However, for many actual diatomics this model is too restrictive since distances are usually not completely fixed.
Scale factors are of importance for quantum mechanical applications since they enter the Laplacian expressed in curvilinear coordinates.
The linear rigid rotor model can be used in quantum mechanics to predict the rotational energy of a diatomic molecule.
According to quantum mechanics, the energy levels of a system can be determined by solving the Schrödinger equation:
This operator appears also in the Schrödinger equation of the hydrogen atom after the radial part is separated off.
A typical rotational absorption spectrum consists of a series of peaks that correspond to transitions between levels with different values of the angular momentum quantum number (
Pure rotational transitions, in which the vibronic (= vibrational plus electronic) wave function does not change, occur in the microwave region of the electromagnetic spectrum.
Typically, rotational transitions can only be observed when the angular momentum quantum number changes by
This selection rule arises from a first-order perturbation theory approximation of the time-dependent Schrödinger equation.
is the direction of the electric field component of the incoming electromagnetic wave, the transition moment is,
After integration over the vibronic coordinates the following rotational part of the transition moment remains,
This effect can be accounted for by introducing a correction factor known as the centrifugal distortion constant
where The non-rigid rotor is an acceptably accurate model for diatomic molecules but is still somewhat imperfect.
In microwave spectroscopy—the spectroscopy based on rotational transitions—one usually classifies molecules (seen as rigid rotors) as follows: This classification depends on the relative magnitudes of the principal moments of inertia.
If the body lacks cylinder (axial) symmetry, a last rotation around its z-axis (which has polar coordinates
The classical kinetic energy T of the rigid rotor can be expressed in different ways: Since each of these forms has its use and can be found in textbooks we will present all of them.
on the left hand side contains the components of the angular velocity of the rotor expressed with respect to the body-fixed frame.
[2] The dots over the time-dependent Euler angles on the right hand side indicate time derivatives.
into T gives the kinetic energy in Lagrange form (as a function of the time derivatives of the Euler angles).
This angular momentum is a conserved (time-independent) quantity if viewed from a stationary space-fixed frame.
with respect to the stationary space-fixed frame, we would find time independent expressions for its components.
The Hamilton form of the kinetic energy is written in terms of generalized momenta
As usual quantization is performed by the replacement of the generalized momenta by operators that give first derivatives with respect to its canonically conjugate variables (positions).
The quantization rule is sufficient to obtain the operators that correspond with the classical angular momenta.
The explicit form of the rigid rotor angular momentum operators is given here (but beware, they must be multiplied with
The quantization rule is not sufficient to obtain the kinetic energy operator from the classical Hamiltonian.
This operator has the general form (summation convention: sum over repeated indices—in this case over the three Euler angles
Given the inverse of the metric tensor above, the explicit form of the kinetic energy operator in terms of Euler angles follows by simple substitution.
Only measurement techniques with atomic resolution made it possible to detect the rotation of a single molecule.
This could be directly visualized by scanning tunneling microscopy, i.e., the stabilization could be explained at higher temperatures by the rotational entropy.