# 5.6.2 User-Defined RSH Functionals

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 B3LYP68, 873 and PBE022 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 ($c_{x,\rm LR}=1$ in Eq. (5.13)) are known as “long-range-corrected” (LRC) functionals. An LRC version of PBE0 would, for example, have $c_{x,\rm SR}=0.25$.

To fully specify an LRC functional, one must choose a value for the range separation parameter $\omega$ in Eq. (5.12). In the limit $\omega\rightarrow 0$, 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 $\omega\rightarrow\infty$ limit corresponds to a new functional, $E_{xc}^{\rm RSH}=E_{c}+E_{x}^{\rm HF}$. 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, $E_{x,\rm SR}^{\rm DFT}$ 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.

Example 5.4  Application of LRC-BOP to $\rm(H_{2}O)_{2}^{-}$.

$comment The value of omega is 0.47 by default but can be overwritten by specifying OMEGA.$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 $c_{x,\rm SR}=0.2$ (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.

Example 5.5  Application of LRC-$\omega$PBEh to the $\rm C_{2}H_{4}$$\rm C_{2}F_{4}$ dimer at 5 Å separation.

$comment This example uses the "optimal" parameter set discussed above. It can also be run by setting METHOD = LRC-wPBEh.$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 discontinuity183, 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 $n<0$, a fraction of an electron is removed from the system.

Example 5.6  Example of a DFT job with a fractional number of electrons. Here, we make the $-1.x$ anion of fluoride by subtracting a fraction of an electron from the HOMO of F${}^{2-}$.

$comment Subtracting a whole electron recovers the energy of F-. Adding electrons to the LUMO is possible as well.$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