Asymmetric tops

Quack M and Sutcliffe E 1983 Quantum interference in the IR-multiphoton excitation of small asymmetric-top molecules ozone Chem. Phys. Lett. 99 167-72... 

The rotational eigenfunctions and energy levels of a molecule for which all three principal moments of inertia are distinct (a so-called asymmetric top) can not easily be expressed in terms of the angular momentum eigenstates and the J, M, and K quantum numbers. However, given the three principal moments of inertia la, Ib, and Ic, a matrix representation of each of the three contributions to the rotational Hamiltonian... 

Finally, an asymmetric top is one in which all three principal moments of inertia are different. The energy levels are given by... 

This simple relaxation theory becomes invalid, however, if motional anisotropy, or internal motions, or both, are involved. Then, the rotational correlation-time in Eq. 30 is an effective correlation-time, containing contributions from reorientation about the principal axes of the rotational-diffusion tensor. In order to separate these contributions, a physical model to describe the manner by which a molecule tumbles is required. Complete expressions for intramolecular, dipolar relaxation-rates for the three classes of spherical, axially symmetric, and asymmetric top molecules have been evaluated by Werbelow and Grant, in order to incorporate into the relaxation theory the appropriate rotational-diffusion model developed by Woess-ner. Methyl internal motion has been treated in a few instances, by using the equations of Woessner and coworkers to describe internal rotation superimposed on the overall, molecular tumbling. Nevertheless, if motional anisotropy is present, it is wiser not to attempt a quantitative determination of interproton distances from measured, proton relaxation-rates, although semiquantitative conclusions are probably justified by neglecting motional anisotropy, as will be seen in the following Section. 

Figure 4.9 Spherical top (methane), symmetric tops (benzene, decane), asymmetric top (water), clockwise from lower left...

Each of the elements of Jc2, Ja2, and Jb2 must, of course, be multiplied, respectively, by l/2Ic, l/2Ia, and l/2Ib and summed together to form the matrix representation of Hrot- The diagonalization of this matrix then provides the asymmetric top energies and wavefunctions. 

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 ... 

By equation 74, we see that [ ], depending on the form of the wavefunction, and by equation 75 that ) can be linearly expanded (hence linearly varied) in some set of N functions. This is directly analogous to expanding the asymmetric top rotational wavefunctions in a complete set of symmetric top rotational wavefunctions. The task is to minimize E subject to the single constraint that the wavefunction remain normalized, or... 

The rotational dynamics of nitrogen and carbon dioxide were recorded by Akhmanov and Koroteev [7]. The transients look similar to the transients by Morgen et al. [8], recorded with time resolved Raman induced polarization spectroscopy [9]. A fs-DFWM experiment was performed by Frey et al. [10] on diatomics and linear polyatomics. To prevent collisional dephasing, they transferred the method into the expansion zone of a molecular beam. In succession, experiments on linear molecules and symmetric tops were performed on molecules like CHCI3 [11] and CgHf, [12], Transients of asymmetric tops like the near oblate pyrimidine, pyrazine and pyridine [13] and SO2 [11] were reported in the following years. 

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 ... 

Since (Section 5.3) the asymmetric-top function is an eigenfunction of P2 with eigenvalue J(J+ 1)A2, we only include in the sum those symmetric-top functions that have the same J value as f/rJSee the paragraph after (1.37).] Similarly, since is an eigenfunction of P2 with eigenvalue Mh, we only include

quantum numbers J K and M therefore reduces to a finite sum over the 2J 4-1 possible values of K ... 

Vibration-rotation interaction causes the rotational constants to vary with the vibrational quantum numbers [Eq. (5.72)]. Correction terms for centrifugal distortion can also be added to the energy expressions for asymmetric tops. 

For asymmetric tops with a dipole moment, the J and M selection rules are... 

For the asymmetric top, K is not a good quantum number. Asymmetric-top A/ = 0 transitions correspond to lines of nonzero frequency, in contrast to the symmetric top. Moreover, it is possible for an asymmetric-top level with quantum number J — 1 to lie above a level with quantum number J (see Fig. 5.4) asymmetric-top microwave absorption lines with Ay= —1 are thus possible. Microwave transitions with Ay= -1, 0, and +1 are called / -, Q-, and / -branch lines, respectively, although these lines are interspersed among one another and do not really form branches as in the infrared spectrum. The dipole moment of an asymmetric top can be resolved into components along the three principal axes ... 

Solve the secular equation for the 7 = 2 asymmetric-top levels. (Be sure to group even K values together.)... 

The asymmetric-top eigenfunctions are linear combinations of symmetric-top eigenfunctions, which allows the asymmetric-top selection rules to be found. Since the dipole moment can have an arbitrary orientation, we get the three kinds of selection rules in (5.89) details are omitted. 

The pure-rotational Raman spectrum of a polyatomic molecule provides information on the moments of inertia, hence allowing a structural determination. For a molecule to exhibit a pure-rotational Raman spectrum, the polarizability must be anisotropic that is, the polarizability ellipsoid must not be a sphere. As noted in Section 5.2, a spherical top has a spherical polarizability ellipsoid, and so gives no pure-rotational Raman spectrum. Symmetric and asymmetric tops have asymmetric polarizabilities. The structures of several nonpolar molecules (which cannot be studied by microwave spectroscopy) have been determined from their pure-rotational Raman spectra these include F2, C2H4, and C6H6. 

Asymmetric tops have no degenerate vibrational modes (this will be shown in Chapter 9) hence we need not worry about vibrational angular momentum for them. 

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