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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: [email protected] (Y. Dou)
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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
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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)
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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 " .
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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
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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
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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
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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
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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
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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.
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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.
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FIGURES
Fig. 1
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Fig. 2
17
Fig. 3
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Fig. 4
19
Fig. 5
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Fig. 6