5.6.3 User-Defined RSH Functionals‣ 5.6 Range-Separated Hybrid Density Functionals ‣ Chapter 5 Density Functional Theory ‣ Q-Chem 6.4 User’s Manual

As pointed out in Ref. 343 Dreuw A., Weisman J. L., Head-Gordon M.
J. Chem. Phys.
(2003), 119, pp. 2943. Link 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 ⁹⁵ Becke A. D.
J. Chem. Phys.
(1993), 98, pp. 5648. Link ^, ¹²⁶⁵ Stephens P. J. et al.
J. Phys. Chem.
(1994), 98, pp. 11623. Link and PBE0 ²⁰ Adamo C., Scuseria G. E., Barone V.
J. Chem. Phys.
(1999), 111, pp. 2889. Link 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. ⁷³² Lange A., Herbert J. M.
J. Chem. Theory Comput.
(2007), 3, pp. 1680. Link ^, ⁷³⁴ Lange A. W., Herbert J. M.
J. Am. Chem. Soc.
(2009), 131, pp. 124115. Link 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) ${}_{4}$ . ⁷³² Lange A., Herbert J. M.
J. Chem. Theory Comput.
(2007), 3, pp. 1680. Link 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. ⁷³⁹ Lange A. W., Rohrdanz M. A., Herbert J. M.
J. Phys. Chem. B
(2008), 112, pp. 6304. Link ^, ¹¹³⁹ Rohrdanz M. A., Herbert J. M.
J. Chem. Phys.
(2008), 129, pp. 034107. Link

Evaluation of the short- and long-range HF exchange energies is straightforward, ²⁴ Adamson R. D., Dombroski J. P., Gill P. M. W.
J. Comput. Chem.
(1999), 20, pp. 921. Link 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, ⁵⁹⁴ Iikura H. et al.
J. Chem. Phys.
(2001), 115, pp. 3540. Link ^, ¹²⁴¹ Song J. W. et al.
J. Chem. Phys.
(2007), 126, pp. 154105. Link called $\mu$ B88 and $\mu$ PBE in Q-Chem, ¹¹³⁰ Richard R. M., Herbert J. M.
J. Chem. Theory Comput.
(2011), 7, pp. 1296. Link and an alternative formulation of short-range PBE exchange proposed by Scuseria and co-workers, ⁵³⁸ Henderson T. M., Janesko B. G., Scuseria G. E.
J. Chem. Phys.
(2008), 128, pp. 194105. Link which is known as $\omega$ PBE. These functionals are available in Q-Chem thanks to the efforts of the Herbert group. ¹¹³⁹ Rohrdanz M. A., Herbert J. M.
J. Chem. Phys.
(2008), 129, pp. 034107. Link ^, ¹¹⁴⁰ Rohrdanz M. A., Martins K. M., Herbert J. M.
J. Chem. Phys.
(2009), 130, pp. 054112. Link 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, ⁵⁶⁰ Heyd J., Scuseria G. E., Ernzerhof M.
J. Chem. Phys.
(2003), 118, pp. 8207. Link 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. ⁷³⁹ Lange A. W., Rohrdanz M. A., Herbert J. M.
J. Phys. Chem. B
(2008), 112, pp. 6304. Link However, certain results depend sensitively upon the value of the range-separation parameter $\omega$ , ⁷³⁹ Lange A. W., Rohrdanz M. A., Herbert J. M.
J. Phys. Chem. B
(2008), 112, pp. 6304. Link ^, ¹¹³⁹ Rohrdanz M. A., Herbert J. M.
J. Chem. Phys.
(2008), 129, pp. 034107. Link ^, ¹¹⁴⁰ Rohrdanz M. A., Martins K. M., Herbert J. M.
J. Chem. Phys.
(2009), 130, pp. 054112. Link ^, ⁷³⁴ Lange A. W., Herbert J. M.
J. Am. Chem. Soc.
(2009), 131, pp. 124115. Link ^, ¹³³⁴ Uhlig F. et al.
J. Phys. Chem. A
(2014), 118, pp. 7507. Link 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

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

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

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

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

HFK_LR_COEF
       Sets the coefficient for long-range HF exchange
TYPE:
       INTEGER
DEFAULT:
       100000000
OPTIONS:
       $n$ Corresponding to $n/100000000$
RECOMMENDATION:
       None

Example 5.6 Application of LRC- $\mu$ 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         6-311(1+,2+)G*
   XC_GRID       2
   LRC_DFT       TRUE
   OMEGA         300      ! = 0.300 bohr**(-1)
$end

Rohrdanz et al. ¹¹⁴⁰ Rohrdanz M. A., Martins K. M., Herbert J. M.
J. Chem. Phys.
(2009), 130, pp. 054112. Link 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, ¹¹⁴⁰ Rohrdanz M. A., Martins K. M., Herbert J. M.
J. Chem. Phys.
(2009), 130, pp. 054112. Link 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 $\omega$ . ⁷³⁴ Lange A. W., Herbert J. M.
J. Am. Chem. Soc.
(2009), 131, pp. 124115. Link ^, ¹¹⁴⁰ Rohrdanz M. A., Martins K. M., Herbert J. M.
J. Chem. Phys.
(2009), 130, pp. 054112. Link In such cases (and maybe in general), the “tuning” procedure described in Section 5.6.4 is recommended.

