^{1,a)}, R. Spesyvtsev

^{2}, O. M. Kirkby

^{2}, R. S. Minns

^{2,b)}, D. S. N. Parker

^{2,c)}, H. H. Fielding

^{2}and G. A. Worth

^{1,d)}

### Abstract

Time-resolved photoelectron spectroscopy can obtain detailed information about the dynamics of a chemical process on the femtosecond timescale. The resulting signal from such detailed experiments is often difficult to analyze and therefore theoretical calculations are important in providing support. In this paper we continue our work on the competing pathways in the photophysics and photochemistry of benzene after excitation into the “channel 3” region [R. S. Minns, D. S. N. Parker, T. J. Penfold, G. A. Worth, and H. H. Fielding, Phys. Chem. Chem. Phys.12, 15607 (2010)]10.1039/c001671c with details of the calculations shown previously, building on a vibronic coupling Hamiltonian [T. J. Penfold and G. A. Worth, J. Chem. Phys.131, 064303 (2009)]10.1063/1.3197555 to include the triplet manifold. New experimental data are also presented suggesting that an oscillatory signal is due to a hot band excitation. The experiments show that signals are obtained from three regions of the potential surfaces, three open channels, which are assigned with the help of simulations showing that following excitation into vibrationally excited-states of S_{1} the wavepacket not only crosses through the prefulvenoid conical intersection back to the singlet ground state, but also undergoes ultrafast intersystem crossing to low lying triplet states. The model is, however, not detailed enough to capture the full details of the oscillatory signal due to the hot band.

The authors would like to thank Benjamin Lasorne, Michael Robb, and Michael Bearpark for useful discussion throughout this work.

I. INTRODUCTION

II. THEORY

A. Vibronic coupling Hamiltonian model

B. Quantum chemistry calculations: Obtaining the Hamiltonians

C. Quantum dynamics: Multi configuration time dependent Hartree (MCTDH) method

D. Experimental TRPES

III. RESULT AND DISCUSSION

A. The vibronic coupling Hamiltonian

1. Zero order parameters

2. First order parameters

3. Second order parameters

4. Spin orbit coupling parameters

B. The dynamics

1. Experimental TRPES

2. The quantum dynamics

IV. CONCLUSION

### Key Topics

- Manifolds
- 18.0
- Potential energy surfaces
- 17.0
- Normal modes
- 12.0
- Photoelectron spectra
- 11.0
- Self organized systems
- 10.0

## Figures

A cut through the potential energy surfaces for benzene along the prefulvene mode. The singlet states are in black and triplet in red. Note the near degeneracy of the potential energy leading to the CI, with the lines for S_{1} and a triplet state superimposed. In addition we have labelled either side of the barrier ( and ) according to the diabatic character of the S_{1} state. This is important for the discussion of the experimental results shown in Sec. III.

A cut through the potential energy surfaces for benzene along the prefulvene mode. The singlet states are in black and triplet in red. Note the near degeneracy of the potential energy leading to the CI, with the lines for S_{1} and a triplet state superimposed. In addition we have labelled either side of the barrier ( and ) according to the diabatic character of the S_{1} state. This is important for the discussion of the experimental results shown in Sec. III.

Cuts along the normal modes through the triplet potential energy surface for benzene. In order of energy at **Q** = 0 these states are T_{1} (B_{1u }) and T_{2} (E_{1u }). (a) ν_{1} (1a_{1g }), the breathing mode, (b) ν_{4} (1b_{2g }), the chair mode, (c) ν_{6a } (1e_{2g }), the quinoid mode, (d) ν_{16a } (1e_{2u }), the boat mode, (e) ν_{9a } (4e_{1g }), and (f) the prefulvene combination mode (ν_{4,16a }). The solid black dots are the *ab initio* points for the triplet states, the open circles are the S_{1} state, and the black line is the fit to the triplet states.

Cuts along the normal modes through the triplet potential energy surface for benzene. In order of energy at **Q** = 0 these states are T_{1} (B_{1u }) and T_{2} (E_{1u }). (a) ν_{1} (1a_{1g }), the breathing mode, (b) ν_{4} (1b_{2g }), the chair mode, (c) ν_{6a } (1e_{2g }), the quinoid mode, (d) ν_{16a } (1e_{2u }), the boat mode, (e) ν_{9a } (4e_{1g }), and (f) the prefulvene combination mode (ν_{4,16a }). The solid black dots are the *ab initio* points for the triplet states, the open circles are the S_{1} state, and the black line is the fit to the triplet states.

