limit of $\varepsilon_{\text{c}}^{0}(\rho_{\uparrow},\rho_{\downarrow},s)$
reproduces the exact correlation energy of the Helium isoelectronic series in
the large-nuclear charge limit, i.e.
$\lim_{Z\to\infty}E_{\text{c}}[\rho_{N=2,Z}]=E_{\text{c}}^{\text{GL2}}[\rho_{N=2,Z=1}^{\text{H}}]=-0.0467$
hartree [Eq. (2.72)]. The parameter $b_{3\text{c}}=0.125541$ is determined to
saturate the lower bound on the exchange-correlation energies of two-electron
densities [Eq. (2.77)]. The parameter $b_{2\text{c}}=0.0889$ is determined to
reproduce the exact exchange-correlation energy of the He atom.

The remaining seven parameters ($k_{1}=0.065$, $c_{1\text{x}}=0.667$,
$c_{2\text{x}}=0.8$, $d_{\text{x}}=1.24$, $c_{1\text{c}}=0.64$,
$c_{2\text{c}}=1.5$, and $d_{\text{c}}=0.7$) are determined by fitting to the
asymptotic expansions of the exchange and correlation energies of neutral
atoms in large nuclear charge limit [Eqs. (2.74) and (2.75)], the binding
energy curve of compressed Ar2, and jellium surface exchange-correlation
energies.

##  4 Single-determinant hybrid approximations

###  4.1 Hybrid approximations

Based on arguments relying on the adiabatic-connection formalism, in 1993
Becke [130] proposed to mix a fraction of the exact or Hartree-Fock (HF)
exchange energy $E_{\text{x}}^{\text{HF}}$ with GGA functionals. In
particular, he proposed a three-parameter hybrid (3H) approximation [131] of
the form, written here in its spin-independent version,

$E_{\text{xc}}^{\text{3H}}[\Phi]=a\;E_{\text{x}}^{\text{HF}}[\Phi]+b\;E_{\text{x}}^{\text{GGA}}[\rho_{\Phi}]+(1-a-b)\;E_{\text{x}}^{\text{LDA}}[\rho_{\Phi}]+c\;E_{\text{c}}^{\text{GGA}}[\rho_{\Phi}]+(1-c)\;E_{\text{c}}^{\text{LDA}}[\rho_{\Phi}],$
(4.1)

with empirical parameters $a$, $b$, and $c$. The functional
$E_{\text{xc}}^{\text{3H}}[\Phi]$ is thought of as a functional of a single-
determinant wave function $\Phi\in{\cal S}^{N}$ since
$E_{\text{x}}^{\text{HF}}[\Phi]$ is itself a functional of $\Phi$

$\displaystyle E_{\text{x}}^{\text{HF}}[\Phi]$ $\displaystyle=$
$\displaystyle\langle\Phi|\hat{W}_{\text{ee}}|\Phi\rangle-
E_{\text{H}}[\rho_{\Phi}]$ (4.2) $\displaystyle=$
$\displaystyle-\frac{1}{2}\sum_{\sigma\in\\{\uparrow,\downarrow\\}}\sum_{i=1}^{N_{\sigma}}\sum_{j=1}^{N_{\sigma}}\int_{\mathbb{R}^{3}\times\mathbb{R}^{3}}\frac{\varphi_{i\sigma}^{*}(\mathbf{r}_{1})\varphi_{j\sigma}(\mathbf{r}_{1})\varphi_{j\sigma}^{*}(\mathbf{r}_{2})\varphi_{i\sigma}(\mathbf{r}_{2})}{|\mathbf{r}_{1}-\mathbf{r}_{2}|}\text{d}\mathbf{r}_{1}\text{d}\mathbf{r}_{2},$

where $\\{\varphi_{i\sigma}\\}_{i=1,...,N_{\sigma}}$ are the orbitals occupied
in $\Phi$. In 1996, Becke proposed a simpler one-parameter hybrid (1H)
approximation [132]

$\displaystyle
E_{\text{xc}}^{\text{1H}}[\Phi]=a\;E_{\text{x}}^{\text{HF}}[\Phi]+(1-a)\;E_{\text{x}}^{\text{GGA}}[\rho_{\Phi}]+E_{\text{c}}^{\text{GGA}}[\rho_{\Phi}],$
(4.3)

where the fraction $a$ of HF exchange has to be determined. For simplicity, we
considered GGA functionals $E_{\text{x}}^{\text{GGA}}[\rho_{\Phi}]$ and
$E_{\text{c}}^{\text{GGA}}[\rho_{\Phi}]$ in Eq. (4.3) but we can more
generally use meta-GGA functionals
$E_{\text{x}}^{\text{mGGA}}[\rho_{\Phi},\tau_{\Phi}]$ and
$E_{\text{c}}^{\text{mGGA}}[\rho_{\Phi},\tau_{\Phi}]$.

These hybrid approximations should be considered as approximations of the GKS
exchange-correlation functional $E_{\text{xc}}^{S}[\Phi]$ in Eq. (1.47) with
$S[\Phi]=a\;E_{\text{x}}^{\text{HF}}[\Phi]$. The corresponding GKS equations
[Eq. (1.4)] then include the term

$\frac{\delta
S[\Phi]}{\delta\varphi_{i\sigma}^{*}(\mathbf{r})}=a\int_{\mathbb{R}^{3}}v_{\text{x},\sigma}^{\text{HF}}(\mathbf{r},\mathbf{r}^{\prime})\varphi_{i\sigma}(\mathbf{r}^{\prime})\text{d}\mathbf{r}^{\prime},$
(4.4)

where $v_{\text{x},\sigma}^{\text{HF}}(\mathbf{r},\mathbf{r}^{\prime})$ is the
nonlocal HF exchange potential 777The possibility of combining a nonlocal HF
potential with a local correlation potential was mentioned already in 1965 in
the paper by Kohn and Sham [19].

$v_{\text{x},\sigma}^{\text{HF}}(\mathbf{r},\mathbf{r}^{\prime})=-\sum_{j=1}^{N_{\sigma}}\frac{\varphi_{j\sigma}(\mathbf{r})\varphi_{j\sigma}^{*}(\mathbf{r}^{\prime})}{|\mathbf{r}-\mathbf{r}^{\prime}|}.$
(4.5)

The main benefit of adding a fraction of HF exchange is to decrease the self-
interaction error (see Section 2.3) introduced by semilocal exchange
functionals which tends to favor too much delocalized electron densities over
localized electron densities. The fraction of HF exchange should however be
small enough to keep the compensation of errors usually occurring between the
approximate semilocal exchange and correlation functionals. First, Becke used
the value $a=0.5$ in the so-called Becke Half-and-Half functional [130], but
then fits to various experimental data often repeatedly gave an optimal
parameter $a$ around $0.20$-$0.25$. A rationale has been proposed in favor of
the value $0.25$ [133]. By decreasing self-interaction errors in the exchange
energy, hybrid approximations are often a big improvement over semilocal
approximations for molecular systems with sufficiently large electronic gaps.
However, for systems with small HOMO-LUMO gaps, such as systems with stretched
chemical bonds or with transition metal elements, they tend to increase
static-correlation errors.

An interesting extension of the hybrid approximations are the so-called local
hybrids which uses a position-dependent fraction $a(\mathbf{r})$ of a (non-
uniquely defined) HF exchange energy density
$e_{\text{x}}^{\text{HF}}(\mathbf{r})$ [134] (see, Ref. [135] for a recent
review), and which belong to the wider family of hyper-GGA functionals in
which the correlation energy can also be expressed as a function of
$e_{\text{x}}^{\text{HF}}(\mathbf{r})$ [6]. The local-hybrid approximations
are much more flexible than the global hybrid approach exposed in this Section
but require more complicated and computationally expensive implementations.
For this reason, they have not been often used yet and we will not consider
them any further here.

We now describe some of the most used hybrid approximations.

B3LYP exchange-correlation functional

The B3LYP exchange-correlation functional [136] is the most famous and widely
used three-parameter hybrid approximation [Eq. (4.1)]. It uses the B88
exchange functional and the LYP correlation functional

$\displaystyle E_{\text{xc}}^{\text{B3LYP}}[\Phi]$ $\displaystyle=$
$\displaystyle
a\;E_{\text{x}}^{\text{HF}}[\Phi]+b\;E_{\text{x}}^{\text{B88}}[\rho_{\uparrow,\Phi},\rho_{\downarrow,\Phi}]+(1-a-b)\;E_{\text{x}}^{\text{LDA}}[\rho_{\uparrow,\Phi},\rho_{\downarrow,\Phi}]$
(4.6)
$\displaystyle+c\;E_{\text{c}}^{\text{LYP}}[\rho_{\uparrow,\Phi},\rho_{\downarrow,\Phi}]+(1-c)\;E_{\text{c}}^{\text{LDA}}[\rho_{\uparrow,\Phi},\rho_{\downarrow,\Phi}],$

and the parameters $a=0.20$, $b=0.72$, and $c=0.81$ were found by optimizing
on a set of atomization energies, ionization energies, proton affinities of
small molecules and first-row total atomic energies [131]. A caveat is that
the VWN parametrization of the RPA correlation energy (sometimes referred to
as VWN3) of the UEG was actually used for
$E_{\text{c}}^{\text{LDA}}[\rho_{\uparrow},\rho_{\downarrow}]$ instead of the
VWN parametrization of the accurate correlation energy (sometimes referred to
as VWN5) of the UEG [75].

B97 exchange-correlation functional

The Becke 97 (B97) exchange-correlation functional [101] is a GGA hybrid of
the form

$\displaystyle
E_{\text{xc}}^{\text{B97}}[\Phi]=a\;E_{\text{x}}^{\text{HF}}[\Phi]+(1-a)\;E_{\text{x}}^{\text{B97-GGA}}[\rho_{\uparrow,\Phi},\rho_{\downarrow,\Phi}]+E_{\text{c}}^{\text{B97-GGA}}[\rho_{\uparrow,\Phi},\rho_{\downarrow,\Phi}],$
(4.7)

where the form of the B97-GGA exchange and correlation functionals were given
in Eqs. (4.24) and (3.39). The fraction of HF exchange $a=0.1943$ and the
remaining parameters $c_{\text{x},0}=1.00459$, $c_{\text{x},1}=0.629639$,
$c_{\text{x},2}=0.928509$, $c_{\text{c},0}^{\uparrow\downarrow}=0.9454$,
$c_{\text{c},1}^{\uparrow\downarrow}=0.7471$,
$c_{\text{c},2}^{\uparrow\downarrow}=-4.5961$,
$c_{\text{c},0}^{\sigma\sigma}=0.1737$,
$c_{\text{c},1}^{\sigma\sigma}=2.3487$, and
$c_{\text{c},2}^{\sigma\sigma}=-2.4868$ for a polynomial degree $m=2$ in Eqs.
(3.38), (3.42), and (3.43) were optimized on a set of total energies,
atomization energies, ionization energies, and proton affinities. Note that,
for $x_{\sigma}=0$, the UEG limit is not imposed, which would require the
parameters $c_{\text{x},0}$, $c_{\text{c},0}^{\uparrow\downarrow}$, and
$c_{\text{c},0}^{\sigma\sigma}$ to be all strictly equal to $1$. With the
above optimized parameters, we see that it is nearly satisfied for the
exchange energy and the opposite-spin correlation energy, but very far from it
for the same-spin correlation energy which is drastically reduced compared to
the LDA.

PBE0 exchange-correlation functional

The PBE0 exchange-correlation functional [137, 138] is a GGA hybrid using the
PBE exchange and correlation functionals

$\displaystyle
E_{\text{xc}}^{\text{PBE0}}[\Phi]=a\;E_{\text{x}}^{\text{HF}}[\Phi]+(1-a)\;E_{\text{x}}^{\text{PBE}}[\rho_{\uparrow,\Phi},\rho_{\downarrow,\Phi}]+E_{\text{c}}^{\text{PBE}}[\rho_{\uparrow,\Phi},\rho_{\downarrow,\Phi}],$
(4.8)

and the fraction of the HF exchange is fixed at $a=0.25$ according to the
rationale of Ref. [133]. This functional is also known under the name PBE1PBE.
The “1” in the latter name emphasizes that there is one parameter, $a$, while
the “0” in the more common name PBE0 emphasizes that this parameter is not
found by fitting.

TPSSh exchange-correlation functional

The TPSSh exchange-correlation functional [139] is a meta-GGA hybrid using the
TPSS exchange and correlation functionals

$\displaystyle
E_{\text{xc}}^{\text{TPSSh}}[\Phi]=a\;E_{\text{x}}^{\text{HF}}[\Phi]+(1-a)\;E_{\text{x}}^{\text{TPSS}}[\rho_{\uparrow,\Phi},\rho_{\downarrow,\Phi},\tau_{\uparrow,\Phi},\tau_{\downarrow,\Phi}]+E_{\text{c}}^{\text{TPSS}}[\rho_{\uparrow,\Phi},\rho_{\downarrow,\Phi},\tau_{\uparrow,\Phi},\tau_{\downarrow,\Phi}],\;$
(4.9)

and the fraction of the HF exchange $a=0.10$ was determined by optimizing on a
large set of atomization energies.

M06 and M06-2X exchange-correlation functionals

The M06 exchange-correlation functional [140] is a meta-GGA hybrid using the
M06-L exchange and correlation functionals

$\displaystyle
E_{\text{xc}}^{\text{M06}}[\Phi]=a\;E_{\text{x}}^{\text{HF}}[\Phi]+(1-a)\;E_{\text{x}}^{\text{M06-L}}[\rho_{\uparrow,\Phi},\rho_{\downarrow,\Phi},\tau_{\uparrow,\Phi},\tau_{\downarrow,\Phi}]+E_{\text{c}}^{\text{M06-L}}[\rho_{\uparrow,\Phi},\rho_{\downarrow,\Phi},\tau_{\uparrow,\Phi},\tau_{\downarrow,\Phi}],\;$
(4.10)

and the parameters in the M06-L exchange and correlation functionals were
reoptimized together with the fraction of HF exchange $a=0.27$ on the same
large set of diverse physicochemical properties used for the M06-L functional.
In the M06-2X exchange-correlation functional the fraction of HF exchange is
doubled, i.e. $a=0.54$, and the parameters were reoptimized with the function
$h_{\text{x}}(x_{\sigma},Z_{\sigma})$ in Eq. (3.65) set to zero and excluding
transition metal properties in the training set. With this large fraction of
HF exchange, the M06-2X functional is designed for systems without transition
metal elements.

