How Method-Dependent Are Calculated Differences between Vertical, Adiabatic, and 0–0 Excitation Energies?

Abstract

ABSTRACT:Through a large number of benchmark studies, the performance of different quantum chemical methods in calculating vertical excitation energies is today quite well established. Furthermore, these efforts have in recent years been complemented by a few benchmarks focusing instead on adiabatic excitation energies. However, it is much less well established how calculated differences between vertical, adiabatic and 0-0 excitation energies vary between methods, which may be due to the cost of evaluating zero-point vibrational energy corrections for excited states. To fill this gap, we have calculated vertical, adiabatic and 0-0 excitation energies for a benchmark set of molecules covering both organic and inorganic systems. Considering in total 96 excited states and using both TD-DFT with a variety of exchange-correlation functionals and the ab initio CIS and CC2 methods, it is found that while the vertical excitation energies obtained with the various methods show an average (over the 96 states) standard deviation of 0.39 eV, the corresponding standard deviations for the differences between vertical, adiabatic and 0-0 excitation energies are much smaller: 0.10 (difference between adiabatic and vertical) and 0.02 eV (difference between 0-0 and adiabatic). These results provide a quantitative measure showing that the calculation of such quantities in photochemical modeling is well amenable to low-level methods. In addition, we also report on how these energy differences vary between chemical systems and assess the performance of TD-DFT, CIS and CC2 in reproducing experimental 0-0 excitation energies.

Assessment of a composite CC2/DFT procedure for calculating 0–0 excitation energies of organic molecules

“…Indeed, for such states, CC2 typically shows excellent agreement (∼0.1 eV) with experimental ΔE 00 energies [25,26,29].…”

“…we investigated to what extent calculated differences between ΔE ve , ΔE ad and ΔE 00 energies vary with the choice of method [26]. To the best of our knowledge, this was the first investigation of its kind to be reported for a large set of excited states and methods.…”

“…Instead of calculating ΔE ve energies and comparing these with experimental absorption maxima, a few excited-state benchmarks have rather focused on adiabatic or 0-0 excitation energies [15][16][17][18][19][20][21][22][23][24][25][26][27][28][29], herein denoted ΔE ad and ΔE 00 , respectively. As illustrated in Figure 1, these are energy differences between ground and excited states at their respective equilibrium geometries, without (ΔE ad ) or with (ΔE 00 ) inclusion of the lowest vibrational level.…”

“…Specifically, for a slightly reduced version of Furche's 109-member set of excited states [21], ΔE ve , ΔE ad and ΔE 00 energies were calculated using TD-DFT with seven different functionals, CC2 and the configuration interaction singles (CIS) method [31]. Thereby, it was found that while the average standard deviations for the ΔE ve , ΔE ad and ΔE 00 energies obtained with the nine methods amount to ∼0.4 eV, the corresponding values for the ΔΔE ad = ΔE ad -ΔE ve and ΔΔE 00 = ΔE 00 -ΔE ad energy differences are only 0.10 and 0.02 eV, respectively [26]. These results are a clear, quantitative indication that while calculating excitation energies is a notoriously challenging and methoddependent task in quantum chemistry, determining the photochemically relevant ΔΔE ad (excitedstate relaxation energy) and ΔΔE 00 (difference in ZPVE corrections between the excited state and the ground state) quantities is much less sensitive to the choice of method.…”

“…A frequent observation regarding single-reference organic excited states in ΔE 00 -based benchmarks is that TD-DFT, which is currently the most popular tool for modeling excited states, cannot fully match the performance of CC2 [25,26,29]. Indeed, for such states, CC2 typically shows excellent agreement (∼0.1 eV) with experimental ΔE 00 energies [25,26,29].…”