Photoelectron spectra of benzene from the continuous (a) and pulsed (b) expansion, following excitation with a 243 nm pump pulse and a 235 nm probe pulse as a function of pump-probe delay.

Photoelectron spectra of benzene from the continuous (a) and pulsed (b) expansion, following excitation with a 243 nm pump pulse and a 235 nm probe pulse as a function of pump-probe delay.

Photoelectron signal as a function of the pump-probe delay from the continuous expansion: (a) total integrated photoelectron signal, (b) 0.34–0.37 eV, (c) 0.65–1.0 eV, and (d) 1.11–1.46 eV. Experimental data (open circles) and fit (bold solid line) together with the contributions from the decays, τ_{1} ∼ 233 fs (widely spaced dots), τ_{2} ∼ 1.8 ps (closely spaced dots), τ_{3} ∼ ∞ (dashes), and the cosinusoidal oscillation (solid line) obtained using the global fitting procedure described in the text.

Photoelectron signal as a function of the pump-probe delay from the continuous expansion: (a) total integrated photoelectron signal, (b) 0.34–0.37 eV, (c) 0.65–1.0 eV, and (d) 1.11–1.46 eV. Experimental data (open circles) and fit (bold solid line) together with the contributions from the decays, τ_{1} ∼ 233 fs (widely spaced dots), τ_{2} ∼ 1.8 ps (closely spaced dots), τ_{3} ∼ ∞ (dashes), and the cosinusoidal oscillation (solid line) obtained using the global fitting procedure described in the text.

Photoelectron signal as a function of the pump-probe delay from the continuous expansion: (a) total integrated photoelectron signal, (b) 0.34–0.37 eV, (c) 0.65–1.0 eV, and (d) 1.11–1.46 eV. Experimental data (open circles) and fit (bold solid line) together with the contributions from the decays, τ1 ∼ 256 fs (widely spaced dots) and τ3 ∼ ∞ (dashes) obtained using the global fitting procedure described in the text.

Photoelectron signal as a function of the pump-probe delay from the continuous expansion: (a) total integrated photoelectron signal, (b) 0.34–0.37 eV, (c) 0.65–1.0 eV, and (d) 1.11–1.46 eV. Experimental data (open circles) and fit (bold solid line) together with the contributions from the decays, τ1 ∼ 256 fs (widely spaced dots) and τ3 ∼ ∞ (dashes) obtained using the global fitting procedure described in the text.

State populations following excitation to the Franck-Condon point on with (a) (red) and S_{1} (black) using the 2-state Hamiltonian with . The blue line is for a cold simulation for which ν_{16} was not initially excited. (b) (red) and S_{1} plus triplets (black) using the 4-state ISC Hamiltonian with . (c) (red) and S_{1} plus triplets (black) using the 4-state ISC Hamiltonian with . (d) (red) and S_{1} plus triplets (black) using the 4-state ISC Hamiltonian with no hot band, . In (a) and (b) the purple lines correspond to the double exponential fit of the decay curves. In (b)–(d) the green line is the sum of the triplet populations. (e) The population of the *T* _{1} (red), *T* _{2} (blue), and total triplet population (green) for simulation (c). (f) The population of the *T* _{1} (red), *T* _{2} (blue), and total triplet population (green) for simulation (d).

State populations following excitation to the Franck-Condon point on with (a) (red) and S_{1} (black) using the 2-state Hamiltonian with . The blue line is for a cold simulation for which ν_{16} was not initially excited. (b) (red) and S_{1} plus triplets (black) using the 4-state ISC Hamiltonian with . (c) (red) and S_{1} plus triplets (black) using the 4-state ISC Hamiltonian with . (d) (red) and S_{1} plus triplets (black) using the 4-state ISC Hamiltonian with no hot band, . In (a) and (b) the purple lines correspond to the double exponential fit of the decay curves. In (b)–(d) the green line is the sum of the triplet populations. (e) The population of the *T* _{1} (red), *T* _{2} (blue), and total triplet population (green) for simulation (c). (f) The population of the *T* _{1} (red), *T* _{2} (blue), and total triplet population (green) for simulation (d).

Expectation values of the position of the wavepacket during propagations for first 2000 fs using the 2-state Hamiltonian. (a) ⟨*q*⟩ of ν_{4} (blue) and ν_{16a } (green) on S_{0}. (b) ⟨*q*⟩ of ν_{1} (red) and ν_{6a } (black) on S_{0}. (c) ⟨*q*⟩ of ν_{4} and ν_{16a } on S_{1}. (d) ⟨*q*⟩ of ν_{1} and ν_{6a } on S_{1}.

