# Assessment of the second-order statically screened exchange correction to
the random phase approximation for correlation energies

Arno Förster<EMAIL_ADDRESS>Theoretical Chemistry, Vrije Universiteit,
De Boelelaan 1083, NL-1081 HV, Amsterdam, The Netherlands

###### Abstract

With increasing inter-electronic 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 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($W$,$v_{c}$)). 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($W$,$v_{c}$), 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($W$,$v_{c}$)
for a wide range of chemical reactions. While both methods fail for barrier
heights, SOSEX($W(0)$,$W(0)$) agrees very well with SOSEX($W$,$v_{c}$) for
charged excitations and non-covalent interactions where they lead to major
improvements over RPA.

###### keywords:

GW, SOSEX, RPA, vertex corrections, non-covalent interactions

##  1 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 (particle-hole) RPA can be derived
in the framework of the adiabatic connection (AC) fluctuation-dissipation
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-Ward26 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 GWA31, 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 The pair bubble
approximation is typically also denoted as RPA. To avoid potential confusion
with the expression for the correlation energy, we will use the term bubble
approximation when referring to the screening.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 $\mathcal{O}\left(N^{4}\right)$ 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.46, 47, 4, 10, 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 Macke1 and
has been demonstrated later on for metals56 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
regularizers59, 60, 61, 62, 63, 64, 65, 66, 67, 68, 69 or diagrammatically
motivated modifications70, 34, 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 short-range 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,
99, 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 RPAx103, 104, 105, 106 a local 4-point vertex obtained from
the functional derivative of the _local_ exact exchange potential calculated
within the optimized effective potential method107, 108, 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.111,
112, 86, 6, 87, 113, 50 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 recently114 and
is the main topic of this work.

In our recent work, we have assessed the accuracy of the statically screened
$G3W2$ 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 electron-electron 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.

##  2 Theory

The central object of MBPT is the one-particle irreducible (1PI) electronic
self-energy $\Sigma$. It is the sum of all 1PI skeleton diagrams (diagrams
which do not contain any self-energy insertions) of $n$th order in the
electron-electron interaction $v_{c}$. It maps the interacting single-particle
Green’s function $G$ to its non-interacting counterpart $G^{(0)}$ by means of
Dyson’s equation,116

$G(1,2)=G^{(0)}(1,2)+G^{(0)}(1,3)\Sigma(3,4)G(4,2)\;.$ (1)

Space, spin, and imaginary time indices are collected as
$1=(\bm{r}_{1},\sigma_{1},i\tau_{1})$. One can always switch between imaginary
time and imaginary frequency using the Laplace transforms117

$f(i\tau)=\frac{i}{2\pi}\int d\omega F(i\omega)e^{i\omega\tau}$ (2)

and

$f(i\omega)=-i\int d\tau F(i\tau)e^{-i\omega\tau}\;.$ (3)

In (1), $G=G_{1}$ is defined by

$G_{n}(1,\dots
2n)=\left\langle\Psi^{(N)}_{0}\Big{|}\mathcal{T}\left[\hat{\psi}^{\dagger}(1)\hat{\psi}(2)\dots\hat{\psi}^{\dagger}(2n-1)\hat{\psi}(2n)\right]\Big{|}\Psi^{(N)}_{0}\right\rangle\;.$
(4)

Here, $\Psi^{(N)}_{0}$ is the ground state of an $N$-electron system,
$\mathcal{T}$ is the time-ordering operator and $\hat{\psi}$ is the field
operator. $\Sigma$ is given by

$\Sigma(1,2)=v_{H}(1)\delta(1,2)+\Sigma_{xc}(1,2)\;,$ (5)

where the second term on the _r.h.s._ can be written as

$\Sigma_{xc}(1,2)=iG(1,2)W(1,2)+iG(1,3)W^{(0)}(1,4)\chi(6,4,5,4^{+})\Gamma_{xc}^{(0)}(6,5,2,3)\;.$
(6)

For a detailed deviation we refer to the supporting information. We note, that
Maggio and Kresse118 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

$\Gamma_{Hxc}^{(0)}(1,2,3,4)=\Gamma_{H}^{(0)}(1,2,3,4)+\Gamma_{xc}^{(0)}(1,2,3,4)=i\frac{\delta\Sigma_{H}(1,3)}{\delta
G(4,2)}+i\frac{\delta\Sigma_{xc}(1,3)}{\delta G(4,2)}\;,$ (7)

the non-interacting generalized susceptibility,

$\chi^{(0)}(1,2,3,4)=-iG(1,4)G(2,3)\;,$ (8)

and the screened (bare) Coulomb interaction $W$ ($W^{(0)}$). These quantities
are related by the Dyson equation

$W(1,2)=W^{(0)}(1,2)+W^{(0)}(1,3)P(3,4)W^{(0)}(4,2)\;,$ (9)

with

$W^{(0)}(1,2)=v_{c}(\bm{r}_{1},\bm{r}_{2})\delta_{\sigma,\sigma^{\prime}}\delta(t_{1}-t_{2})\;,$
(10)

given in terms of the bare coulomb interaction $v_{c}$ and the reducible
polarizability

$P(1,2)=\chi(1,2,3,4)\delta(1,4)\delta(2,3)\;,$ (11)

with

$\chi(1,2,3,4)=-iG_{2}(1,2,3,4)-iG(1,2)G(3,4)\;.$ (12)

$\chi$ is related to its non-interacting counterpart $\chi^{(0)}$ by a Bethe-
Salpeter equation (BSE),120, 119

$\chi(1,2,3,4)=\chi^{(0)}(1,2,3,4)+\chi^{(0)}(1,8,3,7)\Gamma_{Hxc}^{(0)}(7,5,8,6)\chi(6,2,5,4)\;,$
(13)

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

$P^{(0)}(1,2)=\chi^{(0)}(1,2,3,4)\delta(1,4)\delta(2,3)\;.$ (14)

Using this quantity, (9) can also be written as

$W(1,2)=W^{(0)}(1,2)+W^{(0)}(1,3)P^{(0)}(3,4)W(4,2)\;.$ (15)

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 derivative

$\Sigma_{xc}=\frac{\delta\Phi[G]}{\delta G}\;.$ (16)

$\Phi$ is a universal functional of the interacting $G$ and is defined by26,
123, 24

$\Phi[G]=\frac{1}{2}\sum_{n}\frac{1}{n}\int
d1d2G(1,2)\Sigma_{xc}^{(n)}[G](2,1)\;.$ (17)

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

$G^{s}(1,2)=G^{(0)}(1,2)+G^{(0)}(1,3)v_{s}(3,4)G^{s}(4,1)\;,$ (18)

where

$v_{s}(1,2)=v_{H}(1,2)\delta(1,2)+v_{xc}(\bm{r}_{1},\bm{r}_{2})\delta(\tau_{12})$
(19)

and with $v_{xc}$ being a KS xc-potential mixed with a fraction of HF exchange
and $\tau_{12}=\tau_{1}-\tau_{2}$. The correlation energy

$E_{c}[G^{s}]=E_{Hxc}[G^{s}]-E_{Hx}[G^{s}]$ (20)

is then given by30

$E_{c}=\frac{1}{2}\sum_{n=2}\frac{1}{n}\int
d1d2G^{s}(1,2)\Sigma^{(n)}(2,1)[G^{s}]\;.$ (21)

The $Hx$ contribution to the electron-electron interaction energy is obtained
as

$E_{Hx}[G^{s}]=\frac{1}{2}\int d1d2G^{s}(1,2)\Sigma^{(1)}(2,1)[G^{s}]\;.$ (22)

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
$\Sigma\approx\Sigma_{H}+iGW$. $W$ is typically calculated within the RPA
which consists in approximating $\Gamma^{(0)}_{Hxc}\approx\Gamma^{(0)}_{H}$ in
the BSE (13). Making both approximations and using eqs. 14 and 9, the RPA
exchange-correlation energy

$\displaystyle E^{RPA}_{xc}=$ $\displaystyle i\frac{1}{2}\int
d1d2G^{s}(1,2)G^{s}(2,1)W(2,1)$ (23) $\displaystyle=$
$\displaystyle-\frac{1}{2}\int
d1d2P^{(0)}(1,2)\;\left\\{W^{(0)}(2,1)+\frac{1}{2}W^{(0)}(2,3)P^{(0)}(3,4)W^{(0)}(4,1)+\dots\right\\}$

is obtained.123 Isolating the exchange contribution to the Hartree-exchange
energy,

$E_{x}=\int d1d2\;\delta(\tau_{12})G^{s}(1,2)W^{(0)}(2,1)G^{s}(2,1)\;,$ (24)

we obtain the RPA correlation energy

$\displaystyle E^{RPA}_{c}=$ $\displaystyle-\frac{1}{2}\sum_{n}\frac{1}{n}\int
d1d2\left\\{\left[\int d3P^{(0)}(1,3)W^{(0)}(3,2)\right]^{n}+\int
d3P^{(0)}(1,3)W^{(0)}(3,2)\right\\}$ (25) $\displaystyle=$
$\displaystyle\frac{1}{2}\int d1d2\left\\{\ln\left[\delta(1,2)-\int
d3P^{(0)}(1,3)W^{(0)}(3,2)\right]+\int d3P^{(0)}(1,3)W^{(0)}(3,2)\right\\}\;,$

and using (2) as well as the symmetry of the polarizability on the imaginary
frequency axis, its well-known representation due to Langreth and Perdew12 is
obtained,

$\displaystyle E^{RPA}_{c}=$ $\displaystyle\frac{1}{2\pi}\int
d\bm{r}_{1}\int^{\infty}_{0}d\omega\left\\{\ln\left[\delta(1,2)-\int
d\bm{r}^{\prime}P^{(0)}(\bm{r}_{1},\bm{r}^{\prime},i\omega)v_{c}(\bm{r}^{\prime},\bm{r}_{1})\right]\right.$
(26) $\displaystyle+\left.\int
d\bm{r}^{\prime}P^{(0)}(\bm{r}_{1},\bm{r}^{\prime},i\omega)v_{c}(\bm{r}^{\prime},\bm{r}_{1})\right\\}\;.$

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 well-known 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,

$\Gamma_{xc}^{(0),HF}(1,2,3,4)=i\frac{\delta\Sigma^{HF}_{xc}(1,3)}{\delta
G^{s}(4,2)}=-W^{(0)}(1,2)\delta(1,4)\delta(3,2)\;.$ (27)

Using this expression in (6) with (8) yields the AC-SOSEX contribution to the
self-energy.127, 118 We first notice that within the pair bubble approximation
for $W$, (6) becomes

$\Sigma^{\textrm{SOSEX}(W,v_{c})}(1,2)=-\int
d3d4G^{s}(1,3)W(1,4)G^{s}(3,4)G^{s}(4,2)W^{(0)}(3,2)\;,$ (28)

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 $W\chi^{(0)}=W^{(0)}\chi$ in (6) (see supporting
information) which is only valid if $W$ is calculated within the RPA. 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,

$\Gamma_{xc}^{(0),GW}(1,2,3,4)=i\frac{\delta\Sigma^{GW}_{xc}(1,3)}{\delta
G^{s}(4,2)}=-W(1,2)\delta(1,4)\delta(3,2)\;.$ (29)

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

$\Sigma^{\textrm{G3W2}}(1,2)=-\int
d3d4G^{s}(1,3)W(1,4)G^{s}(3,4)G^{s}(4,2)W(3,2)$ (30)

and contains the AC-SOSEX self-energy.

Figure 1: Diagrammatic representation of the different contributions to the
second order exchange (SOX) term. pluses and minuses denote the different
branches on the Keldysh contour. The double and single wiggly lines are
screened and bare electron-electron interactions, respectively a) Greater and
lesser contributions to the full $G3W2$ 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 $G3W2$ 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.

