalent Interactions S66 Interaction Energies
We now turn to our benchmark results for non-covalent interactions. As for the previous datasets, we also assess the dependence of RPA and RPA+SOSEX correlation energies on the choice of the KS Green's function G s . In figure 6 the interaction energies in the S66 database 173 obtained using different G s are compared to each other as well as to the

S66x8 Interaction Energy
The S66x8 dataset contains the complexes in the S66 database at 8 different geometries. 173 The separations of the monomers in the complexes are given relative to their equilibrium distances, i.e. a relative separation of 2.0 means that the monomers separation in the complex is twice as large as the equilibrium separation. For our assessment of the SOSEX(W (0), W (0)) correction, we divide the separations of the potential energy curve in three regions, which we denote as short (equilibrium distance scaled by a factor 0.9-0.   W (0)). This can be rationalized by observing that for large electron-electron distances the correlation contributions to the interaction energies quickly decay to zero. This is shown in figure 9 where we have plotted three of  In all three plots, the potential energy curves are dominated by the difference of the correlation energy of the dimer and the sum of correlation energies of the monomers. Therefore, the approximation used for the calculation of the correlation energy plays a large role.
However, this difference quickly goes to zero for larger separations. At two times of the equilibrium distance, the correlation contributions to the potential energy curves are almost zero in all three considered examples. Therefore, the expression used for the correlation energy becomes less and less important with increasing monomer separation. This argument also holds if one expresses the contributions in % of the total interaction energy.
One would expect the SOSEX contribution to decay faster than the RPA one, since the former is of exchange nature and therefore fundamentally short-ranged. 52 However, the plots in the lower part of figure 9 shows that this is only the case for the potential energy curve on the right, but not for the two curves on the left, where SOSEX and RPA contributions seem to decay equally fast.

Conclusions
The accuracy of the RPA can in principle be improved by including vertex correction in the self-energy. This can be done either directly, or indirectly through the solution of the BSE.
Including the first-order vertex in the self-energy, different variants of SOSEX are obtained.
These are the well-known AC-SOSEX, herein termed SOSEX(W, v c ), first introduced by Jansen et. al, 111 in which only one of the Coulomb interaction lines is dynamically screened, as well as an energy expression which is obtained from the statically screened G3W 2 correction to the GW self-energy. 114,115 This energy expression has already been introduced in our earlier work, 114 albeit without a rigorous derivation. Especially, we have implicitly assumed that the integral over the coupling strength is evaluated using a trapezoidal rule. Here, we have derived this expression (referred to as SOSEX(W (0), W (0)) in this work) taking into account its λ-dependence and highlighted the differences to SOSEX(W, v c ). We have then assessed the accuracy of the SOSEX(W (0), W (0)) correction to RPA correlation energies for a wide range of chemical problems including bond dissociation, thermochemistry, kinetics, and non-covalent interactions.
The main conclusion we can draw from our work is that in situation where the addition of SOSEX(W, v c ) leads to major improvements over the RPA, the addition of SOSEX(W (0), W (0)) does as well. This is the case for the calculation of ionization potentials and electron affinities where RPA+SOSEX approaches challenge the accuracy of modern double-hybrid functionals. 149 Also for non-covalent interactions both SOSEX variants lead to the same substantial improvements over RPA. SOSEX(W, v c ) is most accurate for the hydrogen-bonded com-plexes while SOSEX(W (0), W (0)) is slightly more accurate for dispersion interactions. We also showed that the frequency-dependence of the screened interactions does seem to be an important factor for hydrogen-bonding but not for dispersion interactions.
Differences between both SOSEX variants have been observed in the dissociation of diatomic molecules. As RPA and unlike RPA+SOSEX(W ,v c ), 112,145 RPA+SOSEX(W (0), W (0)) dissociates the Hydrogen molecule correctly. RPA does so because the self-correlation error effectively describes static correlation. 145 The situation seems to be similar for RPA+SOSEX (W (0), W (0)) since in contrast to RPA+SOSEX(W ,v c ) it is not completely self-correlation free for 1-electron systems. We have also shown that this qualitative difference is due to the screening of the second electron-electron interaction line.
The incomplete cancellation of self-correlation error does however negatively affect the dissociation of charged dimers for which RPA+SOSEX(W ,v c ) fixes most of the deficiencies of RPA. 112,150 Here, RPA+SOSEX(W (0), W (0)) performs even worse than RPA. Furthermore, the good dissociation of diatomic molecules does not automatically carry over to accurate barrier heights 153,154 where both SOSEX variants considerably worsen the RPA results.
In summary, our results suggest that the statically screened SOSEX is a suitable alternative to dynamically screened SOSEX. While both formally scale as N 5 with system size, the computation of the SOSEX(W, v c ) correction requires a numerical imaginary frequency integration. The calculation of the SOSEX(W (0), W (0)) correction is therefore much cheaper, comparable to MP2. MP2 is however inadequate for large molecules since it neglects screening effects entirely. 1,48 RPA+SOSEX(W (0), W (0)) is in principle applicable also to large molecules. A stochastic linear scaling implementation of the SOSEX self-energy has already been developed 179 and a recent RPA+SOSEX implementation by Ochsenfeld and co-workers 180 allowed applications to the L7 dataset, 181 albeit with small basis sets. Other low-scaling MP2 implementations 182-184 could potentially be generalized to SOSEX as well.
Finally, it should be mentioned that the accuracy of the dynamically screened SOSEX correction to the RPA can be improved upon by the addition of renormalized single exci-tations. 6,87 Other methods which have been shown to outperform SOSEX, in particular for barrier heights, are the AXK kernel method 94,150,185 or a SOSEX variant in which the terms of RPA and SOSEX beyond second order in v c are scaled down. 185 It remains to be investigated whether the concept of static screening can also be combined with those approaches and leads to good results.