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Molecular Dynamics of Stilbene Molecule under Laser Excitation V. D. Vachev, V. N. Zadkov

International Laser Center, Moscow State University Lenin's Hills, 119899 Moscow, USSR

1. INTRODUCTION
A detailed analysis of photo-stimulated structural transitions plays an important role in studying of organic
molecules. This problem can be successfully solved by using optical methods since both linear and non-linear properties of organic molecules strongly depend on their structure.1 Recent progress in the field of short laser pulse generation in pico- and femtosecond range2 makes it possible to obtain detailed information on the structural dynamics of macromolecules.3 The computer molecular dynamics (MD) simulation method47 enables one to interpret the experimental information available and in many cases, to deepen and broaden understanding of the physics of processes taking place in molecules. We report here on the MD study of the structure dynamics of a stilbene molecule excited by a laser pulse.8 Note, that the molecule of stilbene which was studied experimentally in detail3' 10 13 is of particular interest being a simple model for studying the mechanism of laser-induced molecular photoisomerization.

2. FORMULATION OF THE MD EXPERIMENT
The energetic parameters of various possible conformations of macromolecules can be studied using molecular mechanics calculations. The latter are based upon molecular energy dependence on the structure of molecules.14 By minimizing the potential energy of the molecule as a function of coordinates of it's atoms one can determine both possible conformers (in potential energy local minima) and transition states (the saddle points) . These methods are well developed and are widely used.15 At the same time, molecular structure dynamics, both in ground and excited electronic states, and the processes of electronic excitation have not yet been studied in detail. An approach to this problem was made by Birge el al.57 who have used semi-classical MD simulation to study conformational transitions in rhodopsin and bacteriorhodopsin. The isomerization reaction quantum yield and molecular trajectories of (including that in the presence of a solvent) motion have been calculated in those works. However, the development of the photo-isomerization reaction has been considered by the authors only in a direction along the corresponding torsion angle. In so doing, the processes of the internal energy redistribution in molecules have not been taken into account.

The same semiclassical MD method was used in the present work for studying the structural dynamics of the stilbene molecule (Fig. 1) excited by laser pulse. In so doing, consideration of all contributions to molecular
potential energy U(ri , . . . , rN) (energy of chemical bonds, valence and torsion angles, Van-der-Waals interactions, except electrostatic forces) enabled us to get detailed information on internal energy redistribution in a molecule and the dynamics of the structure of the latter.
Isolated molecule without any influence from the molecules of solvent was considered in order to make the model simpler. Experimentally this condition can be fulfilled for stilbene molecules in a supersonic molecular jet (see Ref. 10). The electronic excitation was assumed to be instantaneous and was defined by initial conditions imposed on stilbene geometry'.
detailed quantum theory of the single photon excitation of an electronic transition in multi-atomic molecule with subsequent calculation of dynamics of the molecule in excited state in phase space using the MD method can be found in Ref. 16.

SPIE Vol. 1403 Laser Applications in Life Sciences (1990) / 487


(a)

(b)

Fig. 1. Ground state equilibrium geometries of stilbene: (a) -- trans-stilbene, (b) -- cis-stilbene.
370 360 350 340

N

E 320

310 300
>290

L 0 c 270 U

cT)

260
250-

--25 0

25 50 75 100 125 150 175 200

ItIIIII;IIlII1IIuIII

tors.angle, degr
Fig. 2. Potential energy surfaces for the ground and excited electronic states as a function of the torsional angle (ethylene

coordinate) 13

488 / SPIE Vol. 1403 Laser Applications in Life Sciences(1990)


Trajectories of motion for all atoms were obtained by numerical solution of classical Newton's equations:

m2(d2r2/dt2) --(OU/0r1) i = 1, . . . , N,
where rnj and r1 represent the mass and the position of the i-th atom, correspondingly.

(1)

The initial condition (Ro {r1, . . . , rN}, V0 = {vi, . . . , vN}) was defined by coordinates found using special procedure of the potential energy surface minimization, by the temperature of the system and by type and the degree of electronic-vibrational excitation of the molecule. The found solutions R(t), V(t) provide information on the dynamics of the molecular structure and on the intra-molecular excitation energy redistribution.

2.1. Force field
The force F1 = --OU(ri , . . . , rN)/0r2 , applied to the i-tb atom of the molecule is completely determined by the

potential energy of the latter

U(ri, . . , rN) = U(qi, . . , qN) =
. .

