Asymmetric top energies

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 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... 

Townes, C. H., and A. L. Schawlow, Microwave Spectroscopy, McGraw-Hill, New York, 1955. Comprehensive discussions of theory and experimental techniques useful appendices for calculating asymmetric-top energy levels. For internal rotation, however, consult Sugden and Kenney or Wollrab. 

Figure 7 Rotation-vibration mixing in SO, for states of a, symmetry and 7 = 12. The columns correspond to TV, = 2, 10, 16 (cf. Eq. (31)). The standard deviation of the symmetric and asymmetric top energies are plotted against the expectation value of the corresponding energies in the upper and lower row, respectively. (From Ref. 49.)...

Figure 5.7 Correlation diagram for asymmetric rotor state energies between the oblate A= B > C) and prolate A > B = C) limits. Here the rotational constants A and C are fixed, while B (the horizontal coordinate) is varied continously between A and C. The oblate and prolate top energies are given by Eqs. 5.25 and 5.26, respectively the asymmetric top energies for intermediate rotational constants B must be found by explicit diagonalization of the asymmetric top Hamiltonian [2,6].

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... 

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 ... 

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. 

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.

The energy and symmetry properties of rotational levels have been discussed comprehensively by Herzberg (1945). The principles are unchanged as between the different classes, although the details vary somewhat from one class to another. Here we shall consider only the case of a near-prolate asymmetric top (lcxlb >la), a class which contains formaldehyde, propynal, and (raws-bent acetylene among other examples. In first approximation the rotational energy levels are expressed in terms of the rotational constants A, B, and C, where... 

The manifold of energy levels for small J then has the appearance shown in Fig. 7. In higher approximation it would be necessary to take account of centrifugal distortion which causes the effective moments of inertia to increase slightly with J. For highly asymmetric tops the evaluation of the energy levels is difficult but approximate values can be interpolated from published tables (Erlandsson, 1956 King ef al., 1943, 1949). 

A rigorous modelling of thermal broadening is — in practice — quite cumbersome and tedious. Let us consider a general asymmetric top molecule such as H2O, for example. Each total angular momentum state, specified by the quantum number J, splits into (2 J + 1) nondegenerate substates with energies E 0f (K = 1,..., 2J + 1). Every one of these (2J + 1) rotational states corresponds to a different type of rotational motion and is described by a distinct rotational wavefunction (see Section 11.3). 

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

We note that this argument is independent of the p values of the energy levels. As such the arguments apply equally well to the asymmetric top where the p degeneracy does not exist [88]. 

Discussion of selection rules must be modified for NH2D (ND2H) because they are asymmetric top molecules The convention for inertia moments in asymmetric, tops is/ [Pg.83]

Figure 4.3-12 Schematic diagram of the lowest rotational energy levels of an asymmetric top molecule.

Experimental [23, 24] probabilities for the many possible collision-induced transitions in this molecule show no apparent dependence on energy gap between initial and final state. However, the probabilities fall exponentially with the magnitude of the AM gap measured as shown in Fig. 3. This static model (Fig. 3) of an asymmetric top 7ka kc AM vector is not wholly representative since the vector s natural precession creates an AM trajectory. Coupling of this vector to an external AM introduced on collision can cause the vector to hop to an adjacent trajectory of different y ka kc on the molecule s AM surface. When this dynamical picture is considered, the conclusions above remain valid [24, 25]. The energy gap generally becomes more dominant as initial j state increases [25], an effect also found in... 

We now turn to the order of magnitude considerations that will eventually lead to the neglect of the off-diagonal matrix elements in most asymmetric top molecules. From second order perturbation theory, the contribution of the off diagonal elements to the rotational energies is given by... 

Now we return to Eq. (III.13) which, as we have discussed, gives the rotational energy levels within the experimental uncertainties for many asymmetric top molecules including ethylene oxide. Since, according to this equation, the Zee-man perturbation of the levels is linear with respect to the g-values and susceptibility anisotropies, the same must be true for the Zeeman splittings of the rotational transition frequencies. Thus, from each measured Zeeman satellite with a frequency shift Av H) with respect to the zero field frequency ... 

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