###  4.2 Range-separated hybrid approximations

Based on earlier ideas of Savin [141] (exposed in details in Section 5.2), in
2001, Iikura, Tsuneda, Yanai, and Hirao [142] proposed a long-range correction
(LC) scheme in which the exchange-correlation energy is written as, in its
spin-independent version,

$\displaystyle
E_{\text{xc}}^{\text{LC}}[\Phi]=E_{\text{x}}^{\text{lr},\mu,\text{HF}}[\Phi]+E_{\text{x}}^{\text{sr},\mu,\text{GGA}}[\rho_{\Phi}]+E_{\text{c}}^{\text{GGA}}[\rho_{\Phi}].$
(4.11)

This scheme has also been referred to as the range-separated hybrid exchange
(RSHX) scheme [143]. In Eq. (4.11),
$E_{\text{x}}^{\text{lr},\mu,\text{HF}}[\Phi]$ is the HF exchange energy for a
long-range electron-electron interaction
$w_{\text{ee}}^{\text{lr},\mu}(r_{12})=\operatorname{erf}(\mu r_{12})/r_{12}$
(where $\operatorname{erf}$ is the error function and the parameter
$\mu\in[0,+\infty)$ controls the range of the interaction)

$\displaystyle
E_{\text{x}}^{\text{lr},\mu,\text{HF}}[\Phi]=-\frac{1}{2}\sum_{\sigma\in\\{\uparrow,\downarrow\\}}\sum_{i=1}^{N_{\sigma}}\sum_{j=1}^{N_{\sigma}}\int_{\mathbb{R}^{3}\times\mathbb{R}^{3}}\varphi_{i\sigma}^{*}(\mathbf{r}_{1})\varphi_{j\sigma}(\mathbf{r}_{1})\varphi_{j\sigma}^{*}(\mathbf{r}_{2})\varphi_{i\sigma}(\mathbf{r}_{2})w_{\text{ee}}^{\text{lr},\mu}(r_{12})\text{d}\mathbf{r}_{1}\text{d}\mathbf{r}_{2},\;\;\;$
(4.12)

and $E_{\text{x}}^{\text{sr},\mu,\text{GGA}}[\rho]$ is a GGA exchange energy
functional for the complementary short-range interaction
$w_{\text{ee}}^{\text{sr},\mu}(r_{12})=1/r_{12}-w_{\text{ee}}^{\text{lr}}(r_{12})$.
This latter functional can be thought of as an approximation to the short-
range exchange functional

$E_{\text{x}}^{\text{sr},\mu}[\rho]=\frac{1}{2}\int_{\mathbb{R}^{3}\times\mathbb{R}^{3}}\rho(\mathbf{r}_{1})h_{\text{x}}(\mathbf{r}_{1},\mathbf{r}_{2})w_{\text{ee}}^{\text{sr},\mu}(r_{12})\text{d}\mathbf{r}_{1}\text{d}\mathbf{r}_{2},$
(4.13)

where $h_{\text{x}}(\mathbf{r}_{1},\mathbf{r}_{2})$ is the KS exchange hole of
Section 2.1. For $\mu=0$, the long-range HF exchange energy vanishes, i.e.
$E_{\text{x}}^{\text{lr},\mu=0,\text{HF}}[\Phi]=0$, and the short-range
exchange functional reduces to the standard KS exchange functional, i.e.
$E_{\text{x}}^{\text{sr},\mu=0}[\rho]=E_{\text{x}}[\rho]$. Reversely, for
$\mu\to\infty$, the long-range HF exchange energy reduces to the full-range HF
exchange energy, i.e.
$E_{\text{x}}^{\text{lr},\mu\to\infty,\text{HF}}[\Phi]=E_{\text{x}}^{\text{HF}}[\Phi]$,
and the short-range exchange functional vanishes, i.e.
$E_{\text{x}}^{\text{sr},\mu\to\infty}[\rho]=0$. Significantly, for large
$\mu$, the short-range exchange functional becomes a local functional of the
density [144, 145]

$E_{\text{x}}^{\text{sr},\mu}[\rho]\underset{\mu\to\infty}{\sim}-\frac{\pi}{4\mu^{2}}\int_{\mathbb{R}^{3}}\rho(\mathbf{r})^{2}\text{d}\mathbf{r}.$
(4.14)

Like the hybrid approximations of Section 4.1, Eq. (4.11) should be considered
as an approximation of the GKS exchange-correlation functional
$E_{\text{xc}}^{S}[\Phi]$ in Eq. (1.47) with
$S[\Phi]=E_{\text{x}}^{\text{lr},\mu,\text{HF}}[\Phi]$, and the corresponding
GKS equations [Eq. (1.4)] then includes a long-range nonlocal HF exchange
potential
$v_{\text{x},\sigma}^{\text{lr},\mu,\text{HF}}(\mathbf{r}_{1},\mathbf{r}_{2})=-\sum_{j=1}^{N_{\sigma}}\varphi_{j\sigma}(\mathbf{r}_{1})\varphi_{j\sigma}^{*}(\mathbf{r}_{2})w_{\text{ee}}^{\text{lr},\mu}(r_{12})$.
Similarly to the hybrid approximations, the introduction of a fraction of
long-range HF exchange reduces the self-interaction error (see, e.g., Ref.
[146]). In addition, the short-range exchange part is easier to approximate
with semilocal density-functional approximations, as Eq. (4.14) strongly
suggests. In particular, the $-1/r$ asymptotic behavior of the exchange
potential [Eq. (1.33)], which is difficult to satisfy with semilocal
approximations, does not apply anymore to the short-range exchange potential.

In 2004, Yanai, Tew, and Handy [147], introduced a more flexible scheme called
the Coulomb-attenuating method (CAM) [147] in which fractions of HF exchange
are added at both short range and long range

$E_{\text{xc}}^{\text{CAM}}[\Phi]=a\;E_{\text{x}}^{\text{sr},\mu,\text{HF}}[\Phi]+b\;E_{\text{x}}^{\text{lr},\mu,\text{HF}}[\Phi]+(1-a)\;E_{\text{x}}^{\text{sr},\mu,\text{GGA}}[\rho_{\Phi}]+(1-b)\;E_{\text{x}}^{\text{lr},\mu,\text{GGA}}[\rho_{\Phi}]+E_{\text{c}}^{\text{GGA}}[\rho_{\Phi}],$
(4.15)

where
$E_{\text{x}}^{\text{sr},\mu,\text{HF}}[\Phi]=E_{\text{x}}^{\text{HF}}[\Phi]-E_{\text{x}}^{\text{lr},\mu,\text{HF}}[\Phi]$
is the short-range HF exchange energy and
$E_{\text{x}}^{\text{lr},\mu,\text{GGA}}=E_{\text{x}}^{\text{GGA}}-E_{\text{x}}^{\text{sr},\mu,\text{GGA}}$
is a long-range GGA exchange energy. The reintroduction of HF exchange at
short range further reduces the self-interaction error and improves
thermodynamic properties such as atomization energies. Again, Eq. (4.15)
should be considered as an approximation of the GKS exchange-correlation
functional $E_{\text{xc}}^{S}[\Phi]$ in Eq. (1.47) with
$S[\Phi]=a\;E_{\text{x}}^{\text{sr},\mu,\text{HF}}[\Phi]+b\;E_{\text{x}}^{\text{lr},\mu,\text{HF}}[\Phi]$.
Other forms of modified electron-electron interactions are also possible (see,
e.g., Refs. [148, 145, 149]).

The approximations in Eqs. (4.11) and (4.15) are usually collectively referred
to as range-separated hybrid approximations. Range-separated hybrids in the
form of Eq. (4.15) are more flexible that the hybrid approximations of Section
4.1, and consequently are potentially more accurate, in particular for long-
range electronic excitations. However, like the hybrid approximations, the
presence of HF exchange tends to induce static-correlation errors for systems
with small HOMO-LUMO gaps.

The range-separation parameter $\mu$ (also sometimes denoted as $\omega$) is
generally chosen empirically, e.g. by fitting to experimental data. In
practice, a value around $\mu\approx 0.3-0.5$ bohr-1, fixed for all systems,
is often found to be optimal. It has also been proposed to adjust the value of
$\mu$ in each system, e.g. by requiring that the opposite of the HOMO energy
be equal to the ionization energy calculated by total energy differences [150,
151, 152]. These so-called optimally tuned range-separated hybrids are well
suited for the calculation of charge-transfer electronic excitations but have
the disadvantage of being non size consistent [153].

A natural idea is to use a position-dependent range-separation parameter
$\mu(\mathbf{r})$ which allows the range of the modified interaction to adapt
to the local average electron-electron distance in the diverse spatial regions
of the system. These locally range-separated hybrids [154, 155, 156] are
promising but they induced computational complications and are still in the
early stages of development. We will thus not consider them any further here.

We now describe some of the most used approximations in the context of the
range-separated hybrids.

Short-range LDA exchange functional

The short-range LDA exchange functional [141, 144] can be obtained by using in
Eq. (4.13) the LDA exchange hole [Eq. (3.14)], which leads to

$E_{\text{x}}^{\text{sr},\mu,\text{LDA}}[\rho]=\int_{\mathbb{R}^{3}}e_{\text{x}}^{\text{sr},\mu,\text{UEG}}(\rho(\mathbf{r}))\text{d}\mathbf{r},$
(4.16)

with the short-range UEG exchange energy density

$e_{\text{x}}^{\text{sr},\mu,\text{UEG}}(\rho)=e_{\text{x}}^{\text{UEG}}(\rho)\left[1-\frac{8\tilde{\mu}}{3}\left(\sqrt{\pi}\operatorname{erf}\left(\frac{1}{2\tilde{\mu}}\right)+(2\tilde{\mu}-4\tilde{\mu}^{3})e^{-1/(4\tilde{\mu}^{2})}-3\tilde{\mu}+4\tilde{\mu}^{3}\right)\right],$
(4.17)

where $\tilde{\mu}=\mu/(2k_{\text{F}})$ is a dimensionless range-separation
parameter. The spin-dependent version is obtained from the same spin-scaling
relation as in the standard case [Eq. (1.45)]. The short-range LDA exchange
functional becomes exact for large $\mu$ [Eq. (4.14)] and is the first
building block for constructing short-range exchange GGA functionals.

CAM-B3LYP exchange-correlation functional

The CAM-B3LYP exchange-correlation functional [147] uses Eq. (4.15) with
short- and long-range versions of the B88 exchange functional and the same
correlation functional used in B3LYP (i.e.,
$0.81\;E_{\text{c}}^{\text{LYP}}+0.19\;E_{\text{c}}^{\text{LDA}}$)

$\displaystyle
E_{\text{xc}}^{\text{CAM-B3LYP}}[\Phi]=a\;E_{\text{x}}^{\text{sr},\mu,\text{HF}}[\Phi]+b\;E_{\text{x}}^{\text{lr},\mu,\text{HF}}[\Phi]+(1-a)\;E_{\text{x}}^{\text{sr},\mu,\text{B88}}[\rho_{\uparrow,\Phi},\rho_{\downarrow,\Phi}]\phantom{xxxxxxxx}$
$\displaystyle\phantom{xxxxxxxx}+(1-b)\;E_{\text{x}}^{\text{lr},\mu,\text{B88}}[\rho_{\uparrow,\Phi},\rho_{\downarrow,\Phi}]+0.81E_{\text{c}}^{\text{LYP}}[\rho_{\uparrow,\Phi},\rho_{\downarrow,\Phi}]+0.19E_{\text{c}}^{\text{LDA}}[\rho_{\uparrow,\Phi},\rho_{\downarrow,\Phi}],$
(4.18)

where the parameters $a=0.19$ and $b=0.65$ were optimized on atomization
energies and the range-separation parameter $\mu=0.33$ bohr-1 was taken from
Ref. [157] where it was optimized on equilibrium distances of diatomics
molecules. In this expression, the short-range B88 exchange functional
$E_{\text{x}}^{\text{sr},\mu,\text{B88}}$ is defined by using in Eq. (4.13)
the following generic GGA model for the exchange hole [142] (given here in its
spin-independent version)

$h_{\text{x}}^{\text{GGA}}(\rho,\nabla\rho,r_{12})=-\rho\;\frac{9}{2}\left(\frac{j_{1}(k_{\text{GGA}}r_{12})}{k_{\text{GGA}}r_{12}}\right)^{2},$
(4.19)

with
$k_{\text{GGA}}=k_{\text{F}}/\sqrt{e_{\text{x}}^{\text{GGA}}(\rho,\nabla\rho)/e_{\text{x}}^{\text{UEG}}(\rho)}$.
The exchange-hole model of Eq. (4.19) properly yields to the GGA exchange
energy density $e_{\text{x}}^{\text{GGA}}(\rho,\nabla\rho)$ for $\mu=0$ and
thus allows one to extend any standard GGA exchange functional to a short-
range GGA exchange functional. Note however that it does not fulfil the sum
rule [Eq. (2.11)]. The long-range B88 exchange functional is then simply
$E_{\text{x}}^{\text{lr},\mu,\text{B88}}=E_{\text{x}}^{\text{B88}}-E_{\text{x}}^{\text{sr},\mu,\text{B88}}$.

LC-$\omega$PBE exchange-correlation functional

The LC-$\omega$PBE exchange-correlation functional [158, 159] uses a short-
range version of the PBE exchange functional as well as the standard PBE
correlation functional

$\displaystyle
E_{\text{xc}}^{\text{LC-$\omega$PBE}}[\Phi]=E_{\text{x}}^{\text{lr},\mu,\text{HF}}[\Phi]+E_{\text{x}}^{\text{sr},\mu,\text{PBE}}[\rho_{\uparrow,\Phi},\rho_{\downarrow,\Phi}]+E_{\text{c}}^{\text{PBE}}[\rho_{\uparrow,\Phi},\rho_{\downarrow,\Phi}].$
(4.20)

The short-range PBE exchange functional is obtained by using in Eq. (4.13) the
following GGA exchange hole model constructed to yield the PBE exchange energy
[160]

$\displaystyle h_{\text{x}}^{\text{PBE}}(\rho,\nabla\rho,r_{12})=\rho
J^{\text{PBE}}(s,k_{\text{F}}r_{12}),$ (4.21)

where $s=|\nabla\rho|/(2k_{\text{F}}\rho)$ and

$\displaystyle J^{\text{PBE}}(s,u)$ $\displaystyle=$
$\displaystyle\Biggl{[}-\frac{{\cal A}}{u^{2}}\frac{1}{1+(4/9){\cal A}u^{2}}+$
(4.22) $\displaystyle\left(\frac{{\cal A}}{u^{2}}+{\cal B}+{\cal
C}[1+s^{2}{\cal F}(s)]u^{2}+{\cal E}[1+s^{2}{\cal G}(s)]u^{4}\right)e^{-{\cal
D}u^{2}}\Biggl{]}e^{-s^{2}{\cal H}(s)u^{2}},$

where ${\cal A}$, ${\cal B}$, ${\cal C}$, ${\cal D}$, and ${\cal E}$ are
constants chosen to obtain an oscillation-averaged UEG exchange hole for
$s=0$, and ${\cal F}(s)$, ${\cal G}(s)$ and ${\cal H}(s)$ are functions
determined so that the hole yields the PBE exchange density for $\mu=0$, and
satisfies the sum rule [Eq. (2.11)] and the small-$r_{12}$ expansion [Eq.
(3.52)] using the gradient expansion of $\tau$ of Eq. (3.56). The range-
separation parameter is fixed at $\mu=\omega=0.4$ bohr-1 which has been found
to be close to optimal for atomization energies, reaction barrier heights, and
ionization energies [158].

$\omega$B97X exchange-correlation functional

The $\omega$B97X exchange-correlation functional [161] has the form of Eq.
(4.15) with $b=1$

$\displaystyle E_{\text{xc}}^{\text{$\omega$B97X}}[\Phi]$ $\displaystyle=$
$\displaystyle
a\;E_{\text{x}}^{\text{sr},\mu,\text{HF}}[\Phi]+E_{\text{x}}^{\text{lr},\mu,\text{HF}}[\Phi]+(1-a)\;E_{\text{x}}^{\text{sr},\mu,\text{B97-GGA}}[\rho_{\uparrow,\Phi},\rho_{\downarrow,\Phi}]$
(4.23)
$\displaystyle+E_{\text{c}}^{\text{B97-GGA}}[\rho_{\uparrow,\Phi},\rho_{\downarrow,\Phi}].$

The short-range B97-GGA exchange density is defined as

$\displaystyle
e_{\text{x}}^{\text{sr},\mu,\text{B97-GGA}}(\rho_{\uparrow},\rho_{\downarrow},\nabla\rho_{\uparrow},\nabla\rho_{\downarrow})=\sum_{\sigma\in\\{\uparrow,\downarrow\\}}e_{\text{x},\sigma}^{\text{sr},\mu,\text{UEG}}(\rho_{\sigma})\;g_{\text{x}}(x_{\sigma}),$
(4.24)

where
$e_{\text{x},\sigma}^{\text{sr},\mu,\text{UEG}}(\rho_{\sigma})=e_{\text{x}}^{\text{sr},\mu,\text{UEG}}(\rho_{\sigma},0)$
is the spin-$\sigma$ contribution to the short-range UEG exchange energy
density [Eq. (4.17)] and the gradient correction $g_{\text{x}}(x_{\sigma})$
where $x_{\sigma}=|\nabla\rho_{\sigma}|/\rho_{\sigma}^{4/3}$ has the same form
as in Eq. (3.38) with polynomial degree $m=4$. In Eq. (4.23), the correlation
functional has the same form as the B97-GGA correlation functional but again
with polynomial degree $m=4$ in Eqs. (3.42) and (3.43). The fraction of short-
range HF exchange $a\approx 0.16$, the range-separation parameter
$\mu=\omega=0.3$ bohr-1, and the linear coefficients in Eqs. (3.38), (3.42),
and (3.43) were optimized on sets of atomic energies, atomization energies,
ionization energies, electron and proton affinities, reaction barrier heights,
and noncovalent interactions, with the constraints $a+c_{\text{x},0}=1$,
$c_{\text{c},0}^{\uparrow\downarrow}=1$, and $c_{\text{c},0}^{\sigma\sigma}=1$
to enforce the correct UEG limit.

