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(May 16, 2021)

As pointed out in Ref. 286 and elsewhere, the description
of charge-transfer excited states within density functional theory (or more
precisely, time-dependent DFT, which is discussed in Section 7.3)
requires full (100%) non-local HF exchange, at least in the limit of large
donor–acceptor distance. Hybrid functionals such as
B3LYP
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J. Chem. Phys.

(1993),
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pp. 5648.
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J. Phys. Chem.

(1994),
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pp. 11623.
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and PBE0
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J. Chem. Phys.

(1999),
111,
pp. 2889.
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that are
well-established and in widespread use, however, employ only 20% and 25% HF
exchange, respectively. While these functionals provide excellent results for
many ground-state properties, they cannot correctly describe the distance
dependence of charge-transfer excitation energies, which are enormously
underestimated by most common density functionals. This is a serious problem
in any case, but it is a *catastrophic* problem in large molecules and in
non-covalent clusters, where TDDFT often predicts a near-continuum of spurious,
low-lying charge transfer states.
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J. Chem. Theory Comput.

(2007),
3,
pp. 1680.
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^{,}
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J. Am. Chem. Soc.

(2009),
131,
pp. 124115.
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The problems with
TDDFT’s description of charge transfer are not limited to large donor–acceptor
distances, but have been observed at $\sim $2 Å separation, in systems as
small as uracil–(H${}_{2}$O)${}_{4}$.
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J. Chem. Theory Comput.

(2007),
3,
pp. 1680.
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Rydberg excitation energies
also tend to be substantially underestimated by standard TDDFT.

One possible avenue by which to correct such problems is to parameterize
functionals that contain 100% HF exchange, though few such functionals exist
to date. An alternative option is to attempt to preserve the form of common
GGAs and hybrid functionals at short range (*i.e.*, keep the 25% HF exchange in
PBE0) while incorporating 100% HF exchange at long range, which provides a
rigorously correct description of the long-range distance dependence of
charge-transfer excitation energies, but aims to avoid contaminating
short-range exchange-correlation effects with additional HF exchange. The
separation is accomplished using the range-separation *ansatz* that was
introduced in Section 5.3. In particular, functionals that use
100% HF exchange at long range (${c}_{x,\mathrm{LR}}=1$ in Eq. (5.13))
are known as “long-range-corrected” (LRC) functionals. An LRC version of
PBE0 would, for example, have ${c}_{x,\mathrm{SR}}=0.25$.

To fully specify an LRC functional, one must choose a value for the range
separation parameter $\omega $ in Eq. (5.12). In the limit
$\omega \to 0$, the LRC functional in Eq. (5.13) reduces to
a non-RSH functional where there is no “SR” or “LR”, because all exchange
and correlation energies are evaluated using the full Coulomb operator,
${r}_{12}^{-1}$. Meanwhile the $\omega \to \mathrm{\infty}$ limit corresponds to a
new functional, ${E}_{xc}^{\mathrm{RSH}}={E}_{c}+{E}_{x}^{\mathrm{HF}}$. Full HF exchange
is inappropriate for use with most contemporary GGA correlation functionals, so
the latter limit is expected to perform quite poorly. Values of $\omega >1.0$ bohr${}^{-1}$ are likely not worth considering, according to benchmark
tests.
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J. Phys. Chem. B

(2008),
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pp. 6304.
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J. Chem. Phys.

(2008),
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pp. 034107.
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Evaluation of the short- and long-range HF exchange energies is
straightforward,
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J. Comput. Chem.

(1999),
20,
pp. 921.
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so the crux of any RSH functional is the
form of the short-range GGA exchange functional, and several such functionals
are available in Q-Chem. These include short-range variants of the B88 and
PBE exchange described by Hirao and co-workers,
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J. Chem. Phys.

(2001),
115,
pp. 3540.
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^{,}
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(2007),
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called $\mu $B88 and $\mu $PBE in Q-Chem,
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J. Chem. Theory Comput.

(2011),
7,
pp. 1296.
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and an
alternative formulation of short-range PBE exchange proposed by Scuseria and
co-workers,
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J. Chem. Phys.

(2008),
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pp. 194105.
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which is known as $\omega $PBE. These
functionals are available in Q-Chem thanks to the efforts of the Herbert
group.
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J. Chem. Phys.

