Rigid Rotor Wave Functions

In the rigid rotor approximation, the radial wave function is independent of /. The total wave function is... 

The derivation above may be generalized to wave functions other than electronic ones. By evaluation of transition dipole matrix elements for rigid-rotor and harmonic-oscillator rotational and vibrational wave functions, respectively, one arrives at the well-known selection rules in those systems that absorptions and emissions can only occur to adjacent levels, as previously noted in Chapter 9. Of course, simplifications in the derivations lead to many forbidden transitions being observable in the laboratory as weakly allowed, both in the electronic case and in the rotational and vibrational cases. 

The nuclear function %a(R) is usually expanded in terms of a wave function describing the vibrational motion of the nuclei, and a rotational wave function [36, 37]. Analysis of the vibrational part of the wave function usually assumes that the vibrational motion is harmonic, such that a normal mode analysis can be applied [36, 38]. The breakdown of this approximation leads to vibrational coupling, commonly termed intramolecular vibrational energy redistribution, IVR. The rotational basis is usually taken as the rigid rotor basis [36, 38 -0]. This separation between vibrational and rotational motions neglects centrifugal and Coriolis coupling of rotation and vibration [36, 38—401. Next, we will write the wave packet prepared by the pump laser in terms of the zeroth-order BO basis as... 

The coordinate system used in the close-coupling method is the space-fixed frame. For simplicity we consider the atom-diatom scattering. The wave function iM(.R,r,R) for an atom-rigid rotor system corresponding to the total energy E, total angular momentum J, and its projection M on the space-fixed z axis can be written as an expansion,... 

From Eqs. (3.40) and (3.35) it is obvious that the inversion—rotation wave functions i//°. (0,, X, p) of NH3 which are the eigenfunctions of the operator, , can be written as a product of the rigid-rotor symmetric top wave functions depending on the Euler angles 0,4>, x and the inversion wave functions, depending on the variable p. Integration of the Schrodinger equation... 

To obtain the selection rules for the rigid rotor we must look at the symmetry of the problem in slightly greater detail. The rotor is described by the two angles 6 and 0 and by a wave function having the form (omitting normalization constants)... 

The wave functions for the hydrogen atom have the form of a product of a radial function (a function of r only) and the rigid rotor functions. The selection rules for the rotor functions must be the same as those obtained above, namely. 

For a symmetric rotor, in the present approximation, only the z component of in, the vibrational angular momentum, needs to be considered. The problem may be treated as a perturbation employing zero-order wave functions which are products of rigid rotor and harmonic oscillator functions. When the molecule is in a state such that vka + Vkb — 1, where Qka and Qw> are degenerate, it is necessary to solve the secular determinant... 

Ja -H Jb + Jc It would also be useful to know whether any space-fixed components of J commute with particular body-fixed components of J, so that commuting sets of observables may be constructed as an aid in visualizing the physical significance of rigid rotor wave functions. 

The rotational wave function is the same spherical harmonic function that occurred with the hydrogen atom or the rigid rotor ... 

There is a polynomial solution for the radial part of the H atom solution of the Schrodinger equation. The functions are related to previously studied Laguerre polynomials. The total solution for the H atom is the product of the rigid rotor (0, < ) wave functions with the Laguerre radial functions. The eigenvalues for the orbitals are exactly the same as for the... 

---