The rotational coupling

In the equations above we have just considered the vibrational and the hy-perspherical degrees of freedom. Coupling to the over-all rotational motion is most important for small systems but can be treated also within the linear approximation as follows. The two terms are given as the rotational/vibrational and the rotational contributions to the kinetic energy, i.e. 

Similar expressions follow for Xq, with p replaced by 6 etc. The reason for this partitioning is only to pull out the correct hamiltonian for the iV=3 case. We have furthermore used the notation 

Introducing the momenta Px = dTjduox etc. we obtain the following connection between the momenta and the derivatives  

Notice that the derivation is carried to first order in pk or Qk If second order terms are included the above coupling matrix will also contain coupling terms to the vibrational modes. The derivation including second order terms is in principle straightforward, but the final expression is rather lengthy. 

It is furthermore convenient to pull out the moment of inertia connected to the three-atomic system. Thus if we ignore the contribution from the vibrational displacements we have  

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The rotational coupling matrix elements between Z-IT and fl-A states have been evaluated analytically by use of the L. and L. operators. 

E.E.Nikitin and K.Taulbjerg, Effect of the rotational coupling on charge transfer into Coulomb channels, J. Phys. B 27, 2259 (1994)... 

Figure 5.18b The dependence of the (3II. L+1 3E+) matrix element of NO+ on the internuclear separation. The rotational coupling matrix element is shown (from Hutter, et al., 1994). The matrix elements that are actually plotted are of the spherical tensor form of the operator, L+i = — 2-1/2L+, which has sign opposite to that for L+ discussed in the text.

The value of the rotational coupling between the B EJ and C II states which are the two upper states arising in the Lyman and Werner band systems of H, has been calculated by Ford (1974). He subsequently calculated the effect of this coupling on the rotational line strengths of the first Lyman (V jx = 25) and Werner (V gx systems for J = 0 to... 

The effect of the rotational coupling is readily seen on the ratio of the emission intensities of P and R lines connected to the same v J upper level, since in this case one eliminates the density of the upper level and one can compare directly theoretical results to the intensity measurements. Table 2 reproduces a part of Table VI of Abgrall et al (1987) in order to show the quantitative importance of this effect. The HL column refers to the ratios of the H6nl-London factors which appear when one neglects coupling effects and the role of... 

Most of the present information on the coupling of the rotational degrees of freedom to the helium environment come from the line widths of infra-red ro-vibrational transitions of small mostly diatomic and triatomic molecules. In view of the small changes in vibrational amplitudes accompanying vibrational fundamentals of only about 0.05 — 0.10 A, the effect on the line width is considered to be negligible and the rotational coupling is the dominant mechanism determining the linewidth. This conclusion has been confirmed... 

Launay and Le Dourneuf use essentially, although not exactly, the same PA hyperspherical coordinates as Pack and Parker. In particular, they choose the quantisation axis for the internal motion as the axis of least inertia, so that the rotational couplings about this axis are minimised in a variational way [49, 52]. This leads to rapid convergence of reaction probabilities with respect to the total angular momentum projection quantum number i , and so allows one to use far smaller basis sets for large J values than are required when all possible projections are retained. 

If one transforms to the manifold R+ x x R then one can consider explicitly the rotational coupling of electronic and angular motions. The fact that the transformation is to a manifold rather than a vector space means that any operator built using coordinates defined on the manifold will be well defined only where the Jacobian for the transformation does not vanish. This does not cause great problems here because the only places where the Jacobian vanishes are when R = 0 and where j8 = 0 and = 7t. The region around ft = 0 is inaccessible because of the nuclear repulsion term and the exact angular wavefunctions take care of the problem with j3. 

The rotational coupling matrix elements (i/rs- % i//L) between Z - n molecular states were determined directly from the quadrupole moment tensor which allows the consideration of translation effects in the collision dynamics [26]. In the approximation of the common translation factor [27], the radial and rotational coupling matrix elements between states y/K and y/L may indeed be transformed respectively into ... 

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