Rotor quantum number

Figure 8. Lowest adiabatic channel potential curves [33] for the interaction of electronic ground state N2 with ions (q = ionic charge, Q = N2 quadnipole moment N, M = free-rotor quantum numbers k,v = harmonic oscillator quantum numbers for more details, see Ref. 33).

J. Troe My answer to Prof. Herman is that the high-Stark-field description of the close approach of a dipole to an ion can very well be represented in terms of the relevant quantum numbers. The linear dipole-free rotor quantum numbers j and m are converted to the oscillating dipole quantum number v with the identity v - 2j - m. ... 

Here j labels the diabatic electronic state, while k is the free rotor quantum numbers. As should be clear from eq. (16) the doorway state has exactly the same distribution. It is however much more interesting to look at the corresponding partition with respect to the adiabatic states. This may be easily derived in terms of the orthogonal matrix U(9) which diagonalizes the electronic Hamiltonian, eq. (5). 

The microwave spectram exhibits a tunnelling sphtting due to a large amphtude internal rotation of the H2O subunit that exchanges bonded and non-bonded atoms. The internal rotor states are labelled with the asymmetric rotor quantum number for the rotational levels of free water jkak. ... 

The only atom-nonlinear molecule system whose excited states have been studied in any detail is Ar-H20. This system is actually quite weakly anisotropic the anisotropy of the potential splits and shifts the H2O free-rotor levels, but the free-rotor quantum numbers are... 

K is the symmetric-rotor quantum number to Ja, see section 1.1 of this Introduction ... 

T =the asymmetric rotor quantum number can also be expressed as K, ... 

For asymmetric rotors the selection mle inJisAJ = 0, 1, 2, but the fact that K is not a good quantum number results in the additional selection mles being too complex for discussion here. 

Thus, the operators H and have the same eigenfunctions, namely, the spherical harmonics Yj iO, q>) as given in equation (5.50). It is customary in discussions of the rigid rotor to replace the quantum number I by the index J m the eigenfunctions and eigenvalues. 

In order to appreciate the size of the basis sets required for fully converged calculations, consider the interaction of the simplest radical, a molecule in a electronic state, with He. The helium atom, being structureless, does not contribute any angular momentum states to the coupled channel basis. If the molecule is treated as a rigid rotor and the hyperfine structure of the molecule is ignored, the uncoupled basis for the collision problem is comprised of the direct products NMf ) SMg) lnii), where N = is the quantum number... 

Low energy photons in the far IR can only modify the term ERol. This leads to pure rotational spectra that can be easily studied for small diatomic gases. However, in the mid IR, photons have sufficient energy to modify Vib and Fr,. This leads to vibrational-rotational spectra (Fig. 10.6). Each vibrational transition is accompanied by tens of individual rotational transitions. The molecule becomes an oscillating rotor for which energy VR approximately corresponds to the following values, where 7 (./ = 0, 1, 2, 3,...) and V (V = 0, 1, 2) are the rotational and vibrational quantum numbers, respectively. 

It is conventional to use J and Af, rather than / and m, for the rotational angular-momentum quantum numbers. Also, the moment of inertia / of the rotor about an axis passing through the center of mass and perpendicular to the interparticle line is found to equal jmd2. (The moment of inertia of a system of particles about an axis is defined by I = 2imirf, where r-t is the perpendicular distance from particle i to the axis.) Hence we write... 

The two-particle rotor levels are (2J + l)-fold degenerate, since for each J there are 2J + 1 possible values of the quantum number M (ranging from — J to +. /), and E is independent of M. 

Up to this point, the molecule has been considered to be a rigid rotor, but the work in Chapter 4 on diatomics shows that we must add corrections for rotation-vibration interaction and centrifugal distortion. For a polyatomic molecule, there are several normal modes of vibration, each with its own vibrational quantum number (see Chapter 6). By analogy to (4.75), we write for polyatomic molecules... 

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