The $G3W2$ self-energy can be decomposed into 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 $G3W2$ 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,

$\Sigma^{\textrm{SOSEX}(W(0),W(0))}(1,2)=-\int
d3d4G^{s}(1,3)W(1,4)G^{s}(3,4)G^{s}(4,2)W(3,2)\delta(\tau_{32})\delta(\tau_{14})\;,$
(31)

whose diagrammatic form is shown in figure 1c). Due to the presence of the two
$\delta$-functions, only two out of the eight diagrams of the $G3W2$ 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 $\frac{1}{n}$ in (17). Using (9),
eq. (31) can be written as

$\Sigma^{\textrm{SOSEX}(W(0),W(0))}(1,2)=\Sigma^{\textrm{MP2-SOX}}(1,2)+\Sigma^{\delta\textrm{MP2-SOX}}(1,2)\;,$
(32)

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

$\delta W(1,2)=\int d3d4W^{(0)}(1,3)P(3,4)W^{(0)}(4,2)\;,$ (33)

it can be written as

$\displaystyle\Sigma^{\delta\textrm{MP2-SOX}}(1,2)=$ $\displaystyle-\int
d3d4G^{s}(1,3)G^{s}(3,4)G^{s}(4,2)\left[W^{(0)}(1,4)\delta
W(3,2)\delta(\tau_{32})\right.$ (34) $\displaystyle\left.\qquad+\delta
W(1,4)\delta(\tau_{14})W^{(0)}(3,2)+\delta W(1,4)\delta(\tau_{14})\delta
W(3,2)\delta(\tau_{32})\right]\;.$

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 $\left\\{\phi_{i}(\bm{r,\sigma})\right\\}$ in which the time-ordered KS
Green’s function is diagonal,

$\displaystyle G^{s}_{kk^{\prime}}(i\tau_{12})=$
$\displaystyle\delta_{kk^{\prime}}\Theta(\tau_{12})G^{>}_{kk^{\prime}}(i\tau_{12})-\delta_{kk^{\prime}}\Theta(\tau_{21})G^{<}_{kk^{\prime}}(i\tau_{21})$
(35) $\displaystyle G^{>}_{kk}(i\tau_{12})=$ $\displaystyle
i\left(1-f(\epsilon_{k})\right)e^{-\epsilon_{k}\tau_{12}}$ $\displaystyle
G^{<}_{kk}(i\tau_{12})=$ $\displaystyle
if(\epsilon_{k})e^{-\epsilon_{k}\tau_{12}}\;.$

The $\epsilon_{k}$ denote KS eigenvalues which are understood to be measured
relative to the chemical potential $\mu$ and $f(\epsilon_{k})$ denotes the
occupation number of the $k$th orbital. One can now obtain energy expressions
analogous to (26). For example, inserting the AC-SOSEX self-energy (28) into
(21), we obtain

$\displaystyle E^{\textrm{SOSEX}}_{c}=$ $\displaystyle\frac{1}{2}\int
d1d2d3d4\;G^{s}(1,2)G^{s}(2,3)G^{s}(3,4)G^{s}(4,1)$ (36)
$\displaystyle\times\left\\{\frac{1}{2}W^{(0)}(3,1)W^{(0)}(2,4)+\frac{1}{3}\int
d5d6W^{(0)}(3,1)W^{(0)}(2,5)P^{(0)}(5,6)W^{(0)}(6,4)+\dots\right\\}\;.$

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

$\Sigma_{Hxc}^{\lambda}=\sum_{n=1}^{\infty}\lambda^{n}\Sigma_{Hxc}^{(n)}\left[G^{s},v_{c}\right]$
(37)

we can rewrite (21) as an integral over a coupling constant $\lambda$,

$E_{c}=\frac{1}{2}\sum_{n=2}\frac{1}{n}\int
d1d2G^{s}(1,2)\Sigma_{Hxc}^{(n)}(2,1)[G^{s}]=\frac{1}{2}\int^{1}_{0}\frac{d\lambda}{\lambda}\int
d1d2G^{s}(1,2)\Sigma_{Hxc}^{(\lambda)}(2,1)[G^{s}]\;,$ (38)

Therefore, (37) becomes

$\Sigma_{Hxc}^{\lambda}=\sum_{n=1}^{\infty}\Sigma_{Hxc}^{(n)}\left[G^{s},\lambda
v_{c}\right]=\sum_{n=1}^{\infty}\Sigma_{Hxc}^{(n)}\left[G^{s},W^{(\lambda)}\right]\;,$
(39)

where $W^{(\lambda)}$ is defined as in (15), with $W^{(0)}$ replaced by
$\lambda W^{(0)}$. Defining

$\overline{W}=\int^{1}_{0}d\lambda W^{(\lambda)}\;,$ (40)

and

$\overline{\Sigma}=\Sigma\left[\overline{W}\right]$ (41)

the correlation energy becomes

$E_{c}=\frac{1}{2}\int d1d2\;G^{s}(1,2)\overline{\Sigma}_{c}(2,1)\;.$ (42)

The integral in (40) needs to be computed numerically, but converges typically
very fast when Gauss-Legendre grids are employed.87 In ref. 130 a trapezoidal
rule for the solution of this integral has been used and also ref. 3 suggests
that this choice is often suitable for the calculation of correlation energies
within the RPA and beyond. This choice is very well justified for weakly
correlated systems for which the adiabatic connection is approximately a
straight line.131, 132 Below, we will assess the effect of such approximate
coupling constant integration on absolute and relative correlation energies
for non-covalent interactions. Notice, that using a trapezoidal rule, (42)
reduces to

$E_{c}=\frac{1}{4}\int d1d2\;G^{s}(1,2)\Sigma_{c}(2,1)\;,$ (43)

and when the statically screened $G3W2$ self-energy (31) is used in this
expression, the energy expression of ref. 114 is obtained. When additionally
both $W(0)$ are replaced by $v_{c}$, (43) 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. 42, 28 and 35 we have

$\displaystyle E^{\textrm{SOSEX}(W,v_{c})}=$
$\displaystyle\frac{i}{2}\sum_{pqrs}\int
d\tau_{12}d\tau_{3}G^{s}_{p}(\tau_{13})G^{s}_{q}(\tau_{31})G^{s}_{r}(\tau_{12})G^{s}_{s}(\tau_{21})W^{(0)}_{spqr}\overline{W}_{rspq}(\tau_{23})$
(44) $\displaystyle=$ $\displaystyle-\frac{1}{4\pi}\sum_{pqrs}\int
d\omega^{\prime}W^{(0)}_{spqr}\overline{W}_{rspq}(i\omega^{\prime})\int
d\tau_{12}G^{s}_{r}(\tau_{12})G^{s}_{s}(\tau_{21})$
$\displaystyle\times\underbrace{\int
d\tau_{3}e^{-i\omega^{\prime}\tau_{23}}G^{s}_{p}(\tau_{13})G^{s}_{q}(\tau_{31})}_{I(i\tau_{12})}\;.$

In going from the second equations, we have used (2) to transform $W$ to the
imaginary frequency axis. The integral over $\tau_{3}$ can be evaluated by
splitting it at $\tau_{1}$ and using the definition of the KS Green’s function
(35),

$I(i\tau_{12})=\frac{\left[\left(1-f(\epsilon_{p})\right)f(\epsilon_{q})-\left(1-f(\epsilon_{q})\right)f(\epsilon_{p})\right]e^{i\omega^{\prime}\tau_{12}}}{\epsilon_{p}-\epsilon_{q}+i\omega^{\prime}}=-e^{i\omega^{\prime}\tau_{12}}\frac{f(\epsilon_{p})-f(\epsilon_{q})}{\epsilon_{p}-\epsilon_{q}+i\omega^{\prime}}$
(45)

The remaining integral over $\tau_{12}$ is

$I_{\tau_{12}}=-\int
G^{(0)}_{r}(\tau_{12})G^{(0)}_{s}(\tau_{21})e^{i\omega^{\prime}\tau_{12}}d\tau_{12}=\frac{f(\epsilon_{r})-f(\epsilon_{s})}{\epsilon_{r}-\epsilon_{s}-i\omega^{\prime}}\;,$
(46)

so that the correlation energy becomes

$E^{\textrm{SOSEX}(W,v_{c})}=-\frac{1}{4\pi}\sum_{pqrs}\int
d\omega^{\prime}W^{(0)}_{spqr}\overline{W}_{rspq}(i\omega^{\prime})\frac{f(\epsilon_{r})-f(\epsilon_{s})}{\epsilon_{r}-\epsilon_{s}-i\omega^{\prime}}\frac{f(\epsilon_{p})-f(\epsilon_{q})}{\epsilon_{p}-\epsilon_{q}+i\omega^{\prime}}$
(47)

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

$E^{\textrm{SOSEX}(W,v_{c})}=-\frac{1}{4\pi}\sum^{occ}_{ij}\sum^{virt}_{ab}\int^{\infty}_{0}d\omega\overline{W}_{iajb}(i\omega)W^{(0)}_{jaib}\frac{4(\epsilon_{i}-\epsilon_{a})(\epsilon_{j}-\epsilon_{b})}{\left[(\epsilon_{i}-\epsilon_{a})^{2}+\omega^{2}\right]\left[(\epsilon_{j}-\epsilon_{b})^{2}+\omega^{2}\right]}\;.$
(48)

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

$\displaystyle E^{\textrm{SOSEX}(W(0),W(0))}=$
$\displaystyle-\frac{1}{4}\sum_{pqrs}\int
d\tau_{12}G^{s}_{p}(\tau_{12})G^{s}_{q}(\tau_{21})G^{s}_{r}(\tau_{12})G^{s}_{s}(\tau_{21})$
(49)
$\displaystyle\times\overline{W}_{spqr}(i\omega=0)\overline{W}_{rspq}(i\omega=0)$
$\displaystyle=$
$\displaystyle-\frac{1}{2}\sum^{occ}_{ij}\sum^{virt}_{ab}\frac{\overline{W}_{spqr}(i\omega=0)\overline{W}_{rspq}(i\omega=0)}{\epsilon_{i}+\epsilon_{j}-\epsilon_{a}-\epsilon_{b}}\;.$

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.

##  3 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) is calculated in imaginary
time with quadratic scaling with system size in the atomic orbital basis. The
implementation is described in detail in ref. 39. In all calculations, we
expand the KS Green’s functions in correlation consistent bases of Slater-type
orbitals of triple- and quadruple-$\zeta$ quality (TZ3P and QZ6P,
respectively).134 All 4-point correlation functions (screened and unscreened
Coulomb interactions as well as polarizabilities) are expressed in auxiliary
basis sets of Slater type functions which are usually 5 to 10 times larger
than the primary bases. In all calculations, we use auxiliary basis sets of
_VeryGood_ quality. The transformation between primary and auxiliary basis
(for the polarizability) is implemented with quadratic scaling with system
size using the pair-atomic density fitting (PADF) method for products of
atomic orbitals.135, 136 For an outline of the implementation of this method
in ADF, we refer to ref. 137. Eq. (26) is then evaluated in the basis of
auxiliary fit functions with cubic scaling with system size. Equations 48 and
49 are evaluated with quintic scaling with system size in the canonical basis
of KS states. This implementation is completely equivalent to the canonical
MP2 implementation outlined in ref. 137.

Eq. (40) is evaluated using Gauss-Legendre grids with 8 points, except for the
potential energy curves where 20 points have been used. At stretched bonds,
the integrands become increasingly non-linear and a large number of
integration points are necessary. As discussed in detail in the supporting
information, for non-covalent interactions a single integration point does
generally suffice and therefore we have used a single integration point only
for all calculations for the S66 and S66x8 database. In case of a single
$\lambda-$point, a trapezoidal rule is used for integration.

