1

Ultrafast laser excitation and rotational de-excitation of cis-stilbene

Yusheng Doua,b,*, Weifeng Wua,, Hong Tanga, and Roland E. Allenc

a Bio-Informatics Institute, Chongqing University of Posts and Telecommunications, Chongqing, 400065, P. R. China b Department of Physical Sciences, Nicholls State University, PO Box 2022, Thibodaux, LA 70310, U. S. A. c Department of Physics, Texas A&M University, College Station, Texas 77843, U. S. A.

ABSTRACT

A realistic dynamical simulation is reported for ultrafast laser excitation and rotational

de-excitation of cis-stilbene. Upon irradiation by a laser pulse with a FWHM of 100 fs,

the molecule first rotates around its vinyl bond to about 90°, where an avoided crossing

leads to electronic de-excitation. The molecule then immediately twists back to about

- 60° within only about 80 fs, avoiding isomerization. The other principal dihedral angles,

within the phenyl rings, lag behind in time, so there is significant large strain between the

phenyl rings and vinyl bond. The importance of these rotations indicates that the

experimentally observed solvent dependence of excited state lifetime is associated with

the torsional dynamics of the phenyl rings.

_____________________________________________________ *Corresponding author. Fax: 985 448 4927 Email address: Yusheng.Dou@nicholls.edu (Y. Dou)

2

INTRODUCTION

The photoisomerization of stilbene is a prototypical reaction [1] related to the

natural cis-trans isomerization process of retinal [2] (the chromophore of rhodopsin), and

the effects of different environments on the photoisomerization process [3-6]. In stilbene

isomerization can start from either the cis or the trans isomer. However, the reaction

proceeds much faster from the cis geometry, on a time scale of roughly 0.3-0.5

picosecond as compared to 10-20 ps [7,8]. The presently accepted mechanism [8] is as

follows: Once in the electronically excited state, the molecule rotates about its vinyl bond

toward the minimum of the potential energy surface. From this minimum, or “phantom

state”, the molecule decays nonadiabatically to the ground state, and finally emerges as

either the product trans-stilbene or the original reactant cis-stilbene. For the gas phase, an

energy barrier of about 0.15 eV has been determined experimentally for an initial trans

geometry [9, 10], whereas a much smaller barrier, no more than 0.05 eV, was found for

an initial cis structure [5, 11].

Experimental evidence [1, 12, 13] shows that the photoisomerization process is

more complicated than mere torsional relaxation around the bond linking the two phenyl

rings, in part because of the steric interaction between these rings [14] which results from

their nonplanar arrangement in the molecule.

The dynamical photoisomerization processes for both cis- and trans-stilbene are

strongly affected by the solvent properties. The excited molecule in cis-trans

photoisomerization has a shorter lifetime in polar solvents than in nonpolar. It was found

experimentally that the lifetime of an excited molecule is 0.32 ps in the gas phase [15],

but 0.5 ps in methanol [4], 1.0 ps in isopentane, 1.6 ps in hexadecane [11], and 2.1 ps in

3

hexadecane [4]. In order to understand the experimental observations, solvent viscosity

models have been proposed, which include the influence of the solvent on the internal

rotation about the central vinyl bond [16, 17]. We will see below that, in addition to

rotation about the vinyl bond, the rotations about two vinyl-phenyl bonds also play a

significant role, and we expect all these rotations to be influenced by solvent-solute

interactions. It should be mentioned that involvement of the phenyl-vinyl torsion in the

photoisomerization has been suggested earlier [18, 19], although no specific conclusions

can be drawn from these previous investigations.

METHODOLOGY

We use a semiclassical method in which the states of the valence electrons are

calculated using the time-dependent Schrödinger equation, but both the radiation field

and the motion of the nuclei are treated classically. According to time-dependent

perturbation theory, such a semiclassical treatment effectively includes effective “n-

photon” and “n-phonon” processes in absorption and stimulated emission. This allows us

to examine nontrivial processes such as multi-electron and multi-photon excitations, the

indirect excitation of vibrational modes, intra-molecular vibrational energy redistribution,

and interdependence of the various electronic and vibrational degrees of freedom.

