Assessing Excited State Methods by Adiabatic Excitation Energies

Abstract

We compile a 109-membered benchmark set of adiabatic excitation energies (AEEs) from high-resolution gas-phase experiments. Our data set includes a variety of organic chromophores with up to 46 atoms, radicals, and inorganic transition metal compounds. Many of the 91 molecules in our set are relevant to atmospheric chemistry, photovoltaics, photochemistry, and biology. The set samples valence, Rydberg, and ionic states of various spin multiplicities. As opposed to vertical excitation energies, AEEs are rigorously defined by energy differences of vibronic states, directly observable, and insensitive to errors in equilibrium structures. We supply optimized ground state and excited state structures, which allows fast and convenient evaluation of AEEs with two single-point energy calculations per system. We apply our benchmark set to assess the performance of time-dependent density functional theory using common semilocal functionals and the configuration interaction singles method. Hybrid functionals such as B3LYP and PBE0 yield the best results, with mean absolute errors around 0.3 eV. We also investigate basis set convergence and correlations between different methods and between the magnitude of the excited state relaxation energy and the AEE error. A smaller, 15-membered subset of AEEs is introduced and used to assess the correlated wave function methods CC2 and ADC(2). These methods improve upon hybrid TDDFT for systems with single-reference ground states but perform less well for radicals and small-gap transition metal compounds. None of the investigated methods reaches "chemical accuracy" of 0.05 eV in AEEs.

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

“…29,57 To this end, Table 2 shows the complete set (96 excited states, nine methods) of calculated ΔE 00 energies. Furthermore, for each group of molecules as well as for data sets containing either all of the 96 excited states or all systems in groups I-III (inorganic molecules) and groups IV-IX (organic molecules), for the inorganic molecules is likely to reflect the relative unimportance of multireference correlation effects for most of the organic excited states studied, whereas for the inorganic systems, at least a few states (e.g., the 1 1 Π state of SiO 86 ) have some double-excitation character or are populated from a multi-reference ground state (as identifiable by various diagnostics [87][88][89] excitation energies compared with experimental absorption maxima, the full inclusion of geometric relaxation effects may render the present analysis oppositely inclined, despite that the comparison is here with experimental ΔE 00 energies.…”

“…29 While that set also includes a few doublet states of organic and inorganic radicals and higher spin-multiplicity states of transition-metal compounds, such states oftentimes require more elaborate methods than TD-DFT and CC2 that explicitly account for multi-reference correlation effects, 73 and were therefore not considered in this work.…”

“…Furthermore, for most of the current excited states, expanding the basis set from double-ζ to triple-ζ quality, or including diffuse basis functions, has a marginal effect on the excitation energies obtained with TD-DFT. 29 Second, excited-state geometries were optimized using analytic TD-DFT [45][46][47] and CC2 43 excited-state gradients. Then, adiabatic excitation energies were obtained as electronic energy differences between ground and excited states at their respective equilibrium geometries, to which were subsequently added ZPVE corrections derived from ground and excited-state frequency calculations at optimized ground and excitedstate structures, respectively.…”

“…Following a preliminary investigation along those lines that focused on the chromophores of the photoactive yellow protein and the green fluorescent protein, 59 this study attempts to meet this objective in a much more comprehensive fashion. Specifically, using both TD-DFT and ab initio methods, we calculate vertical and zero-point vibrational energy (ZPVE)-corrected adiabatic excitation energies for most of the molecules and states contained in the large (91 molecules and 109 adiabatic excitation energies) set of reference data collected by Furche and co-workers, 29 and evaluate how the differences between these energies vary with the choice of method. Besides providing valuable insight into how method-sensitive computed excited-state relaxation energies are, such information helps quantifying how large an error might be introduced when a computational analysis of an experimental absorption spectrum is based on vertical excitation energies alone.…”

“…First, with regard to the work by Furche and co-workers, 29 we consider a slightly reduced version of the 109-member set of excited states introduced by them. However, whereas their study was based exclusively on geometries and vibrational frequencies computed with the B3LYP [60][61][62] 69 CAM-B3LYP 70 and ωB97X-D 71,72 ).…”