Imaginary time and imaginary frequency variables are discretized using non-
uniform bases $\mathcal{T}=\left\\{\tau_{\alpha}\right\\}_{\alpha=1,\dots
N_{\tau}}$ and $\mathcal{W}=\left\\{\omega_{\alpha}\right\\}_{\alpha=1,\dots
N_{\omega}}$ of sizes $N_{\tau}$ and $N_{\omega}$, respectively, tailored to
each system. More precisely, (2) and (3) are then implemented by splitting
them into sine- and cosine transformation parts as

$\displaystyle\overline{F}(i\omega_{\alpha})=$
$\displaystyle\sum_{\beta}\Omega^{(c)}_{\alpha\beta}\overline{F}(i\tau_{\beta})$
(50) $\displaystyle\underline{F}(i\omega_{\alpha})=$
$\displaystyle\sum_{\beta}\Omega^{(s)}_{\alpha\beta}\underline{F}(i\tau_{\beta})\;,$

where $\overline{F}$ and $\underline{F}$ denote even and odd parts of $F$,
respectively. The transformation from imaginary frequency to imaginary time
only requires the (pseudo)inversion of $\Omega^{(c)}$ and $\Omega^{(s)}$,
respectively. Our procedure to calculate $\Omega^{(c)}$ and $\Omega^{(s)}$ as
well as $\mathcal{T}$ and $\mathcal{W}$ 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

$E_{CBS}=E_{QZ}+\frac{E_{QZ}*4^{3}-E_{TZ}*3^{3}}{4^{3}-3^{3}}\;,$ (51)

where $E_{QZ}$ ($E_{TZ}$) denotes the total energies at the QZ6P (TZ3P) level.
The extrapolation scheme has been shown to be suitable for correlation
consistent basis sets but cannot be used for KS or HF contributions.141, 142
Therefore, we do not extrapolate the DFT energies, but assume them to be
converged on the QZ level. Since the basis set error is not completely
eliminated with this approach, we also counterpoise correct all energies,
taking into account 100 % of the counterpoise correction. With these settings,
we assume all our calculated values to be converged well enough to be able to
draw quantitative conclusions about the performance of the methods we
benchmark herein. We use the _VeryGood_ numerical quality for integrals over
real space and distance cut-offs. Dependency thresholds39 have been set to
$5e^{-4}$. All Full configuration interaction calculations have been performed
with the code by Knowles and Handy.143, 144 The 1- and 2-electron integral
which are required as input have been generated with ADF.

##  4 Results

### 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: H2 is covalently bound, LiH is an ionic molecule, and He2 has a very
weak, non-covalent bond.

Figure 2: Potential energy curves (in Hartree) of H2 (left) and LiH (middle),
as well as binding energy (in mHartree) as a function of system size for He2
on the right using FCI, RPA@PBE and different variants of RPA+SOSEX@PBE. For
H2 and He2, all calculations have been performed with the TZ3P basis set. For
Lih, all calculations have been performed using the TZP basis set.

The dissociation curve of H2 calculated with RPA+SOSEX($W(0)$,$W(0)$)@PBE is
the red line in figure 2. Our calculations are not converged with respect to
the basis set size but comparison of our dissociation curves calculated with
RPA@PBE and RPA+SOSEX($W$,$v_{c}$)@PBE to the ones presented in ref. 112 and
145 clearly shows that their qualitative behavior is reproduced correctly. It
is known that RPA describes the dissociation of covalently bonded molecules
qualitatively correctly while RPA+SOSEX($W$,$v_{c}$) and other exchange-
corrected RPA approaches fail.112, 145, 91 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 effects145 whose good description is necessary to dissociate H2
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 qualitative 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 H2 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 electron-electron interaction lines are screened. For LiH, we
find a similar behavior than for H2. For He2 (notice that we plotted here the
binding energy and not the total energy) we see that all approaches give the
correct dissociation limit.

method |  H2 |  LiH |  He2 |  F2 |  Be2
---|---|---|---|---|---
exp. |  |  |  | 1.413146 | 2.320147
accurate | 0.741 | 1.601 | 2.626 | 1.413146 | 2.320148
RPA | 0.742 | 1.597 | 2.632 | 1.437 | 2.403
RPA + SOSEX($W(0),W(0)$) | 0.744 | 1.605 | 2.852 | 1.444 | 2.424
RPA + SOSEX($W,v_{c}$) | 0.738 | 1.594 | 2.871 | 1.364 | –
RPA + SOSEX($W(0),v_{c}$) | 0.735 | 1.599 | 3.542 | 1.348 | –
Table 1: Equilibrium bond length of selected molecules. All values are in
$\AA$. The bond lengths for H2, He2, and LiH have been calculated using the
TZ3P and TZP basis sets to make them comparable to the FCI result. The bond
lengths for F2 and Be2 have been obtained using the QZ6P basis set. All
RPA(+SOSEX) calculations have been performed with a PBE Green’s function.

From these potential energy curves, we also extracted the equilibrium bond
lengths via cubic spline interpolation. These are shown in table 1 Around the
equilibrium distances, RPA+SOSEX($W$,$v_{c}$) generally gives the best
energies but this does not necessarily translate into the best bond lengths.
For the covalently bound molecules LiH and F2 as well as LiH
RPA+SOSEX($W$,$v_{c}$) underestimates and RPA+SOSEX($W(0)$,$W(0)$)
overestimates the bond lengths. Again, RPA+SOSEX($W(0)$,$W(0)$) behaves
qualitatively similar to RPA. For He2, both approaches give similar results,
while RPA+SOSEX($W(0)$,$v_{c}$) fails completely. On the other hand, unlike
RPA+SOSEX($W(0)$,$W(0)$), RPA+SOSEX($W$,$v_{c}$) does predict an unbound Be2
dimer.

### Dissociation of charged Dimers

| RPA | SOSEX($W$,$v_{c}$) | SOSEX($W(0)$,$v_{c}$) | SOSEX($W(0)$,$W(0)$)
---|---|---|---|---
H2+ |  |  |  |
1.0 | 5.19 | 0.76 | -2.58 | 3.09
1.25 | 7.59 | -0.26 | -5.33 | 5.19
1.50 | 11.21 | -1.31 | -8.23 | 8.89
1.75 | 16.15 | -2.30 | -11.14 | 14.27
He2+ |  |  |  |
1.0 | 13.23 | 0.23 | -5.30 | 14.34
1.25 | 25.40 | -2.84 | -12.91 | 27.56
1.50 | 40.60 | -5.64 | -20.32 | 44.79
1.75 | 56.76 | -7.65 | -25.76 | 63.38
(NH3)2+ |  |  |  |
1.0 | 5.89 | 15.17 | 24.91 | 16.23
1.25 | 13.00 | 20.08 | 36.23 | 33.50
1.50 | 20.61 | 21.89 | 42.78 | 50.41
1.75 | 30.88 | 15.14 | 28.73 | 61.48
(H2O)2+ |  |  |  |
1.0 | 10.19 | 29.79 | 51.70 | 33.79
1.25 | 20.62 | 12.16 | 21.61 | 38.68
1.50 | 31.88 | 2.35 | 4.58 | 50.58
1.75 | 42.08 | 0.50 | 5.47 | 65.61
MAD | 21.95 | 8.63 | 19.22 | 33.24
Table 2: Errors in kcal/mol for the charger dimers in the SIE4x4 benchmark set
calculated with RPA and different variants of RPA+SOSEX. PBE orbitals have
been used in all calculations

In table 2 we investigate the dissociation of four charged dimers by means of
the SIE4x4 dataset.149 Here, the self-correlation error of RPA leads to
considerable underbinding,112, 145, 6 whereas RPA+SOSEX($W$,$v_{c}$) is
exact,113 the remaining error for H2 being due to basis set errors as well as
the fact that PBE orbitals have been used. Furche and coworkers have observed
a catastrophic failure of RPA+SOX for (NH3)2+ and (H2O)2+150 and also
SOSEX($W(0)$,$W(0)$) considerably deteriorates the RPA results for those
systems. Only for H2+, one finds that the partial cancellation of the RPA
self-correlation leads to small improvements over RPA.

### 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 HTBH38153 and
NHTBH38154 databases for barrier heights by Truhlar and coworkers, which are
typically used in benchmarks of (beyond-)RPA methods.86, 5, 6, 87 The
reference values have been calculated with the W2-F12 protocol.155, 149 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

Figure 3: Mean absolute deviations (MAD) (lower triangle in each plot) and
Maximum deviations (MAX) (upper triangle) with respect to the reference values
as well as using different KS Green’s functions as input for BH76 (left),
G21-IP (middle) and G21-EA (right) for RPA (top) and RPA+SOSEX($W(0)$,$W(0)$)
(bottom). All values are in kcal/mol.

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, 157, 158, 159,
160, 161, 162, 163, 164 (see ref. 165 for a review) This is rarely done in
practice since self-consistent RPA calculations are computationally demanding
and since the resulting energies are often worse than the ones evaluated using
a Green’s function from a generalized gradient approximation (GGA) or hybrid
calculation.159 GGAs like PBE or TPSS are often used to construct $G^{s}$.87,
48, 9 Using hybrid orbitals can be seen as a pragmatic way to compensate for
the lack of self-consistency in the RPA calculation and therefore we assess
here whether they lead to improvements over GGA orbitals.

For W4-11, the differences between different starting points are minor, but
PBE tends to give the best results. For the BH76, G21IP, and G21EA datasets,
we show mean absolute deviations (MAD) and maximum deviations (MAX) with
respect to the reference values and with respect to the different starting
points in figure 3. The RPA results generally improve with increasing amount
of Fock exchange, while 25 % (PBE0) generally seems to work best for
RPA+SOSEX($W(0)$,$W(0)$). The differences are often substantial, for instance
in case of the RPA barrier heights (fig 3a) or the RPA+SOSEX($W(0)$,$W(0)$)
electron affinities (fig 3f).

For charged excitations, this observation aligns very well with the experience
from $G_{0}W_{0}$ calculations where hybrid functional with 25 - 50 % are
typically a much better starting point than GGAs.166, 167 However, when $G3W2$
corrections are added to the $G_{0}W_{0}$ QP energies, using a hybrid
functional with a smaller fraction of exact exchange might often be
beneficial.168, 114 For barrier heights, hybrid functionals with a larger
fraction of exact exchange are usually required to obtain qualitatively
correct barrier heights149, 169 and it therefore not surprising that hybrid
orbitals serve as a suitable starting point for RPA calculations.

Figure 4: Errors of RPA@PBE and RPA+SOSEX($W$,$v_{c}$)@PBE for the atomization
energies in the W4-11 dataset. Black dots denote the individual data points
and the horizonal line in each box denotes the median deviation. the box
contains all data points between the first quartile ($Q1$) and third quartile
($Q2$) and the whiskers are at $Q1\pm|Q1-Q3|$. (in case of a normal
distribution, the whiskers include 99.3% of all data points). All values are
in kcal/mol.

Our atomization energies for the W4-11 dataset are shown in figure 4. It has
first been observed by Furche170 that RPA underestimates atomization energies
(indicated here by negative errors). 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.86, 6 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.Notice,
that our non-counterpoise corrected calculations based on (T,Q) extrapolation
will still include a sizable basis set incompleteness error for atomization
energies. However, our qualitative conclusions will be valid. 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.

Figure 5: Errors of RPA@PBE and different RPA+SOSEX variants for barrier
heights (BH76, left), ionization potentials (G21-IP, middle) and electron
affinities (G21-EA, right). For an explanation of the boxplots, see the
caption of figure 4. All values are in kcal/mol.

