5.6.4 Tuned RSH Functionals

Whereas the range-separation parameters for the functionals described in Section 5.6.2 are wholly empirical in nature and should not be adjusted, for the functionals described in Section 5.6.3 some adjustment was suggested, especially for TDDFT calculations and for any properties that require interpretation of orbital energies, such as HOMO/LUMO gaps. This adjustment can be performed in a non-empirical (albeit system-specific) way,⁴⁸ by “tuning” the value of $\omega$ in order to satisfy the Koopmans-like ionization energy criterion

where $\text{IE}=E(N)-E(N-1)$ is the $\mathrm{\Delta }$SCF value of the ionization energy for the $N$-electron system of interest. The condition ${ϵ}_{\mathrm{HOMO}}=-\text{IE}$ is a theorem in exact DFT,⁸⁷⁰ but this condition is often badly violated by approximate functionals. When an RSH functional is used, both sides of Eq. (5.18) are $\omega$-dependent and this parameter is adjusted until the condition in Eq. (5.18) is met, which requires a sequence of SCF calculations on both the neutral and ionized species, using different values of $\omega$. The value that is obtained has come to be called the “optimally tuned” value of $\omega$. Formally speaking, there is no guarantee that an approximate density functional can be made to satisfy Eq. (5.18) for any given molecule, thus the optimally-tuned value need not exist. In practice it is usually possible to find such a value, although it should be noted that the optimally-tuned value of $\omega$ depends on system size, and as a result this tuning procedure formally violates size-consistency.⁵⁵⁰

A few variations on the simple “IE tuning” criterion in Eq. (5.18) are possible. For proper description of charge-transfer states, Baer and co-workers⁴⁸ suggest finding the value of $\omega$ that (to the extent possible) satisfies Eq. (5.18) for both the neutral donor molecule and (separately) for the anion of the acceptor species. Along similar lines, in an effort to set both the HOMO and LUMO energy levels such that the fundamental gap ($\text{IE}-\text{EA}$) is equal to the HOMO/LUMO gap, Kronik et al.⁵⁹⁴ suggest minimizing the function

with respect to $\omega$, where $\text{EA}=E(N+1)-E(N)$. Minimization of $J(\omega )$ represents an attempt to satisfy the IE theorem of Eq. (5.18) for both the $N$-electron molecule and its ($N+1$)-electron anion, assuming that the latter is bound. Published benchmarks suggest that these system-specific approaches afford the most accurate values of IEs and TDDFT excitation energies.^{710, 988, 48, 594}

A script that optimizes $\omega$, called OptOmegaIPEA.pl, is located in the $QC/bin/tools directory. The script scans $\omega$ over the range 0.1–0.8 bohr${}^{-1}$, corresponding to values of the $rem variable OMEGA in the range 100–800. See the script for the instructions how to modify the script to scan over a wider range. To execute the script, the user must create three inputs for an RSH single-point energy calculation, using the same geometry and basis set: one for a neutral molecule (N.in), one for its anion (M.in), and one for the molecule’s cation (P.in). The user should then run the command

OptOmegaIPEA.pl >& optomega

This command both generates the input files (N_*, P_*, M_*) and also runs Q-Chem on these input files, writing the optimization output into optomega. This script applies the IE condition to both the neutral molecule and its anion, minimizing $J(\omega )$ in Eq. (5.19). A similar script, OptOmegaIP.pl, uses Eq. (5.18) for the neutral molecule only.

Note:  1. If the system does not have positive EA, then the tuning should be done according to the IP condition only. The IP/EA script will yield an incorrect value of $\omega$ in such cases. 2. In order for the scripts to work, one must specify SCF_FINAL_PRINT = 1 in the inputs. The scripts look for specific regular expressions and will not work correctly without this keyword.

Although the tuning procedure was originally developed by Baer and co-workers using the BNL functional,^{710, 988, 48} it can equally well be applied to any RSH functional, as for example LRC-$\omega$PBE (see, Ref. 1141). The aforementioned scripts will work with these other RSH functionals as well.

Unfortunately, optimally-tuned values of “${\omega }_{\text{IE}}$”, obtained using the criterion in Eq. (5.18) exhibit a troubling dependence on system size,^{587, 1141, 348, 1154, 222, 850} leading to a loss of size-extensivity.⁵⁵⁰ For example, the optimally-tuned value for the cluster anion (H${}_2$O)${}^{-}_6$ is very different than the one tuned for (H${}_2$O)${}^{-}_{70}$,¹¹⁴¹ and the optimally-tuned value also varies with conjugation length for $\pi$-conjugated systems.⁵⁸⁷ An alternative to the IE-based criterion in Eq. (5.18) is the global density-dependent (GDD) tuning procedure,⁷⁸⁹ in which the optimal value

is related to the average of the distance $d_{\mathrm{x}}$ between an electron in the outer regions of a molecule and the exchange hole in the region of localized valence orbitals. The quantity $C$ is an empirical parameter for a given LRC functional, which was determined for LRC-$\omega$PBE ($C=0.90$) and LRC-$\omega$PBEh ($C=0.75$) using the def2-TZVPP basis set.⁷⁸⁹ (A slightly different value, $C=0.885$, was determined for Q-Chem’s implementation of LRC-$\omega$PBE.⁶³⁵) Since LRC-$\omega$PBE(${\omega }_{\mathrm{GDD}}$) provides a better description of polarizabilities in polyacetylene as compared to ${\omega }_{\mathrm{IE}}$,⁴²³ it is anticipated that using ${\omega }_{\mathrm{GDD}}$ in place of ${\omega }_{\mathrm{IE}}$ may afford more accurate molecular properties, especially in conjugated systems. GDD tuning of an RSH functional is involving by setting the $rem variable OMEGA_GDD = TRUE. The electron density is obviously needed to compute ${\omega }_{\mathrm{GDD}}$ in Eq. (5.20) and this is accomplished using the converged SCF density computed using the RSH functional with the value of $\omega$ given by the $rem variable OMEGA. The value of ${\omega }_{\mathrm{GDD}}$ therefore depends, in principle, upon the value of OMEGA, although in practice it is not very sensitive to this value.

However the tuning is accomplished, these tuned functionals are generally thought to work by reducing 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^{217, 792}) at integer values of $N$, but in practice these $E(N)$ plots bow away from straight-line segments.²¹⁷ Examination of such plots has been suggested as a means to adjust the fraction of short-range exchange in an RSH functional,⁴⁶ while the $\omega$ parameter is set by tuning.