Ub + Uvj + Utor + Unb,

(2)

, Utor , Uflb are the contributions due to deformations of chemical bonds , deviations of valence and torsional (dihedral) angles and Van-der-Waals interactions , respectively.

where Ub ,

The choice of dependencies of these contributions on coordinates should be in agreement with the electronic structure of the molecule. Thus, the potential parameters are taken from experimental data, for example, from the x-ray structure analysis (the equilibrium values ofcoordinates Ito {r?, . . . , 4}) and from vibrational spectroscopy (Ò2U/i9qÒq I)· All further analysis is made assuming that separate items in (2) in the ground and excited states differ only by the values of parameters.
Harmonic approximation'7 was used for the bonds deformation and valence angles potentials
Ub
Uvai

O.5>kb(r--bo)2,
0.5

:

(3)

k,,(Ão

--

po)2.

(4)

Here are the force constants of deformation, and Ão are equilibrium values of valence angles, correspondingly. The summation is made over all bonds and all equilibrium angles. The homogeneous length b0 and force constant kb are given for each bond (i, i); r = Ir -- is the current length of the bond.

r

The Van-der-Waals (nonbonded) interactions are calculated for all couples of atoms (i, i), which do not belong to the same chemical bond, or for one and the same valence angle. The 6-exp function'7 was used as a potential
function

Ub = f[2.25/rr6 + 8.28 x i05 exp (--rr/O.0736)],

(5)

where rr = r/s, s is the sum of Van-der-Waals radii, f and s are determined by the kinds of atoms i and j.
The following approximation was used for the torsion angle potential:17
Utor = 0.5 > ke[1 + cos (nO + 6)].

(6)

The force constant k9, the multiplicity of the potential surface and the phase shift 6 were defined for each type of bonds.

SPIE Vol. 1403 Laser Applications in Life Sciences(1990) / 489


The expression (6) was used for the description of the ground electronic state of stilbene. However, it was noted in Refs. 9, 18 that it is more preferable to use the potential surface instead of force constants for the excited state description in the case of molecules with delocalized ir-electrons. Thus, the potential energy surfaces obtained by Syage et al.1° were used for the first excited state (Fig. 2).
The values of all parameters in (3)--(6) for stilbene molecule were taken from Ref. 17 and are given in Tables 1--4.

2.2. MD-trajectories and quantum yield of the photoizomerization reaction in stilbene
Semiclassical approach was applied for calculation of the MD-trajectories of isomerization, using dense levels
6

the rate of vibrational relaxation at trajectory time r, kEvib/uIav 15 the absolute (exponential) average vibrational relaxation rate, p is the density of states factor, Ejb(r) is the vibrational energy at trajectory time r, k is the Boltzman constant, T is the temperature.

where

'

(Evjb/it)tT =

IE'vib/1XuIav{1 _ exp [--p(Ejb(T) -- kT)]},

(7)

The theoretical estimae of the quantum yield and the time of isomerization can be obtained from calculation of the transition probability to the ground state a(r) along the MD-trajectory. The following a(r) dependence along trajectory6 is used in semiclassical approximation:

a(r) = exp {--

([4zEio(r)/3h] {2Eio(r)/(O2iEio/O2t2)] 1/2)

}

(8)

where LE1o(r) is the adiabatic potential energy difference between the ground and excited state.

3. MOLECULAR DYNAMICS OF CIS-TRANS PHOTOISOMERIZATION IN STILBENE
The character and the degree of electronic-vibrational excitation of a stilbene by a laser pulse were given by various initial conditions in (1) in our MD-simulations. In so doing, we took into account the results of the stilbene vibrational spectrum analysis9' 10 in the ground (So) and excited (S1) electronic states. The main modes of the stilbene spectrum related to our analysis are given in Table 5.
The system of equations (1) was solved by using the Runge--Kutta method of the 4-th order with variable steps. The potential gradients were calculated analytically. Mean square deviations from the equilibrium values of bond length, valence and torsion angles were calculated every 40 fs and normalized by the corresponding equilibrium parameters.

3. 1. Intramolecular energy redistribution
This series of experiments consists in investigation of the time scale of initial vibrational excitation energy redistribution. It is seen from Table 5 that low frequency out-of-plane modes are effectively excited in stilbene molecule as a result of interaction with a laser pulse. The results of computer experiments on combination modes excitation of the CeCestretch and CCstretch bonds, in-plane modes and low frequency out-of-plane modes are shown in
Figs. 3--10.