As also shown in more detail in figure 5, the performance of
RPA+SOSEX($W(0)$,$W(0)$) is in all cases comparable to RPA+SOSEX($W$,$v_{c}$),
for which the trends presented here are well known:112, 172, 5, 6, 87
RPA+SOSEX($W$,$v_{c}$), fails for barrier heights, where the inclusion of
renormalized singles excitations is necessary to obtain good results86, 6, 87
and works very well for charged excitations.6, 5 We note, that
RPA+SOSEX($W(0)$,$W(0)$)@PBE0 performs very well for charged excitations, with
an accuracy challenging modern double-hybrid functionals.149

### Non-covalent Interactions

#### S66 Interaction Energies

Figure 6: Mean absolute deviations (MAD) (lower triangle in each plot) and
Maximum deviations (MAX) (upper triangle) for a) RPA, b) SOSEX($W(0),W(0)$)
and c) SOSEX($W(0),v_{c}$) interaction energies for the S66 database using
different KS Green’s functions as well as to the CCSD(T) reference values
(ref.). All values are in kcal/mol.

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 database173 obtained using
different $G^{s}$ are compared to each other as well as to the CCSD(T)
reference values by Hobza and coworkers.173 All values have been obtained
using a single integration point for the $\lambda$-integral. As shown in the
supporting information, a few outliers aside the errors arising from this
approximation are small for non-covalent interactions. RPA and
RPA+SOSEX($W(0),W(0)$) are equivalently independent of the choice of the KS
Green’s function, with MADs between 0.20 and 0.39 kcal/mol between the
different functionals. However, individual values can differ by almost 2
kcal/mol which is a sizable difference, given that the largest interaction
energies in the S66 database are of the order of 20 kcal/mol only. The
performance of RPA compared to the CCSD(T) reference is rather insensitive to
the KS Green’s function, even though the hybrid starting points lead to
slightly better results.With 0.52 kcal/mol, the MAD for RPA@PBE is in
excellent agreement with the 0.61 kcal/mol MAD obtained by Nguyen _et. al._ in
ref. 48, which has been obtained with GTO-type basis sets and 50 %
counterpoise correction instead of 100 %. This shows, that our interaction
energies are well converged with respect to the basis set size. The
RPA+SOSEX($W(0),W(0)$) results are much better using the hybrid functionals
than with PBE. RPA+SOSEX($W,v_{c}$)@PBE, is slightly more accurate than
RPA+SOSEX($W,v_{c}$)@PBE0, but unlike for the datasets discussed before, the
differences between the different starting points are negligibly small. Also,
the dependence of SOSEX($W,v_{c}$) on the starting point is smaller than for
SOSEX($W(0),W(0)$).

Figure 7: Deviations of RPA@PBE0 and both RPA + SOSEX@PBE0 variants for the S66 database with respect to the CCSD(T) reference. All values are in kcal/mol. | MAD
---|---
| S66 | hydr. bond | dispersion | mixed
Method | [$\frac{kcal}{mol}$] | [%] | [$\frac{kcal}{mol}$] | [%] | [$\frac{kcal}{mol}$] | [%] | [$\frac{kcal}{mol}$] | [%]
SOSEX($W(0),W(0)$)@PBE0 | 0.32 | 7.28 | 0.45 | 5.76 | 0.29 | 10.33 | 0.21 | 5.50
SOSEX($W(0),v_{c}$)@PBE0 | 0.28 | 6.88 | 0.30 | 3.42 | 0.34 | 11.77 | 0.20 | 5.25
SOSEX($W,v_{c}$)@PBE0 | 0.29 | 6.85 | 0.31 | 3.39 | 0.33 | 11.63 | 0.21 | 5.33
SOSEX($W,v_{c}$)@PBE | 0.26 | 6.25 | 0.23 | 3.51 | 0.33 | 10.16 | 0.17 | 4.26
RPA | 0.46 | 11.54 | 0.55 | 7.19 | 0.47 | 17.74 | 0.34 | 9.41
PBE0-D3(BJ) | 0.28 | 5.09 | 0.47 | 4.80 | 0.18 | 5.09 | 0.18 | 5.42
DSD-PBE-P86-D3(BJ) | 0.23 | 5.07 | 0.31 | 3.71 | 0.21 | 6.99 | 0.16 | 4.43
Table 3: MADs (absolute and in %) of different electronic structure methods
with respect to the CCSD(T) reference values for the whole S66 database and
for its subcategories.

Figure 7 shows the deviations of RPA and both RPA+SOSEX variants with respect
to CCSD(T) for all datapoints in the S66 database. MADs and mean absolute
percentage deviations (MAPD) for the whole database as well as for the
individual categories are presented in table 3. The interactions of the first
22 complexes in the database are dominated by Hydrogen bonds which are
predominantly of electrostatic origin.131 The next 22 complexes are mostly
bound by dispersion interactions and the remaining interactions are of mixed
nature.173 It is useful to distinguish between these different interaction
patterns in the following comparison.

For the whole database, RPA gives a MAPD of 11.5 % and the SOSEX corrections
sizably reduce the MAPDs with respect to the CCSD(T) reference values to in
between 7.3 % and 6.3 %. SOSEX($W,v_{c}$) outperforms SOSEX($W(0),W(0)$) by
far for the hydrogen-bonded complexes, and is even slightly more accurate than
the double-hybrid DSD-PBE-P86-D3(BJ),175 one of the best double hybrid
functionals for weak interactions.176 For dispersion interactions, the
performance of SOSEX($W(0),W(0)$) and SOSEX($W,v_{c}$) is comparable. Here,
the empirically dispersion corrected177, 178 functionals, the hybrid
PBE0-D3(BJ) and DSD-PBE-P86-D3(BJ), are much more accurate than all MBPT based
methods. A few exceptions aside, fig. 7 shows that RPA understabilizes the
complexes in the S66 database (indicated by positive errors). SOSEX
corrections lower the interaction energies, i.e. the complexes are predicted
to be more stable. SOSEX($W,v_{c}$) shows a tendency to overstabilize the
hydrogen-bonded complexes. For these systems, the RPA+SOSEX($W(0),W(0)$)
energies are almost identical to the ones from RPA.

Also from the sizable differences of SOSEX($W,v_{c}$) (green points) to its
static limit (with only a single screened interaction line, blue points) shown
in figure 7 it is clear that the dynamical screening effects are important for
the hydrogen-bonded complexes. As can be seen from the MAPD in table 3, this
does however not improve agreement with the CCSD(T) reference values. For the
dispersion bound complexes, there are only negligible differences between both
variants, demonstrating that the dynamical variations of the screening average
out. For the last 22 complexes in the database the differences are slightly
larger. In all cases, dressing the second electron-electron interaction line
does not alter the results decisively.

#### 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
0.9-0.95), middle (1.0-1.25) and long (1.5-2.0). All RPA (+SOSEX) calculation
discussed here have been performed using a PBE0 Green’s function.

Figure 8: MADs (in percent) for the S66x8 database with respect to the CCSD(T) reference values for RPA, RPA+SOSEX$(W,v_{c})$ and RPA+SOSEX$(W(0),W(0))$. MADs are shown separately for the whole database (columns on the left) and for different monomer-monomer separations. | short [%] | middle [%] | long [%]
---|---|---|---
SOSEX$(W,v_{c})$ | 35.2 | 42.8 | 13.5
SOSEX$(W(0),W(0))$ | 31.0 | 37.9 | 19.1
Table 4: Relative improvements obtained with different SOSEX variants over RPA
for different groups of monomer-monomer separations. Figure 9: Upper plots:
three RPA+SOSEX($W$,$v_{c}$)@PBE0 potential energy curves in the S66x8
database (green), separated in correlation contributions (yellow) and HF
energy (evaluated with PBE0 orbitals). Lower plots: decomposition of the
correlation energies into RPA and SOSEX($W$,$v_{c}$) contributions. All values
are in kcal/mol.

The results of our comparison are shown in figure 8, where the MAPDs with
respect to CCSD(T) for the whole database as well as for the scaled monomer-
monomer separations are shown. For the whole database, the average relative
deviations with respect to the reference values are larger than for S66. With
in between 31 and 43 %, both SOSEX correction lead to sizable improvements
over the RPA in the short and medium regime. For large monomer-monomer
separations, the improvements become much smaller, with 14 % for
SOSEX$(W,v_{c})$ and 19 % for SOSEX$(W(0),W(0))$. 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 the RPA+SOSEX($W$,$v_{c}$)
potential energy curves (Green curves in the upper plots) in the S66x8
database and separated the correlation contributions (The Green curves are the
sums of the red and yellow curves). The lower plots separately show the RPA
and SOSEX($W$,$v_{c}$) contributions to the correlation energy differences.

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.

##  5 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 $G3W2$ correction to the $GW$ self-energy.115, 114 This
energy expression has already been introduced in our earlier work114, 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 $\lambda$-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 complexes 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 heights153,
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 developed179 and a
recent RPA+SOSEX implementation by Ochsenfeld and co-workers180 allowed
applications to the L7 dataset,181 albeit with small basis sets. Other low-
scaling MP2 implementations182, 183, 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 excitations.6, 87 Other methods which have been shown to
outperform SOSEX, in particular for barrier heights, are the AXK kernel
method94, 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.

This research received funding (project number 731.017.417) from the
Netherlands Organization for Scientific Research (NWO) in the framework of the
Innovation Fund for Chemistry and from the Ministry of Economic Affairs in the
framework of the “ _TKI/PPS-Toeslagregeling_ ”. We thank Mauricio Rodríguez-
Mayorga and Timothy Daas for fruitful discussions.

Derivation of (6), discussion of the crossing symmetries of the 4-point
vertices, assessment of the dependence of the $\lambda$-integration on the
number of Gauss-Legendre points for the S66 database, .csv files with all
calculated energies at the extrapolated CBS TZ3P, and QZ6P level and
explanations of those files.