A detailed explanation of this approach has been published elsewhere [20, 21], so

only a brief description is given here. The one-electron states are renewed at each time

step by solving the time-dependent Schrödinger equation in a nonorthogonal basis,

i!!"

j

!t= S

#1$H $"

j, （1）

4

where S is the overlap matrix for the atomic orbitals. The vector potential A of the

radiation field is coupled to the electrons through the time-dependent Peierls substitution

[22]

( ) ( ) ( )!"#

$%& '('=' 'exp''

0XXAXXXX

c

iqHH abab

!. (2)

Here HabX ! X '( ) is the Hamiltonian matrix element for basis functions a and b on

atoms at X and X ' respectively, and q = !e is the charge of the electron.

The Hamiltonian matrix, overlap matrix, and effective nuclear-nuclear repulsion

are based on density-functional calculations [23]. In our previous investigations, this

same model was found to yield good descriptions of molecular response to ultrashort

laser pulses. For example, the explanation of the nonthermal fragmentation of C60 [24] is

in good agreement with experimental observations, the simulation of the formation of the

tetramethylene intermediate diradical [25] is consistent with time-of-flight mass

spectrometry measurements, and the characterization of the geometry changes at some

critical points [26] is compatible with molecular mechanics valence bond calculations.

The nuclear motion is updated by the Ehrenfest equation of motion

Ml

d2Xl!

dt2

= "1

2# j

† $j

% &H&Xl!

" i!&S&Xl!

$&&t

'

()*

+,$# j + h.c."

&Urep

&Xl!

, (3)

where Urep is the effective nuclear-nuclear repulsive potential and !! ll

XX ˆ= is the

expectation value of the time-dependent Heisenberg operator for the ! coordinate of the

nucleus labeled by l (with ! = x, y, z ). Equation (3) is the approximation to the exact

Ehrenfest theorem that results from neglecting terms on the right-hand side that are

second-order in !l

XX ˆˆ " .

5

The time-dependent Schrödinger equation (1) is solved using a unitary algorithm

obtained from the equation for the time evolution operator [27]. Equation (3) is

numerically integrated with the velocity Verlet algorithm (which preserves phase space).

A time step of 50 attoseconds was used, since energy conservation was then found to

hold to better than 1 part in 105 in a one ps simulation for cis-stilbene at 298 K. A few

tens of simulations are run with different choices for the laser parameters, such as fluence

and duration, in order to obtain a clear example of the process being studied.

As indicated in Fig. 1 (a), all nuclear degrees of freedom are included in the

calculation. Before the cis-stilbene molecule is coupled to the vector potential, it is

allowed 1000 fs to relax to its optimized geometry at a temperature of 300 K. The laser

pulse was taken to have a full-width-at-half-maximum (FWHM) duration of 100 fs (with

a profile which is very nearly Gaussian), a fluence of 0.90 kJ/m2, and a wavelength

corresponding to a photon energy of 6.50 eV. This wavelength matches the density-

functional energy gap between the HOMO and LUMO levels of cis-stilbene. The fluence

was chosen such that the forces on the nuclei are large enough to make the isomerization

reaction occur without breaking C-C or C-H bonds. A similar reaction path with a

slightly different time scale has been obtained by Martinez’s group [28].

It should be mentioned that the present “Ehrenfest” approach is complementary to

other +methods based on different approximations. Its weakness is that it amounts to

averaging over all the terms in the Born-Oppenheimer expansion [29-33]

!total

Xn, x

e,t( ) = !

n

iXn,t( )!e

ixe,X

n( )i

"

rather than following the time evolution of a single term – i. e., a single potential energy

surface – which is approximately decoupled from all the others. (Here Xn

and xe

6

represent the sets of nuclear and electronic coordinates respectively, and the !e

i are

eigenstates of the electronic Hamiltonian at fixed Xn.) However, the present approach

also has strengths: It retains all of the 3N nuclear degrees of freedom (instead of only the

2 or 3 that are typically considered in a potential-energy-surface calculation) and it

includes both the excitation due to a laser pulse and the subsequent de-excitation at an

avoided crossing near a conical intersection [31-33,35-37].

RESULTS AND DISCUSSION

Six snapshots from the simulation at different times are shown in Fig. 1, with all

atoms labeled to facilitate the discussion below. Starting from the equilibrium geometry

in the electronic ground state at t = 0 , the cis-stilbene molecule is electronically excited

by the laser pulse (which has a full duration of 200 fs) and then rotates about the vinyl

bond. This torsional angle reaches 45° at 212 fs and 90° at 307 fs. Immediately after 308

fs, the molecule rotates back around the vinyl bond to 43° at 343 fs and - 62° at 384 fs.

