5.6.3 User-Defined RSH Functionals

As pointed out in Ref. 291 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^{78, 1081} and PBE0²⁰ 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.^{621, 623} 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$.⁶²¹ 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.^{628, 973}

Evaluation of the short- and long-range HF exchange energies is straightforward,²⁴ 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,^{500, 1058} called $\mu$B88 and $\mu$PBE in Q-Chem,⁹⁶⁶ and an alternative formulation of short-range PBE exchange proposed by Scuseria and co-workers,⁴⁵⁰ which is known as $\omega$PBE. These functionals are available in Q-Chem thanks to the efforts of the Herbert group.^{973, 974} 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,⁴⁶⁴ 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.⁶²⁸ However, certain results depend sensitively upon the value of the range-separation parameter $\omega$,^{628, 973, 974, 623, 1141} 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.⁹⁷⁴ 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,⁹⁷⁴ 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 $\omega$.^{623, 974} 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,⁵⁹³ and are naturally combined with polarizable continuum models (Section 11.2.3) that employ the same dielectric constant.¹⁰⁷ These have come to be called screened RSH (sRSH) functionals.⁵⁹³ An XC function of this type can be expressed generically as³²

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 $1/r$.²⁹¹ 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.