## References

  * Macke 1950 Macke, W. Über die Wechselwirkungen im Fermi-Gas. _Zeitschrift fur Naturforsch._ 1950, _5_ , 192–208
  * Bohm and Pines 1953 Bohm, D.; Pines, D. A collective description of electron interactions: III. Coulomb interactions in a degenerate electron gas. _Phys. Rev._ 1953, _92_ , 609–625
  * Heßelmann and Görling 2011 Heßelmann, A.; Görling, A. Random-phase approximation correlation methods for molecules and solids. _Mol. Phys._ 2011, _109_ , 2473–2500
  * Eshuis and Furche 2011 Eshuis, H.; Furche, F. A parameter-free density functional that works for noncovalent interactions. _J. Phys. Chem. Lett._ 2011, _2_ , 983–989
  * Eshuis et al. 2012 Eshuis, H.; Bates, J. E.; Furche, F. Electron correlation methods based on the random phase approximation. _Theor. Chem. Acc._ 2012, _131_ , 1084
  * Ren et al. 2012 Ren, X.; Rinke, P.; Joas, C.; Scheffler, M. Random-phase approximation and its applications in computational chemistry and materials science. _J. Mater. Sci._ 2012, _47_ , 7447–7471
  * Chen et al. 2017 Chen, G. P.; Voora, V. K.; Agee, M. M.; Balasubramani, S. G.; Furche, F. Random-Phase Approximation Methods. _Annu. Rev. Phys. Chem._ 2017, _68_ , 421–445
  * Chedid et al. 2018 Chedid, J.; Ferrara, N. M.; Eshuis, H. Describing transition metal homogeneous catalysis using the random phase approximation. _Theor. Chem. Acc._ 2018, _137:158_ , 1–11
  * Kreppel et al. 2020 Kreppel, A.; Graf, D.; Laqua, H.; Ochsenfeld, C. Range-Separated Density-Functional Theory in Combination with the Random Phase Approximation: An Accuracy Benchmark. _J. Chem. Theory Comput._ 2020, _16_ , 2985–2994
  * Modrzejewski et al. 2020 Modrzejewski, M.; Yourdkhani, S.; Klimeš, J. Random Phase Approximation Applied to Many-Body Noncovalent Systems. _J. Chem. Theory Comput._ 2020, _16_ , 427–442
  * Langreth and Perdew 1975 Langreth, D. C.; Perdew, J. P. The Exhange-Correlation Energy of a metalic surface. _Solid State Commun._ 1975, _17_ , 1425–1429
  * Langreth and Perdew 1977 Langreth, D. C.; Perdew, J. P. Exchange-correlation energy of a metallic surface: Wave-vector analysis. _Phys. Rev. B_ 1977, _15_ , 2884–2901
  * Harris and Griffin 1975 Harris, J.; Griffin, A. Correlation energy and van der Waals interaction of coupled metal films. _Phys. Rev. B_ 1975, _11_ , 3669–3677
  * Coester 1958 Coester, F. Bound states of a many-particle system. _Nucl. Phys._ 1958, _7_ , 421–424
  * Coester and Kümmel 1960 Coester, F.; Kümmel, H. Short-range correlations in nuclear wave functions. _Nucl. Phys._ 1960, _17_ , 477–485
  * Čížek 1966 Čížek, J. On the Correlation Problem in Atomic and Molecular Systems. Calculation of Wavefunction Components in Ursell-Type Expansion Using Quantum-Field Theoretical Methods. _J. Chem. Phys._ 1966, _45_ , 4256–4266
  * Čížek 1969 Čížek, J. On the Use of the Cluster Expansion and the Technique of Diagrams in Calculations of Correlation Effects in Atoms and Molecules. _Adv. Chem. Physics, Vol. XIV_ 1969, _XIV_ , 35–89
  * Paldus et al. 1972 Paldus, J.; Čížek, J.; Shavitt, I. Correlation Problems in Atomic and Molecular Systems. IV. Extended Coupled-Pair Many-Electron Theory and Its Application to the BH3 Molecule. _Phys. Rev. A_ 1972, _5_ , 50–67
  * Scuseria et al. 2008 Scuseria, G. E.; Henderson, T. M.; Sorensen, D. C. The ground state correlation energy of the random phase approximation from a ring coupled cluster doubles approach. _J. Chem. Phys._ 2008, _129_ , 231101
  * Scuseria et al. 2013 Scuseria, G. E.; Henderson, T. M.; Bulik, I. W. Particle-particle and quasiparticle random phase approximations: Connections to coupled cluster theory. _J. Chem. Phys._ 2013, _139_ , 104113
  * Abrikosov et al. 1975 Abrikosov, A. A.; Gorkov, P. L.; Dzyaloshinski, I. E. _Methods of quantum field theory in statistical physics_ ; Dover Publications INC. New York, 1975
  * Mattuck 1992 Mattuck, R. D. _A Guide to Feynman Diagrams in the Many-body Problem_ , 2nd ed.; Dover Publications INC. New York, 1992
  * Bruus and Flensberg 2004 Bruus, H.; Flensberg, K. _Many-Body Quantum Theory in Condensed Matter Physics: An Introduction_ ; OUP Oxford: Oxford, 2004
  * Martin et al. 2016 Martin, R. M.; Reining, L.; Ceperley, D. M. _Interacting electrons_ ; Cambridge University Press, 2016
  * Klein 1961 Klein, A. Perturbation theory for an infinite medium of fermions. _Phys. Rev._ 1961, _121_ , 950–956
  * Luttinger and Ward 1960 Luttinger, J. M.; Ward, J. C. Ground-state energy of a many-fermion system. II. _Phys. Rev._ 1960, _118_ , 1417–1427
  * Kohn and Sham. L. J. 1965 Kohn, W.; Sham. L. J., Self-Consistent Equations Including Exchange and Correlation Effects. _Phys. Rev._ 1965, _140_ , A1133
  * Hohenberg and Kohn 1964 Hohenberg, P.; Kohn, W. Inhomogeneous Electron Gas. _Phys. Rev._ 1964, _136_ , 864–871
  * Casida 1995 Casida, M. E. Generalization of the optimized-effective-potential model to include electron correlation: A variational derivation of the Sham-Schlüter equation for the exact exchange-correlation potential. _Phys. Rev. A_ 1995, _51_ , 2005–2013
  * Dahlen et al. 2006 Dahlen, N. E.; van Leeuwen, R.; von Barth, U. Variational energy functionals of the Green function and of the density tested on molecules. _Phys. Rev. A - At. Mol. Opt. Phys._ 2006, _73_ , 012511
  * Hedin 1965 Hedin, L. New method for calculating the one-particle Green’s function with application to the electron-gas problem. _Phys. Rev._ 1965, _139_ , A796
  * 32 The pair bubble approximation is typically also denoted as RPA. To avoid potential confusion with the expression for the correlation energy, we will use the term bubble approximation when referring to the screening.
  * Zhang and Grüneis 2019 Zhang, I. Y.; Grüneis, A. Coupled cluster theory in materials science. _Front. Mater._ 2019, _6_ , 00123
  * Keller et al. 2022 Keller, E.; Tsatsoulis, T.; Reuter, K.; Margraf, J. T. Regularized second-order correlation methods for extended systems. _J. Chem. Phys._ 2022, _156_ , 024106
  * Eshuis et al. 2010 Eshuis, H.; Yarkony, J.; Furche, F. Fast computation of molecular random phase approximation correlation energies using resolution of the identity and imaginary frequency integration. _J. Chem. Phys._ 2010, _132_ , 234114
  * Wilhelm et al. 2016 Wilhelm, J.; Seewald, P.; Del Ben, M.; Hutter, J. Large-Scale Cubic-Scaling Random Phase Approximation Correlation Energy Calculations Using a Gaussian Basis. _J. Chem. Theory Comput._ 2016, _12_ , 5851–5859
  * Wilhelm et al. 2018 Wilhelm, J.; Golze, D.; Talirz, L.; Hutter, J.; Pignedoli, C. A. Toward GW Calculations on Thousands of Atoms. _J. Phys. Chem. Lett._ 2018, _9_ , 306–312
  * Wilhelm et al. 2021 Wilhelm, J.; Seewald, P.; Golze, D. Low-scaling GW with benchmark accuracy and application to phosphorene nanosheets. _J. Chem. Theory Comput._ 2021, _17_ , 1662–1677
  * Förster and Visscher 2020 Förster, A.; Visscher, L. Low-Order Scaling G0W0 by Pair Atomic Density Fitting. _J. Chem. Theory Comput._ 2020, _16_ , 7381–7399
  * Duchemin and Blase 2019 Duchemin, I.; Blase, X. Separable resolution-of-the-identity with all-electron Gaussian bases: Application to cubic-scaling RPA. _J. Chem. Phys._ 2019, _150_ , 174120
  * Duchemin and Blase 2021 Duchemin, I.; Blase, X. Cubic-Scaling All-Electron GW Calculations with a Separable Density-Fitting Space-Time Approach. _J. Chem. Theory Comput._ 2021, _17_ , 2383–2393
  * Schurkus and Ochsenfeld 2016 Schurkus, H. F.; Ochsenfeld, C. Communication: An effective linear-scaling atomic-orbital reformulation of the random-phase approximation using a contracted double-Laplace transformation. _J. Chem. Phys._ 2016, _144_ , 031101
  * Luenser et al. 2017 Luenser, A.; Schurkus, H. F.; Ochsenfeld, C. Vanishing-Overhead Linear-Scaling Random Phase Approximation by Cholesky Decomposition and an Attenuated Coulomb-Metric. _J. Chem. Theory Comput._ 2017, _13_ , 1647–1655
  * Vlček et al. 2017 Vlček, V.; Rabani, E.; Neuhauser, D.; Baer, R. Stochastic GW Calculations for Molecules. _J. Chem. Theory Comput._ 2017, _13_ , 4997–5003
  * Graf et al. 2018 Graf, D.; Beuerle, M.; Schurkus, H. F.; Luenser, A.; Savasci, G.; Ochsenfeld, C. Accurate and Efficient Parallel Implementation of an Effective Linear-Scaling Direct Random Phase Approximation Method. _J. Chem. Theory Comput._ 2018, _14_ , 2505–2515
  * Lu et al. 2010 Lu, D.; Nguyen, H. V.; Galli, G. Power series expansion of the random phase approximation correlation energy: The role of the third- and higher-order contributions. _J. Chem. Phys._ 2010, _133_
  * Lebègue et al. 2010 Lebègue, S.; Harl, J.; Gould, T.; Ángyán, J. G.; Kresse, G.; Dobson, J. F. Cohesive properties and asymptotics of the dispersion interaction in graphite by the random phase approximation. _Phys. Rev. Lett._ 2010, _105_ , 1–4
  * Nguyen et al. 2020 Nguyen, B. D.; Chen, G. P.; Agee, M. M.; Burow, A. M.; Tang, M. P.; Furche, F. Divergence of Many-Body Perturbation Theory for Noncovalent Interactions of Large Molecules. _J. Chem. Theory Comput._ 2020, _16_ , 2258–2273
  * Irmler et al. 2019 Irmler, A.; Gallo, A.; Hummel, F.; Grüneis, A. Duality of Ring and Ladder Diagrams and Its Importance for Many-Electron Perturbation Theories. _Phys. Rev. Lett._ 2019, _123_ , 156401
  * Hummel et al. 2019 Hummel, F.; Grüneis, A.; Kresse, G.; Ziesche, P. Screened Exchange Corrections to the Random Phase Approximation from Many-Body Perturbation Theory. _J. Chem. Theory Comput._ 2019, _15_ , 3223–3236
  * Singwi et al. 1968 Singwi, K. S.; Tosi, M. P.; Land, R. H.; Sjölander, A. Electron correlations at metallic densities. _Phys. Rev._ 1968, _176_ , 589–599
  * Jiang and Engel 2007 Jiang, H.; Engel, E. Random-phase-approximation-based correlation energy functionals: Benchmark results for atoms. _J. Chem. Phys._ 2007, _127_ , 184108
