Noncovalently bound excited-state dimers: a perspective on current time-dependent density functional theory approaches applied to aromatic excimer models

Excimers are supramolecular systems whose binding strength is influenced by many factors that are ongoing challenges for computational methods, such as charge transfer, exciton coupling, and London dispersion interactions. Treating the various intricacies of excimer binding at an adequate level is expected to be particularly challenging for Time-Dependent Density Functional Theory (TD-DFT) methods. In addition to well-known limitations for some TD-DFT methods in the description of charge transfer or exciton coupling, the inherent London dispersion problem from ground-state DFT translates to TD-DFT. While techniques to appropriately treat dispersion in DFT are well-developed for electronic ground states, these dispersion corrections remain largely untested for excited states. Herein, we aim to shed light on current TD-DFT methods, including some of the newest developments. The binding of four model excimers is studied across nine density functionals with and without the application of additive dispersion corrections against a wave function reference of SCS-CC2/CBS(3,4) quality, which approximates select CCSDR(3)/CBS data adequately. To our knowledge, this is the first study that presents single-reference wave function dissociation curves at the complete basis set level for the assessed model systems. It is also the first time range-separated double-hybrid density functionals are applied to excimers. In fact, those functionals turn out to be the most promising for the description of excimer binding followed by global double hybrids. Range-separated and global hybrids—particularly with large fractions of Fock exchange—are outperformed by double hybrids and yield worse dissociation energies and inter-molecular equilibrium distances. The deviation between each assessed functional and reference increases with system size, most likely due to missing dispersion interactions. Additive dispersion corrections of the DFT-D3(BJ) and DFT-D4 types reduce the average errors for TD-DFT methods but do so inconsistently and therefore do not offer a black-box solution in their ground-state parametrised form. The lack of appropriate description of dispersion effects for TD-DFT methods is likely hindering the practical application of the herein identified more efficient methods. Dispersion corrections parametrised for excited states appear to be an important next step to improve the applicability of TD-DFT methods and we hope that our work assists with the future development of such corrections.


SI.1 Explicit definition of coordinates defining intermonomer distance
Figure S1: Diagrams of excimer model structures illustrating the atomic coordinates used to define the inter-monomer distance for dissociation energy curves.Z-matrix templates used for defining the inter-monomer distance from these coordinates is given in the other supplementary files.Refer to README file for usage; xyz of optimised dimers are also provided.

SI.2.1 Numerical Data for Wavefunction Methods
Table S1: Explicit minima of SCS-CC2 and CC2 interaction energy curves for the basis set study.The distance of the minimum is denoted re ( Å) and the associated interaction energy at that position ∆E (kcal/mol).

SI.3 The ωB97X Problem
There were some unanticipated technical problems for ωB97X resulting in wobbly interaction energy curves as shown in the figures below.

SI.4.3 Numerical Data For Binding Minima and Associated Percentage errors
Table S5: Binding of the benzene excimer described by the dissociation energy (De; kcal/mol), equilibrium inter-monomer distance (re; Å) and the unsigned percentage errors of each compared to that of the SCS-CC2/CBS(3,4) reference.S6, which is the bestpossible approximation to the true minimum given the above-mentioned difficulties with this functional.We assessed the pure ωB97X.Its various dispersion-corrected variants all depend on slightly different underlying XC expressions, S1-S6 which is why dispersion-corrected results are not provided.

Functional
Table S6: Binding of the naphthalene described by the dissociation energy (De; kcal/mol), equilibrium inter-monomer distance (re; Å) and the unsigned percentage errors of each compared to that of the SCS-CC2/CBS(3,4) reference.12.9 12.9 12.9 1.0 1.0 1.0 a Based on the observed minimum for the curve shown in Fig. S6, which is the bestpossible approximation to the true minimum given the above-mentioned difficulties with this functional.We assessed the pure ωB97X.Its various dispersion-corrected variants all depend on slightly different underlying XC expressions, S1-S6 which is why dispersion-corrected results are not provided.

Functional
Table S7: Binding of the anthracene excimer described by the dissociation energy (De; kcal/mol), equilibrium inter-monomer distance (re; Å) and the unsigned percentage errors of each compared to that of the SCS-CC2/CBS (3,4)  1.6 a Based on the observed minimum for the curve shown in Fig. S6, which is the bestpossible approximation to the true minimum given the above-mentioned difficulties with this functional.We assessed the pure ωB97X.Its various dispersion-corrected variants all depend on slightly different underlying XC expressions, S1-S6 which is why dispersion-corrected results are not provided.
Table S8: Binding of the pyrene excimer described by the dissociation energy (De; kcal/mol), equilibrium inter-monomer distance (re; Å) and the unsigned percentage errors of each compared to that of the SCS-CC2/CBS(3,4) reference.1.9 1.9 1.9 a Based on the observed minimum for the curve shown in Fig. S6, which is the bestpossible approximation to the true minimum given the above-mentioned difficulties with this functional.We assessed the pure ωB97X.Its various dispersion-corrected variants all depend on slightly different underlying XC expressions, S1-S6 which is why dispersion-corrected results are not provided.

