In Silico Ultrafast Nonlinear Spectroscopy Meets Experiments: The Case of Perylene Bisimide Dye

Spectroscopy simulations are of paramount importance for the interpretation of experimental electronic spectra, the disentangling of overlapping spectral features, and the tracing of the microscopic origin of the observed signals. Linear and nonlinear simulations are based on the results drawn from electronic structure calculations that provide the necessary parameterization of the molecular systems probed by light. Here, we investigate the applicability of excited-state properties obtained from linear-response time-dependent density functional theory (TDDFT) in the description of nonlinear spectra by employing the pseudowavefunction approach and compare them with benchmarks from highly accurate RASSCF/RASPT2 calculations and with high temporal resolution experimental results. As a test case, we consider the prediction of femtosecond transient absorption and two-dimensional electronic spectroscopy of a perylene bisimide dye in solution. We find that experimental signals are well reproduced by both theoretical approaches, showing that the computationally cheaper TDDFT can be a suitable option for the simulation of nonlinear spectroscopy of molecular systems that are too large to be treated with higher-level RASSCF/RASPT2 methods.


This PDF includes:
• Summary of the parameters employed in the spectroscopy simulations; • Configurations of TDDFT states: comparison of different representations; • PBI and PBI-(R + ) 2 linear absorption spectra; • PBI and PBI-(R + ) 2 : RASSCF active space orbitals and excited state energies; • PBI relevant DFT molecular orbitals; • Normal modes at the MP2 and DFT level of theory; • Spectral densities of the bright transitions; • Shape of the experimental laser pulses and their overlap with the (experimental) linear absorption spectrum; • TA spectrum for GSB, SE, and GSB+SE contributions; • Fourier Analysis of TA data with finite and infinite resolution; • Fourier Analysis of TA data for GSB, SE, and GSB+SE contributions; • Cuts of the 2DES spectra at selected t 2 times. S2 S1 Summary of the parameters employed in the spectroscopy simulations Simulations were carried on with t 2 in the interval (0, 600) fs. In fact, the relevant dynamics happens before 600 fs, and 600 fs upper limit is long enough to safely perform the Fourier transform of the low frequency modes.
The time delay between consecutive t 2 points was set to 4 fs. This is short enough to allow proper sampling of the high-frequency modes (so that they can be revealed by Fourier transform of the data), but also long enough to reduce the number of t 2 snapshots to be evaluated.
Other parameters include the OBO λ and Λ −1 values, set to 240 cm −1 and 40 fs respectively, and whose meaning has been explained in the main text.
As discussed in the main text, the lifetime of the S 1 state was set to infinity (i.e. very long compared to the time scale of the simulations reported here). This allows to use analytic expressions for the response functions.

S2 Configurations of TDDFT states: comparison of different representations
The following Table reports on the leading configuration (weight > 0.05) of the states involved in the bright transitions. Two different representations are compared: the one directly obtained from the calculations (in which the HOMO→LUMO configuration is used as a reference), and one in which the closed shell GS configuration is used as a reference. The latter was also employed in Table 1 of the main text, as it makes the comparison with RASSCF/RASPT2 simpler. To make an example, the HOMO → LUMO transition from the HOMO → LUMO reference, corresponds to a HOMO ⇒ LUMO from the closed-shell reference; the HOMO−3 → HOMO transition from the HOMO → LUMO reference, instead, corresponds to a HOMO−3 → LUMO transition. Table S1: Comparison of TDDFT leading configurations (weights > 0.06) of the states involved in the transitions, for a) the GS (closed-shell configuration) as reference, and b) the S 1 state (HOMO → LUMO configuration) as reference. Single and double arrows denote single and double occupied to virtual transitions, respectively. The orbital are labeled consistently to their depiction if Figure S4.

Wavelength [nm]
Figure S1: Linear absorption (vibronic) spectra of PBI (red) and PBI-(R + ) 2 ion (blue) computed at RASSCF/RASPT2 level of theory; due to the computational cost of the MP2 method, frequencies has been computed at DFT level (using ωB97X-D functional and 6-31G * * basis-set). To match the experimental λ max of a 1 mM solution of PBI-(R + ) 2 in acetonitrile (black curve), the computed spectra have been blue-shifted by +1200 cm −1 (PBI) and +400 cm −1 (PBI-(R + ) 2 ). Upon normalization to the maximum of the first vibronic band, the successive vibronic bands of PBI-(R + ) 2 ion are more intense with respect to those of the neutral PBI and closer to the experimental spectra. The total reorganization energy for the S 1 state in PBI-(R + ) 2 is 960 cm −1 .

S5
S4 PBI and PBI-(R + ) 2 : RASSCF active space orbitals Figure S2: SA-10-RAS(20,4,4;10,0,10) natural orbitals used for multireference calculations on PBI-(R + ) 2 ion. Natural orbitals are grouped according to the irreducible representations of C 2h symmetry point group. S5 PBI and PBI-(R + ) 2 : RASPT2 excited state energies Table S2: The lowest ten electronic states were considered for all allowed symmetries (four) of the C 2h and D 2h point groups of PBI and PBI-(R + ) 2 . Here we report the states transition energy at the SS-RASPT2 level. The PBI S 0 , S 1 , S 7 and S 12 and the PBI-(R + ) 2 S 0 and S 1 states are highlighted in bold. Transition dipole moments were computed for relevant transitions allowed by symmetry, i.e. A g → B 2u /B 3u and B 3u → A g /B 1g for PBI, and A g → A u /B u and B u → A g /B g for PBI-(R + ) 2 . The brightest transitions have been reported in the main text.  Figure S4: Relevant DFT molecular orbitals for S 1 , S 5 and S 7 excited states of PBI (isovalue 0.01). Molecular orbitals are grouped according to the irreducible representations of D 2h point group. The orbitals are labeled in two ways: the black label below each molecular orbital refers to the RASSCF assignation of Figure S3 (which facilitate the comparison of the states configurations of Table 1

DFT MP2
MP2 DFT Figure S5: Selected normal modes computed at MP2 and DFT level of theory. For each normal mode, the corresponding frequency (in cm −1 ) is also reported in parenthesis. All normal modes reported here are of A g symmetry (i.e. total symmetric modes).

S9
S8 Spectral densities Absorption (a.u.) Wavelength (nm) Figure S8: Experimental linear absorption spectrum of PBI (black) and pulse shapes used in the experiments. The blue pulse shape was used for TA, while the red pulse shape was employed in 2DES experiments. Note how the latter covers a region in which the ESA signals are observed, therefore allowing for their detection.  Figure S12: Comparison of some cuts of the 2DES maps reported in Figure 5 of the main text, for both experiment and simulations. The cuts were performed at Excitation wavelength equal to 530 nm, and for the t 2 times 10, 30, 50, 70 and 90 fs. The main qualitative features of the experiment are reproduced in the simulations, while quantitative differences can be noticed in terms of peak position (minor red-shift of the SE progression at both levels of theory, and blue-shift of the ESA peak at the TDDFT level) and line-shape (excess of broadening of the main GSB/SE peak at early times at both levels of theory, and for the ESA peak at all times at the RASSCF/RASPT2 level).