  * Lochan and Head-Gordon 2007 Lochan, R. C.; Head-Gordon, M. Orbital-optimized opposite-spin scaled second-order correlation: An economical method to improve the description of open-shell molecules. _J. Chem. Phys._ 2007, _126_ , 164101
  * Neese et al. 2009 Neese, F.; Schwabe, T.; Kossmann, S.; Schirmer, B.; Grimme, S. Assessment of orbital-optimized, spin-component scaled second-order many-body perturbation theory for thermochemistry and kinetics. _J. Chem. Theory Comput._ 2009, _5_ , 3060–3073
  * Kossmann and Neese 2010 Kossmann, S.; Neese, F. Correlated ab initio spin densities for larger molecules: Orbital-optimized spin-component-scaled MP2 method. _J. Phys. Chem. A_ 2010, _114_ , 11768–11781
  * Grüneis et al. 2010 Grüneis, A.; Marsman, M.; Kresse, G. Second-order Møller-Plesset perturbation theory applied to extended systems. II. Structural and energetic properties. _J. Chem. Phys._ 2010, _133_ , 074107
  * van Schilfgaarde et al. 2006 van Schilfgaarde, M.; Kotani, T.; Faleev, S. Quasiparticle self-consistent GW theory. _Phys. Rev. Lett._ 2006, _96_ , 226402
  * Tal et al. 2021 Tal, A.; Chen, W.; Pasquarello, A. Vertex function compliant with the Ward identity for quasiparticle self-consistent calculations beyond GW. _Phys. Rev. B_ 2021, _103_ , 161104
  * Jung et al. 2004 Jung, Y.; Lochan, R. C.; Dutoi, A. D.; Head-Gordon, M. Scaled opposite-spin second order moller-plesset correlation energy: An economical electronic structure method. _J. Chem. Phys._ 2004, _121_ , 9793–9802
  * Lochan et al. 2005 Lochan, R. C.; Jung, Y.; Head-Gordon, M. Scaled opposite spin second order Møller-Plesset theory with improved physical description of long-range dispersion interactions. _J. Phys. Chem. A_ 2005, _109_ , 7598–7605
  * Grimme 2003 Grimme, S. Improved second-order Møller-Plesset perturbation theory by separate scaling of parallel- and antiparallel-spin pair correlation energies. _J. Chem. Phys._ 2003, _118_ , 9095–9102
  * Szabados 2006 Szabados, Á. Theoretical interpretation of Grimme’s spin-component-scaled second order Møller-Plesset theory. _J. Chem. Phys._ 2006, _125_ , 214105
  * Pitoňák et al. 2009 Pitoňák, M.; Neogrády, P.; Černý, J.; Grimme, S.; Hobza, P. Scaled MP3 non-covalent interaction energies agree closely with accurate CCSD(T) benchmark data. _ChemPhysChem_ 2009, _10_ , 282–289
  * Sedlak et al. 2013 Sedlak, R.; Riley, K. E.; Řezáč, J.; Pitoňák, M.; Hobza, P. MP2.5 and MP2.X: Approaching CCSD(T) quality description of noncovalent interaction at the cost of a single CCSD iteration. _ChemPhysChem_ 2013, _14_ , 698–707
  * Goldey and Head-Gordon 2012 Goldey, M.; Head-Gordon, M. Attenuating away the errors in inter- and intramolecular interactions from second-order Møller-plesset calculations in the small aug-cc-pVDZ basis set. _J. Phys. Chem. Lett._ 2012, _3_ , 3592–3598
  * Goldey et al. 2013 Goldey, M.; Dutoi, A.; Head-Gordon, M. Attenuated second-order Møller-Plesset perturbation theory: Performance within the aug-cc-pVTZ basis. _Phys. Chem. Chem. Phys._ 2013, _15_ , 15869–15875
  * Goldey et al. 2015 Goldey, M. B.; Belzunces, B.; Head-Gordon, M. Attenuated MP2 with a Long-Range Dispersion Correction for Treating Nonbonded Interactions. _J. Chem. Theory Comput._ 2015, _11_ , 4159–4168
  * Lee and Head-Gordon 2018 Lee, J.; Head-Gordon, M. Regularized Orbital-Optimized Second-Order Møller-Plesset Perturbation Theory: A Reliable Fifth-Order-Scaling Electron Correlation Model with Orbital Energy Dependent Regularizers. _J. Chem. Theory Comput._ 2018, _14_ , 5203–5219
  * Monino and Loos 2022 Monino, E.; Loos, P.-F. Unphysical Discontinuities, Intruder States and Regularization in GW Methods. _J. Chem. Phys._ 2022, _156_ , 231101
  * Pittner 2003 Pittner, J. Continuous transition between Brillouin-Wigner and Rayleigh-Schrödinger perturbation theory, generalized Bloch equation, and Hilbert space multireference coupled cluster. _J. Chem. Phys._ 2003, _118_ , 10876
  * Engel and Jiang 2006 Engel, E.; Jiang, H. Orbital-Dependent Representation of the Correlation Energy Functional: Properties of Second-Order Kohn–Sham Perturbation Expansion. _Int. J. Quantum Chem._ 2006, _106_ , 3242–3259
  * Jiang and Engel 2006 Jiang, H.; Engel, E. Kohn-Sham perturbation theory: Simple solution to variational instability of second order correlation energy functional. _J. Chem. Phys._ 2006, _125_ , 184108
  * Daas et al. 2021 Daas, T. J.; Fabiano, E.; Della Sala, F.; Gori-Giorgi, P.; Vuckovic, S. Noncovalent Interactions from Models for the Møller-Plesset Adiabatic Connection. _J. Phys. Chem. Lett._ 2021, _12_ , 4867–4875
  * Daas et al. 2022 Daas, T. J.; Kooi, D. P.; Grooteman, A. J. A. F.; Seidl, M.; Gori-Giorgi, P. Gradient Expansions for the Large-Coupling Strength Limit of the Møller–Plesset Adiabatic Connection. _J. Chem. Theory Comput._ 2022, _18_ , 1584–1594
  * Kurth et al. 1999 Kurth, S.; Perdew, J. P.; Blaha, P. Molecular and solid-state tests of density functional approximations: LSD, GGAs, and Meta-GGAs. _Int. J. Quantum Chem._ 1999, _75_ , 889–909
  * Yan et al. 2000 Yan, Z.; Perdew, J. P.; Kurth, S. Density functional for short-range correlation: Accuracy of the random-phase approximation for isoelectronic energy changes. _Phys. Rev. B - Condens. Matter Mater. Phys._ 2000, _61_ , 16430–16439
  * Angyan et al. 2005 Angyan, J. G.; Gerber, I. C.; Savin, A.; Toulouse, J. Van der Waals forces in density functional theory: Perturbational long-range electron-interaction corrections. _Phys. Rev. A_ 2005, _72_ , 012510
  * Janesko et al. 2009 Janesko, B. G.; Henderson, T. M.; Scuseria, G. E. Long-range-corrected hybrids including random phase approximation correlation. _J. Chem. Phys._ 2009, _130_ , 081105
  * Janesko et al. 2009 Janesko, B. G.; Henderson, T. M.; Scuseria, G. E. Long-range-corrected hybrid density functionals including random phase approximation correlation: Application to noncovalent interactions. _J. Chem. Phys._ 2009, _131_ , 034110
  * Toulouse et al. 2009 Toulouse, J.; Gerber, I. C.; Jansen, G.; Savin, A.; Ángyán, J. G. Adiabatic-connection fluctuation-dissipation density-functional theory based on range separation. _Phys. Rev. Lett._ 2009, _102_ , 096404
  * Zhu et al. 2010 Zhu, W.; Toulouse, J.; Savin, A.; Ángyán, J. G. Range-separated density-functional theory with random phase approximation applied to noncovalent intermolecular interactions. _J. Chem. Phys._ 2010, _132_ , 244108
  * Toulouse et al. 2010 Toulouse, J.; Zhu, W.; Ángyán, J. G.; Savin, A. Range-separated density-functional theory with the random-phase approximation: Detailed formalism and illustrative applications. _Phys. Rev. A - At. Mol. Opt. Phys._ 2010, _82_ , 032502
  * Toulouse et al. 2011 Toulouse, J.; Zhu, W.; Savin, A.; Jansen, G.; Ángyán, J. G. Closed-shell ring coupled cluster doubles theory with range separation applied on weak intermolecular interactions. _J. Chem. Phys._ 2011, _135_ , 084119
  * Beuerle and Ochsenfeld 2017 Beuerle, M.; Ochsenfeld, C. Short-range second order screened exchange correction to RPA correlation energies. _J. Chem. Phys._ 2017, _147_ , 204107
  * Ren et al. 2011 Ren, X.; Tkatchenko, A.; Rinke, P.; Scheffler, M. Beyond the random-phase approximation for the electron correlation energy: The importance of single excitations. _Phys. Rev. Lett._ 2011, _106_ , 153003
  * Paier et al. 2012 Paier, J.; Ren, X.; Rinke, P.; Scuseria, G. E.; Grüneis, A.; Kresse, G.; Scheffler, M. Assessment of correlation energies based on the random-phase approximation. _New J. Phys._ 2012, _14_ , 043002
  * Ren et al. 2013 Ren, X.; Rinke, P.; Scuseria, G. E.; Scheffler, M. Renormalized second-order perturbation theory for the electron correlation energy: Concept, implementation, and benchmarks. _Phys. Rev. B - Condens. Matter Mater. Phys._ 2013, _88_ , 035120
  * Hellgren and Von Barth 2007 Hellgren, M.; Von Barth, U. Correlation potential in density functional theory at the GWA level: Spherical atoms. _Phys. Rev. B_ 2007, _76_ , 075107
  * Hellgren and Von Barth 2008 Hellgren, M.; Von Barth, U. Linear density response function within the time-dependent exact-exchange approximation. _Phys. Rev. B - Condens. Matter Mater. Phys._ 2008, _78_ , 115107
  * Heßelmann and Görling 2010 Heßelmann, A.; Görling, A. Random phase approximation correlation energies with exact Kohn-Sham exchange. _Mol. Phys._ 2010, _108_ , 359–372
  * Heßelmann and Görling 2011 Heßelmann, A.; Görling, A. Correct description of the bond dissociation limit without breaking spin symmetry by a random-phase-approximation correlation functional. _Phys. Rev. Lett._ 2011, _106_ , 093001
  * Bleiziffer et al. 2015 Bleiziffer, P.; Krug, M.; Görling, A. Self-consistent Kohn-Sham method based on the adiabatic-connection fluctuation-dissipation theorem and the exact-exchange kernel. _J. Chem. Phys._ 2015, _142_ , 244108
  * Mussard et al. 2016 Mussard, B.; Rocca, D.; Jansen, G.; Ángyán, J. G. Dielectric Matrix Formulation of Correlation Energies in the Random Phase Approximation: Inclusion of Exchange Effects. _J. Chem. Theory Comput._ 2016, _12_ , 2191–2202
  * Bates and Furche 2013 Bates, J. E.; Furche, F. Communication: Random phase approximation renormalized many-body perturbation theory. _J. Chem. Phys._ 2013, _139_ , 171103
  * Erhard et al. 2016 Erhard, J.; Bleiziffer, P.; Görling, A. Power Series Approximation for the Correlation Kernel Leading to Kohn-Sham Methods Combining Accuracy, Computational Efficiency, and General Applicability. _Phys. Rev. Lett._ 2016, _117_ , 143002
  * Olsen et al. 2019 Olsen, T.; Patrick, C. E.; Bates, J. E.; Ruzsinszky, A.; Thygesen, K. S. Beyond the RPA and GW methods with adiabatic xc-kernels for accurate ground state and quasiparticle energies. _Nat. Comput. Mater._ 2019, _5_ , 106
