Asymmetric top wave functions

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 ... 

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... 

Equation (5.80) then gives E = 0 for 7 = 0. The 7 = 0 asymmetric-top wave function is, of course, identical with the symmetric-top 7 = 0 wave function. 

R. E. Wyatt, Hindered asymmetric top wave functions for three-dimensional H + H2, J. Chem. Phys. 62 3173 (1975). 

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... 

This is the general expression for the wave function of an asymmetric top molecule. 

As with diatomic molecules, the principal selection rule is that a permanent dipole moment is required for a molecule to produce a microwave spectrum. Linear polyatomic molecules have rotational wave functions exactly like those of diatomic molecules, so their rotational selection rules and spectra are the same as those of diatomic molecules. A symmetric linear molecule such as acetylene (ethyne) has no permanent dipole moment, and does not have a microwave spectrum. The fact that N2O has a microwave spectrum establishes the fact that it is NNO, not NON. Spherical top molecules such as CCI4 and SFe are so symmetrical that they cannot have a nonzero permanent dipole moment, and they have no microwave spectrum. A symmetric top molecule with a permanent dipole moment will have a microwave spectrum. A microwave spectrum is always observed for an asymmetric top molecule, because it has so little symmetry that it must have a nonzero permanent dipole moment. 

There are two contributions to the polarizability of a molecule the distortion of the electronic wave function and the distortion of the nuclear framework. The major contribution is from the electrons, and can be considered to be the sum of contributions from the individual electrons. The contributions of the inner-shell electrons are nearly independent of orientation and these contributions can be ignored. The polarizability of electrons in a bond parallel to the bond direction is different from the polarizability perpendicular to that bond. As a diatomic molecule or linear polyatomic molecule rotates, the components of the polarizability in fixed directions are modulated (fluctuate periodically) as the ellipsoid of polarizability rotates. The rotation of a diatomic or linear polyatomic molecule will be Raman active (produce a Raman spectram). In a nonlinear polyatomic molecule, the polarizabilities of the individual bonds add vectorially to make up the total polarizability. If the molecule is a symmetric top, the total polarizability is the same in all directions and the ellipsoid of polarizability is a sphere. A spherical top molecule has no rotational Raman spectmm. Symmetric tops and asymmetric tops have anisotropic polarizabilities and produce rotational Raman spectra. 

Fig. 1 Top and side views of the pathway connecting the lowest two fullerene isomers of Ceo calculated with a plane-wave density functional approach using the LDA and BLYP functionals [2]. The middle panel is the transition state and the end panels are the two minima in each case. The asymmetric path was calculated using the LDA functional with a cutoff of 30 rydberg and the symmetric path corresponds to the BLYP functional and a cutoff of 40 rydberg [2], Reproduced with permission from D. J. Wales, Energy Landscapes, Cambridge University Press, Cambridge (2003)...

Compaction of polymeric membranes also occurs during gas separation. Reinsch et al. (2000) described the use of UTDR to measure compaction of 175-p.m-thick (with backing) asymmetric cellulose-acetate gas separation membranes provided by Grace Davison (Littleton, CO). Figure 33.8 shows a schematic of the membrane cell used in these characterization smdies of membrane compaction during gas separation and the primary reflections of acoustic waves A and B, which correspond to the cell top-plate-gas interface and the gas-membrane interface, respectively. Compaction was studied as a function of feed gas pressure and composition. Figure 33.9 shows a plot of the membrane strain as a function of time for compaction at a transmembrane nitrogen gas pressure difference 2.8 MPa followed by a recovery cycle at atmospheric pressure for a commercial asymmetric cellulose-acetate membrane. An instantaneous strain of approximately 13% is observed followed by a small time-dependent strain. 

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