: Молекула бензола в сильном лазерном поле

                  Dissociation of Benzene Molecule in a                   
                            Strong Laser Field                            
                                 M. E. Sukharev                                 
                     General Physics Institute of RAS                     
                          117942, Moscow, Russia                          
Dissociation of benzene molecule in a strong low-frequency linearly polarized
laser field is considered theoretically under the conditions of recent
experiments. Analogy with the dissociation of diatomic molecules has been
found. The dissociation probability of benzene molecule has been derived as a
function of time. The three-photon dissociate process is shown to be realized
in experiments.
                               1.   Introduction.                               
The number of articles devoted to the interaction of molecules with a strong
laser field increased considerably in recent years. The main features of
interaction between diatomic molecules and a laser radiation were considered
in a great number of experimental [1-5] and theoretical [6-9] papers.
Classical and quantum investigations of spatial alignment of diatomic
molecules and their molecular ions in a strong laser field, as well as
ionization and dissociation of these molecules and their molecular ions
account for physical pictures of all processes.
However, when considering complex organic molecules, we observe physical
phenomena to be richer, and they are not thoroughly investigated. Most of
results obtained for diatomic molecules can be generalized to the multi-atomic
molecules. This short paper contains the results of theoretical derivations for
dissociation of benzene molecule C6H6 in the field of
linearly polarized Ti:Sapphire laser. Data were taken from experimental results
by ChinТs group, Ref. [4]. We use the atomic system of units throughout the
paper.
2.   Theoretical approach.
Let us consider the benzene molecule C6H6 in the field of
Ti:Sapphire laser with the wavelength l=400 nm, pulse length t=300 fs and
maximum intensity Imax=2´1014 W/cm2.
According to Ref. [4] first electron is ejected from this neutral molecule and
then the dissociation of C6H6+-ion occurs.
The most probable channel for decay of this ion is the separation into the
equal parts :
     
Of course, there is another channel for decay of C6H6
+-ion which includes the ejection of the second electron and subsequent
Coulomb explosion of the C6H6++-ion. We do not
consider the latter process.
The channel (1) is seen to be similar to the dissociation of the hydrogen
molecular ion considered in Ref. [2]. Indeed, the model scheme of energy levels
for C6H6+-ion (see Ref. [4]) reminds the model
scheme of energy levels for H2+ [2] containing only two
low-lying electronic levels: 1sg (even) and 1su (odd).
Therefore we consider the dissociation process of C6H6
+-ion analogously to that for H2+-ion (see Fig. 1).
The benzene molecular ion has the large reduced mass with respect to division
into equal parts. Hence, its wave function is well localized in space (see Fig.
2) and therefore we can apply classical mechanics for description of
the dissociation process (1). However, the solution of Newton equation with the
effective potential (see below) does not produce any dissociation, since laser
pulse length is too small for such large inertial system. In addition to,
effective potential barrier exists during the whole laser pulse and tunneling
of the molecular fragment is impossible due to its large mass ( see Fig. 2).
Thus, we should solve the dissociation problem in the frames of quantum 
mechanics.
     
The ground even electronic term of C6H6+-ion is
presented here in the form of the well-known Morse potential with parameters
b=2k and De=6.2 эВ, where k is approximated by the elastic constant
of C-C coupling in the C6H6-molecule and De is
the dissociation potential for the C2-molecule. The interaction of
the molecular ion with the laser field is given by expression (see. Ref. [9])
Where the strength envelope of the laser radiation is chosen in the simple
Gaussian form F(t)=F0exp(-t2/2t2)  and R
internuclear separation between the fragments C3H3+ 
and C3H3, w is the laser frequency and t is the laser
pulse length. The value½sinwt½ takes into account the repulsion
between the involved ground even electronic term and the first excited odd
repulsive electronic term.
     
Thus, the Hamiltonian of the concerned system is
     
The kinetic energy operator being of the form
Where Re is the equilibrium internuclear separation. When calculating
we make use of Re=1.39 A.
     
