Dissociation of
Benzene Molecule in a Strong Laser Field
M. E. Sukharev, General
Physics Institute of RAS
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.
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 Chin’s group, Ref. [4]. We use the
atomic system of units throughout the paper.
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)=||2 . In Fig. 3
envelope of laser pulse is depicted and the dissociation probability W(t) is
shown in Fig. 4.
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 µ 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 W»0.11 obtained in
the calculations.
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).
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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 C3H3+.
Fig. 3. Envelope of
laser pulse as a function of time (fs).
Fig. 4. The
dissociation probability of benzene molecular ion C6H6+ 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
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