Rigid Rotor Angular Momenta

The rotational Herman-Wallis factor Fv< v(m) of the operator (2.140) is still that of a rigid rotor. In order to describe rotational-vibrational interactions, one must introduce explicitly the angular momentum J. To lowest order, the dipole operator that includes rotational-vibrational interactions is... 

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... 

The operator for the total angular momentum of the n-particle rigid rotor is... 

The coordinate system used in the close-coupling method is the space-fixed frame. For simplicity we consider the atom-diatom scattering. The wave function iM(.R,r,R) for an atom-rigid rotor system corresponding to the total energy E, total angular momentum J, and its projection M on the space-fixed z axis can be written as an expansion,... 

This is an important result. It tells us that the total angular momentum determined in the laboratory frame is the sum of a centre-of-mass contribution and the total angular momentum tiJ measured in the centre-of-mass frame. In a rigid rotor the velocity vector of the kW particle in the centre-of-mass frame is related to the angular velocity, u>, of the rotating particle by... 

The rigid rotor model assumes that the intemuclear distance Risa constant. This is not a bad approximation, since the amplitude of vibration is generally of the order of 1% of i . The Schrbdinger equation for nuclear motion then involves the three-dimensional angular-momentum operator, written J rather than L when it refers to molecular rotation. The solutions to this equation are already known, and we can write... 

In this equation Av, Bv, and Cv are the rotational constants and La, Lb, and Lc are the projections of the rotational angular momentum on the principal inertial axes of the molecule. The Hamiltonian in Eq. (1) is often referred to as a rigid-rotor Hamiltonian, even though significant vibrational effects appear in the rotational constants. To good approximation... 

These operators have the same form as the angular momentum operators of an anisotropic rigid rotor in quantum mechanics (see Edmunds, 1957). / is the total angular momentum operator of the molecule Iz, is the angular momentum operator about the spacefixed Z axis and Iz is the angular momentum about the body-fixed Z axis (usually chosen as the axis of highest symmetry) in the molecule. 

It is less obvious how to assign quantum variables. While, for instance, the total classical angular momentum of a collision fragment is clearly defined, in the case of small fragments product rotational quantum numbers are typically sought. The standard procedure for linear molecules is to assume rigid rotor conditions and to solve the usual correspondence principle relations to obtain the quasiclassical product quantum numbers (restricted to integer values). 

Centrifugal distortion - An effect in molecular spectroscopy in which rotational levels are lowered in energy, relative to the values of a rigid rotor, as the rotational angular momentum increases. The effect may be understood classically as a stretching of the bonds in the molecule as it rotates faster, thus increasing the moment of inertia. 

Classically, a rigid rotor that has its angular momentum directed exclusively along the z-axis is rotating solely in the xy-plane. Argue that this would violate the uncertainty principle for a quantum rotor. 

If we compare Eq. (4.78) with Eq. (4.73), it is clear that the algebraic three-dimensional model provides the correct rotational spectrum of a rigid linear rotor, where the (vibrational) angular momentum coefficient, ggg, is described by the algebraic parameters A 2 and A j2- The J-rotational band is obtained by recalling in Eq. (4.12), the branching law... 

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