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Classical chaotic behavior in diatomic molecules is studied when chaos is driven by a circularly polarized resonant electric field and expanding up to fourth order of approximation the Morse’s potential and angular momentum of the system. On this double resonant system, we find a weak and a strong stationary (or critical) points where the chaotic characteristics are different with respect to the initial conditions of the system. Chaotic behavior around the weak critical point appears at much weaker intensity on the electric field than the electric field needed for the chaotic behavior around the strong critical point. This classical chaotic behavior is determined through Lyapunov exponent, separation of two nearby trajectories, and Fourier transformation of the time evolution of the system. The threshold of the amplitude of the electric field for appearing the chaotic behavior near each critical point is different and is found for several molecules.

Beside the clear importance of the study of diatomic molecules [

The study of diatomic molecule is a typical two bodies problem with radial force as shown in

where

Due to the symmetry under rotation of the system, the relative motion is reduced to 1-D problem and its

equation is given in spherical coordinates by

where the effective potential is

being l the angular moment of the system with

The constant of motion (energy) associated to this system is

and its Lagrangian is

or

Therefore, its Hamiltonian is

Let

and

Since one has that_{o} (the natural frequency of oscillation of the molecule) from the

relation

where one has that

The potential associated to the molecular interaction

where D,

Then, the above Hamiltonian can be written of the form

where

and

Let us recall that

Then, Hamilton’s equations of motion are

From the last equation one has that

The set of critical points for this system,

that is, the critical points are located over the

Since one has that

hyperbolic points(if

and its characteristic frequency is

The set critical points is

and one has

Therefore,

The values

Diatomic molecules with a dipolar moment

moment is just the charge times the distance between atoms,

Let

In this way, using (14), (15) and (16), the full Hamiltonian is

and the equations of motion are now

By choosing the study of motion at

These equation are solved numerically to find the dynamical behavior of the system. What we are interested in is on the threshold of the intensity of the electric field

Let us consider the diatomic molecule BeO and the initial conditions near the weak critical point

tioned range of values near

For an intensity of the electric field such that

component. As we can see clearly from these figures, this trajectory is chaotic and the system behaves as chaotic system (the same was done for the other nine trajectories).

When initial conditions

Now, choosing the initial conditions close to the critical value

For an intensity field such that

These figures show that the trajectories are chaotic with this aptitude of electric field. In fact we checked that the same happen independently of the initial conditions chosen and for higher values of the magnitude of electric field. The transition region (just for

course, if a trajectory is chaotic around the critical point

tical point

We presented the study of the classical chaotic behavior of a diatomic molecule driven by a circularly polarized resonant electric field. The double resonance system appears from expanding up to fourth order of approximation the Morse’s potential and angular momentum. Chaotic behavior of trajectories around the weak critical point appears at much weaker electric field strength than the strength of the electric field needed to appear the chaotic behavior of trajectories around the strong critical points. This result points out the possible chaotic behavior of double nonlinear resonant systems depending on its initial condition. The exact transition region to chaotic behavior will be presented in other articles. The gap (weak-strong) on the thresholds of the electric field strength to occur the chaotic behavior may be important for the study of diatomic molecules in different environments and for quantum dynamical studies.