The time dependent Schrodinger equation with Hamiltonian (3) has been solved
numerically by the split-operator method. The wave function has been derived
by the iteration procedure according to formula
The initial wave function Y(R,0) was chosen as the solution of the
unperturbed problem for a particle in the ground state of Morse potential.
The dissociation probability  has been derived  as a function  of time according
to formula W(t)=|<Y(R,0)|Y(R,t)>|2 . In Fig. 3 envelope of
laser pulse is depicted and the dissociation probability W(t) is shown in Fig.
4.
3.   Results.
The quantity W(t) is seen from Fig. 4 increase exponentially with time and it
is equal to 0.11 after the end of laser pulse. It should be noted that the
dissociation process can not be considered as a tunneling of a fragment
through the effective potential barrier (see Fi. 2). Indeed, the
     
tunneling probability is on the order of magnitude of
Where Veff is substituted for maximum value of the field strength and
the integral is derived over the classically forbidden region under the
effective potential barrier. The tunneling effect is seen to be negligibly
small due to large reduced mass of the molecular fragment m>>1. The
Keldysh parameter g=w(2mE)1/2/F>>1. Thus, the dissociation is
the pure multiphoton process. The frequency of laser field is w  2.7 эВ, while
the dissociation potential is De=6 eV. Hence, three-photon 
process of dissociation takes place. The dissociation rate of three-photon
process is proportional to m-1/2. The total dissociation probability
is obtained by means of multiplying of this rate by the pulse length t.
Therefore the probability of three-photon process can be large, unlike the
tunneling probability. This is the explanation of large dissociation
probability W0.11 obtained in the calculations.
4.   Conclusions.
Derivations given above of dissociation of benzene molecule show that
approximately 11% of all C3H3+-ions decay on
fragments C3H3 and C3H3+ 
under the conditions of Ref. [4]. The absorption of three photons occurs in this
process.
Author is grateful to N. B. Delone, V. P. Krainov, M. V. Fedorov and S. P.
Goreslavsky for stimulating discussions of this problem. This work was
supported by Russian Foundation Investigations (grant N 96-02-18299).
                                   References                                   
1.     Peter Dietrich, Donna T. Strickland, Michel Laberge and Paul B. Corkum,
Phys. Rev. A, 47, N3, 2305 (1993). M. Ivanov, T. Siedeman, P. Corkum,
Phys. Rev. A, 54, N2, 1541 (1996).
2.     F. A. Ilkov, T. D. G. Walsh, S. Turgeon and S. L. Chin, Phys. Rev. A, 
51, N4, R2695 (1995). F. A. Ilkov, T. D. G. Walsh, S. Turgeon and S. L.
Chin, Chem. Phys. Lett 247 (1995).
3.     S. L. Chin, Y. Liang, J. E. Decker, F. A. Ilkov, M. V. Amosov, J. Phys.
B: At. Mol. Opt. Phys. 25 (1992), L249.
4.     A. Talebpour, S. Larochelle and S. L. Chin, in press.
5.     D. Normand, S. Dobosz, M. Lezius, P. DТOliveira and M. Schmidt: in 
Multiphoton Processes, 1996, Conf., Garmish-Partenkirchen, Germany, Inst.
Phys. Ser. No 154 (IOPP, Bristol 1997), p. 287.
6.     A. Giusti-Suzor, F. H. Mies, L. F. DiMauro, E. Charon and B. Yang, J.
Phys. B: At. Mol. Opt. Phys. 28 (1995) 309-339.
7.     P. Dietrich, M. Yu. Ivanov, F. A. Ilkov and P. B. Corkum, Phys. Rev.
Lett. 76, 1996.
8.     S. Chelkowski, Tao Zuo, A. D. Bandrauk, Phys. Rev. A, 46, N9,
R5342 (1992)
9.     M. E. Sukharev, V. P. Krainov, JETP, 83, 457,1996. M. E.
Sukharev, V. P. Krainov, Laser Physics, 7, No3, 803, 1997. M. E.
Sukharev, V. P. Krainov, JETP, 113, No2, 573, 1998. M. E. Sukharev, V.
P. Krainov, JOSA B, in press.
                                 Figure captions                                 
Fig. 1. Scheme of dissociation for benzene molecular ion C6H6+.
Fig. 2. The Morse potential (a), the effective potential (b) for maximum value
of the field strength (a.u.), and the square of the wave function of the ground
state for benzene molecular ion (c) as functions of the nuclear separation R
(a.u.) between the fragments C3H3 and C3H
3+.
Fig. 3. Envelope of laser pulse as a function of time (fs).
Fig. 4. The dissociation probability of benzene molecular ion C6H
6+ as a function of time (fs).
     
     
     
     
     
     
     
                                     Fig. 1                                     
Morse potential (a) (a.u.),
effective potential for max. field (b) (a.u),
     
c
     
b
     
a
     
square of the wave function of the ground state for benzene molecular
ion (c)
                                                                         R, a.u.
                                     Fig. 2                                     
     
                              
t, fs
                                     Fig. 3                                     
     
b
     
W(t)
                                                                           t, fs
                                     Fig. 4