Validation of local hybrid functionals for TDDFT calculations of electronic excitation energies

Abstract

The first systematic evaluation of local hybrid functionals for the calculation of electronic excitation energies within linear-response time-dependent density functional theory (TDDFT) is reported. Using our recent efficient semi-numerical TDDFT implementation [T. M. Maier et al., J. Chem. Theory Comput. 11, 4226 (2015)], four simple, thermochemically optimized one-parameter local hybrid functionals based on local spin-density exchange are evaluated against a database of singlet and triplet valence excitations of organic molecules, and against a mixed database including also Rydberg, intramolecular charge-transfer (CT) and core excitations. The four local hybrids exhibit comparable performance to standard global or range-separated hybrid functionals for common singlet valence excitations, but several local hybrids outperform all other functionals tested for the triplet excitations of the first test set, as well as for relative energies of excited states. Evaluation for the combined second test set shows that local hybrids can also provide excellent Rydberg and core excitations, in the latter case rivaling specialized functionals optimized specifically for such excitations. This good performance of local hybrids for different excitation types could be traced to relatively large exact-exchange (EXX) admixtures in a spatial region intermediate between valence and asymptotics, as well as close to the nucleus, and lower EXX admixtures in the valence region. In contrast, the tested local hybrids cannot compete with the best range-separated hybrids for intra- and intermolecular CT excitation energies. Possible directions for improvement in the latter category are discussed. As the used efficient TDDFT implementation requires essentially the same computational effort for global and local hybrids, applications of local hybrid functionals to excited-state problems appear promising in a wide range of fields. Influences of current-density dependence of local kinetic-energy dependent local hybrids, differences between spin-resolved and "common" local mixing functions in local hybrids, and the effects of the Tamm-Dancoff approximation on the excitation energies are also discussed.

Local hybrid functionals: Theory, implementation, and performance of an emerging new tool in quantum chemistry and beyond

“…A similar interrelation between real‐space behavior and performance could be observed for core excitations: while the s‐LMF features a (negative; Figure ) cusp at the position of the nuclei, and thus the corresponding local hybrids again performed more like standard global hybrid functionals, functionals based on a scaled t‐LMF partly showed exceptionally low errors (below 1.0 eV) for core excitations of first‐row elements. This holds in particular for common t‐LMFs with increased prefactors above 0.6, such as for the Lh 12ct‐SsirPW92 and Lh 12ct‐SsifPW92 functionals . These two simple functionals appear to combine sufficiently large EXX admixture in the core and Rydberg regions with the necessary reduced EXX in the valence region.…”

“…In contrast to the scaled t-LMF, the asymptotic value 1.0 of the basic s-LMF is not affected by the single nonlinear empirical parameter (e.g., b, see above) introduced by the mapping. Therefore, this type of s-LMF satisfies the asymptotic limit within all parametrizations, but the initial increase in the intermediate region is significantly slower than for the t-LMF (likely too slow 44 ). The one-electron limit is not correctly described by a simple s-LMF.…”

“…This has been discussed for charge transfer in long-chain donor-acceptor systems, 75 for the energies of d-orbitals in transition-metal systems, 76 or for lower-lying Rydberg excitations. 44 Local hybrids have in principle more freedom to enable the abovementioned balance. In fact, the description of nondynamical correlation is closely related to the mitigation of many-electron SIE (MESIE) and spinpolarization errors (SPEs).…”

“…As a consequence, common LMFs introduce some additional oppositespin nondynamical correlation contributions (cross terms) according to Equation 3, shown to be beneficial for various properties. 44,60 However, since spin-channel and common LMFs in general satisfy the same exact constraints and have a similar real-space behavior, we will not distinguish them in the discussion in this subsection. The spatial behavior of several LMF types is graphically shown along the molecular axis of the NO molecule (Figures 2 and 3).…”

“…That is, a substantial variety of LMFs have been suggested and tested, the first so‐called calibration functions (CFs) to address the gauge problem have been constructed, and substantial progress has been made in the efficient implementation of local hybrids into computer programs, using in particular semi‐numerical integration techniques. Several local hybrids have turned out to provide enhanced accuracy compared to global hybrid functionals for various properties such as thermochemical kinetics and excitation energies . While local hybrids have so far not yet entered the mainstream application of XC functionals in a similar way as global or range‐separated hybrids have, we believe they are on the brink of doing so.…”