Symmetric top energies

Since the symmetric-top energy eigenfunctions are eigenfunctions of Pc and of Pz, they have the form... 

Fig. 49. Correlation between the energy levels of (1) free rotation of the symmetric top, and (2) torsion vibrations in the potential with symmetry Cj. Quantum numbers J and K enumerate rotational levels, n vibrational levels. Relative positions of A and E levels are shown on the right.

With these results for the angular-momentum operators it is possible to obtain die Hamiltonian for the rotation of a symmetric top by direct substitution in Eq. (13). The leader is warned that care must be taken in this substitution, as die order of the derivatives is to be rigorously respected. However, given sufficient patience one can show that the classical energy becomes the Hamiltonian operator in the form (problem 12)... 

Show that the rotational energy of a rigid symmetric top is given by... 

Hougen, J. T. (1962), Classification of Rotational Energy Levels for Symmetric-Top Molecules, 7. Chem. Phys. 37, 1433. 

The eigenfunctions of J2, Ja (or Jc) and Jz clearly play important roles in polyatomic molecule rotational motion they are the eigenstates for spherical-top and symmetric-top species, and they can be used as a basis in terms of which to expand the eigenstates of asymmetric-top molecules whose energy levels do not admit an analytical solution. These eigenfunctions IJ,M,K> are given in terms of the set of so-called "rotation matrices" which are denoted Dj m,K ... 

Prolate symmetric top, 199, 211 Propane, dipole moment of, 225 Proper axis of symmetry, 53 Proper rotation, 395-396 Proton, 178 Pseudovector, 434 Pulse laser, 137,139 Purcell, E. M., 328, 360 Purely electronic energy, 57 Pure-rotation spectra, 165... 

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... 

Fig. 5.4 Correlation diagram for asymmetric-top levels. Ray s asymmetry parameter k is defined as k = (2B — A — C)/(A - C). The symmetric-top levels have been taken from Fig. 5.3 thus it is assumed that A =4.5C. Note that the 1], energy is independent of B and hence gives a horizontal line.

A spherical top is a special case of a symmetric top the rotational energy depends only on 7, and the J selection rule for the vibration-rotation transitions is the same as for symmetric tops ... 

A linear molecule in a vibrational level with vibrational angular momentum quantum number / can be viewed as a symmetric top with Ih replacing K h as the absolute value of the component of nuclear angular momentum about the symmetry axis. Replacing A with / in (5.67), we have as the rotational energy of a linear molecule for a degenerate vibrational level... 

We omit consideration of the rotational energies of symmetric tops and spherical tops in degenerate vibrational levels.14... 

Symmetrical tops are of two types. A prolate spheroid (football shape) in which /A < /b, and an oblate spheroid (pancake shape) in which /A > /b. Again, there are 2J + 1 sets of energy levels for each J, but they are no longer degenerate, as can be seen from equation (A6.9), which includes both the J and K quantum number. 

Fig. 7. Energy levels of a prolate near-symmetric top. For small asymmetry the splitting of the levels with K > 1 is negligible, thus pairs of levels (of different symmetry species) coincide at each energy value.

The extension of the trajectory calculations to a system with any number of atoms is straightforward except for the quantization of the vibrational and rotational states of the molecules. For a molecule with three different principal moments of inertia, there does not exist a simple analytical expression for the quantized rotational energy. This is only the case for molecules with some symmetry like a spherical top molecule, where all moments of inertia are identical, and a symmetric top, where two moments of inertia are identical and different from the third. For the vibrational modes, we may use a normal coordinate analysis to determine the normal modes (see Appendix E) and quantize those as for a one-dimensional oscillator. 

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