Symmetric top wave functions

The Hamiltonian which represents the rotational kinetic energy of an asymmetric top with three unequal moments of inertia is 

Let us consider the behaviour of this function, with a particular value pJM(w ) at (cf)], 6, xi), under a rotation of the body through (02. Under the rotation, the eigenfunction is transformed to a new function 4V where 

We know that the value of the transformed wave function in the new orientation is the same as that of the original wave function pJM in the original orientation, i.e. 

The second equation states that the value of the new function at a particular coordinate position ( / , 6, x)is the same as the valueofthe original function at the point ((/ , O, x ) which transforms into ( p, 6, / ) under the rotation through ( fi2, 92, xi) (see figure 5.1). From equations (5.59) and (5.60) we have 

If we start out with the principal inertial axes coincident with (.X, Y, Z), that is, 

Let us consider the behaviom of this fimction, with a particular value at 

under a rotation of the body through C02. Under the rotation, the eigenfunction is transformed to a new fimction (j) where 

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Let i//, be an asymmetric-top wave function. A convenient complete orthonormal set to use here is the symmetric-top wave functions, which are functions of the same coordinates (the Eulerian angles) and satisfy the same boundary conditions as the asymmetric-top functions ... 

For a symmetric top, symmetry requires the dipole moment to lie along the symmetry axis, so that two of the three principal-axis components of d must vanish. In deriving the symmetric-top wave functions in Section 5.5, we assumed that the c axis was the symmetry axis hence to use the eigenfunctions (5.68) to find the selection rules, we must take da = db — 0, dcJ= 0. For a symmetric top, we thus must evaluate only the three integrals IXOc, lYoc anc Azof The three relevant direction cosines are given in (6.64) and Problem 5.15 they are independent of x- Since the integral... 

The theta factor (which turns out to be a rather complicated function) can be found5 by direct solution of the Schrodinger equation or by the use of ladder operators. The work is involved and we omit it. The spherical-top wave functions (5.60) are identical with the symmetric-top wave functions (5.68)—see Problem 5.14. 

The convention is that r ranges from - 7 to 7 as E increases. The index t is a bookkeeping number, rather than a true quantum number. The degeneracy of the asymmetric-top energy levels is 27+ 1, corresponding to the 27 +1 values of M, which do not affect the energy. Each asymmetric-top wave function is a linear combination of the 27+1 symmetric-top wave functions with the same value of 7 and of M as i Let us consider some examples. For 7 = 0, the only value of K is 0, and the secular equation (5.78) is... 

The first term represents a statistical weighting factor, the second term a solid angle factor for the probability of getting the angle 0, and the third term is a symmetric top wave function. 

With our choice of phases, the symmetric top wave function is related to the corresponding rotation matrix element by... 

The above discussion does not consider the role of angular momentum, shown here to be crucial to overall control. Specifically, to consider the effect of averaging over M [84], the situation that would exist in a nonpolarized medium, we associate with each of the D), L), 1), and 2) states a parity-adapted symmetric-top wave function of the type... 

Using the same formulation of the Hamiltonian as in Sec. VII [specifically Eqs. (67)—(70)], the two-step process makes use of five pairs of rovibrational states (specified explicitly below). The vibrational eigenstates correspond to the combined torsional and S-D asymmetric stretching modes. The rotational eigenfunctions are the parity-adapted symmetric top wave functions. Each eigenstate has additionally an Si A label denoting its symmetry with respect to inversion. Within the pairs used, the observable chiral states are composed as... 

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... 

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