Example 5.7 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

Recently, Li and Chai reoptimized the B97-type range-separated functional with the latest version of the semi-empirical dispersion correction D4 ⁷⁹⁴ Li S., Chai J.-D.
J. Chem. Theory Comput.
(2025), 21, pp. 9538. Link . The resulting functional is denoted as KS- $\omega$ B97X-D4, which performs comparably to $\omega$ B97M-V ⁸⁷⁸ Mardirossian N., Head-Gordon M.
J. Chem. Phys.
(2016), 144, pp. 214110. Link , and consistently outperforms $\omega$ B97X-D3 ⁸¹¹ Lin Y.-S. et al.
J. Chem. Theory Comput.
(2013), 9, pp. 263. Link and $\omega$ B97X-V ⁸⁷⁶ Mardirossian N., Head-Gordon M.
Phys. Chem. Chem. Phys.
(2014), 16, pp. 9904. Link for the entire GMTKN55 database ⁷⁹⁴ Li S., Chai J.-D.
J. Chem. Theory Comput.
(2025), 21, pp. 9538. Link .

Example 5.8 KS- $\omega$ B97X-D4 calculation on H ${}_{2}$ molecule

$molecule
0 1
H 0.00 0.00 0.00
H 1.00 0.00 0.00
$end

$rem
JOBTYPE              = sp
unrestricted         = FALSE
scf_guess            = SAD
scf_guess_mix        = 0
scf_algorithm        = diis
max_scf_cycles       = 300
GEN_SCFMAN           = FALSE
xc_grid              = 000075000302
scf_convergence      = 8
thresh               = 14
BASIS                = 6-31G(d)
TAO_DFT              = true
TAO_DFT_THETA        = 0
TAO_DFT_THETA_NDP    = 0
EXCHANGE             = gen
LRC_DFT              = TRUE
COMBINE_K            = TRUE
OMEGA                = 300
symmetry             = false
sym_ignore           = true
basis_lin_dep_thresh = 8
DFT_D                = D4
DFT_D4_A1            = 35000000
DFT_D4_A2            = 500000000
DFT_D4_S6            = 100000000
DFT_D4_S8            = 139483100
DFT_D4_S9            = 100000000
DFT_D4_S10           = 0
$end

$XC_functional
X HF                  0.186861
X KS_wB97X            1.000000
C KS_wB97X            1.000000
$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 $1/(\varepsilon r)$ , where $\varepsilon$ is the (static) dielectric constant of the material. In conjunction with “optimal tuning” of the range-separation parameter, as described in Section 5.6.4, such functionals have been shown to afford accurate fundamental gaps for organic photovoltaic materials, ⁷⁰³ Kronik L., Kümmel S.
Adv. Mater.
(2018), 30, pp. 1706560. Link and are naturally combined with polarizable continuum models (Section 11.2.3) that employ the same dielectric constant. ¹²⁷ Bhandari S. et al.
J. Chem. Theory Comput.
(2019), 14, pp. 6287. Link These have come to be called screened RSH (sRSH) functionals. ⁷⁰³ Kronik L., Kümmel S.
Adv. Mater.
(2018), 30, pp. 1706560. Link An XC function of this type can be expressed generically as ³³ Alam B., Morrison A. F., Herbert J. M.
J. Phys. Chem. C
(2020), 124, pp. 24653. Link

E_{xc}^{\textrm{sRSH}}=c_{x,{\rm SR}}E_{x,{\rm SR}}^{\textrm{HF}}+\varepsilon^% {-1}E_{x,\rm LR}^{\textrm{HF}}+(\varepsilon^{-1}-c_{x,\rm SR})E_{x,\rm SR}^{% \textrm{DFT}}+(1-\varepsilon^{-1})E_{x,\rm LR}^{\textrm{DFT}}+E_{c}^{\textrm{% 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 $c_{x,\rm LR}$ for the long-range Hartree-Fock exchange term, as in Eq. (5.13), this implies that the asymptotic electron–hole interaction has the form $c_{x,\rm LR}/r$ rather than $1/r$ . ³⁴³ Dreuw A., Weisman J. L., Head-Gordon M.
J. Chem. Phys.
(2003), 119, pp. 2943. Link As such, LRC functionals are a particular class of RSH functionals where $c_{x,\rm LR}=1$ , ensuring proper asymptotic behavior in vacuum. Along the same lines, sRSH functionals set $c_{x,\rm LR}=\varepsilon^{-1}$ 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.

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5.6.3 User-Defined RSH Functionals