HSE exchange-correlation functional

The Heyd-Scuseria-Ernzerhof (HSE) exchange-correlation functional [162] is of
the form of Eq. (4.15) with $b=0$ (i.e., no long-range HF exchange)

$\displaystyle E_{\text{xc}}^{\text{HSE}}[\Phi]$ $\displaystyle=$
$\displaystyle
aE_{\text{x}}^{\text{sr},\mu,\text{HF}}[\Phi]+(1-a)E_{\text{x}}^{\text{sr},\mu,\text{PBE}}[\rho_{\uparrow,\Phi},\rho_{\downarrow,\Phi}]+E_{\text{x}}^{\text{lr},\mu,\text{PBE}}[\rho_{\uparrow,\Phi},\rho_{\downarrow,\Phi}]$
(4.25)
$\displaystyle+E_{\text{c}}^{\text{PBE}}[\rho_{\uparrow,\Phi},\rho_{\downarrow,\Phi}],$

and involves the long-range PBE exchange functional
$E_{\text{x}}^{\text{lr},\mu,\text{PBE}}=E_{\text{x}}^{\text{PBE}}-E_{\text{x}}^{\text{sr},\mu,\text{PBE}}$
complementary to the short-range PBE exchange functional constructed from the
PBE exchange hole model [Eqs. (4.21) and (4.22)]. In order to reproduce
reliable values for the band gap in semiconducting solids, the range-
separation parameter is fixed at $\mu=0.15$ bohr-1, which is a very small
value compared to the other range-separated hybrids. It means that the range
of electron-electron distances covered by HF exchange is large, and the HSE
functional could be thought of as a regular hybrid approximation but with the
very long-range contribution of the HF exchange removed. This is particularly
appropriate for solids since in these systems the very long-range HF exchange
is effectively balanced by the correlation effects (a phenomenon known as
screening). The fraction of (short-range) HF exchange is fixed at $a=0.25$
like in the PBE0 hybrid functional.

##  5 Multideterminant hybrid approximations

###  5.1 Double-hybrid approximations

In 2006, Grimme [163] introduced a two-parameter double-hybrid (2DH)
approximation, written here in its spin-independent version,

$E_{\text{xc}}^{\text{2DH}}=a_{\text{x}}\;E_{\text{x}}^{\text{HF}}[\Phi]+(1-a_{\text{x}})\;E_{\text{x}}^{\text{GGA}}[\rho_{\Phi}]+(1-a_{\text{c}})E_{\text{c}}^{\text{GGA}}[\rho_{\Phi}]+a_{\text{c}}E_{\text{c}}^{\text{MP2}},$
(5.1)

mixing a fraction $a_{\text{x}}$ of the HF exchange energy with a GGA exchange
functional, and a fraction $a_{\text{c}}$ of the second-order Møller-Plesset
(MP2) correlation energy $E_{\text{c}}^{\text{MP2}}$ with a GGA correlation
functional. In Eq. (5.1), the first three terms are first calculated in a
self-consistent manner, and then the last term $E_{\text{c}}^{\text{MP2}}$ is
added perturbatively using the orbitals determined in the first step. The
expression of $E_{\text{c}}^{\text{MP2}}$ is [164]

$E_{\text{c}}^{\text{MP2}}=-\frac{1}{4}\sum_{i=1}^{N}\sum_{j=1}^{N}\;\sum_{a\geq
N+1}\sum_{b\geq
N+1}\frac{|\langle\phi_{i}\phi_{j}||\phi_{a}\phi_{b}\rangle|^{2}}{\varepsilon_{a}+\varepsilon_{b}-\varepsilon_{i}-\varepsilon_{j}},$
(5.2)

where $i,j$ and $a,b$ run over occupied and virtual spin orbitals,
respectively, $\varepsilon_{k}$ are spin orbital energies, and
$\langle\phi_{i}\phi_{j}||\phi_{a}\phi_{b}\rangle=\langle\phi_{i}\phi_{j}|\phi_{a}\phi_{b}\rangle-\langle\phi_{i}\phi_{j}|\phi_{b}\phi_{a}\rangle$
are antisymmetrized two-electron integrals with (in physicists’ notation)

$\displaystyle\langle\phi_{p}\phi_{q}|\phi_{r}\phi_{s}\rangle=\int_{(\mathbb{R}^{3}\times\\{\uparrow,\downarrow\\})^{2}}\frac{\phi_{p}^{*}(\mathbf{x}_{1})\phi_{q}^{*}(\mathbf{x}_{2})\phi_{r}(\mathbf{x}_{1})\phi_{s}(\mathbf{x}_{2})}{|\mathbf{r}_{1}-\mathbf{r}_{2}|}\text{d}\mathbf{x}_{1}\text{d}\mathbf{x}_{2}.$
(5.3)

Note that the notation in Eq. (5.2) assumes that the one-electron wave-
function space is spanned by a discrete set of spin orbitals. In the exact
theory, the continuum limit of the set of virtual spin orbitals is implied.

The rigorous framework underlying these double-hybrid approximations was
established by Sharkas, Toulouse, and Savin [165]. The idea is to decompose
the universal density functional of Eq. (1.8) as

$\displaystyle F[\rho]=\min_{\Psi\in{\cal
W}^{N}_{\rho}}\langle\Psi|\hat{T}+\lambda\hat{W}_{\text{ee}}|\Psi\rangle+\bar{E}_{\text{Hxc}}^{\lambda}[\rho],$
(5.4)

where $\lambda\in[0,1]$ is a coupling constant and
$\bar{E}_{\text{Hxc}}^{\lambda}[\rho]$ is a complementary density functional
defined to make Eq. (5.4) exact. From Eqs. (1.13) and (2.24), we see that
$\bar{E}_{\text{Hxc}}^{\lambda}[\rho]=E_{\text{Hxc}}[\rho]-E_{\text{Hxc}}^{\lambda}[\rho]$
where $E_{\text{Hxc}}[\rho]$ is the standard Hartree-exchange-correlation
functional of the KS scheme and $E_{\text{Hxc}}^{\lambda}[\rho]$ is the
Hartree-exchange-correlation functional along the adiabatic connection. The
Hartree and exchange contributions are simply linear in $\lambda$

$\displaystyle\bar{E}_{\text{H}}^{\lambda}[\rho]=(1-\lambda)E_{\text{H}}[\rho],$
(5.5)
$\displaystyle\bar{E}_{\text{x}}^{\lambda}[\rho]=(1-\lambda)E_{\text{x}}[\rho],$
(5.6)

where $E_{\text{H}}[\rho]$ and $E_{\text{x}}[\rho]$ are the standard Hartree
and exchange functionals of the KS scheme. Moreover, from the uniform
coordinate scaling relation of Eq. (2.61), we have

$\displaystyle\bar{E}_{\text{c}}^{\lambda}[\rho]=E_{\text{c}}[\rho]-\lambda^{2}E_{\text{c}}[\rho_{1/\lambda}],$
(5.7)

where $E_{\text{c}}[\rho]$ is the standard correlation functional of the KS
scheme and
$\rho_{1/\lambda}(\mathbf{r})=(1/\lambda)^{3}\rho(\mathbf{r}/\lambda)$ is the
scaled density. The decomposition in Eq. (5.4) leads to the following
expression of the exact ground-state energy

$\displaystyle E_{0}=\inf_{\Psi\in{\cal
W}^{N}}\left\\{\langle\Psi|\hat{T}+\hat{V}_{\text{ne}}+\lambda\hat{W}_{\text{ee}}|\Psi\rangle+\bar{E}_{\text{Hxc}}^{\lambda}[\rho_{\Psi}]\right\\},$
(5.8)

where the infimum is over general multideterminant wave functions
$\Psi\in{\cal W}^{N}$. This constitutes a multideterminant extension of the KS
scheme. Note that this multideterminant KS scheme can trivially be extended to
spin-dependent density functionals and functionals depending on the kinetic-
energy density [112].

The double-hybrid ansatz can be seen as a particular approximation within this
multideterminant KS scheme [165]. To see this, we define a density-scaled one-
parameter hybrid (DS1H) approximation by restricting the minimization in Eq.
(5.8) to single-determinant wave functions $\Phi\in{\cal S}^{N}$

$\displaystyle E^{\text{DS1H},\lambda}_{0}$ $\displaystyle=$
$\displaystyle\inf_{\Phi\in{\cal
S}^{N}}\Bigl{\\{}\langle\Phi|\hat{T}+\hat{V}_{\text{ne}}+\lambda\hat{W}_{\text{ee}}|\Phi\rangle+\bar{E}_{\text{Hxc}}^{\lambda}[\rho_{\Phi}]\Bigl{\\}},$
(5.9)

obtaining an energy which necessarily depends on $\lambda$. A minimizing
single-determinant wave function $\Phi^{\lambda}$ must satisfy the self-
consistent eigenvalue equation

$\displaystyle\left(\hat{T}+\hat{V}_{\text{ne}}+\lambda\hat{V}_{\text{Hx}}^{\text{HF}}[\Phi^{\lambda}]+\hat{\bar{V}}_{\text{Hxc}}^{\lambda}[\rho_{\Phi^{\lambda}}]\right)|\Phi^{\lambda}\rangle={\cal
E}_{0}^{\lambda}|\Phi^{\lambda}\rangle,$ (5.10)

where $\hat{V}_{\text{Hx}}^{\text{HF}}[\Phi^{\lambda}]$ is the nonlocal HF
potential operator evaluated with the DS1H wave function $\Phi^{\lambda}$ and
$\hat{\bar{V}}_{\text{Hxc}}^{\lambda}[\rho_{\Phi^{\lambda}}]$ is the local
Hartree-exchange-correlation potential operator generated by the energy
functional $\bar{E}_{\text{Hxc}}^{\lambda}[\rho]$ and evaluated at the DS1H
density $\rho_{\Phi^{\lambda}}$. If written explicitly in terms of spin
orbitals, Eq. (5.10) would have the form of the GKS equations [Eq. (1.49)].
The DS1H ground-state energy can be finally written as

$\displaystyle E^{\text{DS1H},\lambda}_{0}$ $\displaystyle=$
$\displaystyle\langle\Phi^{\lambda}|\hat{T}+\hat{V}_{\text{ne}}|\Phi^{\lambda}\rangle+E_{\text{H}}[\rho_{\Phi^{\lambda}}]+\lambda
E_{\text{x}}^{\text{HF}}[\Phi^{\lambda}]+(1-\lambda)E_{\text{x}}[\rho_{\Phi^{\lambda}}]+\bar{E}^{\lambda}_{\text{c}}[\rho_{\Phi^{\lambda}}],$
(5.11)

where the full Hartree energy $E_{\text{H}}[\rho]$ has been recomposed. The
exchange-correlation energy in Eq. (5.11) is of the form of a hybrid
approximation [Eq. (4.3)].

All what is missing in Eq. (5.11) is the correlation energy associated with
the scaled interaction $\lambda\hat{W}_{\text{ee}}$. It can be calculated by a
nonlinear Rayleigh-Schrödinger perturbation theory [166, 167, 168] starting
from the DS1H reference. Consider the following energy expression with the
perturbation parameter $\alpha\in[0,1]$

$\displaystyle E^{\lambda,\alpha}_{0}$ $\displaystyle=$
$\displaystyle\inf_{\Psi\in{\cal
W}^{N}}\Bigl{\\{}\langle\Psi|\hat{T}+\hat{V}_{\text{ne}}+\lambda\hat{V}_{\text{Hx}}^{\text{HF}}[\Phi^{\lambda}]+\alpha\lambda\hat{W}|\Psi\rangle+\bar{E}_{\text{Hxc}}^{\lambda}[\rho_{\Psi}]\Bigl{\\}},$
(5.12)

where
$\lambda\hat{W}=\lambda\left(\hat{W}_{\text{ee}}-\hat{V}_{\text{Hx}}^{\text{HF}}[\Phi^{\lambda}]\right)$
is the scaled Møller-Plesset perturbation operator. For $\alpha=0$, the
stationary equation associated with Eq. (5.12) reduces to the DS1H eigenvalue
equation [Eq. (5.10)]. For $\alpha=1$, Eq. (5.12) reduces to Eq. (5.8), so
$E^{\lambda,\alpha=1}_{0}$ is the exact energy, independently of $\lambda$.
The sum of the zeroth-order energy and first-order energy correction gives
simply the DS1H energy,
$E^{\text{DS1H},\lambda}_{0}=E^{\lambda,(0)}_{0}+E^{\lambda,(1)}_{0}$. Thanks
to the existence of a Brillouin theorem just like in standard Møller-Plesset
perturbation theory (see Refs. [166, 167, 168]), only double excitations
contribute to the first-order wave-function correction $\Psi^{\lambda,(1)}$
and the second-order energy correction has a standard MP2 form

$\displaystyle
E^{\lambda,(2)}_{0}=\lambda^{2}\langle\Phi^{\lambda}|\hat{W}|\Psi^{\lambda,(1)}\rangle=\lambda^{2}E_{\text{c}}^{\text{MP2}},$
(5.13)

where $E_{\text{c}}^{\text{MP2}}$ has the expression in Eq. (5.2) with DS1H
spin orbitals and associated orbital eigenvalues (which implicitly depend on
$\lambda$). This second-order perturbation theory defines a density-scaled
one-parameter double-hybrid (DS1DH) approximation

$\displaystyle
E^{\text{DS1DH},\lambda}_{0}=E^{\text{DS1H},\lambda}_{0}+E^{\lambda,(2)}_{0},$
(5.14)

which contains the exchange-correlation energy contribution

$\displaystyle E^{\text{DS1DH},\lambda}_{\text{xc}}$ $\displaystyle=$
$\displaystyle\lambda
E_{\text{x}}^{\text{HF}}[\Phi^{\lambda}]+(1-\lambda)E_{\text{x}}[\rho_{\Phi^{\lambda}}]+\bar{E}^{\lambda}_{\text{c}}[\rho_{\Phi^{\lambda}}]+\lambda^{2}E_{\text{c}}^{\text{MP2}}.$
(5.15)

To make connection with the double-hybrid ansatz of Eq. (5.1), we can also
define a one-parameter double-hybrid (1DH) approximation, obtained by
neglecting the density scaling in the correlation functional, i.e.
$E_{\text{c}}[\rho_{1/\lambda}]\approx E_{\text{c}}[\rho]$ in Eq. (5.7),

$\displaystyle E^{\text{1DH},\lambda}_{\text{xc}}$ $\displaystyle=$
$\displaystyle\lambda
E_{\text{x}}^{\text{HF}}[\Phi^{\lambda}]+(1-\lambda)E_{\text{x}}[\rho_{\Phi^{\lambda}}]+(1-\lambda^{2})E_{\text{c}}[\rho_{\Phi^{\lambda}}]+\lambda^{2}E_{\text{c}}^{\text{MP2}},$
(5.16)

which, after using semilocal approximations for $E_{\text{x}}[\rho]$ and
$E_{\text{c}}[\rho]$, has the form of Eq. (5.1) with parameters
$a_{\text{x}}=\lambda$ and $a_{\text{c}}=\lambda^{2}$. In this rigorous
formulation of the double-hybrid approximations, the fraction of HF exchange
is thus connected to the fraction of MP2 correlation. Taking into account
approximately the scaling of the density in $E_{\text{c}}[\rho_{1/\lambda}]$,
it has also been proposed to use $a_{\text{c}}=\lambda^{3}$ [169]. Fromager
[170] also proposed an extension of this rigorous formulation in order to
justify the use of double-hybrid approximations with two parameters such that
$a_{\text{c}}\leq a_{\text{x}}^{2}=\lambda^{2}$.

An essential advantage of double-hybrid approximations is that the presence of
nonlocal MP2 correlation allows one to use a larger fraction of nonlocal HF
exchange, which helps decreasing the self-interaction error. This usually
provides an improvement over hybrid approximations for molecular systems with
sufficiently large electronic gaps. However, a large fraction of HF exchange
and a fraction of MP2 correlation also generally means large static-
correlation errors in systems with small HOMO-LUMO gaps.

The first and still best known double-hybrid approximation is B2PLYP [163]
which is based on the B88 exchange functional and the LYP correlation
functional

$E_{\text{xc}}^{\text{B2PLYP}}=a_{\text{x}}\;E_{\text{x}}^{\text{HF}}[\Phi]+(1-a_{\text{x}})\;E_{\text{x}}^{\text{B88}}[\rho_{\uparrow,\Phi},\rho_{\downarrow,\Phi}]+(1-a_{\text{c}})E_{\text{c}}^{\text{LYP}}[\rho_{\uparrow,\Phi},\rho_{\downarrow,\Phi}]+a_{\text{c}}E_{\text{c}}^{\text{MP2}},$
(5.17)

and the parameters $a_{\text{x}}=0.53$ and $a_{\text{c}}=0.27$ have been
optimized on a set of atomization energies. Interestingly, even though the two
parameters have been optimized without any constraint, we have
$a_{\text{c}}\approx a_{\text{x}}^{2}=0.28$ as predicted by Eq. (5.16).

