The Two-Particle Rigid Rotor

Equation (6.44) is a special case of Eq. (6.41), and we may therefore use the results of the last section to separate off the translational motion of the system as a whole. We will concern ourselves only with the rotational energy. The Hamiltonian operator for the rotation is given by the terms in brackets in (6.43) as 

Instead of the relative Cartesian coordinates x, y, z, it will prove more fruitful to use the relative spherical coordinates r, d, / . The r coordinate is equal to the magnitude of 

Since V = 0 is a special case of V = V r), the results of Section 6.1 tell us that the eigenfunctions are given by (6.16) with the r factor omitted  

The Hamiltonian operator is given by Eq. (6.8) with the r derivatives omitted and V r) = 0. Thus 

The moment of inertia / of a system of n particles about some particular axis in space as defined as 

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The rotational levels of a diatomic molecule can be well approximated by the two-particle rigid-rotor energies (6.52). It is found Levine, Molecular Spectroscopy, Section 4.4) that when a diatomic molecule absorbs or emits radiation the allowed pure-rotational transitions are... 

FIGURE 6.3 Coordinate system for the two-particle rigid rotor. 

Evaluation of the transition moment for the harmonic oscillator and for the two-particle rigid rotor gives the selection rules Au = 1 and A/ = 1 stated in Sections 4.3 and 6.4. 

Because molecules are not rigid, the rotational energy levels for diatomic molecules differ slightly from rigid-rotor levels. From (6.52) and (6.55), the two-particle rigid-rotor levels are = BhJ J -l-1). Because of the anharmonicity of molecular vibration (Fig. 4.6), the average internuclear distance increases with increasing vibrational quantum number v, so as v increases, the moment of inertia I increases and the rotational constant B decreases. To allow for the dependence of B on v, one replaces B in E by The mean rotational constant B for vibrational level v is - Ug v + 1/2),... 

The two-particle rigid rotor consists of particles of masses wii and separated by a fixed distance d. Its energy is the sum of the energy of translation and the energy of... 

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