integral is non-zero. By separating the rotational part of the molecular wavefunction from the vibronic part, one can show that this means that the molecule must have a permanent dipole moment. After integration over the vibronic coordinates the following rotational part of the transition moment remains, Here is the z component of the permanent dipole moment. The moment is the vibronically averaged component of the dipole operator. Only the component of the permanent dipole along the axis of a heteronuclear molecule is non-vanishing. By the use of the orthogonality of the spherical harmonics it is possible to determine which values of , , , and will result in nonzero values for the dipole transition moment integral. This constraint results in the observed selection rules for the rigid rotor: Non-rigid linear rotor The rigid rotor is commonly used to describe the rotational energy of diatomic molecules but it is not a completely accurate description of such molecules. This is because molecular bonds (and therefore the interatomic distance ) are not completely fixed; the bond between the atoms stretches out as the molecule rotates faster (higher values of the rotational quantum number ). This effect can be accounted for by introducing a correction factor known as the centrifugal distortion constant (bars on top of various quantities indicate that these quantities are expressed in cm−1): where is the fundamental vibrational frequency of the bond (in cm−1). This frequency is related to the reduced mass and the force constant (bond strength) of the molecule according to The non-rigid rotor is an acceptably accurate model for diatomic molecules but is still somewhat imperfect. This is because, although the model does account for bond stretching due to rotation, it ignores any bond stretching due to vibrational energy in the bond (anharmonicity in the potential). Arbitrarily shaped rigid rotor An arbitrarily shaped rigid rotor is a rigid body of arbitrary shape with its center of mass fixed (or in uniform rectilinear motion) in field-free space R3, so that its energy consists only of rotational kinetic energy (and possibly constant translational energy that can be ignored). A rigid body can be (partially) characterized by the three eigenvalues of its moment of inertia tensor, which are real nonnegative values known as principal moments of inertia. In microwave spectroscopy—the spectroscopy based on rotational transitions—one usually classifies molecules (seen as rigid rotors) as follows: spherical rotors symmetric rotors oblate symmetric rotors