It has also been proposed to use the spin-component-scaled (SCS) version of
MP2 [171] to construct spin-component-scaled double-hybrid approximations of
the form [172, 173]

$\displaystyle E_{\text{xc}}^{\text{SCS-DH}}$ $\displaystyle=$ $\displaystyle
a_{\text{x}}\;E_{\text{x}}^{\text{HF}}[\Phi]+(1-a_{\text{x}})\;E_{\text{x}}^{\text{GGA}}[\rho_{\Phi}]+(1-a_{\text{c}})E_{\text{c}}^{\text{GGA}}[\rho_{\Phi}]+c_{\text{OS}}E^{\text{MP2}}_{\text{c},\text{OS}}+c_{\text{SS}}E^{\text{MP2}}_{\text{c},\text{SS}},$

which contains four empirical parameters $a_{\text{x}}$, $a_{\text{c}}$,
$c_{\text{OS}}$, and $c_{\text{SS}}$. In this expression,
$E^{\text{MP2}}_{\text{c},\text{OS}}$ and
$E^{\text{MP2}}_{\text{c},\text{SS}}$ are the opposite-spin (OS) and same-spin
(SS) contributions to the MP2 correlation energy obtained by restricting the
sums over $i$ and $j$ in Eq. (5.2) to spin orbitals of opposite and same
spins, respectively. Since in MP2 the same-spin component is usually
overestimated relative to the opposite-spin component, this SCS variant is a
simple way to achieve higher accuracy without increasing computational cost.

For reviews on different flavors of double hybrids and their assessments, the
reader may consult Refs. [174, 175, 176, 177]. It has also been proposed to
construct double-hybrid approximations where the MP2 correlation term is
extended to a higher-order correlation method such as RPA [178, 179, 180, 181,
182] or coupled-cluster [183, 184]. More generally, the multideterminant
extension of the KS scheme of Eq. (5.8) allows one to define hybrids combining
any wave-function method with density functionals. For example, a
multiconfiguration hybrid approximation based on Eq. (5.8) which combines a
multiconfiguration self-consistent-field (MCSCF) wave function with density
functionals has been proposed in the goal of tackling strongly correlated
systems [185]. This approach has also been used to combine valence-bond (VB)
theory [186] or variational two-electron reduced-density-matrix theory [187]
with DFT.

###  5.2 Range-separated double-hybrid approximations

####  5.2.1 Range-separated one-parameter double-hybrid approximations

In 2005, Ángyán, Gerber, Savin, and Toulouse [166] introduced what could be
called the first range-separated one-parameter double-hybrid approximation,
i.e. combining HF exchange and MP2 correlation with density functionals using
a one-parameter decomposition of the electron-electron interaction. This is
based of the range-separated multideterminant extension of the KS scheme
introduced earlier by Savin [141] (see, also, Refs. [148, 188, 145]) and which
actually predates and inspired the multideterminant extension of the KS scheme
of Eq. (5.8).

The idea is to decompose the universal density functional of Eq. (1.8) as

$\displaystyle F[\rho]=\min_{\Psi\in{\cal
W}^{N}_{\rho}}\langle\Psi|\hat{T}+\hat{W}_{\text{ee}}^{\text{lr},\mu}|\Psi\rangle+\bar{E}_{\text{Hxc}}^{\text{sr},\mu}[\rho],$
(5.19)

where $\hat{W}_{\text{ee}}^{\text{lr},\mu}$ is the long-range electron-
electron interaction operator (associated with the pair potential
$w_{\text{ee}}^{\text{lr},\mu}(r_{12})=\operatorname{erf}(\mu r_{12})/r_{12}$
as already used in the range-separated hybrids of Section 4.2) and
$\bar{E}_{\text{Hxc}}^{\text{sr},\mu}[\rho]$ is the complementary short-range
density functional defined to make Eq. (5.19) exact. As before, the parameter
$\mu\in[0,+\infty)$ controls the range of the separation. The complementary
short-range functional can be written as
$\bar{E}_{\text{Hxc}}^{\text{sr},\mu}[\rho]=E_{\text{Hxc}}[\rho]-E_{\text{Hxc}}^{\text{lr},\mu}[\rho]$
where $E_{\text{Hxc}}[\rho]$ is the standard Hartree-exchange-correlation
functional of the KS scheme and $E_{\text{Hxc}}^{\text{lr},\mu}[\rho]$ is the
Hartree-exchange-correlation functional associated with the long-range
interaction $w_{\text{ee}}^{\text{lr},\mu}(r_{12})$. It is often convenient to
decompose the short-range functional as (see Refs. [189, 190, 191] for an
alternative decomposition)

$\displaystyle\bar{E}_{\text{Hxc}}^{\text{sr},\mu}[\rho]=E_{\text{H}}^{\text{sr},\mu}[\rho]+E_{\text{x}}^{\text{sr},\mu}[\rho]+\bar{E}_{\text{c}}^{\text{sr},\mu}[\rho],$
(5.20)

where $E_{\text{H}}^{\text{sr},\mu}[\rho]$ is the short-range Hartree
functional

$E_{\text{H}}^{\text{sr},\mu}[\rho]=\frac{1}{2}\int_{\mathbb{R}^{3}\times\mathbb{R}^{3}}\rho(\mathbf{r}_{1})\rho(\mathbf{r}_{2})w_{\text{ee}}^{\text{sr},\mu}(r_{12})\text{d}\mathbf{r}_{1}\text{d}\mathbf{r}_{2},$
(5.21)

with the short-range interaction
$w_{\text{ee}}^{\text{sr},\mu}(r_{12})=1/r_{12}-w_{\text{ee}}^{\text{lr},\mu}(r_{12})$,
$E_{\text{x}}^{\text{sr},\mu}[\rho]$ is the short-range exchange functional
[Eq. (4.13)] which can also be written as

$\displaystyle E_{\text{x}}^{\text{sr}}[\rho]$ $\displaystyle=$
$\displaystyle\langle\Phi[\rho]|\hat{W}_{\text{ee}}^{\text{sr},\mu}|\Phi[\rho]\rangle-
E_{\text{H}}^{\text{sr},\mu}[\rho],$ (5.22)

with the KS single-determinant wave function $\Phi[\rho]$, and
$\bar{E}_{\text{c}}^{\text{sr},\mu}[\rho]$ is the complementary short-range
correlation functional. Just like for Eq. (5.8), the decomposition in Eq.
(5.19) leads to the following expression of the exact ground-state energy

$\displaystyle E_{0}=\inf_{\Psi\in{\cal
W}^{N}}\left\\{\langle\Psi|\hat{T}+\hat{V}_{\text{ne}}+\hat{W}_{\text{ee}}^{\text{lr},\mu}|\Psi\rangle+\bar{E}_{\text{Hxc}}^{\text{sr},\mu}[\rho_{\Psi}]\right\\},$
(5.23)

where the infimum is over general multideterminant wave functions
$\Psi\in{\cal W}^{N}$.

To obtain a MP2/DFT hybrid scheme, we proceed analogously to Section 5.1.
First, we define the following range-separated hybrid (RSH) approximation by
restricting the minimization in Eq. (5.23) to single-determinant wave
functions $\Phi\in{\cal S}^{N}$

$\displaystyle E^{\text{RSH},\mu}_{0}$ $\displaystyle=$
$\displaystyle\inf_{\Phi\in{\cal
S}^{N}}\Bigl{\\{}\langle\Phi|\hat{T}+\hat{V}_{\text{ne}}+\hat{W}_{\text{ee}}^{\text{lr},\mu}|\Phi\rangle+\bar{E}_{\text{Hxc}}^{\text{sr},\mu}[\rho_{\Phi}]\Bigl{\\}},$
(5.24)

obtaining an energy which necessarily depends on $\mu$. A minimizing single-
determinant wave function $\Phi^{\mu}$ must satisfy the self-consistent
eigenvalue equation

$\displaystyle\left(\hat{T}+\hat{V}_{\text{ne}}+\hat{V}_{\text{Hx}}^{\text{lr},\mu,\text{HF}}[\Phi^{\mu}]+\hat{\bar{V}}_{\text{Hxc}}^{\text{sr},\mu}[\rho_{\Phi^{\mu}}]\right)|\Phi^{\mu}\rangle={\cal
E}_{0}^{\mu}|\Phi^{\mu}\rangle,$ (5.25)

where $\hat{V}_{\text{Hx}}^{\text{lr},\mu,\text{HF}}[\Phi^{\mu}]$ is the
nonlocal long-range HF potential operator evaluated with the RSH wave function
$\Phi^{\mu}$ and
$\hat{\bar{V}}_{\text{Hxc}}^{\text{sr},\mu}[\rho_{\Phi^{\mu}}]$ is the local
short-range Hartree-exchange-correlation potential operator generated by the
energy functional $\bar{E}_{\text{Hxc}}^{\text{sr},\mu}[\rho]$ and evaluated
at the RSH density $\rho_{\Phi^{\mu}}$. The RSH ground-state energy can be
finally written as

$\displaystyle E^{\text{RSH},\mu}_{0}$ $\displaystyle=$
$\displaystyle\langle\Phi^{\mu}|\hat{T}+\hat{V}_{\text{ne}}|\Phi^{\mu}\rangle+E_{\text{H}}[\rho_{\Phi^{\mu}}]+E_{\text{x}}^{\text{lr},\mu,\text{HF}}[\Phi^{\mu}]+E_{\text{x}}^{\text{sr},\mu}[\rho_{\Phi^{\mu}}]+\bar{E}^{\text{sr},\mu}_{\text{c}}[\rho_{\Phi^{\mu}}],$
(5.26)

where the full Hartree energy $E_{\text{H}}[\rho]$ has been recomposed. The
exchange-correlation energy in Eq. (5.26) has a similar form as in the LC
scheme of Eq. (4.11).

To calculate the missing long-range correlation energy in Eq. (5.26), we can
define a nonlinear Rayleigh-Schrödinger perturbation theory [166, 167, 168]
starting from the RSH reference. We start from the following energy expression
with the perturbation parameter $\alpha\in[0,1]$

$\displaystyle E^{\mu,\alpha}_{0}$ $\displaystyle=$
$\displaystyle\inf_{\Psi\in{\cal
W}^{N}}\Bigl{\\{}\langle\Psi|\hat{T}+\hat{V}_{\text{ne}}+\hat{V}_{\text{Hx}}^{\text{lr},\mu,\text{HF}}[\Phi^{\mu}]+\alpha\hat{W}^{\text{lr},\mu}|\Psi\rangle+\bar{E}_{\text{Hxc}}^{\text{lr},\mu}[\rho_{\Psi}]\Bigl{\\}},$
(5.27)

where
$\hat{W}^{\text{lr},\mu}=\left(\hat{W}_{\text{ee}}^{\text{lr},\mu}-\hat{V}_{\text{Hx}}^{\text{lr},\mu,\text{HF}}[\Phi^{\mu}]\right)$
is the long-range Møller-Plesset perturbation operator. For $\alpha=0$, the
stationary equation associated with Eq. (5.27) reduces to the RSH eigenvalue
equation [Eq. (5.25)]. For $\alpha=1$, Eq. (5.27) reduces to Eq. (5.23), so
$E^{\mu,\alpha=1}_{0}$ is the exact energy, independently of $\mu$. The sum of
the zeroth-order energy and first-order energy correction gives simply the RSH
energy, $E^{\text{RSH},\mu}_{0}=E^{\mu,(0)}_{0}+E^{\mu,(1)}_{0}$. As in
Section 5.1, only double excitations contribute to the first-order wave-
function correction $\Psi^{\mu,(1)}$ and the second-order energy correction
has a standard MP2 form

$\displaystyle
E^{\mu,(2)}_{0}=\langle\Phi^{\mu}|\hat{W}^{\text{lr},\mu}|\Psi^{\mu,(1)}\rangle=E_{\text{c}}^{\text{lr},\mu,\text{MP2}},$
(5.28)

where $E_{\text{c}}^{\text{lr},\mu,\text{MP2}}$ has the same expression as in
Eq. (5.2) with RSH spin orbitals and associated orbital eigenvalues (which
implicitly depend on $\mu$) but using the long-range two-electron integrals

$\displaystyle\langle\phi_{p}\phi_{q}|\phi_{r}\phi_{s}\rangle^{\text{lr},\mu}=\int_{(\mathbb{R}^{3}\times\\{\uparrow,\downarrow\\})^{2}}\phi_{p}^{*}(\mathbf{x}_{1})\phi_{q}^{*}(\mathbf{x}_{2})\phi_{r}(\mathbf{x}_{1})\phi_{s}(\mathbf{x}_{2})w_{\text{ee}}^{\text{lr},\mu}(r_{12})\text{d}\mathbf{x}_{1}\text{d}\mathbf{x}_{2},$
(5.29)

instead of the standard two-electron integrals of Eq. (5.3). This second-order
perturbation theory defines a RSH+MP2 approximation

$\displaystyle
E^{\text{RSH+MP2},\mu}_{0}=E^{\text{RSH},\mu}_{0}+E_{\text{c}}^{\text{lr},\mu,\text{MP2}},$
(5.30)

which contains the exchange-correlation energy contribution

$\displaystyle E^{\text{RSH+MP2},\mu}_{\text{xc}}$ $\displaystyle=$
$\displaystyle
E_{\text{x}}^{\text{lr},\mu,\text{HF}}[\Phi^{\mu}]+E_{\text{x}}^{\text{sr},\mu}[\rho_{\Phi^{\mu}}]+\bar{E}^{\text{sr},\mu}_{\text{c}}[\rho_{\Phi^{\mu}}]+E_{\text{c}}^{\text{lr},\mu,\text{MP2}}.$
(5.31)

When using semilocal density-functional approximations for the short-range
functionals $E_{\text{x}}^{\text{sr},\mu}[\rho]$ and
$\bar{E}^{\text{sr},\mu}_{\text{c}}[\rho]$, the RSH+MP2 exchange-correlation
energy expression of Eq. (5.31) thus constitutes range-separated double-hybrid
approximations similar to the double hybrids of Section 5.1. The optimal value
for the range-separation parameter is often around $\mu\approx 0.5$ bohr-1
[143, 192]. This scheme has the advantage to drop the long-range part of both
the exchange and correlation density functionals which are usually not well
described by semilocal density-functional approximations. Moreover, using a
long-range MP2 correlation energy has the advantage of leading to a correct
qualitative description of London dispersion interaction energies [166, 193,
194, 195], while displaying a fast convergence with the one-electron basis
size [196]. Similar to the SCS double hybrids [Eq. (LABEL:2DHSCS)], a SCS
variant of the RSH+MP2 scheme has also been proposed [197].

The range-separated multideterminant extension of the KS scheme of Eq. (5.23)
allows one to define various hybrid schemes combining any wave-function method
with density functionals. For example, one can go beyond second order by using
long-range coupled-cluster [198, 199, 200, 201] or random-phase approximations
[202, 203, 204, 205, 200]. To describe strongly correlated systems, one can
also use for the long-range part wave-function methods such as configuration
interaction (CI) [206, 207, 208, 209], MCSCF [210, 211, 212], density-matrix
renormalization group (DMRG) [213], or multireference perturbation theory
[214]. Density-matrix functional theory (DMFT) [215, 216, 217] and Green-
function methods [218, 219] have also been used for the long-range part.

We now consider the approximations used for
$E_{\text{x}}^{\text{sr},\mu}[\rho]$ and
$\bar{E}^{\text{sr},\mu}_{\text{c}}[\rho]$. In Section 4.2, we have already
described the short-range exchange LDA [Eq. (4.16)] and some short-range
exchange GGAs for $E_{\text{x}}^{\text{sr},\mu}[\rho]$. Here, we describe the
short-range LDA correlation functional and another short-range GGA exchange-
correlation functional.