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(2009),
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By way of notation, the terms
“$\mu $PBE”, “$\omega $PBE”, *etc.*, refer only to the short-range exchange
functional, ${E}_{x,\mathrm{SR}}^{\mathrm{DFT}}$ in Eq. (5.13). These
functionals could be used in “screened exchange” mode, as described in
Section 5.3, as for example in the HSE03
functional,
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J. Chem. Phys.

(2003),
118,
pp. 8207.
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therefore the designation “LRC-$\omega $PBE”, for
example, should only be used when the short-range exchange functional
$\omega $PBE is combined with 100% Hartree-Fock exchange in the long range.

In general, LRC-DFT functionals have been shown to remove the near-continuum of
spurious charge-transfer excited states that appear in large-scale TDDFT
calculations.
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J. Phys. Chem. B

(2008),
112,
pp. 6304.
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However, certain results depend sensitively upon
the value of the range-separation parameter
$\omega $,
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J. Phys. Chem. B

(2008),
112,
pp. 6304.
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^{,}
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(2008),
129,
pp. 034107.
Link
^{,}
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(2009),
130,
pp. 054112.
Link
^{,}
^{
605
}
J. Am. Chem. Soc.

(2009),
131,
pp. 124115.
Link
^{,}
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J. Phys. Chem. A

(2014),
118,
pp. 7507.
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especially in TDDFT calculations (Section 7.3) and therefore the
results of LRC-DFT calculations must therefore be interpreted with caution, and
probably for a range of $\omega $ values. This can be accomplished by
requesting a functional that contains some short-range GGA exchange functional
($\omega $PBE or $\mu $PBE, in the examples mentioned above), in combination with
setting the *$rem* variable LRC_DFT = TRUE, which requests
the addition of 100% Hartree-Fock exchange in the long-range. Basic
job-control variables and an example can be found below. The value of the
range-separation parameter is then controlled by the variable OMEGA,
as shown in the examples below.

LRC_DFT

Controls the application of long-range-corrected DFT

TYPE:

LOGICAL

DEFAULT:

FALSE

OPTIONS:

FALSE (or 0)
Do not apply long-range correction.
TRUE (or 1)
Add 100% long-range Hartree-Fock exchange to the requested functional.

RECOMMENDATION:

The *$rem* variable OMEGA must also be specified, in order to set
the range-separation parameter.

OMEGA

Sets the range-separation parameter, $\omega $, also known as $\mu $, in functionals based on Hirao’s RSH scheme.

TYPE:

INTEGER

DEFAULT:

No default

OPTIONS:

$n$
Corresponding to $\omega =n/1000$, in units of bohr${}^{-1}$

RECOMMENDATION:

None

COMBINE_K

Controls separate or combined builds for short-range and long-range K

TYPE:

LOGICAL

DEFAULT:

FALSE

OPTIONS:

FALSE (or 0)
Build short-range and long-range K separately (twice as expensive as a global hybrid)
TRUE (or 1)
Build short-range and long-range K together ($\approx $ as expensive as a global hybrid)

RECOMMENDATION:

Most pre-defined range-separated hybrid functionals in Q-Chem use this
feature by default. However, if a user-specified RSH is desired, it is
necessary to manually turn this feature on.

HFK_SR_COEF

Sets the coefficient for short-range HF exchange

TYPE:

INTEGER

DEFAULT:

0

OPTIONS:

$n$
Corresponding to $n/100000000$

RECOMMENDATION:

None

HFK_LR_COEF

Sets the coefficient for long-range HF exchange

TYPE:

INTEGER

DEFAULT:

100000000

OPTIONS:

$n$
Corresponding to $n/100000000$

RECOMMENDATION:

None

$comment The value of omega is 0.47 by default but can be overwritten by specifying OMEGA. $end $molecule -1 2 O 1.347338 -0.017773 -0.071860 H 1.824285 0.813088 0.117645 H 1.805176 -0.695567 0.461913 O -1.523051 -0.002159 -0.090765 H -0.544777 -0.024370 -0.165445 H -1.682218 0.174228 0.849364 $end $rem EXCHANGE LRC-BOP BASIS 6-311(1+,2+)G* XC_GRID 2 LRC_DFT TRUE OMEGA 300 ! = 0.300 bohr**(-1) $end

