1. TOWARD THE ACCURATE SIMULATION OF TWO-DIMENSIONAL ELECTRONIC SPECTRA
International Symposium on Molecular Spectroscopy
70TH MEETING - JUNE 22-26, CHAMPAIGN-URBANA, ILLINOIS
TOWARD THE ACCURATE SIMULATION OF TWO-DIMENSIONAL ELECTRONIC SPECTRA
Angelo Giussani, Artur Nenov, Javier Segarra-Martí, Vishal K. Jaiswal, Ivan Rivalta, Elise Dumont, Shaul Mukamel, Marco Garavelli
Dipartimento di Chimica G. Ciamician, Universit di Bologna, Via F. Selmi 2, Bologna, Italy.

2. Outline Introduction on two-dimensional (2D) electronic spectroscopy
Basic principles and potentialities
How to compute 2D electronic spectra?
Static approach
Single-trajectory approach
Static and single-trajectory approach at work: the 2D spectra of pyrene
Conclusions

3. Introduction on two-dimensional electronic spectroscopy

4. 2D electronic spectroscopy
2D electronic spectroscopy is a nonlinear optical technique that measures the full nonlinear polarization of a quantum system in third order (R3) with respect to the field-matter interaction.
Schematic representation of experimental setup
for heterodyne detected
three-pulses photon echo
k1,k2: Pump signal split in two pulses. Varying the t1 time and through a subsequent Fourier transformation the spectral resolution for the pump frequency is obtained. k3: Probe signal, probing the sample after t2 time of the pump interaction.
Rivalta I, Nenov A, Cerullo G, Mukamel S, Garavelli M, Int. J. Quantum Chem., 2014, 114, 85

5. 2D electronic spectra f La Lb e g Pyrene 2D spectrum at t2=0
Red; excitation/de-excitation to/from the pump
reachable excited states
Blue ; excitation to a higher excited state
pyrene electronic states f
pump La Lb e
bright
dark g
La trace

6. 2D electronic spectra f La Lb e g Pyrene 2D spectrum at t2=0
Red; excitation/de-excitation to/from the pump
reachable excited states
Blue ; excitation to a higher excited state GS
bleaching
pyrene electronic states f
pump La Lb e
bright
dark g
La trace

7. 2D electronic spectra f La Lb e g Pyrene 2D spectrum at t2=0
Red; excitation/de-excitation to/from the pump
reachable excited states
Blue ; excitation to a higher excited state ES
absorption GS
bleaching
pyrene electronic states f ES
absorption ES
absorption
pump La Lb e
bright
dark g
La trace

8. 2D electronic spectroscopy; main potentialities
Model peptide containing benzene and phenol*
Potentially, 2D electronic spectroscopy provides both high spectral and temporal resolution and can clearly distinguish between different chromophores and/or bright states present in a system. This in turn will in principle allow to:
resolve chromophore interactions
disentangles de-excitation pathways
resolves population transfer 1D 2D
benzene
phenol
*Nenov A, Beccara S, Rivalta I, Cerullo G, Mukamel S, Garavelli M Chem. Phys. Chem. 2014, 15, 1-10

9. How to compute 2D electronic spectra

10. Nonlinear optical spectroscopy theory*
In semi-impulsive limit (laser pulses shorter than time separation between them) the emitting signal is proportional to the third order response function, R(3)
𝜇 is the transition dipole moment operator, G(t) is the retarded Green’s function, which governs the field-free propagation of the density matrix r(t)
* S. Mukamel, Principles of Nonlinear Optics and Spectroscopy, Oxford Unversity Press, Oxford, UK, 1995.