  * Görling 2019 Görling, A. Hierarchies of methods towards the exact Kohn-Sham correlation energy based on the adiabatic-connection fluctuation-dissipation theorem. _Phys. Rev. B_ 2019, _99_ , 235120
  * Maggio and Kresse 2016 Maggio, E.; Kresse, G. Correlation energy for the homogeneous electron gas: Exact Bethe-Salpeter solution and an approximate evaluation. _Phys. Rev. B_ 2016, _93_ , 235113
  * Holzer et al. 2018 Holzer, C.; Gui, X.; Harding, M. E.; Kresse, G.; Helgaker, T.; Klopper, W. Bethe-Salpeter correlation energies of atoms and molecules. _J. Chem. Phys._ 2018, _149_ , 144106 (2018);
  * Loos et al. 2020 Loos, P.-F.; Scemama, A.; Duchemin, I.; Jacquemin, D.; Blase, X. Pros and Cons of the Bethe-Salpeter Formalism for Ground-State Energies. _J. Phys. Chem. Lett._ 2020, _11_ , 3536–3545
  * Trushin et al. 2021 Trushin, E.; Thierbach, A.; Görling, A. Toward chemical accuracy at low computational cost: Density-functional theory with $\sigma$-functionals for the correlation energy. _J. Chem. Phys._ 2021, _154_ , 014104
  * Fauser et al. 2021 Fauser, S.; Trushin, E.; Neiss, C.; Görling, A. Chemical accuracy with $\sigma$-functionals for the Kohn-Sham correlation energy optimized for different input orbitals and eigenvalues. _J. Chem. Phys._ 2021, _155_ , 134111
  * Colonna et al. 2014 Colonna, N.; Hellgren, M.; De Gironcoli, S. Correlation energy within exact-exchange adiabatic connection fluctuation-dissipation theory: Systematic development and simple approximations. _Phys. Rev. B - Condens. Matter Mater. Phys._ 2014, _90_ , 1–10
  * Colonna et al. 2016 Colonna, N.; Hellgren, M.; De Gironcoli, S. Molecular bonding with the RPAx: From weak dispersion forces to strong correlation. _Phys. Rev. B_ 2016, _93_ , 1–11
  * Hellgren et al. 2018 Hellgren, M.; Colonna, N.; De Gironcoli, S. Beyond the random phase approximation with a local exchange vertex. _Phys. Rev. B_ 2018, _98_ , 1–12
  * Hellgren and Baguet 2021 Hellgren, M.; Baguet, L. Random phase approximation with exchange for an accurate description of crystalline polymorphism. _Phys. Rev. Res._ 2021, _3_
  * Sharp and Horton 1953 Sharp, T.; Horton, G. A Variational Approach to the Unipotential Many-Electron Problem. _Phys. Rev._ 1953, _90_ , 317
  * Talman and Shadwick 1976 Talman, J. D.; Shadwick, W. F. Optimized effective atomic central potential. _Phys. Rev. A_ 1976, _14_ , 36–40
  * Engel, Eberhard and Dreizler 2013 Engel, Eberhard and Dreizler, R. M. _Density functional theory An Advanced Course_ ; Springer, 2013
  * Freeman 1977 Freeman, D. Coupled-cluster expansion applied to the electron gas: Inclusion of ring and exchange effects. _Phys. Rev. B_ 1977, _15_ , 5512–5521
  * Jansen et al. 2010 Jansen, G.; Liu, R. F.; Ángyán, J. G. On the equivalence of ring-coupled cluster and adiabatic connection fluctuation-dissipation theorem random phase approximation correlation energy expressions. _J. Chem. Phys._ 2010, _133_ , 154106
  * Paier et al. 2010 Paier, J.; Janesko, B. G.; Henderson, T. M.; Scuseria, G. E.; Grüneis, A.; Kresse, G. Hybrid functionals including random phase approximation correlation and second-order screened exchange. _J. Chem. Phys._ 2010, _132_ , 094103
  * Grüneis et al. 2009 Grüneis, A.; Marsman, M.; Harl, J.; Schimka, L.; Kresse, G. Making the random phase approximation to electronic correlation accurate. _J. Chem. Phys._ 2009, _131_ , 154115
  * Förster and Visscher 2022 Förster, A.; Visscher, L. Exploring the statically screened G3W2 correction to the GW self-energy : Charged excitations and total energies of finite systems. _Phys. Rev. B_ 2022, _105_ , 125121
  * Grüneis et al. 2014 Grüneis, A.; Kresse, G.; Hinuma, Y.; Oba, F. Ionization potentials of solids: The importance of vertex corrections. _Phys. Rev. Lett._ 2014, _112_ , 096401
  * Dyson 1949 Dyson, F. J. The S matrix in quantum electrodynamics. _Phys. Rev._ 1949, _75_ , 1736–1755
  * Rieger et al. 1999 Rieger, M. M.; Steinbeck, L.; White, I. D.; Rojas, H. N.; Godby, R. W. GW space-time method for the self-energy of large systems. _Comput. Phys. Commun._ 1999, _117_ , 211–228
  * Maggio and Kresse 2017 Maggio, E.; Kresse, G. GW Vertex Corrected Calculations for Molecular Systems. _J. Chem. Theory Comput._ 2017, _13_ , 4765–4778
  * Baym and Kadanoff 1961 Baym, G.; Kadanoff, L. P. Conservation laws and correlation functions. _Phys. Rev._ 1961, _124_ , 287–299
  * Salpeter and Bethe 1951 Salpeter, E. E.; Bethe, H. A. A relativistic equation for bound-state problems. _Phys. Rev._ 1951, _84_ , 1232–1242
  * Starke and Kresse 2012 Starke, R.; Kresse, G. Self-consistent Green function equations and the hierarchy of approximations for the four-point propagator. _Phys. Rev. B_ 2012, _85_ , 075119
  * Held et al. 2011 Held, K.; Taranto, C.; Rohringer, G.; Toschi, A. In _LDA+DMFT approach to strongly Correl. Mater._ ; Pavarini, E., Koch, E., Vollhardt, D., Lichtenstein, A., Eds.; 2011; Chapter 13, pp 13.1–13.22
  * Caruso et al. 2013 Caruso, F.; Rohr, D. R.; Hellgren, M.; Ren, X.; Rinke, P.; Rubio, A.; Scheffler, M. Bond breaking and bond formation: How electron correlation is captured in many-body perturbation theory and density-functional theory. _Phys. Rev. Lett._ 2013, _110_ , 146403
  * Rohringer et al. 2012 Rohringer, G.; Valli, A.; Toschi, A. Local electronic correlation at the two-particle level. _Phys. Rev. B - Condens. Matter Mater. Phys._ 2012, _86_ , 125114
  * Rohringer et al. 2018 Rohringer, G.; Hafermann, H.; Toschi, A.; Katanin, A. A.; Antipov, A. E.; Katsnelson, M. I.; Lichtenstein, A. I.; Rubtsov, A. N.; Held, K. Diagrammatic routes to nonlocal correlations beyond dynamical mean field theory. _Rev. Mod. Phys._ 2018, _90_ , 25003
  * Krien et al. 2021 Krien, F.; Kauch, A.; Held, K. Tiling with triangles: Parquet and GW$\gamma$ methods unified. _Phys. Rev. Res._ 2021, _3_ , 13149
  * Ren et al. 2015 Ren, X.; Marom, N.; Caruso, F.; Scheffler, M.; Rinke, P. Beyond the GW approximation: A second-order screened exchange correction. _Phys. Rev. B - Condens. Matter Mater. Phys._ 2015, _92_ , 081104(R)
  * van Leeuwen et al. 2015 van Leeuwen, R.; Dahlen, N. E.; Stefanucci, G.; Almbladh, C. O.; Von Barth, U. In _Time-Dependent Density Funct. Theory_ ; Marques, M. A., Ullrich, C. A., Nogueira, F., Rubio, A., Burke, K., Gross, E. K., Eds.; Springer Heidelberg, 2015; pp 185–217
  * Stefanucci et al. 2014 Stefanucci, G.; Pavlyukh, Y.; Uimonen, A. M.; van Leeuwen, R. Diagrammatic expansion for positive spectral functions beyond GW: Application to vertex corrections in the electron gas. _Phys. Rev. B_ 2014, _90_ , 115134
  * Rodríguez-Mayorga et al. 2021 Rodríguez-Mayorga, M.; Mitxelena, I.; Bruneval, F.; Piris, M. Coupling Natural Orbital Functional Theory and Many-Body Perturbation Theory by Using Nondynamically Correlated Canonical Orbitals. _J. Chem. Theory Comput._ 2021, _17_ , 7562–7574
  * Becke 2014 Becke, A. D. Perspective: Fifty years of density-functional theory in chemical physics. _J. Chem. Phys._ 2014, _140_ , 18A301
  * Vuckovic et al. 2020 Vuckovic, S.; Fabiano, E.; Gori-Giorgi, P.; Burke, K. MAP: An MP2 Accuracy Predictor for Weak Interactions from Adiabatic Connection Theory. _J. Chem. Theory Comput._ 2020, _16_ , 4141–4149
  * Baerends et al. 2022 Baerends, E.; Ziegler, T.; Atkins, A.; Autschbach, J.; Baseggio, O.; Bashford, D.; Bérces, A.; Bickelhaupt, F.; Bo, C.; Boerrigter, P.; Cavallo, L.; Daul, C.; Chong, D.; Chulhai, D.; Deng, L.; Dickson, R.; Dieterich, J.; Ellis, D.; van Faassen, M.; Fan, L.; Fischer, T.; Förster, A.; Guerra, C. F.; Franchini, M.; Ghysels, A.; Giammona, A.; van Gisbergen, S.; Goez, A.; Götz, A.; Groeneveld, J.; Gritsenko, O.; Grüning, M.; Gusarov, S.; Harris, F.; van den Hoek, P.; Hu, Z.; Jacob, C.; Jacobsen, H.; Jensen, L.; Joubert, L.; Kaminski, J.; van Kessel, G.; König, C.; Kootstra, F.; Kovalenko, A.; Krykunov, M.; van Lenthe, E.; McCormack, D.; Michalak, A.; Mitoraj, M.; Morton, S.; Neugebauer, J.; Nicu, V.; Noodleman, L.; Osinga, V.; Patchkovskii, S.; Pavanello, M.; Peeples, C.; Philipsen, P.; Post, D.; Pye, C.; Ramanantoanina, H.; Ramos, P.; Ravenek, W.; Reimann, M.; Rodríguez, J.; Ros, P.; Rüger, R.; Schipper, P.; Schlüns, D.; van Schoot, H.; Schreckenbach, G.; Seldenthuis, J.; Seth, M.; Snijders, J.; Solà, M.; Stener, M.; Swart, M.; Swerhone, D.; Tognetti, V.; te Velde, G.; Vernooijs, P.; Versluis, L.; Visscher, L.; Visser, O.; Wang, F.; Wesolowski, T.; van Wezenbeek, E.; Wiesenekker, G.; Wolff, S.; Woo, T.; Yakovlev, A. ADF2022.1, locally modified development version. 2022
  * Förster and Visscher 2021 Förster, A.; Visscher, L. GW100: A Slater-Type Orbital Perspective. _J. Chem. Theory Comput._ 2021, _17_ , 5080–5097
  * Krykunov et al. 2009 Krykunov, M.; Ziegler, T.; Van Lenthe, E. Hybrid density functional calculations of nuclear magnetic shieldings using slater-type orbitals and the zeroth- order regular approximation. _Int. J. Quantum Chem._ 2009, _109_ , 1676–1683
  * Wirz et al. 2017 Wirz, L. N.; Reine, S. S.; Pedersen, T. B. On Resolution-of-the-Identity Electron Repulsion Integral Approximations and Variational Stability. _J. Chem. Theory Comput._ 2017, _13_ , 4897–4906
  * Förster et al. 2020 Förster, A.; Franchini, M.; van Lenthe, E.; Visscher, L. A Quadratic Pair Atomic Resolution of the Identity Based SOS-AO-MP2 Algorithm Using Slater Type Orbitals. _J. Chem. Theory Comput._ 2020, _16_ , 875 – 891
  * Kaltak et al. 2014 Kaltak, M.; Klimeš, J.; Kresse, G. Low scaling algorithms for the random phase approximation: Imaginary time and laplace transformations. _J. Chem. Theory Comput._ 2014, _10_ , 2498–2507
  * Kaltak et al. 2014 Kaltak, M.; Klimeš, J.; Kresse, G. Cubic scaling algorithm for the random phase approximation: Self-interstitials and vacancies in Si. _Phys. Rev. B_ 2014, _90_ , 054115