Rohrdanz *et al.*
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(2009),
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pp. 054112.
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published a thorough benchmark study of
both ground- and excited-state properties using the LRC-$\omega $PBEh
functional, in which the “h” indicates a short-range hybrid (*i.e.*, the
presence of some short-range HF exchange). Empirically-optimized parameters of
${c}_{x,\mathrm{SR}}=0.2$ (see Eq. (5.13)) and $\omega =0.2$ bohr${}^{-1}$ were obtained,
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(2009),
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and these parameters are
taken as the defaults for LRC-$\omega $PBEh. Caution is warranted, however,
especially in TDDFT calculations for large systems, as excitation energies for
states that exhibit charge-transfer character can be rather sensitive to the
precise value of $\omega $.
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J. Am. Chem. Soc.

(2009),
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pp. 124115.
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(2009),
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In such cases (and
maybe in general), the “tuning” procedure described in
Section 5.6.3 is recommended.

$comment This example uses the "optimal" parameter set discussed above. It can also be run by setting METHOD = LRC-wPBEh. $end $molecule 0 1 C 0.670604 0.000000 0.000000 C -0.670604 0.000000 0.000000 H 1.249222 0.929447 0.000000 H 1.249222 -0.929447 0.000000 H -1.249222 0.929447 0.000000 H -1.249222 -0.929447 0.000000 C 0.669726 0.000000 5.000000 C -0.669726 0.000000 5.000000 F 1.401152 1.122634 5.000000 F 1.401152 -1.122634 5.000000 F -1.401152 -1.122634 5.000000 F -1.401152 1.122634 5.000000 $end $rem EXCHANGE GEN BASIS 6-31+G* LRC_DFT TRUE OMEGA 200 ! = 0.2 a.u. CIS_N_ROOTS 4 CIS_TRIPLETS FALSE $end $xc_functional C PBE 1.00 X wPBE 0.80 X HF 0.20 $end

By adding 100% Hartree-Fock exchange to the asymptotic Coulomb operator, LRC functionals
guarantee that an electron and hole experience an asymptotic interaction potential $1/r$.
This is correct for a molecule in the gas phase, but to simulate a material one might desire an asymptotic
behavior of $1/(\epsilon r)$, where $\epsilon $ is the (static) dielectric constant of the material.
In conjunction with “optimal tuning” of the range-separation parameter, as described in Section 5.6.3,
such functionals have been shown to afford accurate fundamental gaps for organic photovoltaic
materials,
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Adv. Mater.

(2018),
30,
pp. 1706560.
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and are naturally combined with polarizable continuum models
(Section 11.2.2)
that employ the same dielectric constant.
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J. Chem. Theory Comput.

(2019),
14,
pp. 6287.
Link
These have come to be called screened RSH (sRSH) functionals.
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Adv. Mater.

(2018),
30,
pp. 1706560.
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An XC function of this type can be expressed generically as
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J. Phys. Chem. C

(2020),
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pp. 24653.
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$${E}_{xc}^{\text{sRSH}}={c}_{x,\mathrm{SR}}{E}_{x,\mathrm{SR}}^{\text{HF}}+{\epsilon}^{-1}{E}_{x,\mathrm{LR}}^{\text{HF}}+({\epsilon}^{-1}-{c}_{x,\mathrm{SR}}){E}_{x,\mathrm{SR}}^{\text{DFT}}+(1-{\epsilon}^{-1}){E}_{x,\mathrm{LR}}^{\text{DFT}}+{E}_{c}^{\text{DFT}},$$ | (5.17) |

which should be compared to Eq. (5.13) that provides the generic form for an RSH functional.
Although the RSH formalism allows for an arbitrary coefficient ${c}_{x,\mathrm{LR}}$ for the long-range Hartree-Fock
exchange term, as in Eq. (5.13),
this implies that the asymptotic electron–hole interaction has the form ${c}_{x,\mathrm{LR}}/r$ rather than $1/r$.
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J. Chem. Phys.

(2003),
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pp. 2943.
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As such, LRC functionals are a particular class of RSH functionals where ${c}_{x,\mathrm{LR}}=1$, ensuring proper
asymptotic behavior in vacuum. Along the same lines, sRSH functionals set ${c}_{x,\mathrm{LR}}={\epsilon}^{-1}$
to ensure proper asymptotic behavior in a dielectric material. Using Eq. (5.17), users may construct
sRSH functionals by means of a *$xc_functional* input section.