Short-range LDA correlation functional

The complementary short-range LDA correlation functional is

$\bar{E}_{\text{c}}^{\text{sr},\mu,\text{LDA}}[\rho_{\uparrow},\rho_{\downarrow}]=\int_{\mathbb{R}^{3}}\bar{e}_{\text{c}}^{\text{sr},\mu,\text{UEG}}(\rho_{\uparrow}(\mathbf{r}),\rho_{\downarrow}(\mathbf{r}))\text{d}\mathbf{r},$
(5.32)

where
$\bar{e}_{\text{c}}^{\text{sr},\mu,\text{UEG}}(\rho_{\uparrow},\rho_{\downarrow})=\rho\;\bar{\varepsilon}_{\text{c}}^{\text{sr},\mu,\text{UEG}}(\rho_{\uparrow},\rho_{\downarrow})$
is the complementary short-range UEG correlation energy density. In this
expression,
$\bar{\varepsilon}_{\text{c}}^{\text{sr},\mu,\text{UEG}}(\rho_{\uparrow},\rho_{\downarrow})$
is defined by

$\bar{\varepsilon}_{\text{c}}^{\text{sr},\mu,\text{UEG}}(\rho_{\uparrow},\rho_{\downarrow})=\varepsilon_{\text{c}}^{\text{UEG}}(\rho_{\uparrow},\rho_{\downarrow})-\varepsilon_{\text{c}}^{\text{lr},\mu,\text{UEG}}(\rho_{\uparrow},\rho_{\downarrow}),$
(5.33)

where $\varepsilon_{\text{c}}^{\text{UEG}}(\rho_{\uparrow},\rho_{\downarrow})$
and
$\varepsilon_{\text{c}}^{\text{lr},\mu,\text{UEG}}(\rho_{\uparrow},\rho_{\downarrow})$
are the correlation energies per particle of the UEG with the standard Coulomb
and long-range electron-electron interactions, respectively. A simple spin-
independent parametrization of
$\bar{\varepsilon}_{\text{c}}^{\text{sr},\mu,\text{UEG}}$ was given in Ref.
[220]. A better spin-dependent parametrization was constructed in Ref. [221]
which uses the PW92 parametrization for
$\varepsilon_{\text{c}}^{\text{UEG}}(\rho_{\uparrow},\rho_{\downarrow})$ [Eq.
(3.8)] and the following parametrization for
$\varepsilon_{\text{c}}^{\text{lr},\mu,\text{UEG}}(\rho_{\uparrow},\rho_{\downarrow})$
in terms of $r_{\text{s}}=(3/(4\pi\rho))^{1/3}$ and
$\zeta=(\rho_{\uparrow}-\rho_{\downarrow})/\rho$

$\displaystyle\varepsilon_{\text{c}}^{\text{lr},\mu,\text{UEG}}(\rho_{\uparrow},\rho_{\downarrow})=\phantom{xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx}$
$\displaystyle\frac{\phi_{2}(\zeta)^{3}Q\left(\frac{\mu\sqrt{r_{\text{s}}}}{\phi_{2}(\zeta)}\right)+a_{1}(r_{\text{s}},\zeta)\mu^{3}+a_{2}(r_{\text{s}},\zeta)\mu^{4}+a_{3}(r_{\text{s}},\zeta)\mu^{5}+a_{4}(r_{\text{s}},\zeta)\mu^{6}+a_{5}(r_{\text{s}},\zeta)\mu^{8}}{(1+b_{0}(r_{\text{s}})^{2}\mu^{2})^{4}}.$
(5.34)

In this expression, $\phi_{2}(\zeta)$ is a spin-scaling function defined by
Eq. (3.5), $Q(x)$ is a function determined from the small-$\mu$ and/or
small-$r_{\text{s}}$ limit

$Q(x)=\frac{2\ln(2)-2}{\pi^{2}}\ln\left(\frac{1+ax+bx^{2}+cx^{3}}{1+ax+dx^{2}}\right),$
(5.35)

with $a=5.84605$, $c=3.91744$, $d=3.44851$, $b=d-3\pi\alpha/[4\ln(2)-4]$,
$\alpha=4/(9\pi)^{1/3}$, and the functions $a_{i}(r_{\text{s}},\zeta)$ are

$a_{1}(r_{\text{s}},\zeta)=4b_{0}(r_{\text{s}})^{6}C_{3}(r_{\text{s}},\zeta)+b_{0}(r_{\text{s}})^{8}C_{5}(r_{\text{s}},\zeta),$
(5.36a)
$a_{2}(r_{\text{s}},\zeta)=4b_{0}(r_{\text{s}})^{6}C_{2}(r_{\text{s}},\zeta)+b_{0}(r_{\text{s}})^{8}C_{4}(r_{\text{s}},\zeta)+6b_{0}(r_{\text{s}})^{4}\varepsilon_{\text{c}}^{\text{PW92}}(r_{\text{s}},\zeta),$
(5.36b)
$a_{3}(r_{\text{s}},\zeta)=b_{0}(r_{\text{s}})^{8}C_{3}(r_{\text{s}},\zeta),$
(5.36c)
$a_{4}(r_{\text{s}},\zeta)=b_{0}(r_{\text{s}})^{8}C_{2}(r_{\text{s}},\zeta)+4b_{0}(r_{\text{s}})^{6}\varepsilon_{\text{c}}^{\text{PW92}}(r_{\text{s}},\zeta),$
(5.36d)
$a_{5}(r_{\text{s}},\zeta)=b_{0}(r_{\text{s}})^{8}\varepsilon_{\text{c}}^{\text{PW92}}(r_{\text{s}},\zeta),$
(5.36e)

where $\varepsilon_{\text{c}}^{\text{PW92}}(r_{\text{s}},\zeta)$ is the PW92
parametrization of the UEG correlation energy par particle, and the functions
$C_{i}(r_{\text{s}},\zeta)$ are determined from the large-$\mu$ limit

$C_{2}(r_{\text{s}},\zeta)=-\frac{3g_{\text{c}}(0,r_{\text{s}},\zeta)}{8r_{\text{s}}^{3}},$
(5.37a)
$C_{3}(r_{\text{s}},\zeta)=-\frac{g(0,r_{\text{s}},\zeta)}{\sqrt{2\pi}r_{\text{s}}^{3}},$
(5.37b)
$C_{4}(r_{\text{s}},\zeta)=-\frac{9[g_{\text{c}}^{\prime\prime}(0,r_{\text{s}},\zeta)+(1-\zeta^{2})D_{2}(r_{\text{s}})]}{64r_{\text{s}}^{3}},$
(5.37c)
$C_{5}(r_{\text{s}},\zeta)=-\frac{9[g^{\prime\prime}(0,r_{\text{s}},\zeta)+(1-\zeta^{2})D_{3}(r_{\text{s}})]}{40\sqrt{2\pi}r_{\text{s}}^{3}},$
(5.37d)

where $g(0,r_{\text{s}},\zeta)$ is the on-top pair-distribution function888For
a general system, the pair-distribution function
$g(\mathbf{r}_{1},\mathbf{r}_{2})$ is defined from the pair density
$\rho_{2}(\mathbf{r}_{1},\mathbf{r}_{2})$ [Eq. (2.1)] as
$\rho_{2}(\mathbf{r}_{1},\mathbf{r}_{2})=\rho(\mathbf{r}_{1})\rho(\mathbf{r}_{2})g(\mathbf{r}_{1},\mathbf{r}_{2})$.
The on-top value is the value at electron coalescence, i.e. for
$\mathbf{r}_{1}=\mathbf{r}_{2}$. of the Coulombic UEG and
$g^{\prime\prime}(0,r_{\text{s}},\zeta)$ is its second-order derivative with
respect to $r_{12}$ at $r_{12}=0$, and similarly for their correlation parts
$g_{\text{c}}(0,r_{\text{s}},\zeta)=g(0,r_{\text{s}},\zeta)-(1-\zeta^{2})/2$
and
$g_{\text{c}}^{\prime\prime}(0,r_{\text{s}},\zeta)=g^{\prime\prime}(0,r_{\text{s}},\zeta)-\phi_{8}(\zeta)/(5\alpha^{2}r_{\text{s}}^{2})$
with $\phi_{8}(\zeta)$ defined by Eq. (3.5). The $\zeta$-dependence of the
latter quantities is assumed to be exchange-like, i.e.
$g(0,r_{\text{s}},\zeta)\approx(1-\zeta^{2})g(0,r_{\text{s}},\zeta=0)$ and
$g^{\prime\prime}(0,r_{\text{s}},\zeta)\approx\zeta_{+}^{2}g^{\prime\prime}(0,r_{\text{s}}/\zeta_{+}^{1/3},\zeta=1)+\zeta_{-}^{2}g^{\prime\prime}(0,r_{\text{s}}/\zeta_{-}^{1/3},\zeta=1)$
where $\zeta_{\pm}=(1\pm\zeta)/2$. The on-top pair-distribution function has
been parametrized in Ref. [222]

$g(0,r_{\text{s}},\zeta=0)=(1-Br_{\text{s}}+Cr_{\text{s}}^{2}+Dr_{\text{s}}^{3}+Er_{\text{s}}^{4})e^{-Fr_{\text{s}}},$
(5.38)

with $B=0.7317-d$, $C=0.08193$, $D=-0.01277$, $E=0.001859$, and $F=0.7524$.
The remaining functions were determined by fitting to QMC data

$b_{0}(r_{\text{s}})=0.784949r_{\text{s}},$ (5.39a)
$g^{\prime\prime}(0,r_{\text{s}},\zeta=1)=\frac{2^{5/3}}{5\alpha^{2}r_{\text{s}}^{2}}\frac{1-0.02267r_{\text{s}}}{1+0.4319r_{\text{s}}+0.04r_{\text{s}}^{2}},$
(5.39b)
$D_{2}(r_{\text{s}})=\frac{e^{-0.547r_{\text{s}}}}{r_{s}^{2}}(-0.388r_{\text{s}}+0.676r_{\text{s}}^{2}),$
(5.39c)
$D_{3}(r_{\text{s}})=\frac{e^{-0.31r_{\text{s}}}}{r_{s}^{3}}(-4.95r_{\text{s}}+r_{\text{s}}^{2}).$
(5.39d)

Short-range PBE(GWS) exchange-correlation functional

The Goll-Werner-Stoll (GWS) variant of the short-range PBE exchange-
correlation functional [198, 199] is a slight modification of the short-range
PBE functional developed in Ref. [223]. The exchange energy density is

$\displaystyle e_{\text{x}}^{\text{sr},\mu,\text{PBE(GWS)}}(\rho,\nabla\rho)$
$\displaystyle=$ $\displaystyle
e_{\text{x}}^{\text{sr},\mu,\text{UEG}}(\rho)F_{\text{x}}(s,\tilde{\mu}),$
(5.40)

with an enhancement factor of the same form as in the standard PBE exchange
functional

$\displaystyle
F_{\text{x}}(s,\tilde{\mu})=1+\kappa-\frac{\kappa}{1+b(\tilde{\mu})s^{2}/\kappa},$
(5.41)

with $s=|\nabla\rho|/(2k_{\text{F}}\rho)$ and
$\tilde{\mu}=\mu/(2k_{\text{F}})$. In this expression, $\kappa=0.840$, as in
the standard PBE exchange functional, to saturate the local Lieb-Oxford bound
(for $\mu=0$) and
$b(\tilde{\mu})=b^{\text{PBE}}[b^{\text{T}}(\tilde{\mu})/b^{\text{T}}(0)]e^{-\alpha_{\text{x}}\tilde{\mu}^{2}}$
where $b^{\text{PBE}}=0.21951$ is the second-order gradient-expansion
coefficient of the standard PBE exchange functional,
$b^{\text{T}}(\tilde{\mu})$ is a function coming from the second-order GEA of
the short-range exchange energy [189, 223]

$b^{\text{T}}(\tilde{\mu})=\frac{-c_{1}(\tilde{\mu})+c_{2}(\tilde{\mu})e^{1/(4\tilde{\mu}^{2})}}{c_{3}(\tilde{\mu})+54c_{4}(\tilde{\mu})e^{1/(4\tilde{\mu}^{2})}},$
(5.42)

with $c_{1}(\tilde{\mu})=1+22\tilde{\mu}^{2}+144\tilde{\mu}^{4}$,
$c_{2}(\tilde{\mu})=2\tilde{\mu}^{2}(-7+72\tilde{\mu}^{2})$,
$c_{3}(\tilde{\mu})=-864\tilde{\mu}^{4}(-1+2\tilde{\mu}^{2})$, and
$c_{4}(\tilde{\mu})=\tilde{\mu}^{2}[-3-24\tilde{\mu}^{2}+32\tilde{\mu}^{4}+8\tilde{\mu}\sqrt{\pi}\operatorname{erf}(1/(2\tilde{\mu}))]$.
Finally, $\alpha_{\text{x}}=19.0$ is a damping parameter optimized on the He
atom.

Similarly, the correlation energy density has the same form as the standard
PBE correlation functional

$\bar{e}_{\text{c}}^{\text{sr},\mu,\text{PBE(GWS)}}(\rho_{\uparrow},\rho_{\downarrow},\nabla\rho_{\uparrow},\nabla\rho_{\downarrow})=\rho\left[\bar{\varepsilon}_{\text{c}}^{\text{sr},\mu,\text{UEG}}(\rho_{\uparrow},\rho_{\downarrow})+H^{\mu}(\rho_{\uparrow},\rho_{\downarrow},t)\right],$
(5.43)

with $t=|\nabla\rho|/(2\phi_{2}(\zeta)k_{\text{s}}\rho)$ and the gradient
correction

$\displaystyle
H^{\mu}(\rho_{\uparrow},\rho_{\downarrow},t)=A(0)\phi_{2}(\zeta)^{3}\ln\left[1+\frac{\beta(\mu)}{A(0)}t^{2}\frac{1+{\cal
A}(\mu)t^{2}}{1+{\cal A}(\mu)t^{2}+{\cal A}(\mu)^{2}t^{4}}\right],$ (5.44)

where

$\displaystyle{\cal
A}(\mu)=\frac{\beta(\mu)}{A(0)}\left[\exp(-\bar{\varepsilon}_{\text{c}}^{\text{sr},\mu,\text{UEG}}(\rho_{\uparrow},\rho_{\downarrow})/(A(0)\phi_{2}(\zeta)^{3}))-1\right]^{-1},$
(5.45)

and

$\displaystyle\beta(\mu)=\beta^{\text{PBE}}\left(\frac{\bar{\varepsilon}_{\text{c}}^{\text{sr},\mu,\text{UEG}}(\rho_{\uparrow},\rho_{\downarrow})}{\bar{\varepsilon}_{\text{c}}^{\text{sr},\mu=0,\text{UEG}}(\rho_{\uparrow},\rho_{\downarrow})}\right)^{\alpha_{\text{c}}},$
(5.46)

and the value of $A(0)$ is given after Eq. (3.6). In Eq. (5.46),
$\beta=0.066725$ is the second-order gradient coefficient of the standard PBE
correlation functional and $\alpha_{\text{c}}=2.78$ is a damping parameter
optimized on the He atom.

For $\mu=0$, this short-range PBE exchange-correlation functional reduces to
the standard PBE exchange-correlation functional and for large $\mu$ it
reduces to the short-range LDA exchange-correlation functional.

####  5.2.2 Range-separated two-parameter double-hybrid approximations

In 2018, Kalai and Toulouse [224] introduced what we will call range-separated
two-parameter double-hybrid approximations, combining HF exchange and MP2
correlation with density functionals using a two-parameter decomposition of
the electron-electron in a way reminiscent of the CAM decomposition [Eq.
(4.15)] (see, also, Refs. [225, 201]). This is based on a multideterminant
extension of the KS scheme which generalizes the schemes of Sections 5.1 and
5.2.1.

We first decompose the universal density functional of Eq. (1.8) as

$\displaystyle F[\rho]=\min_{\Psi\in{\cal
W}^{N}_{\rho}}\langle\Psi|\hat{T}+\hat{W}_{\text{ee}}^{\text{lr},\mu}+\lambda\hat{W}_{\text{ee}}^{\text{sr},\mu}|\Psi\rangle+\bar{E}_{\text{Hxc}}^{\text{sr},\mu,\lambda}[\rho],$
(5.47)

where the parameter $\mu\in[0,+\infty)$ controls the range of the separation
as always, the parameter $\lambda\in[0,1]$ corresponds to the fraction of the
short-range electron-electron interaction in the wave-function part, and
$\bar{E}_{\text{Hxc}}^{\text{sr},\mu,\lambda}[\rho]$ is the complementary
short-range density functional defined to make this decomposition exact. As
before, the latter functional can be decomposed as

$\displaystyle\bar{E}_{\text{Hxc}}^{\text{sr},\mu,\lambda}[\rho]=E_{\text{H}}^{\text{sr},\mu,\lambda}[\rho]+E_{\text{x}}^{\text{sr},\mu,\lambda}[\rho]+\bar{E}_{\text{c}}^{\text{sr},\mu,\lambda}[\rho].$
(5.48)

The Hartree and exchange contributions are linear in $\lambda$

$\displaystyle
E_{\text{H}}^{\text{sr},\mu,\lambda}[\rho]=(1-\lambda)E_{\text{H}}^{\text{sr},\mu}[\rho],$
(5.49) $\displaystyle
E_{\text{x}}^{\text{sr},\mu,\lambda}[\rho]=(1-\lambda)E_{\text{x}}^{\text{sr},\mu}[\rho],$
(5.50)

where $E_{\text{H}}^{\text{sr},\mu}[\rho]$ and
$E_{\text{x}}^{\text{sr},\mu}[\rho]$ are the short-range Hartree and exchange
functionals introduced in Section 5.2.1, and the correlation contribution can
be written as

$\displaystyle\bar{E}_{\text{c}}^{\text{sr},\mu,\lambda}[\rho]=E_{\text{c}}[\rho]-E_{\text{c}}^{\mu,\lambda}[\rho],$
(5.51)

where $E_{\text{c}}[\rho]$ is the standard KS correlation functional and
$E_{\text{c}}^{\mu,\lambda}[\rho]$ is the correlation functional associated
with the interaction $w_{\text{ee}}^{\text{lr},\mu}(r_{12})+\lambda
w_{\text{ee}}^{\text{sr},\mu}(r_{12})$. The exact ground-state energy can then
be expressed as

$\displaystyle E_{0}=\inf_{\Psi\in{\cal
W}^{N}}\left\\{\langle\Psi|\hat{T}+\hat{V}_{\text{ne}}+\hat{W}_{\text{ee}}^{\text{lr},\mu}+\lambda\hat{W}_{\text{ee}}^{\text{sr},\mu}|\Psi\rangle+\bar{E}_{\text{Hxc}}^{\text{sr},\mu,\lambda}[\rho_{\Psi}]\right\\},$
(5.52)

which constitutes a generalization of Eqs. (5.8) and (5.23).