11. Mixed quantum-classical dynamics
In mixed quantum-classical dynamics, the following coupled equations govern the nuclei and electrons respectively: ;
Making the approximation of using one single trajectory, instead of a swarm, is equivalent to neglect electronic inter-state interactions, which translates in:

12. Mixed quantum-classical dynamics
In mixed quantum-classical dynamics, the following coupled equations govern the nuclei and electrons respectively: ;
Making the approximation of using one single trajectory, instead of a swarm, is equivalent to neglect electronic inter-state interactions, which translates in:
coherence dynamics

13. Mixed quantum-classical dynamics
In mixed quantum-classical dynamics, the following coupled equations govern the nuclei and electrons respectively: ;
Making the approximation of using one single trajectory, instead of a swarm, is equivalent to neglect electronic inter-state interactions, which translates in:
coherence dynamics

14. Mixed quantum-classical dynamics
In mixed quantum-classical dynamics, the following coupled equations govern the nuclei and electrons respectively: ;
Making the approximation of using one single trajectory, instead of a swarm, is equivalent to neglect electronic inter-state interactions, which translates in:
coherence dynamics g f e
pump
probe

15. Brownian harmonic oscillator model
Since the calculation of the coherence dynamics using a single trajectory can cause significant problems, it is preferable to use the analytical solution provided by the Brownian harmonic oscillator model (BHO)*:
BHO
Where wij is adiabatic transition between the involved electronic states,
* S. Mukamel, Principles of Nonlinear Optics and Spectroscopy, Oxford Unversity Press, Oxford, UK, 1995.

16. Brownian harmonic oscillator model
Since the calculation of the coherence dynamics using a single trajectory can cause significant problems, it is preferable to use the analytical solution provided by the Brownian harmonic oscillator model (BHO)*:
BHO
Where wij is adiabatic transition between the involved electronic states,
and g(t) is a function of the Franck-Condon coefficients wk and Dk
g(t) can be obtained using a Fourier series to fit the temporal evolution of the electronic gap Ei(t)-Ej(t) along a computed dynamics on the populated excited state. Since dynamics are expensive, the BHO model is only used for coherence dynamics among ground and first bright state (i.e. along t1)
* S. Mukamel, Principles of Nonlinear Optics and Spectroscopy, Oxford Unversity Press, Oxford, UK, 1995.

17. Static and single-trajectory approaches
Using the presented approximations, (i.e. quantum-classical description of the system, neglecting electronic inter-state interactions, Brownian harmonic oscillator model along t1) the third order response is written as follows, which is what we referred as the single-trajectory approach:

18. Static and single-trajectory approaches
Using the presented approximations, (i.e. quantum-classical description of the system, neglecting electronic inter-state interactions, Brownian harmonic oscillator model along t1) the third order response is written as follows, which is what we referred as the single-trajectory approach:
Making the further assumption that the coherence dynamics is slower than the duration of the experiment (i.e. that during t1 and t3 the system has no enough time to evolve) leads to the following simplification, which constitutes the static approach:

19. Static and single-trajectory approaches at work: the 2D spectra of pyrene

20. Pyrene photophysics Pyrene
Pyrene is characterized by two low-lying excited states: the dark 1pp* Lb state (S1) and the bright 1pp* La state (S2). Experimental evidences show that from the La state the system decays with a time constant of 85 fs to the Lb states, where it remains trapped in the Lb minimum up to the picosecond time scale.*
Pyrene
La pp*
Lb pp* gs
85 fs
1 ps hn
* N. Krebs, I. Pugliesi, J. Hauer and E. Riedle, New J. Phys., 2013, 15,

21. Pyrene 2D electronic spectrum at t2=0
1° pulse
2° pulse
3° pulse
La pp*
Lb pp* gs
85 fs
1 ps
2°,3°, pulses
1° pulse
During t1 the system is excited into the bright La state and evolves along it. Setting t2=0 means that the third (probe) pulse arrive at the same time that the second pumping pulse, which in this case translate in probing the system along its evolution on the La potential energy hypersurface.