Finally this angle stays in the vicinity of 5° for the remainder of the simulation, indicating

a return to the cis-stilbene geometry.

The variations with time of the torsional angles for both the vinyl bond and the two

vinyl-phenyl bonds are shown in Fig. 2. It can be seen that, after reaching 90° at about

300 fs, the vinyl-bond torsional angle (C6-C7-C8-C9) decreases to about - 60° within less

than 100 fs, and then changes back to about 5° shortly after 400 fs, finally fluctuating

about this value until the end of the simulation. On the other hand, the two torsional

angles about the vinyl-phenyl bonds (C5-C6-C7-C8 and C7-C8-C9-C10) decrease slowly

7

before 350 fs, and then rise quickly to again oscillate around their initial values.

It can be seen in Fig. 2 that all these torsional angles vary significantly only after 200

fs (when the laser pulse radiation has ended), demonstrating that the torsional modes are

not directly excited by laser absorption, and that the energy for the torsional motion must

come from other vibrational modes through intramolecular vibrational energy

redistribution. Both the decrease before 300 fs in the vinyl-phenyl torsional angles and

the increase afterward are associated with global geometric relaxation of the stilbene

molecule, with this phenyl-ring motion lagging behind the vinyl-bond rotation. The large

strain between the phenyl rings and vinyl bond is a driving force for the torsional

dynamics of the phenyl rings to achieve new equilibrium positions. The importance of all

three rotations indicates that the experimentally observed solvent dependence of excited-

state lifetimes is associated with the torsional dynamics of the phenyl rings, which

produces de-excitation of the molecule and which is affected by the properties of the

solvent.

The most remarkable feature in Fig. 2 is the sharp decrease in the vinyl-bond

torsional angle immediately after 300 fs, as it changes by 152° within about 100 fs. This

rapid rotation results from a nonadiabatic process: electrons undergo a LUMO to HOMO

transition which will be discussed below.

The variations of the lengths of the vinyl (C7-C8) and vinyl-phenyl (C6-C7, and C8-C9)

bonds with time are presented in Fig. 3. The vinyl bond stretches from about 1.35 Å to

an average length of 1.47 Å (typical for a single bond) shortly after 100 fs, keeps this

length until after 300 fs, and then shrinks to less than 1.40 Å. On the other hand, the two

vinyl-phenyl bonds shorten to about 1.40 Å by 100 fs, remain near this length until after

8

300 fs, and then return to an average length of 1.45 Å. The stretching vibrations of the

two vinyl-phenyl bonds are strongly activated after 300 fs, while that of the vinyl bond is

more active after 400 fs.

The variations with time of the bending angles of the vinyl bond and two vinyl-

phenyl bonds are presented in Fig. 4. Both angles are initially about 130°. They sharply

drop until after 350 fs, and finally vibrate about 125° with an amplitude of as large as 15°.

The energies of the HOMO and LUMO are shown in Fig. 5. The energy gap

decreases dramatically after the laser pulse is applied, and two couplings (avoided

crossings): one shortly before and one shortly after 300 fs. Then the gap reopens as the

molecule moves away from the geometry at 300 fs which is responsible for these avoided

crossings (i.e., a value of 90° for angle of rotation about the central bond, as can be seen

in Fig. 2).

The time-dependent population of the LUMO is presented in Fig. 6. The laser pulse

excites about 0.8 electrons from the HOMO to LUMO (a π → π* excitation) during the

200 fs full duration of the pulse. A further increase at about 280 fs results from an upward

transition at the first HOMO-LUMO avoided crossing mentioned above. Depopulation

takes place soon afterward at the second avoided crossing, and there are about 0.3

electrons remaining in the LUMO after 300 fs. Another, smaller decrease in the

population of the LUMO shortly after 900 fs is induced by coupling between the LUMO

and LUMO+1.

The transfer of electrons from LUMO to HOMO at about 300 fs nearly brings the

molecule back to the electronic ground state. This is confirmed by the changes in the

lengths of the C6-C7 and C8-C9 bonds at about 300 fs, as shown in Fig. 3, where

9

essentially two double bonds become single bonds. This de-excitation is also responsible

for the rapid rotation of the molecule about the vinyl bond which is shown in Fig. 2.