To obtain a MP2/DFT hybrid scheme, we proceed in full analogy to Sections 5.1
and 5.2.1. First, we define the following single-determinant range-separated
two-parameter hybrid (RS2H) approximation

$\displaystyle E^{\text{RS2H},\mu,\lambda}_{0}$ $\displaystyle=$
$\displaystyle\inf_{\Phi\in{\cal
S}^{N}}\Bigl{\\{}\langle\Phi|\hat{T}+\hat{V}_{\text{ne}}+\hat{W}_{\text{ee}}^{\text{lr},\mu}+\lambda\hat{W}_{\text{ee}}^{\text{sr},\mu}|\Phi\rangle+\bar{E}_{\text{Hxc}}^{\text{sr},\mu,\lambda}[\rho_{\Phi}]\Bigl{\\}},$
(5.53)

and used it as a reference for defining a perturbation theory similarly to
Eqs. (5.12) and (5.27). At second order, we obtain

$\displaystyle
E^{\text{RS2H+MP2},\mu,\lambda}_{0}=E^{\text{RS2H},\mu,\lambda}_{0}+E^{\mu,\lambda,\text{MP2}}_{\text{c}},$
(5.54)

where $E^{\mu,\lambda,\text{MP2}}_{\text{c}}$ is the MP2 correlation energy
expression evaluated with RS2H spin orbitals and orbital eigenvalues, and the
two-electron integrals associated with the interaction
$w_{\text{ee}}^{\text{lr},\mu}(r_{12})+\lambda
w_{\text{ee}}^{\text{sr},\mu}(r_{12})$. This RS2H+MP2 scheme thus contains the
exchange-correlation energy contribution

$\displaystyle E^{\text{RS2H+MP2},\mu,\lambda}_{\text{xc}}$ $\displaystyle=$
$\displaystyle
E_{\text{x}}^{\text{lr},\mu,\text{HF}}[\Phi^{\mu,\lambda}]+\lambda
E_{\text{x}}^{\text{sr},\mu,\text{HF}}[\Phi^{\mu,\lambda}]+(1-\lambda)E_{\text{x}}^{\text{sr},\mu}[\rho_{\Phi^{\mu,\lambda}}]$
(5.55)
$\displaystyle+\bar{E}^{\text{sr},\mu,\lambda}_{\text{c}}[\rho_{\Phi^{\mu,\lambda}}]+E_{\text{c}}^{\mu,\lambda,\text{MP2}},$

where $\Phi^{\mu,\lambda}$ is a minimizing single-determinant wave function in
Eq. (5.53).

A good approximation for the $\lambda$-dependence of the complementary
correlation functional $\bar{E}^{\text{sr},\mu,\lambda}_{\text{c}}[\rho]$ is
[224]

$\displaystyle\bar{E}^{\text{sr},\mu,\lambda}_{\text{c}}[\rho]\approx\bar{E}^{\text{sr},\mu}_{\text{c}}[\rho]-\lambda^{2}\bar{E}^{\text{sr},\mu\sqrt{\lambda}}_{\text{c}}[\rho],$
(5.56)

where $\bar{E}_{\text{c}}^{\text{sr},\mu}[\rho]$ is the short-range
correlation functional introduced in Section 5.2.1. In particular, the
$\lambda$-dependence in Eq. (5.56) is correct both in the high-density limit,
for a non-degenerate KS system, and in the low-density limit. Thanks to Eqs.
(5.50) and (5.56), the semilocal density-functional approximations for
$E^{\text{sr},\mu}_{\text{x}}[\rho]$ and
$\bar{E}^{\text{sr},\mu}_{\text{c}}[\rho]$ of Section 5.2.1 can be reused here
without developing new ones. In Ref. [224], the short-range PBE(GWS) exchange
and correlation functionals were used, and the optimal parameters $\mu=0.46$
bohr-1 and $\lambda=0.58$ were found on small sets of atomization energies and
reaction barrier heights, i.e. values similar to the ones usually used
separately in range-separated hybrids and double hybrids.

The RS2H+MP2 scheme improves a bit over the RSH+MP2 scheme of Section 5.2.1,
particularly for interaction energies of hydrogen-bonded systems. Even if the
presence of short-range MP2 correlation deteriorates in principle the
convergence rate with the one-electron basis size, in practice the fraction of
pure short-range MP2 correlation ($\lambda^{2}\approx 0.34$) is small enough
to keep a fast basis convergence. Accuracy can be improved, particularly for
dispersion interactions, by supplanting the MP2 term by coupled-cluster or
random-phase approximations [226]. Like for the approach of Section 5.2.1,
many wave-function methods could be used in the general scheme of Eq. (5.52).

##  6 Semiempirical dispersion corrections and nonlocal van der Waals density
functionals

Among the previously considered exchange-correlation approximations, only the
range-separated double hybrids of Section 5.2, thanks to their long-range
nonlocal correlation component, are capable of fully describing London
dispersion interactions, crucial for describing weakly bonded systems. To
improve the other approximations (semilocal functionals, single-determinant
hybrids, double hybrids without range separation) for weakly bonded systems,
it has been proposed to add to them a semiempirical dispersion correction or a
nonlocal van der Waals density functional. We describe now these approaches.

###  6.1 Semiempirical dispersion corrections

To explicitly account for London dispersion interactions, it has been proposed
in the 2000s to add to the standard approximate functionals a semiempirical
dispersion correction of the form [227, 228, 229, 230]

$\displaystyle
E_{\text{disp}}=-s\sum_{\alpha=1}^{N_{\text{n}}}\sum_{\begin{subarray}{c}\beta=1\\\
\beta>\alpha\end{subarray}}^{N_{\text{n}}}f(R_{\alpha\beta})\frac{C_{6}^{\alpha\beta}}{R_{\alpha\beta}^{6}},$
(6.1)

where $R_{\alpha\beta}$ is the distance between each pair of atoms,
$C_{6}^{\alpha\beta}$ is the London dispersion coefficient between these
atoms, $f(R_{\alpha\beta})$ is a damping function which tends to 1 at large
$R_{\alpha\beta}$ and tends to zero at small $R_{\alpha\beta}$, e.g.

$\displaystyle
f(R_{\alpha\beta})=\frac{1}{1+e^{-d(R_{\alpha\beta}/R_{\alpha\beta}^{\text{vdW}}-1)}},$
(6.2)

with the sum of tabulated atomic van der Waals radii
$R_{\alpha\beta}^{\text{vdW}}=R_{\alpha}^{\text{vdW}}+R_{\beta}^{\text{vdW}}$
and a constant $d$, and $s$ is a scaling parameter that can be adjusted for
each approximate functional. The dispersion coefficient $C_{6}^{\alpha\beta}$
for any pair of atoms is empirically calculated from tabulated same-atom
dispersion coefficients $C_{6}^{\alpha\alpha}$ and/or atomic polarizabilities.
This approach was named “DFT-D” by Grimme [229].

The last version of DFT-D (referred to as DFT-D3) also includes
$C_{8}^{\alpha\beta}$ two-body terms and $C_{9}^{\alpha\beta\gamma}$ three-
body terms [231]. There have also been various proposals to make the
determination of dispersion coefficients less empirical, such as the scheme of
Becke and Johnson [232] based on the exchange-hole dipole moment, the scheme
of Tkatchenko and Scheffler [233] based on a Hirshfeld atomic partitioning, or
the scheme of Sato and Nakai [234] based on the local-response approximation
[235].

The “DFT-D” approach provides a big and inexpensive improvement for the
description of weakly bonded systems. One limitation is that the semiempirical
dispersion correction, being just a force field in its simplest variant,
affects only the molecular geometry of the system but not directly its
electronic structure. Some of the most used DFT-D functionals are:

  * •

The PBE-D exchange-correlation functional [230], based on the PBE functional
with a scaling parameter $s=0.75$;

  * •

The B97-D exchange-correlation functional [230], based on the B97-GGA
functional with a scaling parameter $s=1.25$ and reoptimized linear
coefficients in Eqs. (3.38), (3.42), and (3.43) in the presence of the
semiempirical dispersion correction;

  * •

The B3LYP-D exchange-correlation functional [230], based on the B3LYP hybrid
functional with a scaling parameter $s=1.05$;

  * •

The $\omega$B97X-D exchange-correlation functional [236], based on the
$\omega$B97X range-separated hybrid functional with a scaling parameter $s=1$,
a modified damping function, and reoptimized parameters in $\omega$B97X in the
presence of the semiempirical dispersion correction.

The semiempirical dispersion correction can also be added to double-hybrid
approximations. For example, B2PLYP-D [237] is based on the B2PLYP double
hybrid with a scaling parameter $s=0.55$. The scaling parameter is small since
the fraction of MP2 correlation in B2PLYP already partially takes into account
dispersion interactions. It has also been proposed to add a semiempirical
dispersion correction to the SCS version of the double hybrids [Eq.
(LABEL:2DHSCS)], resulting in a family of dispersion-corrected spin-component-
scaled double-hybrid (DSD) approximations [172, 173, 238]. An example of
double hybrid is this latter family is DSD-BLYP [172] which uses the B88
exchange functional and the LYP correlation functional.

###  6.2 Nonlocal van der Waals density functionals

Another approach to describe dispersion interactions is to add to the standard
approximate functionals a so-called nonlocal van der Waals density functional
of the form [239, 240, 241, 242, 243]

$\displaystyle
E^{\text{nl}}_{\text{c}}[\rho]=\frac{1}{2}\int_{\mathbb{R}^{3}\times\mathbb{R}^{3}}\rho(\mathbf{r}_{1})\rho(\mathbf{r}_{2})\phi(\mathbf{r}_{1},\mathbf{r}_{2})\text{d}\mathbf{r}_{1}\text{d}\mathbf{r}_{2},$
(6.3)

where $\phi(\mathbf{r}_{1},\mathbf{r}_{2})$ is a correlation kernel. Two main
families of such nonlocal correlation functionals exist: the “van der Waals
density functionals” (vdW-DF) [239, 242] and the Vydrov-Van Voorhis (VV)
functionals [240, 241, 243].

We will only describe the last version of the VV functionals, i.e. the VV10
nonlocal correlation functional [243]. In this functional, the correlation
kernel is taken as

$\displaystyle\phi^{\text{VV10}}(\mathbf{r}_{1},\mathbf{r}_{2})=-\frac{3}{2g(\mathbf{r}_{1},r_{12})g(\mathbf{r}_{2},r_{12})(g(\mathbf{r}_{1},r_{12})+g(\mathbf{r}_{2},r_{12}))}+\beta\delta(\mathbf{r}_{1}-\mathbf{r}_{2}),$
(6.4)

where $r_{12}=|\mathbf{r}_{2}-\mathbf{r}_{1}|$ is the interelectronic
distance, $\beta$ is a constant determining the local (delta-distribution)
part of the kernel, and the function $g$ is defined as

$\displaystyle
g(\mathbf{r},r_{12})=\omega_{0}(\mathbf{r})r_{12}^{2}+\kappa(\mathbf{r}).$
(6.5)

In Eq. (6.5),
$\omega_{0}(\mathbf{r})=\sqrt{\omega_{\text{g}}(\mathbf{r})^{2}+\frac{\omega_{\text{p}}(\mathbf{r})^{2}}{3}}$
involves the square of the local plasma frequency
$\omega_{\text{p}}(\mathbf{r})^{2}=4\pi\rho(\mathbf{r})$ and the square of the
local band gap
$\omega_{\text{g}}(\mathbf{r})^{2}=C|\nabla\rho(\mathbf{r})|^{4}/\rho(\mathbf{r})^{4}$
where $C$ is an adjustable parameter controlling the large-$r_{12}$ asymptotic
dispersion coefficients, and
$\kappa(\mathbf{r})=b\;k_{\text{F}}(\mathbf{r})^{2}/\omega_{\text{p}}(\mathbf{r})$
where $k_{\text{F}}(\mathbf{r})=(3\pi^{2}\rho(\mathbf{r}))^{1/3}$ is the local
Fermi wave vector and $b$ is an adjustable parameter controlling the short-
range damping of the large-$r_{12}$ asymptote. As expected for dispersion
interactions, in the large-$r_{12}$ limit,
$\phi^{\text{VV10}}(\mathbf{r}_{1},\mathbf{r}_{2})$ behaves as $1/r_{12}^{6}$

$\displaystyle\phi^{\text{VV10}}(\mathbf{r}_{1},\mathbf{r}_{2})\underset{r_{12}\to\infty}{\sim}-\frac{3}{2\omega_{0}(\mathbf{r}_{1})\omega_{0}(\mathbf{r}_{2})(\omega_{0}(\mathbf{r}_{1})+\omega_{0}(\mathbf{r}_{2}))r_{12}^{6}}.$
(6.6)

The constant $\beta=(3/b^{2})^{3/4}/16$ is chosen to make
$E^{\text{nl}}_{\text{c}}[\rho]$ vanish in the uniform density limit, leaving
thus this limit unchanged when $E^{\text{nl}}_{\text{c}}[\rho]$ is added to
another density functional. The adjustable parameters $C\approx 0.009$ and
$b\approx 6$ are found by optimization on $C_{6}$ dispersion coefficients and
on weak intermolecular interaction energies, respectively, the precise values
depending on which exchange-correlation functional the VV10 correction is
added to.

Non local van der Waals density functionals are necessarily more
computationally expensive than semiempirical dispersion corrections. However,
they have the advantage of being less empirical and, since they are
functionals of the density, of impacting directly on the electronic structure
of the system. The VV10 nonlocal functional has been incorporated in a number
of recently developed exchange-correlation functionals, for example:

  * •

The $\omega$B97X-V exchange-correlation functional [244], based on the
$\omega$B97X range-separated hybrid [Eq. (4.23)] with reoptimized linear
coefficients in Eq. (3.38) with polynomial degree $m=2$ and in Eqs. (3.42) and
(3.43) with polynomial degree $m=1$, as well as reoptimized VV10 parameters
$C=0.01$ and $b=6.0$;

  * •

The $\omega$B97M-V exchange-correlation functional [245], based on the
$\omega$B97X range-separated hybrid [Eq. (4.23)] but with more general and
combinatorially optimized meta-GGA exchange and correlation enhancement
factors and the same VV10 parameters $C=0.01$ and $b=6.0$ as in
$\omega$B97X-V.

##  7 Orbital-dependent exchange-correlation density functionals

We discuss here some exchange-correlation density functionals explicitly
depending on the KS orbitals. Since the KS orbitals are themselves functionals
of the density, these exchange-correlation expressions are thus implicit
functionals of the density (for notational simplicity, this dependence on the
density of the orbitals and other intermediate quantities will not be
explicitly indicated). In fact, the single-determinant and multideterminant
hybrid approximations of Sections 4 and 5 already belong to this family, with
the caveat that the orbitals are obtained with a nonlocal potential. In this
Section, we are concerned with orbital-dependent exchange-correlation energy
functionals with orbitals obtained with a local potential, i.e. staying within
the KS scheme999The frontier between the various single-determinant and
multideterminant hybrids of Sections 4 and 5 and the orbital-dependent
functionals of the present Section is however thin. For example, it is
possible to optimize the orbitals using a local potential in hybrids or range-
separated hybrids [246, 247, 248], and in double hybrids or range-separated
double hybrids [249, 250].. These approximations tend to be more
computationally involved than the approximations previously seen and have thus
been much less used so far.

