Angular momentum anharmonicity

From (8.23) and (8.24) one can see two special cases when the potential becomes separable. In the first case c12 = 0, we have two independent anharmonic modes, each having two equilibrium positions. In the second case, the angular part of the potential (8.23) Vr is zero, and the motion breaks up into radial vibration in the double well V0(q) and a free rotation, i.e. propagation of the waves of transverse displacements along the ring. The latter case is called free pseudorotation. Since the displacements of atomic groups in the wave are purely transverse, they do not contribute to the total angular momentum. 

Fig. 2.7. Plot of the ratio of the anharmonic to harmonic state density versus energy relative to the bottom of the well for a range of total angular momentum in CH2CO considering only the anharmonicities in the intermolecular modes of the CH2 + CO channel. The classical dissociation threshold is at about 33,000 cm. The curve for J=Q includes Monte Carlo integration uncertainty error bars.

For the studies on benzene described in the following sections, the Hamiltonian was formulated in rectilinear coordinates. The pure vibrational kinetic energy operator was treated exactly (but nonquadratic vibrational angular momentum terms tt,tt, Cori-olos, and rotational terms were neglected), but the price to be paid is that the anharmonic potential contains a large number of terms. Development of the vibrational anharmonic Hamiltonian is described in the next three sections. 

This corresponds to an anharmonic sequence of levels labeled by m (the total number of states being once more equal to N-l-1). This quantum number, m, should not be confused with the quantum number associated with the projection of the angular momentum operator. The most interesting situations occurs with the particular choice >li=0, A = A2 (i in (2.64) and (2.65). In this case it is possible to put the spectrum (2.65) in a one-to-one correspondence with the bound-state spectrum of the one-dimensional Morse potential. This can be done by choosing in Eq. (2.65) only the nonnegative branch of the quantum... 

The matrix elements of this operator can be obtained explicitly in the local basis. This computation employs the close similarity between the operators and the angular momentum ladder operators, Thus the operator f can be seen as a rotation about a finite angle related to the parameter [3. Consequently, t acts on the single state of the local basis to generate an expansion (in terms of Jacobi polynomials) over the complete set of vibrational states. This is equivalent to saying that the harmonic selection rules are relaxed and transitions An = 0, 1, 2,.. . now have nonzero matrix elements, in accordance with the electric anharmonicity of the dipole operator. A good approximation for these matrix elements is given by the simple formula... 

A detailed study of further properties of the V operator shows that it behaves fairly well compared to realistic situations. In this brief review we have not discussed the dependence of the /-doubling operator on the stretching and quantum numbers or on the angular momentum quantum number A discussion on this dependence can be found in Refs. 86 and 88. Nonetheless, the algebraic realization can lead to results superior to those obtained with standard approaches. Moreover, the algebraic formulation includes vibrational anharmonicity from the very onset of formulating the model for a given system. 

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