Rotational Absorption and Emission Spectroscopy

 

          The energy of rotational transitions corresponds to the microwave region of the electromagnetic spectrum.  The typical frequency range of interest is 3 - 300 GHz (0.1 to 10 cm-1).  The source used to study rotational absorption spectroscopy can be either a klystron (an electronic tube that generates monochromatic coherent radiation at frequencies accurate to a fraction of a MHz) or a farinfrared spectrometer.

          The structural parameters estimated from microwave spectroscopy are averages over the vibrational states.  Usually molecules are in their ground vibrational states so the rotational constant measured is B0.  If we average over the motion of the molecules in their ground state we call this average the average over zero-point motion.  This is because even at T = 0 K there is motion (due to the Uncertainty Principle) in the ground vibrational state of a molecule.  The average B0 is proportional to á1/R2ñ not 1/áR2ñ where the angle brackets represent the average over zero point motion.


          As a concrete example consider a diatomic molecule going from an initial state i to a final state f.  The wavefunction for each state Y is written as a product of electronic and nuclear parts.

The nuclear wavefunction can be separated into vibrational and rotational functions

Where q = R - Re is the displacement of the bond length from its equilibrium value.  Note that YJM(q,f) are the spherical harmonics.  To obtain the allowed transitions we define the transition moment as


The nested integrals represent the procedure of first determining the dipole moment in a particular electronic state and then determining the transition moment between two rovibrational states within this electronic state.  It is important to realize that the dipole moment of the molecule changes during vibration and is therefore a function of the nuclear coordinate Q.  To emphasize this point we write the dipole moment as m(Q) and expand it as a Taylor's series.

The first term is the permanent dipole moment.  Its magnitude is constant, but its orientation in the lab frame is not.  Because of molecular rotation the permanent dipole of the molecule is changing at the frequency of rotation.  Thus, electromagnetic radiation can interact with the rotating permanent dipole moment of the molecule to produce rotational transitions.  Vibrational transitions can occur only if the second (linear) term is considered in the above Taylor's series.  Vibrational wavefunctions are orthogonal and therefore the permanent dipole does not allow for any change dipole moment that will permit interaction with an electromagnetic wave.

          Considering only the permanent dipole moment here we have


The three components of the dipole operator, which allow absorption of radiation polarized in the corresponding directions are given by


The selection rules require that the permanent dipole moment be non-zero m0 ¹ 0.  The selection rules for the allowed levels for transitions JM à J'M' are derived from the properties of spherical harmonics.  These are DJ = ±1 and DM = 0, ±1.  The selection rule for M is not apparent in the spectrum unless there is an applied electric field.  The rotational spectrum consists of a series of lines corresponding to J à J + 1, one for each thermally populated initial state, at frequencies 2B(J+1).  Thus, the pure rotational spectrum consists of a series of lines at frequencies 2B, 4B, 6B, … .  Thus, the line spacing observed in rotational absorption spectra are predicted to be constant Dn = 2B.  While this is indeed true to a first approximation the coupling of vibrations to rotations (breakdown of the rigid-rotator approximation) causes small deviations from this prediction.

         

Selection Rules for Symmetric Top Molecules

          The symmetry of symmetric top molecules imposes additional selection rules on DK.  In the following we use Wigner rotation matrices to determine the selection rules for the transition J,K,M à J',K',M'.  The required condition is


where N = 0 for the Z component and ±1 for X and Y.  To evaluate this expression we need the following identity using Wigner rotation matrices



 The Clebsch-Gordon coefficients vanish unless m1 + m2 = m3 and l3 must take on one of the values l1 + l2, l1 + l2 - 1, … |l1 - l2|.  Based on these sum rules for the Clebsch-Gordon coefficients we find that K = K' or DK = 0 and N + M = M' or DM = 0, ±1.  In addition, DJ = 0, ±1.  The rule for J results from the triangle condition, J' = J+1, J, J-1 where all three values for J' can result only when J > 0.  These selection rules result in an important difference in spectra of closed shell diatomics and symmetric top molecules.  The vibrational spectra of diatomics have a P branch (for J à J - 1, DJ = -1) and an R branch (for J à J + 1, DJ = +1), but no Q branch (DJ = 0).  Symmetric top molecules have a Q branch.   Diatomic molecules with unpaired spins also have a Q branch due to coupling of the electronic and nuclear angular momenta.  These differences can be seen in Figure 9.4 of McHale’s Molecular Spectroscopy (CO has only P and R branches and no Q) and Figure 9.5 (NO has a Q branch as well).   The bending mode of CO2 shown in Figure 10.4 also has a pronounced Q branch as seen in Figure 10.8 on page 294 of McHale’s Molecular Spectroscopy.

          The spectra of asymmetric rotors are more complicated.  There is no selection rule for K, but the rules derived above for J and M are still valid.