Symmetric tops

Marquardt R, Quack M, Stohner J and Sutcliffe E 1986 Quantum-mechanical wavepacket dynamics of the CH group in the symmetric top XgCH compounds using effective Hamiltonians from high-resolution spectroscopy J. Chem. Soc., Faraday Trans. 2 82 1173-87... 

Luckhaus D and Quack M 1993 The role of potential anisotropy in the dynamics of the CH chromophore in CHXg (Cg ) symmetric tops Chem. Phys. Lett. 205 277-84... 

Microwave studies in molecular beams are usually limited to studying the ground vibrational state of the complex. For complexes made up of two molecules (as opposed to atoms), the intennolecular vibrations are usually of relatively low amplitude (though there are some notable exceptions to this, such as the ammonia dimer). Under these circumstances, the methods of classical microwave spectroscopy can be used to detennine the stmcture of the complex. The principal quantities obtained from a microwave spectmm are the rotational constants of the complex, which are conventionally designated A, B and C in decreasing order of magnitude there is one rotational constant 5 for a linear complex, two constants (A and B or B and C) for a complex that is a symmetric top and tliree constants (A, B and C) for an... 

Molecules for which two of the three principal moments of inertia are equal are called symmetric tops. Those for which the unique moment of inertia is smaller than the other two are termed prolate symmetric tops if the unique moment of inertia is larger than the others, the molecule is an oblate symmetric top. 

For molecules that are non-linear and whose rotational wavefunctions are given in terms of the spherical or symmetric top functions D l,m,K, the dipole moment Pave can have components along any or all three of the molecule s internal coordinates (e.g., the three molecule-fixed coordinates that describe the orientation of the principal axes of the moment of inertia tensor). For a spherical top molecule, Pavel vanishes, so El transitions do not occur. 

For symmetric top species, Pave hes along the symmetry axis of the molecule, so the orientation of Pave can again be described in terms of 0 and (j), the angles used to locate the orientation of the molecule s symmetry axis relative to the lab-fixed coordinate system. As a result, the El integral again can be decomposed into three pieces ... 

In a symmetric top molecule such as NH3, if the transition dipole lies along the molecule s symmetry axis, only k = 0 contributes. Such vibrations preserve the molecule s symmetry relative to this symmetry axis (e.g. the totally symmetric N-H stretching mode in NH3). The additional selection rule AK = 0... 

For non-linear molecules of the spherical or symmetric top variety, pf j(Rg) (or dpf j/dRa) may be aligned along or perdendicular to a symmetry axis of the molecule. The selection rules that result are... 

In the symmetric top cases, Hrot can be expressed in terms of J2 and the angular momentum along the axis with the unique moment of inertia (denoted the a-axis for prolate tops and the c-axis of oblate tops) ... 

For a symmetric rotor, or symmetric top as it is sometimes called, two of the principal moments of inertia are equal and the third is non-zero. If... 

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.

Symmetrical tops are of two types. A prolate spheroid (football shape) in which... 

St. Pierre A. G., Steele W. A. Collisional effects upon rotational correlations of symmetrical top molecules. J. Chem. Phys. 57, 4638-48 (1972). 

Benzene is the prototype aromatic Lewis base. It offers formally three pairs of equivalent, conjugated tt bonds as the potential electron donor. Symmetric-top-type rotational spectra have been observed for the benzene HX complexes, where X is F [139], Cl [140] or Br [141], by methods (molecular-beam... 

Although the hypergeometric functions are useful in spectroscopy, as they describe the rotation of a symmetric top molecule (Section 9.2.4), their importance is primarily due to their generality. If, for example, a = 1 and fi say, Eq. (154) becomes a +i — a for all values of n. The result is the ordinary geometric series... 

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... 

Recent experiments11 have shown that for symmetric top molecules, such as methyl chloride, electron impact ionization is more probable at the positive end of the dipole. This follows intuitively from simple electrostatics, since it would be expected that an electron should be more strongly attracted to the positive end of the dipole, leading to a greater ionization probability at this end of the molecule. The effect may be quantified in terms of a steric ratio R. 

Fortunately, in the case of a rotational diffusion tensor with axial symmetry (such molecules are denoted "symmetric top"), some simplification occurs. Let us introduce new notations D// = Dz and D = Dx = Dy. Furthermore, we shall define effective correlation times ... 

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