Vibronic structure of photosynthetic pigments probed by polarized two-dimensional electronic spectroscopy and ab initio calculations

Using polarized 2D spectroscopy and state-of-the-art TDDFT calculations to uncover the vibronic structure of primary photosynthetic pigments and its effect on ultrafast photoexcited dynamics.

Pump and probe spectra, along with absorption spectra of Chl a and Bchl a

S3. Fits of time traces in 2DES
Time traces of cross peaks are fit using the following formula 1 S1 ( ) = - where the first term accounts for the coherent artifact and the second term is to fit the population dynamics. For the coherent artifact, we only consider the instantaneous electronic response since this term is much stronger than the Raman scattering nearby time zero. Here K = D e 0 exp[Ɛ 4 τ pp 2 (ω 2ω 0 ) 2 /(1+2Ɛ 2 )] where D e 0 is the amplitude of the electronic response function. ε = τ prb /τ pp where τ pp , τ prb is the pulse duration of the pump and probe, respectively. The pump duration is determined from a chirp scan. The probe pulse duration (τ prb ) is estimated to be around 10 fs by measuring the 2DES of crystal violet (CV) (see below). , are the center frequency of the probe and the probing frequency, 0 2 respectively. 1 (1/τ 1 ) , 2 (1/τ 2 ) , are internal conversion rates from E 3 or E 2 (or S 2 ) to E 1 (S 1 ) and from E 1 (S 1 ) to the ground electronic state. IRF is the instrument response function which is assumed to be a Gaussian function. To obtain a satisfactory fit, we need to include an empirical phase ( in the term of the sine ) function corresponding to the electronic response for Chl a. This term can be attributed to the high-order phase which can cause a small time zero shift (< 2fs). The fitting results are shown in Table S1.   Figure S4a and S4c depict the dynamics of anisotropy for Bchl a and Chl a. We find that the anisotropic values for both Bchl a and Chl a remain almost unchanged after internal conversion is complete. Since the Stokes shift owing to vibrational relaxation occurs on the time scale of picoseconds, the constant anisotropic values also indicate that ground state bleaching (GSB) and stimulated emission (SE) originate from the same transition dipole moment. We note the potential interference of GSB and excited-state absorption (ESA) can alter our interpretation of anisotropy. To take ESA into account, we calculate the angle using the following formula 2 : where r is the anisotropy, θ ESA is the angle between transition dipole moments corresponding to ESA and photoexcitation, θ da represents the angle between transition dipole moments for photoexcitation and detection, and η is the amplitude ratio of GSB/SE and ESA. We set η to be 0.05 in our calculations for both Bchl a and Chl a since previous studies showed that ESA is likely to be less than 5% of GSB/SE [3][4][5] . Figure  S4b and d show the extracted angle for the limiting cases of θ ESA = 0 o , 90 o , indicating that a weak ESA has a small effect.

S6. Orbitals
All Orbitals shown in the following figures are calculated with the SRSH-PCM(ωPBE) approach in the 6-31++G(d,p) basis set.     Table S3 Chlorophyll a orbital energies (upper rows, in Hartree) and orbital transition contributions (last two rows) are calculated within the RSH(ωB97X-D) approach using the 6-31++G(d,p) basis set. Note that orbital energies and contributions to excited states differ from the optimally-tuned and screened approach reported in Table S2.  Table S5 Bacteriochlorophyll a orbital energies (upper rows, in Hartree) and orbital transition contributions (last two rows) are calculated within the RSH(ωB97X-D) approach using the 6-31++G(d,p) basis set. Note that orbital energies and contributions to excited states differ from the optimally-tuned and screened approach reported in Table S4.

S7. Primary Modes
Figure S11 Illustration of the primary mode (1395 cm -1 ) in the penta-coordinated Chl a S 1 state. Vectors were scaled and the acetone molecule was removed for clarity.
Figure S12 Illustration of the primary mode (1239 cm -1 ) in the penta-coordinated Bchl a S 1 state. Vectors were scaled and the acetone molecule was removed for clarity.