Expectation values of the position of the wavepacket during propagations for first 2000 fs using the 2-state Hamiltonian. (a) ⟨*q*⟩ of ν_{4} (blue) and ν_{16a } (green) on S_{0}. (b) ⟨*q*⟩ of ν_{1} (red) and ν_{6a } (black) on S_{0}. (c) ⟨*q*⟩ of ν_{4} and ν_{16a } on S_{1}. (d) ⟨*q*⟩ of ν_{1} and ν_{6a } on S_{1}.

Expectation values of the position of the wavepacket in the triplet states during propagations for first 2000 fs using the 4-state ISC Hamiltonian. (a) ⟨*q*⟩ of ν_{4} (blue) and ν_{16a } (green) on T_{1}. (b) ⟨*q*⟩ of ν_{1} (red) and ν_{6a } (black) on T_{1}. (c) ⟨*q*⟩ of ν_{4} and ν_{16a } on T_{2, x }. (d) ⟨*q*⟩ of ν_{1} and ν_{6a } on T_{2,x }.

Expectation values of the position of the wavepacket in the triplet states during propagations for first 2000 fs using the 4-state ISC Hamiltonian. (a) ⟨*q*⟩ of ν_{4} (blue) and ν_{16a } (green) on T_{1}. (b) ⟨*q*⟩ of ν_{1} (red) and ν_{6a } (black) on T_{1}. (c) ⟨*q*⟩ of ν_{4} and ν_{16a } on T_{2, x }. (d) ⟨*q*⟩ of ν_{1} and ν_{6a } on T_{2,x }.

## Tables

Computational details for the quantum dynamics simulations. N_{ i },N_{ j } are the number of primitive Harmonic oscillator discrete variable representation (DVR) basis functions used to describe each mode.^{26} n_{ i } are the number of single-particle functions used for the wavepacket on each state. (A) 2-state singlet model. (B) 4-state intersystem crossing model.

Computational details for the quantum dynamics simulations. N_{ i },N_{ j } are the number of primitive Harmonic oscillator discrete variable representation (DVR) basis functions used to describe each mode.^{26} n_{ i } are the number of single-particle functions used for the wavepacket on each state. (A) 2-state singlet model. (B) 4-state intersystem crossing model.

Mode symmetry and vibration energies in eV for all the normal modes of benzene, using Wilson numbering. Calculated using a CAS(6,6) active space and 6-31g* basis set.

Mode symmetry and vibration energies in eV for all the normal modes of benzene, using Wilson numbering. Calculated using a CAS(6,6) active space and 6-31g* basis set.

Vertical excitation energies (in eV) of the lowest four singlet and lowest four triplet states of benzene relative to the benzene singlet ground state, calculated at the equilibrium geometry. The SA-CAS(6,6) used a 6-31g* basis and are averaged over the four states. The CASPT2(6,6) and CASPT2(6,10) calculations use a MOLPRO specific Roos(3s2p1d/2s) basis.

Vertical excitation energies (in eV) of the lowest four singlet and lowest four triplet states of benzene relative to the benzene singlet ground state, calculated at the equilibrium geometry. The SA-CAS(6,6) used a 6-31g* basis and are averaged over the four states. The CASPT2(6,6) and CASPT2(6,10) calculations use a MOLPRO specific Roos(3s2p1d/2s) basis.

On-diagonal κ and off-diagonal, λ, linear coupling constants, (in eV) for the important normal modes of benzene in the triplet manifold obtained by fitting a vibronic coupling Hamiltonian to the adiabatic potential energy surfaces at the CASPT2(6,6) level. Superscript shows the state and subscript the normal mode(s) associated with the parameter. States 1,2,3 refer to and respectively.

On-diagonal κ and off-diagonal, λ, linear coupling constants, (in eV) for the important normal modes of benzene in the triplet manifold obtained by fitting a vibronic coupling Hamiltonian to the adiabatic potential energy surfaces at the CASPT2(6,6) level. Superscript shows the state and subscript the normal mode(s) associated with the parameter. States 1,2,3 refer to and respectively.

First (λ) and second (γ) order vibrational spin orbit coupling terms (in cm^{−1}). Calculations performed with (6,6) active space and MOLPRO specific Roos(3s2p1d/2s) basis. The subscript shows the normal mode(s) associated with the parameter.

First (λ) and second (γ) order vibrational spin orbit coupling terms (in cm^{−1}). Calculations performed with (6,6) active space and MOLPRO specific Roos(3s2p1d/2s) basis. The subscript shows the normal mode(s) associated with the parameter.

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