22. Pyrene 2D electronic spectrum at t2=0
Consequently, in order to compute the spectrum a t2=0, a CASSCF dynamics on the La state from a previously optimized ground state geometry has been performed, and along it the high-lying excited states have been evaluated performing SA-60-RASSCF(4,8|0,0|4,8)/SS-RASPT2 calculations at selected points. (dynamics performed with Gaussian-Cobram, RASSCF-RASPT2 computation with Molcas)
SA-60-RASSCF(4,8|0,0|4,8)
SS-RASPT2
La pp*
Lb pp* gs
85 fs
1 ps
2°,3°, pulses
1° pulse
La CASSCF dynamics f e

23. Pyrene 2D electronic spectrum at t2=0
Using a development version of Spectron 2.7 for simulating the spectra:
Single-
trajectory
approach
Static
approach
Passing from the static to the single-trajectory approach description (which accounts for the evolution of the system during times t1 and t3) causes:

24. Pyrene 2D electronic spectra at t2=0
Using a development version of Spectron 2.7 for simulating the spectra:
Single-
trajectory
approach
Static
approach
Passing from the static to the single-trajectory approach description (which accounts for the evolution of the system during times t1 and t3) causes:
1- appearing of the vibronic structure of the GS bleaching

25. Pyrene 2D electronic spectra at t2=0
Using a development version of Spectron 2.7 for simulating the spectra:
Single-
trajectory
approach
Static
approach
Passing from the static to the single-trajectory approach description (which accounts for the evolution of the system during times t1 and t3) causes:
1- appearing of the vibronic structure of the GS bleaching
2- significant spectral shifts of the peaks

26. Pyrene 2D electronic spectrum at t2=1 ps
1° pulse
2° pulse
3° pulse
There is no need to account for the dynamics evolution during t3, since the system doesn’t evolve, being trapped in the Lb minimum. The two approaches provide the same description for the response of the system with the third pulse, so the only missing data for the construction of the spectrum t2=1ps are the higher excited states at the Lb minimum.
2° pulse
La pp*
Lb pp* gs
85 fs
1 ps
3° pulse
1° pulse

27. Pyrene 2D electronic spectrum at t2=1 ps
Single-
trajectory
approach*
(gas phase)
Experimentally
recorded**
(methanol)
The comparison between theory and experiment shows a remarkable agreement, validating the approximations in the single-trajectory approach.
** Krebs N, Pugliesi I, Hauer J, Riedle E, New J. Phys., 2013, 15,
*Nenov A, Giussani A, Fingerhut B P, Rivalta I, Dumont E, Mukamel S, Garavelli M,
Phys. Chem. Chem. Phys. DOI: /c5cp01167a

28. Pyrene 2D electronic spectrum at t2=1 ps
Theoretical * Experimental** :
The simulation of the spectrum provides
important information not accessible from
experiments, as the characterization of
the electronic nature of the excited states
involved in the appearing peaks.
** Krebs N, Pugliesi I, Hauer J, Riedle E, New J. Phys., 2013, 15,
*Nenov A, Giussani A, Fingerhut B P, Rivalta I, Dumont E, Mukamel S, Garavelli M,
Phys. Chem. Chem. Phys. DOI: /c5cp01167a

29. Conclusions
Two strategies have been presented in order to compute 2D spectra: the static and the single-trajectory approaches, which respectively not consider and consider the evolution of the system during time t1 and t3.
The computation of the pyrene 2D spectrum for t2=0 has shown that the static approximation reproduces the positions of the peaks only qualitatively.
Through the comparison with the experimental spectrum of pyrene at t2=1ps recorded by Krebs et al., the validity of the single-trajectory approach has been proven.
The simulation of 2D spectra provides fundamental information on the origin of the emerging peaks, that can not be extracted from the experiments.

30. Acknowledgments Prof. Marco Garavelli Prof. Shaul Mukamel
ENS-Lyon, FR
Universita di Bologna, IT
Prof. Shaul Mukamel
University of California Irvine, USA
Prof. Giulio Cerullo
Politecnico di Milano, IT
Dr. Artur Nenov
Universita di Bologna, IT
Dr. Ivan Rivalta
ENS-Lyon, FR
Universita di Bologna, IT
Support and funding:

31. Thanks for your attention