S8. Structures
Structure optimizations of Chlorophyll a (Chl a) and Bacteriochlorophyll a (Bchl a) were performed using DFT and [TD]DFT calculations for ground state and excited state equilibrium structures, respectively. The conductor-like polarizable continuum model (C-PCM) was used in all calculations. In all structures, the phytyl-containing side group was replaced by a single hydrogen. Excitation energy differences to the full molecule-calculated for the ground state equilibrium structures of the unligated molecules only were around 0.01eV. If not specifically states otherwise, the 6-31++G(d,p) basis set was used.
Structure optimization based on the SRSH-PCM(ωPBE) functional did not converge properly and are not reported in this study. The poor convergence might originate in the structure-dependence of the tuning parameters α and β. Following the successful protocol of Ref. 6,7, the dispersion-corrected ωB97X-D functional was used instead. Along selected degrees of freedom, single point calculations with the SRSH-PCM(ωPBE) functional were performed in the proximity of the ωB97X-D-ground state equilibrium structure. Results confirm that within the SRSH-PCM(ωPBE) approach the energy minimum is located at the same structure. The popular B3LYP functional was used for comparison as well. In comparison, ωB97X-D-structures are more compact than the B3LYP-structures, resulting in a shorter Mg-N and C-N bondlengths, a weaker doming formation of the central Mg ion, and side groups being further bent towards the porphyrin ring. The total root means square deviations (RMSD) between these two structures of the penta-coordinated Chl a and Bchl a are 0.232 Å and 0.294 Å, respectively. A sensitivity analysis of total energies and excitation energies with respect to these structural changes is outlined in the following section.

S9. Comparison of total energies and excitation energies between wB97X-D-optimized and B3LYPoptimized structures
All energy calculations were performed with [TD]DFT using the 6-31++G(d,p) basis set and the C-PCM. Either the range-separated hybrid functional ωB97XD (in the following ''RSH" abbreviated) or the SRSH-PCM approach with the ωPBE functional (in short "SRSH") were used. Tuning parameters (α, β, γ) were determined for penta-coordinated Bchl a in ωB97X-D structures (0.266, -0.218, 0.129) and in B3LYP structure (0.256, -0.208, 0.122) and for penta-coordinated Chl a in ωB97X-D structures (0.269, -0.219, 0.128) and in B3LYP structure (0.251, -0.201, 0.120).  B3LYP-structures (for experimental references, see main manuscript). The corresponding values based on ωB97X-D structures are listed in Table 1 of the main manuscript. Using B3LYP structures, all excitation energy calculations (RSH and SRSH) show seemingly better agreement with experimental values (see discussion in Table S28). In particular, the Bchl a S 1 excitation energies dropped by approximately 0.5eV correcting the erratic S 1 -S 2 energy gap behavior seen with ωB97X-D-based structures. Bchl a RSH-excitation energies are in perfect agreement with experiments. Chl a RSH-energies are improved compared to ωB97X-D-based structures but are still overestimated and show a largely increased S 1 -S 2 energy gap. SRSH-excitation energies are generally lower, correcting for the overestimation reported for ωB97X-D structures in the main manuscript. SRSH shows consistently good agreement with experiments and particularly in the predicted S 1 -S 2 energy gap.  Table S27 Total SCF ground state energies (in Hartree). Expectedly, using the RSH functional ωB97X-D, the ωB97X-Doptimized structures are more stable than B3LYP-optimized structures. Yet, it is noteworthy that the latter yield better agreement with experimental values in terms of excitation energies (see previous Table). Within the SRSH approach, B3LYPoptimized structures were found to be more stable in the case of Chl a, whereas for Bchl a ωB97X-D-structures are more stable. These findings indicate that mixing functionals for optimization and energy calculation might be problematic. For consistency, we decided to report only ωB97X-D-structures in the main manuscript, despite slightly better experimental agreement (Table S26) Table S28 Energy differences (in eV) between the ground state energy at the S 0 equilibrium geometry and the S n (n=1-2) total energy at the respective excited state optimized geometry of penta-coordinated compounds. The minimum-to-minimum energy difference corresponds approximately to the fundamental line originating from a transition between the lowest vibrational state of the origin and the target electronic state. Interestingly, despite large differences in the structures and in the vertical excitation energies, the minimum-to-minimum energy differences are only marginally affected by the optimization protocol employed and are in good agreement with the experiment. We therefore interpret the overestimation of vertical excitation energies observed for ωB97X-D-based structures (calculated with either approach RSH and SRSH) as a consequence of an overestimation in the coordinate shift between the equilibrium structures of ground and excited states. This can result in an overestimated reorganization energy and thus increased Huang-Rhys factors.