Analyzing the results of these experiments one can reconstruct the following sequence of events. During the first 800 fs the phenil rings excitation is redistributed into bending modes, torsion vibrations and stretching vibrations of the C--H bonds (Figs. 6, 9). After 1 ps weak energy exchange takes place between all modes with a characteristic time of .s 2 ps. Torsion vibrations are excited immediately after laser excitation and are subsequently amplified, so that the torsion angle takes the value close to 90° after about 2 ps (Fig. 10).

490 / SPIE Vol. 1403 Laser Applications in Life Sciences(1990)


Table 6. Stastical analysis of the quantum yield of isoniTable 1. Interaction parameters forchemical bond erization in stilbene after combination modes excitation. potential. Atomi
CÃo

Atom2
CÃo
Ce
CÃo

bo

kb

V

PS

a(t)
0.005
0.15
0.31

1.395
1.520

700.0
2.68
316.8

Si %

s'
0

S81

TO

%

PS

99.5 84.6 41.5
19.5 12.9

Ce Ce
CÃo

0.5
25 25 47 47

--

1.525 1.084

2.76
320.0

14.9

--
4.10 4.10 4.72
--

H

2.89
346.0

33.5
33.5
40.1 40.1

Ce

H

1.100

331.2

3.22

0.53 0.34
0.61

3.55 3.66

5

54.9

Table 2. Interaction parameters for valence angles 3.68 -- 0.64 2.8 43.2 54.9 potentials (measurements units: Ãci in degrees, kp in kcal moF1 degree2). Table 3. Interaction parameters for torsion angles potentials (measurements units: 6 in degrees, kOin kcal/mol). Atomi Atom2 Atom3 ceo ISO
CÃo

CÃo

Op
CÃo

120.0 120.0

0.012

Atomi Atom2 Atom3 Atom4 kO

n
-2

6 180

Ce

Ce

0.013

any

Ce

Ce

any

6.0

Table S. The most prominent vibrational modes in the excited state of t-stilbene1 .
Mode

Frequency, cm1

Relative intensity 25 26
79

Ce--I)tnd (out--of--plane) 83.2

Table 4. Parameters of
nonbonded interactions.
Atom
RVdV

Celorsion
bend

95.1

symmetrical ethylenel9l.6
Ce4 (out--of--plane) 229 280.3

H
1.20
Mi-i

C

1.70

6.1

H
CeCCend

30
C
18
10.6

IHc
fcc

CeCe4end+CCCtend 845.8--851.9

Ce4treich
CCsij-etch

1250

1548.4 1637.8

3.5 2.3

CeCestretch

SPIE Vol. 1403 Laser Applications in Life Sciences(1990) I 491


0.1

0-s

03

C

cM a

10.3
V

c

o.

C

11

--1

0iI3

4

--1 U

340578
tfrns,ps

Lime, ps

Fig. 3. Stretch dependence of Ce Ce bond of time during MD-experiment under the excitation of Ce Ce (combination modes excitation).

Fig. 4. Stretch dependence of C -- C bond of time during MD-experiment under the excitation of Ce CÃ (combination modes excitation).

C
4J

00.4

C

0

I:
0
0.1

I:
-0.0

l

I 24 I 711
time,

-c.'

II

ps

me,

e
ps

Fig. 5. Ce -- Ce CÃp valence angle deformation dependence of time during MD-experiment under the excitation of C -- Ce C bend (cornbination modes excitation) .

Fig. 6. Stretch dependence of Ce H bond of

time during MD-experiment under the excitation of Ce H bend (combination modes excitation).

492 / SPIE Vol. 1403 Laser Applications in Life Sciences(1990)


I
C

0

goA.
t7

oj
0

kI
--I

0

1

2340S
time,

7BI

--1 0

t

2 3 4 51 7 1
time,

ps

Fig. 7. Stretch dependence of C -- C bond of time during MD-experiment under the excitation of C, -- C, bend (combination modes excitation).

-- -- Ce, valence angle deformaFig. 8. tion dependence of time during MD-experiment under the excitation of -- C, -- C, bend (combination modes excitation).

PS

C

O4
'I

O4

0

I)
.J

E

,. --1 0
1

.

