5.6.4 Tuned RSH Functionals Based on the Global Density-Dependent Condition

The value of range-separation parameter based on IP tuning procedure (${\omega }_{\mathrm{IP}}$) exhibits a troublesome dependence on system size.^{937, 286, 950, 189, 697} An alternative method to select $\omega$ is the global density-dependent (GDD) tuning procedure,⁶⁴⁷ 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.⁶⁴⁷ (A slightly different value, $C=0.885$, was determined for Q-Chem’s implementation of LRC-$\omega$PBE.⁵³³) Since LRC-$\omega$PBE(${\omega }_{\mathrm{GDD}}$) provides a better description of polarizabilities in polyacetylene as compared to ${\omega }_{\mathrm{IP}}$³⁴⁶, it is anticipated that using ${\omega }_{\mathrm{GDD}}$ in place of ${\omega }_{\mathrm{IP}}$ 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. (5.18) 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.