It is of interest to compare the dynamics of the cis-cis and cis-trans [26]

isomerization processes. The molecule reaches the phantom state at 307 fs for the cis-cis

reaction and 330 fs for the cis-trans reaction. It decays to the electronic ground state at

310 fs for cis-cis isomerization and 440 fs for cis-trans isomerization. The time delay in

the decay to the electronic ground state for the cis-trans reaction results from the back

and forth motion of the molecule in the vicinity of the minimum of potential energy for

the electronically excited state, before it is de-excited by electronic transitions resulting

from strong interactions of the HOMO and LUMO between 300 fs and 400 fs, as

discussed in Ref. [26]. Perhaps the most notable difference in the two dynamical

processes, after the molecule enters the electronic ground state, is the much faster

movement toward cis-stilbene. In cis-trans isomerization, the molecule rotates about the

vinyl bond from about 90° to 180° in 270 fs, much slower than the rotation from about

90° to - 60° in cis-cis isomerization. The much faster backward rotation of the molecule

indicates that the slope of the reaction pathway along the potential energy surface of the

electronic ground state is much steeper for backward motion than for forward, when the

molecule starts in the phantom state.

CONCLUSION

Ultrafast torsional rotation of cis-stilbene about its ethylenic bond is observed when

the molecule is irradiated by a laser pulse of a FWHM of 100 fs. The excited molecule

decays from to the electronic ground state through nonadiabatic transition induced by a

10

strong interaction between the HOMO and LUMO levels. It is found that both the

torsional motions about the ethylenic bond and the ethylenic-phenyl bonds and the bond

bending vibrations between these two types of bonds play significant rules in this strong

interaction. Compared to the ultrafast rotation of cis-stilene about its ethylenic bond, the

rotations about two ethylenic-phenyl bonds are relatively slow and lag behind in time.

This large strain significantly affects the torsional dynamics of cis-stilbene around its

ethylenic bond. Because of the relatively large size and the fact that the laser excitation in

cis-stilbene is significantly delocalized, the rotation of a phenyl ring in a solution is

viscosity dependent. This indicates that the experimentally observed solvent dependence

of excited-state lifetimes is associated with the torsional dynamics of the phenyl rings.

ACKNOWLEDGMENT

Acknowledgment is made to the donors of The American Chemical Society Petroleum

Research Fund for support of this research at Nicholls State University. The authors also

acknowledge support by the National Natural Science Foundation of China (Grant

20773168), the Natural Science Foundation Project of CQ CSTC, China (Grant

2006BB2367), and the Robert A. Welch Foundation (Grant A-0929). The Supercomputer

Facility at Texas A&M University provided computational assistance.

11

REFERENCES

1. D. H. Waldeck, Chem. Rev. 91, 415 (1991).

2. R. R. Birge, Annu. Rev. Phys. Chem. 41, 683 (1990).

3. R. J. Sension, S. T. Repinec, A. Z. Szarka, R. M. Hochstrasser, J. Chem. Phys.

98, 6291 (1993).

4. J. K. Rice, A. P. Baronavski, J. Phys. Chem. 96, 3359 (1992).

5. D. C. Todd, J. M. Jean, S. J. Rosenthal, A. J. Ruggerio, D. Yang, G. R.

Fleming, J. Chem. Phys. 93, 8658 (1990).

6. L. Nikowa, D. Schwarzer, J. Troe, J. Chem. Phys. 97, 4827 (1992).

7. S. Pedersen, Banares, A.H. Zewail, J. Chem. Phys. 97, 8801 (1992).

8. J. S. Baskin, S. Pedersen, Banares, A.H. Zewail, J. Phys. Chem. 100, 11920

(1996).

9. J. A. Syage, P. M. Felker, A. H. Zewail, J. Chem. Phys. 81, 4706 (1984).

10. T. S. Zwier, E. Carrasquillo M., D. H. Levy, J. Chem. Phys. 78, 5493 (1983).

11. S. Abrash, S. Repinec, R. M. Hochstrasser, J. Chem. Phys. 93, 1041 (1990).

12. H. Petek, K. Yoshihara, Y. Fujiwara, Z. Lin, J. H. Penn, J. H. Frederick, J.

Phys. Chem. 94, 7539 (1990).

13. A. Z. Szarka, N. Pugliano, D. K. Palit, R. M. Hochstrasser, Chem. Phys. Lett.

240, 25 (1995).

