Evaluating the interactions between vibrational modes and electronic transitions using frontier orbital energy derivatives

Vibrations affect molecular optoelectronic properties, even at zero kelvin. Accounting for these effects using computational modelling is costly, as it requires many calculations at geometries distorted from equilibrium. Here, we propose a low-cost method for identifying vibrations most strongly coupled to the electronic structure, based on using orbital energy derivatives as a diagnostic.

Our code for both calculating orbital energy derivatives and generating geometries displaced by σi(T) from Turbomole 2 output is available at https://github.com/lisa-schroeder/mode-resolved-molecular-properties.Briefly, the Turbomole evib 4 module provides derivatives of orbital energies with respect to atomic movements, while aoforce outputs normal modes as linear combinations of atomic movements.The Python script get_dE_dsigma uses the output from these modules to calculate the derivative of orbital energies with respect to normal modes.Geometries displaced along a normal mode i by the standard deviation of the thermal population ±σi(T) at temperature T (zero takes zeropoint vibration energy into account) can be obtained by using generate_coords_tmole.
(2) Static binding approximation (SBA) and the Hubbard model.Let us consider a two-electron two-orbital model described with a Hubbard Hamiltonian, in which U and V represent the on-site and nearest-neighbour Coulombic interaction, and εH and εL the HOMO and LUMO energies, respectively (Figure S3).In such a case, the one-electron terms (orange in Figure S3) of the first vertical excitation energy are equal to Egap, while the two-electron terms (blue in Figure S3) correspond to Ebind (cf. Figure S3 with eq (1) in main text).

Figure S1. First vertical excitation energy in terms of one-(orange) and two-electron (blue) contributions.
When extending the Hubbard model to include electron-vibration coupling, a common approximation is to couple the nuclear degrees of freedom to only the one-electron terms [5][6][7] and treat U and V as independent of geometry, which is equivalent to the SBA (ΔEbind = 0, eq (3) in main text).
(3) Performance of the SBA. Figure S2 lists compounds in the Thiel's set in which the S0→S1 transition can be mostly attributed to the HOMO-LUMO transition, and which are not listed in Figure 1.The SBA shows good performance in the case of first four molecules (Figure S2a-d), mostly good performance for (Figure S2e), while small compounds (S2c,f-h) mostly show poor performance.In the case of tetracene (S2a), anthracene (S2b), and hexatriene (S2b), the first optical transition only corresponds 75-85% to the HOMO-LUMO transition, resulting in the overestimation of ΔEgap relative to ΔEbind.A similar overestimation occurs in benzene (Figure 1o) and zinc porphyrin (Figure 1p).
Electronic Supplementary Material (ESI) for ChemComm.This journal is © The Royal Society of Chemistry 2024 This journal is © The Royal Society of Chemistry 20xx Please do not adjust margins Please do not adjust margins (4) Beyond HOMO-LUMO transitions.The SBA may be extended to more optical transitions by performing a single excitedstate (TD-DFT) calculation, and we recommend doing this for any larger molecule.Such a calculation will characterise the excited states in terms of orbital transitions, enabling us to calculate the ΔEgap for any transition.For example, the first optical transition in adenine (Figure S3a) corresponds to HOMO-1→LUMO, while the second transition (Figure S3e) can be written as roughly 80% HOMO→LUMO + 10% HOMO-2→LUMO+1, and both can be captured using the SBA.Similar good results are obtained in case of thymine (Figure S3b,f) and pyridine (Figure 3c,g), while the results for uracil (Figure 3d,h) are poorer.It is also notable that the SBA performs similarly well for both n→π* and π→π* transitions (cf.top and bottom row in Figure S3).Finally, the SBA is not applicable when the composition of the excited states significantly changes with nuclear motion, as it has no way of predicting how the weights of each contribution change with nuclear movement.

Figure S3 .
Figure S3.Comparison of ΔEgap (vertical axis) and ΔES1 or ΔES2 (horizontal axis) for Thiel's set molecules not included in Figure1nor S2, decomposed by normal modes (triangles).In the case of pyridine (1g), the second optically active transition corresponds to a transition to S3.