As pointed out in Ref. 239 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^{68, 873} and PBE0^{22} 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.^{516, 518} 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}$.^{516} 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.^{522, 788}

Evaluation of the short- and long-range HF exchange energies is
straightforward,^{26} 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,^{405, 856}
called $\mu $B88 and $\mu $PBE in Q-Chem,^{782} and an
alternative formulation of short-range PBE exchange proposed by Scuseria and
co-workers,^{364} which is known as $\omega $PBE. These
functionals are available in Q-Chem thanks to the efforts of the Herbert
group.^{788, 789} By way of notation, the terms
“$\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,^{374} 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.^{522} However, certain results depend sensitively upon
the value of the range-separation parameter
$\omega $,^{522, 788, 789, 518, 927}
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.

$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 6-31(1+,3+)G* LRC_DFT TRUE OMEGA 300 ! = 0.300 bohr**(-1) $end

Rohrdanz *et al.*^{789} 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,^{789} 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 $.^{518, 789} 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

Both LRC functionals and also the asymptotic corrections that will be discussed
in Section 5.10.1 are thought to reduce self-interaction error in
approximate DFT. A convenient way to quantify—or at least depict—this
error is by plotting the DFT energy as a function of the (fractional) number of
electrons, $N$, because $E(N)$ should in principle consist of a sequence of
line segments with abrupt changes in slope (the so-called derivative
discontinuity^{183, 645}) at integer values of $N$,
but in practice these $E(N)$ plots bow away from straight-line
segments.^{183} Examination of such plots has been suggested as a
means to adjust the fraction of short-range exchange in an LRC
functional,^{40} while the range-separation parameter is
tuned as described in Section 5.6.3.

FRACTIONAL_ELECTRON

Add or subtract a fraction of an electron.

TYPE:

INTEGER

DEFAULT:

0

OPTIONS:

0
Use an integer number of electrons.
$n$
Add $n/1000$ electrons to the system.

RECOMMENDATION:

Use only if trying to generate $E(N)$ plots. If $$, a fraction of an
electron is removed from the system.

$comment Subtracting a whole electron recovers the energy of F-. Adding electrons to the LUMO is possible as well. $end $rem EXCHANGE b3lyp BASIS 6-31+G* FRACTIONAL_ELECTRON -500 ! divide by 1000 to get the fraction, -0.5 here. GEN_SCFMAN FALSE ! not yet available in new scf code $end $molecule -2 2 F $end