Lecture 34 Rotational spectroscopy: intensities Rotational spectroscopy In the previous lecture, we have considered the rotational energy levels. In this lecture, we will focus more on selection rules and intensities.
Selection rules and intensities (review) Transition Intensity dipole moment of transition Rotational selection rules
Transition moment Oscillating electric field (microwave) No electronic / vibrational transition x-dipole dipole
Rotational selection rules Gross selection rule: nonzero permanent dipole
Does H2O have microwave spectra? Yes Does N2 have microwave spectra? No Does O2 have microwave spectra? No Quantum in nature
Microwave spectroscopy How could astrochemists know H2O exist in interstellar medium? Public image NASA
Selection rules of atomic spectra (review) From the mathematical properties of spherical harmonics, this integral is zero unless Rotational selection rules
Specific selection rule: Spherical & linear rotors In units of wave number (cm1): Nonrigid rotor: Centrifugal distortion
Diatomic molecule Nonrigid rotor: Centrifugal distortion Diatomic molecule Vibrational frequency Nonrigid rotor: Centrifugal distortion
Nonrigid Rigid Appearance of rotational spectra Rapidly increasing and then decreasing intensities
Transition moment2 Degeneracy Boltzmann distribution (temperature effect) Rotational Raman spectra Gross selection rule:
polarizability changes by rotation Specific selection rule: x +y +z 2 2 2 xy, etc. are essentially Y0,0, Y2,0, Y2,1, Y2,2
Linear rotors: J = 0, 2 Spherical rotors: inactive (rotation cannot change the polarizability) ~ Y0,0 Rotational Raman spectra Anti-Stokes wing slightly less intense than Stokes wing why?
Boltzmann distribution (temperature effect) Rotational Raman spectra Each wings envelope is explained by
Degeneracy Boltzmann distribution (temperature effect) H2 rotational Raman spectra Why does the intensity alternate? H2 rotational Raman spectra Why
does the intensity alternate? Answer: odd J levels are triply degenerate (triplets), whereas even J levels are singlets. Nuclear spin statistics Electrons play no role here; we are concerned with the rotational motion of nuclei. The hydrogens nuclei (protons) are fermions and have / spins .
The rotational wave function (including nuclear spin part) must be antisymmetric with respect to interchange of the two nuclei. The molecular rotation through 180 amounts to interchange. Para and ortho H2 Singlet (para-H2) Sym.
Antisym. Y ( r1 ,r2 ) { spatial part of rotation} { a (1)b (2) - b (1)a (2)} Nuclear (proton) spins Triplet (ortho-H2) Antisym. Sym.
a (1)a (2) Y ( r1 ,r2 ) { spatial part of rotation} b (1)b (2) a (1)b (2) + b (1)a (2) With respect to interchange (180molecular rotation) Spatial part of rotational wave function
By 180 degree rotation, the wave function changes sign as (1)J (cf. particle on a ring) Para and ortho H2 Singlet (para-H2) Sym. Antisym.
Sym. a (1)a (2) Y ( r1 ,r2 ) { J =odd } b (1)b (2) a (1)b (2) + b (1)a (2) Antisym. Y ( r1 ,r2 ) { J =even} { a (1)b (2) - b (1)a (2)}
Triplet (ortho-H2) Summary We have learned the gross and specific selection rules of rotational absorption and Raman spectroscopies. We have explained the typical appearance of rotational spectra where the temperature
effect and degeneracy of states are important. We have learned that nonrigid rotors exhibit the centrifugal distortion effects. We have seen the striking effect of the antisymmetry of proton wave functions in the appearance of H2 rotational Raman spectra.