5.6.3 User-Defined RSH Functionals

As pointed out in Ref.  330 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 ⁸⁹ Becke A. D.
J. Chem. Phys.
(1993), 98, pp. 5648. Link ^{, 1222} 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. ⁷¹⁰ Lange A., Herbert J. M.
J. Chem. Theory Comput.
(2007), 3, pp. 1680. Link ^{, 712} 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$. ⁷¹⁰ 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 ($c_{x,\mathrm{LR}}=1$ 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. ⁷¹⁷ Lange A. W., Rohrdanz M. A., Herbert J. M.
J. Phys. Chem. B
(2008), 112, pp. 6304. Link ^{, 1100} 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, ⁵⁷³ Iikura H. et al.
J. Chem. Phys.
(2001), 115, pp. 3540. Link ^{, 1198} Song J. W. et al.
J. Chem. Phys.
(2007), 126, pp. 154105. Link called $\mu$B88 and $\mu$PBE in Q-Chem, ¹⁰⁹³ 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, ⁵¹⁵ 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. ¹¹⁰⁰ Rohrdanz M. A., Herbert J. M.
J. Chem. Phys.
(2008), 129, pp. 034107. Link ^{, 1101} 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, $E_{x,\mathrm{SR}}^{\mathrm{DFT}}$ 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, ⁵³⁶ 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. ⁷¹⁷ 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 $\omega$, ⁷¹⁷ Lange A. W., Rohrdanz M. A., Herbert J. M.
J. Phys. Chem. B
(2008), 112, pp. 6304. Link ^{, 1100} Rohrdanz M. A., Herbert J. M.
J. Chem. Phys.
(2008), 129, pp. 034107. Link ^{, 1101} Rohrdanz M. A., Martins K. M., Herbert J. M.
J. Chem. Phys.
(2009), 130, pp. 054112. Link ^{, 712} Lange A. W., Herbert J. M.
J. Am. Chem. Soc.
(2009), 131, pp. 124115. Link ^{, 1287} 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.

Rohrdanz et al. ¹¹⁰¹ 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 $c_{x,\mathrm{SR}}=0.2$ (see Eq. (5.13)) and $\omega =0.2$ bohr${}^{-1}$ were obtained, ¹¹⁰¹ 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 $\omega$. ⁷¹² Lange A. W., Herbert J. M.
J. Am. Chem. Soc.
(2009), 131, pp. 124115. Link ^{, 1101} 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.

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 5.6.4, such functionals have been shown to afford accurate fundamental gaps for organic photovoltaic materials, ⁶⁸¹ 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. ¹²¹ Bhandari S. et al.
J. Chem. Theory Comput.
(2019), 14, pp. 6287. Link These have come to be called screened RSH (sRSH) functionals. ⁶⁸¹ 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

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 $1/r$. ³³⁰ 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 $c_{x,\mathrm{LR}}=1$, 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.