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M-degeneracy

The above selection rules for m can be verified by applying a weak external magnetic field to define the z direction and remove the m degeneracy. [Pg.68]

A succession of levels like those of a linear molecule can be calculated for each quantum number K, which in this case describes the quantized component of the angular momentum about the unique a-axis. K cannot exceed 7, the quantum number for the total angular momentum, i.e., K = 0, 1,... dz7. For an oblate symmetric top the rotational constant A j has to be replaced by Q ]. In relation to the case of A" = 0, other K quantum numbers allowed will thus result in lower energies Ejk, which is in contrast to the prolate top with a positive term of (A[ j - 6 ]). Evidently, all rotational levels with 0 are doubly degenerate. It should be noted that each level still possesses an M-degeneracy of (27 -f 1) as discussed in connection with the linear molecule. This is due to space quantization. [Pg.267]

There is, in addition, the (27+ l)-fold M degeneracy. A symmetric rotor is classified as/ ro/are if /a /c andoWareif /a>/b, /c-A linear molecule is a degenerate case of a prolate-symmetric rotor, with Ia and Ja = 0. Dipole-allowed transitions of a symmetric rotor have the additional selection rule AA" = 0, since the electric dipole moment lies in the principal axis and cannot be accelerated to a different orientation within the molecule. [Pg.285]

For asymmetric tops all three expressions apply, while for linear and symmetric-top molecules only the first intensity expression is needed. The P, Q, R coefflcinets are independent of M but depend on the intensity of the unsplit line. Note that the M = 0 component is forbidden for a A 7 = 0 transition, and for a second-order effect (A V M a factor of 1 /2 must be included in the intensity expression for the M = 0 component since the - -M and -M degeneracy is lost. [Pg.321]

Wlien = N/2, the value of g is decreased by a factor of e from its maximum atm = 0. Thus the fractional widtii of the distribution is AOr/A i M/jV)7 For A 10 the fractional width is of the order of 10 It is the sharply peaked behaviour of the degeneracy fiinctions that leads to the prediction that the thennodynamic properties of macroscopic systems are well defined. [Pg.380]

Wall M R, Dieckmann T, Feigon J and Neuhauser D 1998 Two-dimensional filter-diagonalization spectral inversion of 2D NMR time-correlation signals including degeneracies Chem. Phys. Lett. 291 465... [Pg.2328]

From the preceding analysis, it is seen that the coordinate space neai R can be usefully partitioned into the branching space described in tenns of intersection adapted coordinates (p, 9, ) or (x,y,z) and its orthogonal complement the seam space spanned by a set of mutually orthonormal set w, = 4 — M . From Eq. (27), spherical radius p is the parameter that lifts the degeneracy linearly in the branching space spanned by x, y, and z. [Pg.461]

Eqs. (D.5)-(D.7). However, when perturbations occur due to anharmonicity, the wave functions in Eqs. (D.11)-(D.13) will provide the conect zeroth-order ones. The quantum numbers and v h are therefore not physically significant, while V2 arid or V2 and I2 = m, are. It should also be pointed out that the degeneracy in the vibrational levels will be split due to anharmonicity [28]. [Pg.622]

Because the rotational energies now depend on K (as well as on J), the degeneracies are lower than for spherical tops. In particular, because the energies do not depend on M and depend on the square of K, the degeneracies are (2J+1) for states with K=0 and 2(2J+1) for states with K > 0 the extra factor of 2 arises for K > 0 states because pairs of states with K = K and K = -K are degenerate. [Pg.73]

Each set of p orbitals has three distinct directions or three different angular momentum m-quantum numbers as discussed in Appendix G. Each set of d orbitals has five distinct directions or m-quantum numbers, etc s orbitals are unidirectional in that they are spherically symmetric, and have only m = 0. Note that the degeneracy of an orbital (21+1), which is the number of distinct spatial orientations or the number of m-values. [Pg.150]


See other pages where M-degeneracy is mentioned: [Pg.218]    [Pg.212]    [Pg.74]    [Pg.291]    [Pg.72]    [Pg.229]    [Pg.23]    [Pg.24]    [Pg.124]    [Pg.419]    [Pg.429]    [Pg.156]    [Pg.203]    [Pg.149]    [Pg.6]    [Pg.339]    [Pg.26]    [Pg.218]    [Pg.212]    [Pg.74]    [Pg.291]    [Pg.72]    [Pg.229]    [Pg.23]    [Pg.24]    [Pg.124]    [Pg.419]    [Pg.429]    [Pg.156]    [Pg.203]    [Pg.149]    [Pg.6]    [Pg.339]    [Pg.26]    [Pg.23]    [Pg.69]    [Pg.379]    [Pg.379]    [Pg.1135]    [Pg.1554]    [Pg.602]    [Pg.170]    [Pg.191]    [Pg.30]    [Pg.415]    [Pg.664]    [Pg.59]    [Pg.588]    [Pg.214]    [Pg.787]    [Pg.42]    [Pg.92]    [Pg.93]    [Pg.406]    [Pg.131]    [Pg.100]    [Pg.100]   
See also in sourсe #XX -- [ Pg.212 ]




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Degeneracy

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