-----..u --'-i -- '

I

134357
time,

p

0

1

13 4 8 $ ?
tfme,

Fig. 9. Stretch dependence of

--

II bond of

p8

time during MD-experiment under the excitation of C, -- H (combination modes excitation).

Fig. 10. Time dependence of amplitude of torsion vibrations during MD-experiment (combination modes excitation).

SPIE Vol. 1403 Laser Applications in Life Sciences(1990) / 493


370

340

310

U

toriange, der
Fig. 11. Trajectory of molecular dynamics of trans-isomerization in stilbene.

1.0
0.9 0.8

0.7
0.6

>\

0.5

-o 0.4

0 0 0

L

0.3

a

0.2 0.1

--0.0

--0.1 rn-nT--0.5 0.0

1

0.5

1.0 1.5 2.0 2.5 3.0

1 iIj

5.5

4.0

time, ps
Fig. 12. Time dependence of stilbene isomerization probability during MD-experiment (combination mode excitation).

494 / SPIE Vol. 1403 Laser Applications in Life Sciences(1990)


3.2. Photoisomerization
This next series of experiments consisted in studying of isomerization. The following values of the parameters

were substituted in (7): IEvib/tIav

4

eV/ps, T = 1O K, p =

exp

(O.OO42E) (see Ref. 13; E, cm'). Taking

various values of the excitation energy in the range of 3.3--10 kcal/mole we observed the reduction of the isomerization

time (by 200 fs under strong excitation (6--10 kcal/mole). The typical results of the MD experiment are shown in Figs. 11, 12 (the excess energy in excited electronic state was E = 8.2 kcal/mole).
The analysis of some photoisomerization trajectories is given in Table 6. The corresponding time dependence of the transition probability to the ground state is given in Fig. 12. the isomerization time is determined as a time internal necessary for the transition of more than (1 -- e') molecules from excited to ground state. The measured isomerization time is equal to 2.4 Ps.
Besides calculation of the isomerization trajectories, the isomerization reaction quantum yield was also calculated using the following procedure. The first maximum of the transition probability to the ground state appears at 2.68 PS (Table 6), when 0.5% of molecules transfer to the ground state. The remaining 99.5% of molecules continue to move along the trajectories in excited state. The second maximum takes place at 2.76 ps, when the transition probability is about 0.15. Consequently, 15% of remaining molecules transfer from excited to ground state. After seven such maxima more than 98.1% of excited molecules are transfered to the ground state. As a result of it one obtains 54.9% of cis-stilben molecules and 43.2% of trans-stilben molecules.

4. DISCUSSION AND CONCLUSION
The results obtained in this work are in good agreement with experimental results of Ref. 3 except the estimate of isomerization time which has large values in the case of supersonic jet. The reason of this discrepancy might be the overstated value of the parameter IzEvib/tIav in (7).
In conclusion, we would like to stress that processes of intramolecular energy redistribution under laser excitation

and the corresponding time scale were studied in this work by using MD method for the first time. Besides,
isomerization trajectories, time and isomerization reaction quantum yield were calculated.

We believe, that and intramolecular quantum nature of laser pulse must be MD technique (see

further development of the MD approach to analysis of the processes of molecular excitation energy redistribution must be connected with the development of theory taking into account the excitation. In this case the creation of the corresponding distribution in the excited state by a taken into account. The dynamics of this distribution in phase space can be analyzed by usual Ref. 16).

5. ACKNOWLEDGEMENTS
Stimulating discussions with Professor S. Akhmanov and a support of this work provided by Professor N. Koroteev is gratefully acknowledged. This work has been partially supported by a Program of "Engineering Ensimology" via Grant (#1-250).

6. REFERENCES
1. D. C. Chemla, J. Zyss, Nonlinear Oplical Properties of Organic Molecules and Cryjsials, vol. 1--2, Academic Press, 1987. 2. S. A. Akhmanov, V. A. Vysloukh, A. S. Chirkin, Oplics offemiosecond laserpulses, AlP, 1991 (to be published).

3. P. M. Felker, A. H. Zewail, In Advances in Chemical Physics (Eds. I. Prigogine, S. A. Rice), LXX, part 1,
p.2t5S, 1988 and references in it. 4. D. W. Heermann, Computer simulation methods in theoretical physics, Springer, 1990.

5. R. R. Birge, L. M. Hubbard, J. Am. Chem. Soc., 102, p. 2195, 1980.

SPIE Vol. 1403 Laser Applications in Life Sciences (1990) / 495


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