5.6.2 User-Defined RSH Functionals

As pointed out in Ref. 240 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^{69, 882} 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.^{520, 522} 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$.⁵²⁰ 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.^{526, 796}

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,^{408, 864} called $\mu$B88 and $\mu$PBE in Q-Chem,⁷⁹⁰ and an alternative formulation of short-range PBE exchange proposed by Scuseria and co-workers,³⁶⁷ which is known as $\omega$PBE. These functionals are available in Q-Chem thanks to the efforts of the Herbert group.^{796, 797} 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,³⁷⁷ 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.⁵²⁶ However, certain results depend sensitively upon the value of the range-separation parameter $\omega$,^{526, 796, 797, 522, 937} 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.⁷⁹⁷ 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,⁷⁹⁷ 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 $\omega$.^{522, 797} In such cases (and maybe in general), the “tuning” procedure described in Section 5.6.3 is recommended.

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^{184, 650}) at integer values of $N$, but in practice these $E(N)$ plots bow away from straight-line segments.¹⁸⁴ Examination of such plots has been suggested as a means to adjust the fraction of short-range exchange in an LRC functional,⁴¹ while the range-separation parameter is tuned as described in Section 5.6.3.