  * Liu et al. 2016 Liu, P.; Kaltak, M.; Klimeš, J.; Kresse, G. Cubic scaling GW: Towards fast quasiparticle calculations. _Phys. Rev. B_ 2016, _94_ , 165109
  * Helgaker et al. 1997 Helgaker, T.; Klopper, W.; Koch, H.; Noga, J. Basis-set convergence of correlated calculations on water. _J. Chem. Phys._ 1997, _106_ , 9639–9646
  * Jensen 2013 Jensen, F. Atomic orbital basis sets. _Wiley Interdiscip. Rev. Comput. Mol. Sci._ 2013, _3_ , 273–295
  * Knowles and Handy 1984 Knowles, P. J.; Handy, N. C. A new determinant-based full configuration interaction method. _Chem. Phys. Lett._ 1984, _111_ , 315–321
  * Knowles and Handy 1989 Knowles, P. J.; Handy, N. C. A determinant based full configuration interaction program. _Comput. Phys. Commun._ 1989, _54_ , 75–83
  * Henderson and Scuseria 2010 Henderson, T. M.; Scuseria, G. E. The connection between self-interaction and static correlation: A random phase approximation perspective. _Mol. Phys._ 2010, _108_ , 2511–2517
  * Peterson et al. 1993 Peterson, K. A.; Kendall, R. A.; Dunning, T. H. Benchmark calculations with correlated molecular wave functions. III. Configuration interaction calculations on first row homonuclear diatomics. _J. Chem. Phys._ 1993, _99_ , 9790–9805
  * Hönisch et al. 2009 Hönisch, B.; Hemming, G. N.; Archer, D.; Sidall, M.; McManus, J. F. Beryllium Dimer — Caught in the Act of Bonding. _Science (80-. )._ 2009, _324_ , 1548–1552
  * Røeggen and Veseth 2005 Røeggen, I.; Veseth, L. Interatomic potential for the X1g+ state of Be2, revisited. 2005
  * Goerigk et al. 2017 Goerigk, L.; Hansen, A.; Bauer, C.; Ehrlich, S.; Najibi, A.; Grimme, S. A look at the density functional theory zoo with the advanced GMTKN55 database for general main group thermochemistry, kinetics and noncovalent interactions. _Phys. Chem. Chem. Phys._ 2017, _19_ , 32184–32215
  * Chen et al. 2018 Chen, G. P.; Agee, M. M.; Furche, F. Performance and Scope of Perturbative Corrections to Random-Phase Approximation Energies. _J. Chem. Theory Comput._ 2018, _14_ , 5701–5714
  * Karton et al. 2011 Karton, A.; Daon, S.; Martin, J. M. W4-11: A high-confidence benchmark dataset for computational thermochemistry derived from first-principles W4 data. _Chem. Phys. Lett._ 2011, _510_ , 165–178
  * Karton et al. 2006 Karton, A.; Rabinovich, E.; Martin, J. M. W4 theory for computational thermochemistry: In pursuit of confident sub-kJ/mol predictions. _J. Chem. Phys._ 2006, _125_ , 144108
  * Zhao et al. 2005 Zhao, Y.; Lynch, B. J.; Truhlar, D. G. Multi-coefficient extrapolated density functional theory for thermochemistry and thermochemical kinetics. _Phys. Chem. Chem. Phys._ 2005, _7_ , 43–52
  * Zhao et al. 2005 Zhao, Y.; González-Garda, N.; Truhlar, D. G. Benchmark database of barrier heights for heavy atom transfer, nucleophilic substitution, association, and unimolecular reactions and its use to test theoretical methods. _J. Phys. Chem. A_ 2005, _109_ , 2012–2018
  * Karton and Martin 2012 Karton, A.; Martin, J. M. Explicitly correlated Wn theory: W1-F12 and W2-F12. _J. Chem. Phys._ 2012, _136_ , 124114
  * Curtiss et al. 1991 Curtiss, L. A.; Raghavachari, K.; Trucks, G. W.; Pople, J. A. Gaussian-2 theory for molecular energies of first- and second-row compounds. _J. Chem. Phys._ 1991, _94_ , 7221–7230
  * Hellgren et al. 2012 Hellgren, M.; Rohr, D. R.; Gross, E. K. Correlation potentials for molecular bond dissociation within the self-consistent random phase approximation. _J. Chem. Phys._ 2012, _136_ , 034106
  * Verma and Bartlett 2012 Verma, P.; Bartlett, R. J. Increasing the applicability of density functional theory. II. Correlation potentials from the random phase approximation and beyond. _J. Chem. Phys._ 2012, _136_ , 044105
  * Bleiziffer et al. 2013 Bleiziffer, P.; Heßelmann, A.; Görling, A. Efficient self-consistent treatment of electron correlation within the random phase approximation. _J. Chem. Phys._ 2013, _139_ , 084113
  * Klimeš and Kresse 2014 Klimeš, J.; Kresse, G. Kohn-Sham band gaps and potentials of solids from the optimised effective potential method within the random phase approximation. _J. Chem. Phys._ 2014, _140_ , 054516
  * Hellgren et al. 2015 Hellgren, M.; Caruso, F.; Rohr, D. R.; Ren, X.; Rubio, A.; Scheffler, M.; Rinke, P. Static correlation and electron localization in molecular dimers from the self-consistent RPA and GW approximation. _Phys. Rev. B - Condens. Matter Mater. Phys._ 2015, _91_ , 165110
  * Voora et al. 2019 Voora, V. K.; Balasubramani, S. G.; Furche, F. Variational generalized Kohn-Sham approach combining the random-phase-approximation and Green’s-function methods. _Phys. Rev. A_ 2019, _99_ , 012518
  * Graf and Ochsenfeld 2020 Graf, D.; Ochsenfeld, C. A range-separated generalized Kohn-Sham method including a long-range nonlocal random phase approximation correlation potential. _J. Chem. Phys._ 2020, _153_ , 244118
  * Riemelmoser et al. 2021 Riemelmoser, S.; Kaltak, M.; Kresse, G. Optimized effective potentials from the random-phase approximation: Accuracy of the quasiparticle approximation. _J. Chem. Phys._ 2021, _154_ , 154103
  * Yu et al. 2021 Yu, J. M.; Tsai, J.; Hernandez, D. J.; Furche, F.; Tsai, J.; Hernandez, D. J. Selfconsistent random phase approximation methods. _J. Chem. Phys._ 2021, _155_ , 040902
  * Caruso et al. 2016 Caruso, F.; Dauth, M.; Van Setten, M. J.; Rinke, P. Benchmark of GW Approaches for the GW100 Test Set. _J. Chem. Theory Comput._ 2016, _12_ , 5076–5087
  * Zhang et al. 2022 Zhang, L.; Shu, Y.; Xing, C.; Chen, X.; Sun, S.; Huang, Y.; Truhlar, D. G. Recommendation of Orbitals for G 0 W 0 Calculations on Molecules and Crystals. _J. Chem. Theory Comput._ 2022, _18_ , 3523–3537
  * Wang et al. 2021 Wang, Y.; Rinke, P.; Ren, X. Assessing the G 0 W 0 $\Gamma$ 0 (1) Approach: Beyond G 0 W 0 with Hedin’s Full Second-Order Self-Energy Contribution. _J. Chem. Theory Comput._ 2021, _17_ , 5140–5154
  * Hait and Head-Gordon 2018 Hait, D.; Head-Gordon, M. Delocalization Errors in Density Functional Theory Are Essentially Quadratic in Fractional Occupation Number. _J. Phys. Chem. Lett._ 2018, _9_ , 6280–6288
  * Furche 2001 Furche, F. Molecular tests of the random phase approximation to the exchange-correlation energy functional. _Phys. Rev. B_ 2001, _64_ , 195120
  * 171 Notice, that our non-counterpoise corrected calculations based on (T,Q) extrapolation will still include a sizable basis set incompleteness error for atomization energies. However, our qualitative conclusions will be valid.
  * Paier et al. 2010 Paier, J.; Janesko, B. G.; Henderson, T. M.; Scuseria, G. E.; Grüneis, A.; Kresse, G. Erratum: Hybrid functionals including random phase approximation correlation and second-order screened exchange (Journal of Chemical Physics (2010) 132 (094103)). _J. Chem. Phys._ 2010, _133_ , 2009–2011
  * Řezáč et al. 2011 Řezáč, J.; Riley, K. E.; Hobza, P. S66: A well-balanced database of benchmark interaction energies relevant to biomolecular structures. _J. Chem. Theory Comput._ 2011, _7_ , 2427–2438
  * 174 With 0.52 kcal/mol, the MAD for RPA@PBE is in excellent agreement with the 0.61 kcal/mol MAD obtained by Nguyen _et. al._ in ref. 48, which has been obtained with GTO-type basis sets and 50 % counterpoise correction instead of 100 %. This shows, that our interaction energies are well converged with respect to the basis set size.
  * Santra et al. 2019 Santra, G.; Sylvetsky, N.; Martin, J. M. Minimally Empirical Double-Hybrid Functionals Trained against the GMTKN55 Database: RevDSD-PBEP86-D4, revDOD-PBE-D4, and DOD-SCAN-D4. _J. Phys. Chem. A_ 2019, _123_ , 5129–5143
  * Mehta et al. 2018 Mehta, N.; Casanova-Páez, M.; Goerigk, L. Semi-empirical or non-empirical double-hybrid density functionals: Which are more robust? _Phys. Chem. Chem. Phys._ 2018, _20_ , 23175–23194
  * Grimme et al. 2010 Grimme, S.; Antony, J.; Ehrlich, S.; Krieg, H. A consistent and accurate ab initio parametrization of density functional dispersion correction (DFT-D) for the 94 elements H-Pu. _J. Chem. Phys._ 2010, _132_ , 154104
  * Grimme et al. 2011 Grimme, S.; Ehrlich, S.; Goerigk, L. Effect of the damping function in dispersion corrected density functional theory. _J. Comput. Chem._ 2011, _32_ , 1456–1465
  * Vlček 2019 Vlček, V. Stochastic Vertex Corrections: Linear Scaling Methods for Accurate Quasiparticle Energies. _J. Chem. Theory Comput._ 2019, _15_ , 6254–6266
  * Beuerle et al. 2018 Beuerle, M.; Graf, D.; Schurkus, H. F.; Ochsenfeld, C. Efficient calculation of beyond RPA correlation energies in the dielectric matrix formalism. _J. Chem. Phys._ 2018, _148_ , 204104
  * Sedlak et al. 2013 Sedlak, R.; Janowski, T.; Pitoňák, M.; Řezáč, J.; Pulay, P.; Hobza, P. Accuracy of quantum chemical methods for large noncovalent complexes. _J. Chem. Theory Comput._ 2013, _9_ , 3364–3374
  * Doser et al. 2009 Doser, B.; Lambrecht, D. S.; Kussmann, J.; Ochsenfeld, C. Linear-scaling atomic orbital-based second-order Møller-Plesset perturbation theory by rigorous integral screening criteria. _J. Chem. Phys._ 2009, _130_ , 064107
  * Pinski et al. 2015 Pinski, P.; Riplinger, C.; Valeev, E. F.; Neese, F. Sparse maps - A systematic infrastructure for reduced-scaling electronic structure methods. I. An efficient and simple linear scaling local MP2 method that uses an intermediate basis of pair natural orbitals. _J. Chem. Phys._ 2015, _143_ , 034108
  * Nagy et al. 2016 Nagy, P. R.; Samu, G.; Kállay, M. An Integral-Direct Linear-Scaling Second-Order Møller-Plesset Approach. _J. Chem. Theory Comput._ 2016, _12_ , 4897–4914
  * Mezei et al. 2019 Mezei, P. D.; Ruzsinszky, A.; Kállay, M. Reducing the Many-Electron Self-Interaction Error in the Second-Order Screened Exchange Method. _J. Chem. Theory Comput._ 2019, _15_ , 6607–6616