Rotational excitation, effective Hamiltonian

F. Effective Hamiltonian for Rotational Excitations in Diatomic Molecules... 

In Section III.E this partitioning technique is illustrated for two-photon processes in atoms. It is next applied in Section III.F to construct an effective Hamiltonian relevant for the rotational excitation in diatomic molecules. 

To determine an effective dressed Hamiltonian characterizing a molecule excited by strong laser fields, we have to apply the standard construction of the free effective Hamiltonian (such as the Born-Oppenheimer approximation), taking into account the interaction with the field nonperturbatively (if resonances occur). This leads to four different time scales in general (i) for the motion of the electrons, (ii) for the vibrations of the nuclei, (iii) for the rotation of the nuclei, and (iv) for the frequency of the interacting field. It is well known that it is a good strategy to take into account the time scales from the fastest to the slowest one. 

The connection between the full molecular N-particle Hamiltonian of Equation 12.3 including rotational excitations and dressing fields, and the effective Hamiltonian of (Equation 12.1) can be made using a Born-Oppenheimer approximation. The diagonalization of the Hamiltonian [92] Hbo = - d,Ej - -... 

The low-energy effective Hamiltonian for the external motion and internal rotational excitations of a single molecule is... 

See also the reviews by D. J. Kouri, Rotational excitation II Approximation methods, in Atom Molecule Collision Theory A Guide for the Experimentalist", R. B. Bernstein, ed., Plenum Press, New York (1979), p. 301, and H. Rabitz, Effective Hamiltonians in molecular collisions, in "Dynamics of Molecular Collisions, Part B", W. H. Miller, ed., Plenum Press, New York (1976), p. 33. 

The effects of the off-diagonal terms when folded-in by perturbation theory are of two types. They can either produce operators of the same form as those which already exist in the Hamiltonian constructed from the Azl = 0 matrix elements (the zeroth-order Hamiltonian), or they can have a completely novel form. A good example of the former type is the second-order contribution to the rotational constant which arises from admixture of excited and A states,... 

A typical problem of interest at Los Alamos is the solution of the infrared multiple photon excitation dynamics of sulfur hexafluoride. This very problem has been quite popular in the literature in the past few years. (7) The solution of this problem is modeled by a molecular Hamiltonian which explicitly treats the asymmetric stretch ladder of the molecule coupled implicitly to the other molecular degrees of freedom. (See Fig. 12.) We consider the the first seven vibrational states of the mode of SF (6v ) the octahedral symmetry of the SF molecule makes these vibrational levels degenerate, and coupling between vibrational and rotational motion splits these degeneracies slightly. Furthermore, there is a rotational manifold of states associated with each vibrational level. Even to describe the zeroth-order level states of this molecule is itself a fairly complicated problem. Now if we were to include collisions in our model of multiple photon excitation of SF, e wou d have to solve a matrix Bloch equation with a minimum of 84 x 84 elements. Clearly such a problem is beyond our current abilities, so in fact we neglect collisional effects in order to stay with a Schrodinger picture of the excitation dynamics. 

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