###  7.1 Exact exchange

The exact exchange (EXX) energy functional [Eq. (1.21)] can be expressed in
terms of the KS orbitals

$E_{\text{x}}[\rho]=-\frac{1}{2}\sum_{\sigma\in\\{\uparrow,\downarrow\\}}\sum_{i=1}^{N_{\sigma}}\sum_{j=1}^{N_{\sigma}}\int_{\mathbb{R}^{3}\times\mathbb{R}^{3}}\frac{\varphi_{i\sigma}^{*}(\mathbf{r}_{1})\varphi_{j\sigma}(\mathbf{r}_{1})\varphi_{j\sigma}^{*}(\mathbf{r}_{2})\varphi_{i\sigma}(\mathbf{r}_{2})}{|\mathbf{r}_{1}-\mathbf{r}_{2}|}\text{d}\mathbf{r}_{1}\text{d}\mathbf{r}_{2},$
(7.1)

and has exactly the same form as the HF exchange [Eq. (4.2)], but the orbitals
used in both expressions are in general different.

Since the exact exchange energy in Eq. (7.1) is not an explicit functional of
the density, the corresponding exchange potential
$v_{\text{x}}(\mathbf{r})=\delta E_{\text{x}}[\rho]/\delta\rho(\mathbf{r})$
cannot be calculated directly. We can however find an workable equation for
$v_{\text{x}}(\mathbf{r})$ by first considering the functional derivative of
$E_{\text{x}}[\rho]$ with respect to the KS potential
$v_{\text{s}}(\mathbf{r})$ and then applying the chain rule

$\frac{\delta E_{\text{x}}[\rho]}{\delta
v_{\text{s}}(\mathbf{r})}=\int_{\mathbb{R}^{3}}\frac{\delta
E_{\text{x}}[\rho]}{\delta\rho(\mathbf{r}^{\prime})}\frac{\delta\rho(\mathbf{r}^{\prime})}{\delta
v_{\text{s}}(\mathbf{r})}\text{d}\mathbf{r}^{\prime}.$ (7.2)

Introducing the non-interacting KS static linear-response function
$\chi_{0}(\mathbf{r}^{\prime},\mathbf{r})=\delta\rho(\mathbf{r}^{\prime})/\delta
v_{\text{s}}(\mathbf{r})$, we can rewrite Eq. (7.2) as

$\int_{\mathbb{R}^{3}}v_{\text{x}}(\mathbf{r}^{\prime})\chi_{0}(\mathbf{r}^{\prime},\mathbf{r})\text{d}\mathbf{r}^{\prime}=\frac{\delta
E_{\text{x}}[\rho]}{\delta v_{\text{s}}(\mathbf{r})},$ (7.3)

which is known as the optimized-effective-potential (OEP) equation for the
exact-exchange potential [251, 46, 252].

Using first-order perturbation theory on the KS system, explicit expressions
in terms of the orbitals can be derived for
$\chi_{0}(\mathbf{r}^{\prime},\mathbf{r})$ and $\delta
E_{\text{x}}[\rho]/\delta v_{\text{s}}(\mathbf{r})$. The expression of
$\chi_{0}(\mathbf{r}^{\prime},\mathbf{r})$ is

$\chi_{0}(\mathbf{r}^{\prime},\mathbf{r})=-\sum_{\sigma\in\\{\uparrow,\downarrow\\}}\sum_{i=1}^{N_{\sigma}}\sum_{a\geq
N_{\sigma}+1}\frac{\varphi_{i\sigma}^{*}(\mathbf{r}^{\prime})\varphi_{a\sigma}^{*}(\mathbf{r})\varphi_{i\sigma}(\mathbf{r})\varphi_{a\sigma}(\mathbf{r}^{\prime})}{\varepsilon_{a\sigma}-\varepsilon_{i\sigma}}+\text{
c.c. },$ (7.4)

where $i$ and $a$ run over occupied and virtual spatial orbitals,
respectively, and c.c. stands for the complex conjugate. The expression of
$\delta E_{\text{x}}[\rho]/\delta v_{\text{s}}(\mathbf{r})$ is

$\frac{\delta E_{\text{x}}[\rho]}{\delta
v_{\text{s}}(\mathbf{r})}=\sum_{\sigma\in\\{\uparrow,\downarrow\\}}\sum_{i=1}^{N_{\sigma}}\sum_{j=1}^{N_{\sigma}}\sum_{a\geq
N_{\sigma}+1}\langle\varphi_{a\sigma}\varphi_{j\sigma}|\varphi_{j\sigma}\varphi_{i\sigma}\rangle\frac{\varphi_{a\sigma}(\mathbf{r})\varphi_{i\sigma}^{*}(\mathbf{r})}{\varepsilon_{a\sigma}-\varepsilon_{i\sigma}}+\text{
c.c. },$ (7.5)

where
$\langle\varphi_{a\sigma}\varphi_{j\sigma}|\varphi_{j\sigma}\varphi_{i\sigma}\rangle$
are two-electron integrals over KS spatial orbitals

$\displaystyle\langle\varphi_{a\sigma}\varphi_{j\sigma}|\varphi_{j\sigma}\varphi_{i\sigma}\rangle=\int_{\mathbb{R}^{3}\times\mathbb{R}^{3}}\frac{\varphi_{a\sigma}^{*}(\mathbf{r}_{1})\varphi_{j\sigma}^{*}(\mathbf{r}_{2})\varphi_{j\sigma}(\mathbf{r}_{1})\varphi_{i\sigma}(\mathbf{r}_{2})}{|\mathbf{r}_{1}-\mathbf{r}_{2}|}\text{d}\mathbf{r}_{1}\text{d}\mathbf{r}_{2}.$
(7.6)

Applying this OEP method with the EXX energy (and no correlation energy
functional) is an old idea [253, 251], but reasonable efficient calculations
for molecules have been possible only relatively recently [254, 255]. The EXX
occupied orbitals turn out to be very similar to the HF occupied orbitals, and
thus the EXX ground-state properties are also similar to the HF ones. However,
the EXX virtual orbitals (which see a $-1/r$ asymptotic potential for a
neutral system) tend to be much less diffuse than the HF virtual orbitals
(which see an exponentially decaying potential for a neutral system), and may
be more adapted for calculating excited-state properties.

###  7.2 Second-order Görling-Levy perturbation theory

In 1993, Görling and Levy [45, 46] developed a perturbation theory in terms of
the coupling constant $\lambda$ of the adiabatic connection (Section 2.2)
which provides an explicit orbital-dependent second-order approximation for
the correlation energy functional. The Hamiltonian along the adiabatic
connection [Eq. (2.22)] can be written as

$\displaystyle\hat{H}^{\lambda}$ $\displaystyle=$
$\displaystyle\hat{T}+\lambda\hat{W}_{\text{ee}}+\hat{V}^{\lambda}$ (7.7)
$\displaystyle=$
$\displaystyle\hat{H}_{\text{s}}+\lambda(\hat{W}_{\text{ee}}-\hat{V}_{\text{H}\text{x}})-\hat{V}_{\text{c}}^{\lambda},$

where $\hat{H}_{\text{s}}=\hat{H}^{\lambda=0}=\hat{T}+\hat{V}_{\text{s}}$ is
the KS non-interacting reference Hamiltonian (which will be assumed to have a
nondegenerate ground state). Equation (7.7) was obtained by decomposing the
potential operator keeping the density constant as
$\hat{V}^{\lambda}=\hat{V}_{\text{s}}-\lambda\hat{V}_{\text{H}\text{x}}-\hat{V}_{\text{c}}^{\lambda}$
where $\hat{V}_{\text{s}}=\hat{V}^{\lambda=0}$ is the KS potential operator,
$\lambda\hat{V}_{\text{H}\text{x}}$ is the Hartree-exchange potential operator
which is linear in $\lambda$, and $\hat{V}_{\text{c}}^{\lambda}$ is the
correlation potential which starts at second order in $\lambda$, i.e.
$\hat{V}_{\text{c}}^{\lambda}=\lambda^{2}\hat{V}_{\text{c}}^{(2)}+\cdots$.
Using a complete set of orthonormal eigenfunctions $\Phi_{n}$ and eigenvalues
${\cal E}_{n}$ of the KS Hamiltonian,
$\hat{H}_{\text{s}}|\Phi_{n}\rangle={\cal E}_{n}|\Phi_{n}\rangle$, the
normalized ground-state wave function of the Hamiltonian $\hat{H}^{\lambda}$
can be expanded as $\Psi^{\lambda}=\Phi+\lambda\Psi^{(1)}+\cdots$ where
$\Phi=\Phi_{0}$ is the ground-state KS single-determinant wave function and
$\Psi^{(1)}$ is its first-order correction given by

$\displaystyle|\Psi^{(1)}\rangle=-\sum_{n\not=0}\frac{\langle\Phi_{n}|\hat{W}_{\text{ee}}-\hat{V}_{\text{H}\text{x}}|\Phi\rangle}{{\cal
E}_{n}-{\cal E}_{0}}|\Phi_{n}\rangle.$ (7.8)

Using the expression in Eq. (2.27), the correlation energy functional can also
be expanded in powers of $\lambda$

$\displaystyle E_{\text{c}}^{\lambda}[\rho]$ $\displaystyle=$
$\displaystyle\langle\Psi^{\lambda}|\hat{T}+\lambda\hat{W}_{\text{ee}}|\Psi^{\lambda}\rangle-\langle\Phi|\hat{T}+\lambda\hat{W}_{\text{ee}}|\Phi\rangle.$
(7.9) $\displaystyle=$ $\displaystyle E_{\text{c}}^{(0)}+\lambda
E_{\text{c}}^{(1)}+\lambda^{2}E_{\text{c}}^{(2)}+\cdots.$

Since $\Psi^{\lambda=0}=\Phi$, the zeroth-order term vanishes:
$E_{\text{c}}^{(0)}=0$. Using the expression of the first-order derivative of
$E_{\text{c}}^{\lambda}$ with respect to $\lambda$ in Eq. (2.28), i.e.
$\partial
E_{\text{c}}^{\lambda}/\partial\lambda=\langle\Psi^{\lambda}|\hat{W}_{\text{ee}}|\Psi^{\lambda}\rangle-\langle\Phi|\hat{W}_{\text{ee}}|\Phi\rangle$,
we find that the first-order term vanishes as well: $E_{\text{c}}^{(1)}=0$.
The second-order term corresponds to the second-order Görling-Levy (GL2)
correlation energy and is given by

$E_{\text{c}}^{\text{GL2}}[\rho]=E_{\text{c}}^{(2)}=\langle\Phi|\hat{W}_{\text{ee}}|\Psi^{(1)}\rangle=\langle\Phi|\hat{W}_{\text{ee}}-{V}_{\text{H}\text{x}}|\Psi^{(1)}\rangle,$
(7.10)

where the second equality comes the fact that
$\langle\Phi|\hat{V}_{\text{H}\text{x}}|\Psi^{(1)}\rangle=0$ since it is the
derivative with respect to $\lambda$ at $\lambda=0$ of
$\langle\Psi^{\lambda}|\hat{V}_{\text{H}\text{x}}|\Psi^{\lambda}\rangle=\int_{\mathbb{R}^{3}}v_{\text{H}\text{x}}(\mathbf{r})\rho(\mathbf{r})\text{d}\mathbf{r}$
which does not depend on $\lambda$ by virtue of the fact the density
$\rho(\mathbf{r})$ is constant along the adiabatic connection. Using the last
expression in Eq. (7.10) allows one to express the GL2 correlation energy as

$E_{\text{c}}^{\text{GL2}}[\rho]=-\sum_{n\not=0}\frac{|\langle\Phi|\hat{W}_{\text{ee}}-\hat{V}_{\text{H}\text{x}}|\Phi_{n}\rangle|^{2}}{{\cal
E}_{n}-{\cal E}_{0}}.$ (7.11)

It is instructive to decompose the GL2 correlation energy as

$E_{\text{c}}^{\text{GL2}}[\rho]=E_{\text{c}}^{\text{MP2}}+E_{\text{c}}^{\text{S}},$
(7.12)

where $E_{\text{c}}^{\text{MP2}}$ is a MP2-like correlation energy evaluated
with KS spin orbitals

$E_{\text{c}}^{\text{MP2}}=-\frac{1}{4}\sum_{i=1}^{N}\sum_{j=1}^{N}\sum_{a\geq
N+1}\sum_{b\geq
N+1}\frac{|\langle\phi_{i}\phi_{j}||\phi_{a}\phi_{b}\rangle|^{2}}{\varepsilon_{a}+\varepsilon_{b}-\varepsilon_{i}-\varepsilon_{j}},$
(7.13)

and $E_{\text{c}}^{\text{S}}$ is the contribution coming from the single
excitations (which does not vanish here, contrary to HF-based MP2 perturbation
theory)

$E_{\text{c}}^{\text{S}}=-\sum_{i=1}^{N}\sum_{a\geq
N+1}\frac{|\langle\phi_{i}|\hat{V}_{\text{x}}^{\text{HF}}-\hat{V}_{\text{x}}|\phi_{a}\rangle|^{2}}{\varepsilon_{a}-\varepsilon_{i}},$
(7.14)

involving the difference between the integrals over the nonlocal HF exchange
potential
$\langle\phi_{i}|\hat{V}_{\text{x}}^{\text{HF}}|\phi_{a}\rangle=-\sum_{j=1}^{N}\langle\phi_{i}\phi_{j}|\phi_{j}\phi_{a}\rangle$
and over the local KS exchange potential
$\langle\phi_{i}|\hat{V}_{\text{x}}|\phi_{a}\rangle=\int_{\mathbb{R}^{3}\times\\{\uparrow,\downarrow\\}}\phi_{i}^{*}(\mathbf{x})v_{\text{x}}(\mathbf{r})\phi_{a}(\mathbf{x})\text{d}\mathbf{x}$.

Calculations of the GL2 correlation energy using either a non-self-consistent
post-EXX implementation or a more complicated OEP self-consistent procedure
have been tested (see, e.g., Refs. [256, 257, 258]) but the results are often
disappointing. It is preferable to go beyond second order with random-phase
approximations in the adiabatic-connection fluctuation-dissipation approach.

###  7.3 Adiabatic-connection fluctuation-dissipation approach

####  7.3.1 Exact adiabatic-connection fluctuation-dissipation expression

Using the adiabatic-connection formula of Eq. (2.29), the correlation energy
functional can be written as

$\displaystyle E_{\text{c}}[\rho]$ $\displaystyle=$
$\displaystyle\int_{0}^{1}\text{d}\lambda\;\langle\Psi^{\lambda}|\hat{W}_{\text{ee}}|\Psi^{\lambda}\rangle-\langle\Phi|\hat{W}_{\text{ee}}|\Phi\rangle=\frac{1}{2}\int_{0}^{1}\text{d}\lambda\int_{\mathbb{R}^{3}\times\mathbb{R}^{3}}\frac{\rho_{2,\text{c}}^{\lambda}(\mathbf{r}_{1},\mathbf{r}_{2})}{|\mathbf{r}_{1}-\mathbf{r}_{2}|}\text{d}\mathbf{r}_{1}\text{d}\mathbf{r}_{2},$
(7.15)

where
$\rho_{2,\text{c}}^{\lambda}(\mathbf{r}_{1},\mathbf{r}_{2})=\rho_{2}^{\lambda}(\mathbf{r}_{1},\mathbf{r}_{2})-\rho_{2,\text{KS}}(\mathbf{r}_{1},\mathbf{r}_{2})$
is the correlation part of the pair density along the adiabatic connection.
The pair density $\rho_{2}^{\lambda}(\mathbf{r}_{1},\mathbf{r}_{2})$ can be
expressed with the pair-density operator
$\hat{\rho}_{2}(\mathbf{r}_{1},\mathbf{r}_{2})=\hat{\rho}(\mathbf{r}_{1})\hat{\rho}(\mathbf{r}_{2})-\delta(\mathbf{r}_{1}-\mathbf{r}_{2})\hat{\rho}(\mathbf{r}_{1})$
where $\hat{\rho}(\mathbf{r})$ is the density operator

$\displaystyle\rho_{2}^{\lambda}(\mathbf{r}_{1},\mathbf{r}_{2})=\langle\Psi^{\lambda}|\hat{\rho}_{2}(\mathbf{r}_{1},\mathbf{r}_{2})|\Psi^{\lambda}\rangle=\langle\Psi^{\lambda}|\hat{\rho}(\mathbf{r}_{1})\hat{\rho}(\mathbf{r}_{2})|\Psi^{\lambda}\rangle-\delta(\mathbf{r}_{1}-\mathbf{r}_{2})\langle\Psi^{\lambda}|\hat{\rho}(\mathbf{r}_{1})|\Psi^{\lambda}\rangle,$
(7.16)

and the KS pair density $\rho_{2,\text{KS}}(\mathbf{r}_{1},\mathbf{r}_{2})$
simply corresponds to the case $\lambda=0$