SI.4.4 Mean Absolute Deviations
Table S9: Mean absolute deviations (MADs) for the dissociation energy (De; kcal/mol) and equilibrium distance (re; Å) characterising the description of excimer binding by the method relative to SCS-CC2/CBS(3,4) reference.S6, which are the best-possible approximations to the true minima given the above-mentioned difficulties with this functional.We assessed the pure ωB97X.Its various dispersion-corrected variants all depend on slightly different underlying XC expressions, S1-S6  which is why dispersion-corrected results are not provided.

Figure S2 :
Figure S2: Dissociation curves for the lowest-lying excited state of the fully-stacked benzene dimer with SCS-CC2 across two successive basis sets in the Karlsruhe family (def2-nZVP) and their complete basis set extrapolated result, CBS(3,4).A magnified snapshot of the minima is shown to the right.

Figure S3 :
Figure S3: Dissociation curves for the lowest-lying excited state of the fully-stacked naphthalene dimer with spin-component scaled CC2 (SCS-CC2) across two successive basis sets in the Karlsruhe family (def2-nZVP) and their complete basis set extrapolated result, CBS(3,4).A magnified snapshot of the minima is shown to the right.

Figure S4 :
Figure S4: Dissociation curves for the lowest-lying excited state of the fully-stacked anthracene dimer with spin-component scaled CC2 (SCS-CC2) across two successive basis sets in the Karlsruhe family (def2-nZVP) and their complete basis set extrapolated result, CBS(3,4).A magnified snapshot of the minima is shown to the right.

Figure S5 :
Figure S5: Dissociation curves for the lowest-lying excited state of the fully-stacked pyrene dimer with spin-component scaled CC2 (SCS-CC2) across two successive basis sets in the Karlsruhe family (def2-nZVP) and their complete basis set extrapolated result, CBS(3,4).A magnified snapshot of the minima is shown to the right.

Figure S6 :
Figure S6: Dissociation energy curves for ωB97X and SCS-CC2/CBS(3,4) for each excimer.Due to the difficulty of identifying the actual minima for ωB97X we defined each De and re from the observed minimum of each curve.

Figure S7 :
Figure S7: Dissociation energy curves for TD-BLYP and global hybrid variants with varied amounts of Fock exchange.CCS curves are also shown.

Figure S8 :
Figure S8: Dissociation energy curves for TD-PBE and global hybrid variants with varied amounts of Fock exchange.

Figure S10 :
Figure S10: Dissociation energy curves for the naphthalene excimer comparing density functional approximations, with and without DFT-D type dispersion corrections, to the SCS-CC2/CBS(3,4) reference.All TD-DFT results are based on the def2-inter-monomerTZVP basis set.

Figure S11 :
Figure S11: Dissociation energy curves for the anthracene excimer comparing density functional approximations, with and without DFT-D type dispersion corrections, to the SCS-CC2/CBS(3,4) reference.

Figure S12 :
Figure S12: Dissociation energy curves for the pyrene excimer comparing density functional approximations, with and without DFT-D type dispersion corrections, to the SCS-CC2/CBS(3,4) reference.

Figure S13 :
Figure S13: Mid-range region of SCS-CC2 dissociation curves for the benzene excimer with two truncated basis sets and their CBS(3,4) extrapolation.

Figure S14 :
Figure S14: Mid-range region of the dissociation energy curves for the benzene excimer comparing density functional approximations, with and without DFT-D type dispersion corrections, to the SCS-CC2/CBS(3,4) reference.

Figure S15 :
FigureS15: Mid-range region of the dissociation energy curves for the naphthalene excimer comparing density functional approximations, with and without DFT-D type dispersion corrections, to the SCS-CC2/def2-TZVP reference.All TD-DFT results are based on the def2-TZVP basis set.As distances beyond the binding region were not the focus of the paper, some curves may show strange shapes in the mid-range due to the small number of data points however they still showcase the identified positive region.

Figure S16 :
FigureS16: Mid-range region of the dissociation energy curves for the anthracene excimer comparing density functional approximations, with and without DFT-D type dispersion corrections, to the SCS-CC2/def2-TZVP reference.All TD-DFT results are based on the def2-TZVP basis set.As distances beyond the binding region were not the focus of the paper, some curves may show strange shapes in the mid-range due to the small number of data points however they still showcase the identified positive region.

Figure S17 :
Figure S17: Mid-range region of the dissociation energy curves for the pyrene excimer comparing density functional approximations, with and without DFT-D type dispersion corrections, to the SCS-CC2/def2-TZVP reference.All TD-DFT results are based on the def2-TZVP basis set.As distances beyond the binding region were not the focus of the paper, some curves may show strange shapes in the mid-range due to the small number of data points however they still showcase the identified positive region.

Table S4 :
Dissociation energies (De), equilibrium inter-monomer distances (re) and signed percentage errors relative to the SCS-CC2/CBS(3,4) reference for TD-PBE and global hybrid variants with varied amounts of Fock exchange.
Based on the observed minimum for the curve shown in Fig. reference.
Based on the observed minima for the curves shown in Fig.