14. M. J. Bearpark, F. Bernardi, S. Clifford, M. Olivucci, M. A. Robb, T. Vreven,

J. Phys. Chem. A 101, 3841 (1997).

15. B. I. Greene, R. C. Farrow, J. Chem. Phys. 78, 3386 (1983).

12

16. D. Raftery, R. J. Sension, R. M. Hochstrasser, in Activated Barrier Crossing,

G. R. Fleming and P. Hangii, Eds. (World Scientific, Rucer Edge, NJ, 1993),

p. 163.

17. R. M. Hochstrasser, Pure Appl. Chem. 52, 2683 (1980).

18. J. H. Frederick, Y. Fujiwara, J. H. Penn, K. Yoshihara, H. Petek, J. Phys.

Chem. 95, 2845 (1991).

19. J. Quenneville and T. J. Martínez, J. Phys. Chem. A 107, 829 (2003).

20. Y. Dou, B. R. Torralva, R. E. Allen, J. Mod. Optics 50, 2615 (2003).

21. Y. Dou, B. R. Torralva, R. E. Allen, Chem. Phys. Lett. 378, 323 (2003).

22. M. Graf, P. Vogl, Phys. Rev. B 51, 4940 (1995). See also T. B. Boykin, R. C.

Bowen, G. Klimeck, Phys. Rev. B 63, 245314 (2001).

23. M. Elstner, D. Porezag, G. Jungnickel, J. Elsner, M. Haugk, T. Frauenheim, S.

Suhai, G. Seifert, Phys. Rev. B 58, 7260 (1998).

24. B. Torralva, T. A. Niehaus, M. Elstner, S. Suhai, Th. Frauenheim, R. E. Allen,

Phys. Rev. B 64, 153105 (2001).

25. Y. Dou, B. R. Torralva, R. E. Allen, J. Phys. Chem. A 107, 8817 (2003).

26. Y. Dou, R. E. Allen, J. Chem. Phys. 119, 10658 (2003).

27. R. E. Allen, T. Dumitrica, B. R. Torralva, In Ultrafast Physical Processes in

Semiconductors; K. T. Tsen Ed. (Academic Press, New York, 2001), Chapter

7.

28. A. Toniolo, B. Levine, A. L. Thompson, J. Quenneville, M. Ben-Nun, J.

Owens, S. Olsen, L. Manohar, T. J. Martinez, in Computational methods in

13

organic photochemistry, A. Kutateladze Ed. (Marcel-Dekker, New York,

2005).

29. M. Born, J. R. Oppenheimer, Ann. Phys. (Leipzig) 84, 457 (1927).

30. M. Born, K. Huang, The Dynamical Theory of Crystal Lattices (Oxford

University Press, London, 1954).

31. E. Teller, J. Phys. Chem. 41, 109 (1937).

32. Conical Intersections: Electronic Structure, Dynamics, and Spectoscopy, W.

Domcke et al. Eds. (World Scientific, Singapore, 2004).

33. M. Baer, Beyond Born-Oppenheimer: Electronic Nonadiabatic Coupling

Terms and Conical Intersections (Wiley Interscience, Hoboken, 2006).

34. J. C. Tully, J. Chem. Phys. 93, 1061 (1990).

35. D. R. Yarkony, Rev. Mod. Phys. 68, 985 (1996).

36. M. J. Bearpark, F. Bernardi, S. Clifford, M. Olivucci, M. A. Robb, T. Vreven,

J. Phys. Chem. A 101, 3841 (1997).

37. B. G. Levine, T. J. Martínez, Annu. Rev. Phys. Chem. 58, 613 (2007).

14

FIGURE CAPTIONS

FIG. 1. Snapshots taken at different times in a simulation of cis-stilbene responding

to a 100 fs laser pulse, with other parameters given in the text.

FIG. 2. Changes with time in the dihedral angles C6-C7-C8-C9, H1-C6-C7-H8 , and H7-

C8-C9-H10.

FIG. 3. Time dependence of (a) the C7-C8 and (b) the C6-C7 and C8-C9 bonds

FIG. 4. Variation with time of the C6-C7-C8 and C7-C8-C9 bond-bending angles.

FIG. 5. Variation with time of the HOMO and LUMO levels.

FIG. 6. Time-dependent population of the LUMO.

15

FIGURES

Fig. 1

16

Fig. 2

17

Fig. 3

18

Fig. 4

19

Fig. 5

20

Fig. 6