$\displaystyle\rho_{2,\text{KS}}(\mathbf{r}_{1},\mathbf{r}_{2})=\rho_{2}^{\lambda=0}(\mathbf{r}_{1},\mathbf{r}_{2})=\langle\Phi|\hat{\rho}(\mathbf{r}_{1})\hat{\rho}(\mathbf{r}_{2})|\Phi\rangle-\delta(\mathbf{r}_{1}-\mathbf{r}_{2})\langle\Phi|\hat{\rho}(\mathbf{r}_{1})|\Phi\rangle.$
(7.17)

Since the density does not change with $\lambda$, i.e.
$\langle\Psi^{\lambda}|\hat{\rho}(\mathbf{r})|\Psi^{\lambda}\rangle=\langle\Phi|\hat{\rho}(\mathbf{r})|\Phi\rangle=\rho(\mathbf{r})$,
the correlation pair density needed in Eq. (7.15) can thus be expressed as

$\displaystyle\rho_{2,\text{c}}^{\lambda}(\mathbf{r}_{1},\mathbf{r}_{2})=\langle\Psi^{\lambda}|\hat{\rho}(\mathbf{r}_{1})\hat{\rho}(\mathbf{r}_{2})|\Psi^{\lambda}\rangle-\langle\Phi|\hat{\rho}(\mathbf{r}_{1})\hat{\rho}(\mathbf{r}_{2})|\Phi\rangle.$
(7.18)

We would like to calculate
$\rho_{2,\text{c}}^{\lambda}(\mathbf{r}_{1},\mathbf{r}_{2})$ without having to
calculate the complicated many-body wave function $\Psi^{\lambda}$. For this,
we consider the retarded dynamic linear-response function along the adiabatic
connection in frequency space (or the so-called Lehmann representation)

$\displaystyle\chi_{\lambda}(\mathbf{r}_{1},\mathbf{r}_{2};\omega)=\sum_{n\not=0}\frac{\langle\Psi^{\lambda}|\hat{\rho}(\mathbf{r}_{1})|\Psi_{n}^{\lambda}\rangle\langle\Psi_{n}^{\lambda}|\hat{\rho}(\mathbf{r}_{2})|\Psi^{\lambda}\rangle}{\omega-\omega_{n}^{\lambda}+i0^{+}}-\frac{\langle\Psi^{\lambda}|\hat{\rho}(\mathbf{r}_{2})|\Psi_{n}^{\lambda}\rangle\langle\Psi_{n}^{\lambda}|\hat{\rho}(\mathbf{r}_{1})|\Psi^{\lambda}\rangle}{\omega+\omega_{n}^{\lambda}+i0^{+}},$
(7.19)

where the sums are over all eigenstates $\Psi_{n}^{\lambda}$ of the
Hamiltonian $\hat{H}^{\lambda}$, i.e.
$\hat{H}^{\lambda}|\Psi_{n}^{\lambda}\rangle=E_{n}^{\lambda}|\Psi_{n}^{\lambda}\rangle$,
except the ground state $\Psi^{\lambda}=\Psi^{\lambda}_{0}$, and
$\omega_{n}^{\lambda}=E_{n}^{\lambda}-E_{0}^{\lambda}$ are the corresponding
excitation energies. By contour integrating
$\chi_{\lambda}(\mathbf{r}_{1},\mathbf{r}_{2},\omega)$ around the right half
$\omega$-complex plane, we arrive at the (zero-temperature) fluctuation-
dissipation theorem

$\displaystyle
n_{2,\text{c}}^{\lambda}(\mathbf{r}_{1},\mathbf{r}_{2})=-\int_{-\infty}^{+\infty}\frac{\text{d}\omega}{2\pi}\;[\chi_{\lambda}(\mathbf{r}_{1},\mathbf{r}_{2},i\omega)-\chi_{0}(\mathbf{r}_{1},\mathbf{r}_{2},i\omega)],$
(7.20)

which relates ground-state correlations in the time-independent system
$\rho_{2,\text{c}}^{\lambda}(\mathbf{r}_{1},\mathbf{r}_{2})$ to the linear
response of the system due to a time-dependent external perturbation
$\chi_{\lambda}(\mathbf{r}_{1},\mathbf{r}_{2},\omega)$.

Combining Eqs. (7.15) and (7.20), we finally arrive the exact adiabatic-
connection fluctuation-dissipation (ACFD) formula for the correlation energy
[38, 40] (see, also, Ref. [41])

$\displaystyle
E_{\text{c}}[\rho]=-\frac{1}{2}\int_{0}^{1}\text{d}\lambda\int_{-\infty}^{+\infty}\frac{\text{d}\omega}{2\pi}\int_{\mathbb{R}^{3}\times\mathbb{R}^{3}}\frac{\chi_{\lambda}(\mathbf{r}_{1},\mathbf{r}_{2};i\omega)-\chi_{0}(\mathbf{r}_{1},\mathbf{r}_{2};i\omega)}{|\mathbf{r}_{1}-\mathbf{r}_{2}|}\text{d}\mathbf{r}_{1}\text{d}\mathbf{r}_{2}.$
(7.21)

The usefulness of the ACFD formula is due to the fact there are practical ways
of directly calculating $\chi_{\lambda}(\mathbf{r}_{1},\mathbf{r}_{2};\omega)$
without having to calculate the many-body wave function $\Psi^{\lambda}$. In
linear-response time-dependent density-functional theory (TDDFT), one can find
a Dyson-like equation for
$\chi_{\lambda}(\mathbf{r}_{1},\mathbf{r}_{2};\omega)$

$\displaystyle\chi_{\lambda}(\mathbf{r}_{1},\mathbf{r}_{2};\omega)=\chi_{0}(\mathbf{r}_{1},\mathbf{r}_{2};\omega)+\int_{\mathbb{R}^{3}\times\mathbb{R}^{3}}\chi_{0}(\mathbf{r}_{1},\mathbf{r}_{3};\omega)f_{\text{Hxc}}^{\lambda}(\mathbf{r}_{3},\mathbf{r}_{4};\omega)\chi_{\lambda}(\mathbf{r}_{4},\mathbf{r}_{2};\omega)\text{d}\mathbf{r}_{3}\text{d}\mathbf{r}_{4},$
(7.22)

where $f_{\text{Hxc}}^{\lambda}(\mathbf{r}_{3},\mathbf{r}_{4};\omega)$ is the
Hartree-exchange-correlation kernel associated to the Hamiltonian
${H}^{\lambda}$. Here, Eq. (7.22) will be considered as the definition for
$f_{\text{Hxc}}^{\lambda}$. In principle, the exact correlation energy can be
obtained with Eqs. (7.21) and (7.22). In practice, however, we need to use an
approximation for $f_{\text{Hxc}}^{\lambda}$.

####  7.3.2 Random-phase approximations

In the direct random-phase approximation (dRPA, also just referred to as RPA,
or sometimes as time-dependent Hartree), only the Hartree part of the kernel,
which is linear in $\lambda$ and independent from $\omega$, is retained [38,
259]

$\displaystyle
f_{\text{Hxc}}^{\text{dRPA},\lambda}(\mathbf{r}_{1},\mathbf{r}_{2};\omega)=f_{\text{H}}^{\lambda}(\mathbf{r}_{1},\mathbf{r}_{2})=\lambda
w_{\text{ee}}(\mathbf{r}_{1},\mathbf{r}_{2}),$ (7.23)

where
$w_{\text{ee}}(\mathbf{r}_{1},\mathbf{r}_{2})=1/|\mathbf{r}_{1}-\mathbf{r}_{2}|$
is the Coulomb interaction, and the corresponding dRPA linear-response
function then satisfies the equation

$\displaystyle\chi_{\lambda}^{\text{dRPA}}(\mathbf{r}_{1},\mathbf{r}_{2};\omega)=\chi_{0}(\mathbf{r}_{1},\mathbf{r}_{2};\omega)+\lambda\int_{\mathbb{R}^{3}\times\mathbb{R}^{3}}\chi_{0}(\mathbf{r}_{1},\mathbf{r}_{3};\omega)w_{\text{ee}}(\mathbf{r}_{3},\mathbf{r}_{4})\chi_{\lambda}^{\text{dRPA}}(\mathbf{r}_{4},\mathbf{r}_{2};\omega)\text{d}\mathbf{r}_{3}\text{d}\mathbf{r}_{4}.$
(7.24)

The physical contents of this approximation can be seen by iterating Eq.
(7.24) which generates an infinite series

$\displaystyle\chi_{\lambda}^{\text{dRPA}}(\mathbf{r}_{1},\mathbf{r}_{2};\omega)=\chi_{0}(\mathbf{r}_{1},\mathbf{r}_{2};\omega)+\lambda\int_{\mathbb{R}^{3}\times\mathbb{R}^{3}}\chi_{0}(\mathbf{r}_{1},\mathbf{r}_{3};\omega)w_{\text{ee}}(\mathbf{r}_{3},\mathbf{r}_{4})\chi_{0}(\mathbf{r}_{4},\mathbf{r}_{2};\omega)\text{d}\mathbf{r}_{3}\text{d}\mathbf{r}_{4}\;\;\;\;\;\;\;\;\;$
$\displaystyle+\lambda^{2}\int_{\mathbb{R}^{3}\times\mathbb{R}^{3}\times\mathbb{R}^{3}\times\mathbb{R}^{3}}\chi_{0}(\mathbf{r}_{1},\mathbf{r}_{3};\omega)w_{\text{ee}}(\mathbf{r}_{3},\mathbf{r}_{4})\chi_{0}(\mathbf{r}_{4},\mathbf{r}_{5};\omega)w_{\text{ee}}(\mathbf{r}_{5},\mathbf{r}_{6})\chi_{0}(\mathbf{r}_{6},\mathbf{r}_{2};\omega)\text{d}\mathbf{r}_{3}\text{d}\mathbf{r}_{4}\text{d}\mathbf{r}_{5}\text{d}\mathbf{r}_{6}+\cdots,$

which, after plugging it into Eq. (7.21), leads to the dRPA correlation energy
as the following perturbation expansion101010Using the operator viewpoint, the
series in Eq. (7.26) can be formally summed in the form
$E_{\text{c}}^{\text{dRPA}}[\rho]=1/(4\pi)\int_{-\infty}^{+\infty}\text{d}\omega\;\text{Tr}[\ln(1-\chi_{0}(i\omega)w_{\text{ee}})+\chi_{0}(i\omega)w_{\text{ee}}]$
(see, e.g., Ref. [260]).

$\displaystyle
E_{\text{c}}^{\text{dRPA}}[\rho]=-\frac{1}{2}\int_{0}^{1}\text{d}\lambda\int_{-\infty}^{+\infty}\frac{\text{d}\omega}{2\pi}\Biggl{[}\lambda\int_{\mathbb{R}^{3}\times\mathbb{R}^{3}\times\mathbb{R}^{3}\times\mathbb{R}^{3}}\frac{\chi_{0}(\mathbf{r}_{1},\mathbf{r}_{3};i\omega)\chi_{0}(\mathbf{r}_{4},\mathbf{r}_{2};i\omega)}{|\mathbf{r}_{1}-\mathbf{r}_{2}|\;|\mathbf{r}_{3}-\mathbf{r}_{4}|}\text{d}\mathbf{r}_{1}\text{d}\mathbf{r}_{2}\text{d}\mathbf{r}_{3}\text{d}\mathbf{r}_{4}\;\;\;\;\;\;\;\;\;$
$\displaystyle+\lambda^{2}\int_{\mathbb{R}^{3}\times\mathbb{R}^{3}\times\mathbb{R}^{3}\times\mathbb{R}^{3}\times\mathbb{R}^{3}\times\mathbb{R}^{3}}\frac{\chi_{0}(\mathbf{r}_{1},\mathbf{r}_{3};i\omega)\chi_{0}(\mathbf{r}_{4},\mathbf{r}_{5};i\omega)\chi_{0}(\mathbf{r}_{6},\mathbf{r}_{2};i\omega)}{|\mathbf{r}_{1}-\mathbf{r}_{2}|\;|\mathbf{r}_{3}-\mathbf{r}_{4}|\;|\mathbf{r}_{5}-\mathbf{r}_{6}|}\text{d}\mathbf{r}_{1}\text{d}\mathbf{r}_{2}\text{d}\mathbf{r}_{3}\text{d}\mathbf{r}_{4}\text{d}\mathbf{r}_{5}\text{d}\mathbf{r}_{6}+\cdots\Biggl{]}.\;\;\;$
(7.26)

Using now the Lehmann representation [Eq. (7.19)] of the KS dynamic linear-
response function in terms of the KS orbitals and their energies

$\displaystyle\chi_{0}(\mathbf{r}_{1},\mathbf{r}_{2};\omega)$ $\displaystyle=$
$\displaystyle\sum_{\sigma\in\\{\uparrow,\downarrow\\}}\sum_{i=1}^{N_{\sigma}}\sum_{a\geq
N_{\sigma}+1}\Biggl{[}\frac{\varphi_{i\sigma}^{*}(\mathbf{r}_{1})\varphi_{a\sigma}(\mathbf{r}_{1})\varphi_{a\sigma}^{*}(\mathbf{r}_{2})\varphi_{i\sigma}(\mathbf{r}_{2})}{\omega-(\varepsilon_{a\sigma}-\varepsilon_{i\sigma})+i0^{+}}$
(7.27)
$\displaystyle\phantom{xxxxxxxxxxxxxxxxxx}-\frac{\varphi_{i\sigma}^{*}(\mathbf{r}_{2})\varphi_{a\sigma}(\mathbf{r}_{2})\varphi_{a\sigma}^{*}(\mathbf{r}_{1})\varphi_{i\sigma}(\mathbf{r}_{1})}{\omega+(\varepsilon_{a\sigma}-\varepsilon_{i\sigma})+i0^{+}}\Biggl{]},$

one can obtain, after quite some work,

$\displaystyle E_{\text{c}}^{\text{dRPA}}[\rho]$ $\displaystyle=$
$\displaystyle-\frac{1}{2}\sum_{i=1}^{N}\sum_{j=1}^{N}\sum_{a\geq
N+1}\sum_{b\geq
N+1}\frac{|\langle\phi_{i}\phi_{j}|\phi_{a}\phi_{b}\rangle|^{2}}{\varepsilon_{a}+\varepsilon_{b}-\varepsilon_{i}-\varepsilon_{j}}$
$\displaystyle+\sum_{i=1}^{N}\sum_{j=1}^{N}\sum_{k=1}^{N}\sum_{a\geq
N+1}\sum_{b\geq N+1}\sum_{c\geq
N+1}\frac{\langle\phi_{i}\phi_{j}|\phi_{a}\phi_{b}\rangle\langle\phi_{j}\phi_{k}|\phi_{b}\phi_{c}\rangle\langle\phi_{k}\phi_{i}|\phi_{c}\phi_{a}\rangle}{(\varepsilon_{a}+\varepsilon_{b}-\varepsilon_{i}-\varepsilon_{j})(\varepsilon_{a}+\varepsilon_{c}-\varepsilon_{i}-\varepsilon_{k})}+\cdots.$

The dRPA correlation energy is the sum of all the direct terms (i.e., no
exchange terms) of the perturbation expansion up to infinite order. In the
language of diagrammatic perturbation theory, we say that the dRPA correlation
energy is the sum of all direct ring diagrams. Of course, Eq. (LABEL:EcdRPA2)
is not the way to calculate the dRPA correlation energy in practice. This is
done by solving the Dyson equation [Eq. (7.24)] without explicitly expanding
in powers of $\lambda$, e.g. using matrix equations from linear-response TDDFT
[261, 204] or coupled-cluster theory [262, 200].

Most dRPA correlation energy (combined with the EXX energy) calculations are
done in a non-self-consistent way but self-consistent OEP dRPA calculations
have also been performed [263, 264]. One of the main advantage of dRPA is that
it accounts for long-range dispersion interactions [265, 266, 267]. However,
it shows large self-interaction errors. To overcome the latter drawback and
improve the general accuracy, one can add exchange and beyond terms in various
ways (see, e.g., Refs. [202, 268, 204, 269, 270, 271, 272, 273, 274, 275, 260,
276, 277, 278, 279, 280]). This remains an active area of research. For
reviews on random-phase approximations, the reader may consult Refs. [281,
282, 283].

## Acknowledgements

I thank Éric Cancès and Gero Friesecke for discussions and comments on the
manuscript. This review chapter grew out of my lecture notes for DFT courses
given in several summer schools (ISTPC summer schools in June 2015 and 2017 in
France, ICS summer school in July 2015 in France, and ESQC summer school in
September 2019 in Italy).

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