Linear molecules degeneracy

The lnu orbitals in the linear case (these are the non-bonding 2px and 2p orbitals of the oxygen atom) lose their degeneracy when the molecule bends. The 2px orbital retains its non-bonding character, but the 2p orbital makes a very important contribution to the 3a, bonding MO of the bent molecule. It is this factor which is critical in determining the shape of the water molecule. The distortion of the linear molecule... 

To calculate the molecular rotational partition function for an asymmetric, linear molecule, we use Eq. 8.16 for the energy level of rotational state /, and Eq. 8.18 for its degeneracy. As discussed in Section 8.2, rotational energy levels are very closely spaced compared to k/jT unless the molecule s moment of inertia is very small. Therefore, for most molecules, replacing the summation in Eq. 8.50 with an integral introduces little error. Thus the... 

Linear molecules belong to either 6 or the character tables are given in Section 9.12. The operation Ca(z) is a rotation by an angle a about the CM axis. In addition to the Mulliken notation, the spectroscopic symbols 2,n,A,... are also given. As we saw in Section 1.19, apart from spin degeneracy, 2 electronic levels are nondegenerate (A), whereas... 

Fig. 6. Typical potential curves for a linear molecule in a degenerate electronic state. In non-linear configurations the degeneracy is split by the Renner effect. The vertical (most probable) transition from the ground state terminates on an excited level of the bending vibration in the upper state.

The Boltzmann distribution is illustrated in Fig. 1.2.1 for the vibrational states of a one-dimensional harmonic oscillator with the frequency uj = 2itis, where the energy levels are given by Eq. (1.18), and in Fig. 1.2.2 for the rotational states of a linear molecule with the moment of inertia I, where the energy levels are given by Eq. (1.20) with the degeneracy u>j = 2 J + 1. 

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. 

The JT effect is based on the following theorem If a non-linear molecule (or polyatomic ion) has a degenerate electronic level (apart from Kramers degeneracy) it is unstable with respect to displacements of the atoms. 

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. 

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