512 Quantum Mechanical Basis of Molecular Spectra /27 : 3 



forbidden lines.) Absorption corresponds to an increase of J from J x to 

 J 2 = Ji + 1. In this case, the photon energy will be 



F _ A 2 (J 1 + 1)^ + 2) A^QA + 1) _ h 2 (J x + 1) 



Z-^-lL x -- -^ r — ^ " 477 2 7 



(Actually this is a slight oversimplification, because the effective value 

 of / depends on the value of J. Classically, this would be expected for a 

 nonrigid rotor, that is, one which could vibrate as well as rotate. As all 

 molecules vibrate, a better approximation for the molecular energy E K is 



E K = A J {J + 1) - BJ 2 {J + l) 2 



Values for both A and B can be found from spectroscopic measurements 

 for molecules which are asymmetrical rotators.) 



To the best of the author's knowledge, purely rotational spectra have 

 never been used in biophysical research. They have been introduced 

 here because rotational-energy levels affect the vibrational and electronic 

 spectra. Rotational levels are closer in behavior to our ideas of classical 

 macroscopic bodies than are the vibrational and electronic-energy levels. 

 Rotational levels are easier to visualize, and thus these form a good 

 introduction to molecular spectra. 



B. Vibrational Spectra 



A somewhat more complicated mathematical problem arises when one 

 considers the vibrational modes of motion. A quantum mechanical 

 treatment of a simple harmonic vibrator shows its energy is quantized 

 so that 



E = h(v + |) v = 0, 1, 2, 3, • • • (9) 



If the vibrator may also rotate, one should write 



* - *(' + 1) + ^sJj l) (io) 



Note that in the lowest energy state there is still the vibrational energy 



E=\h 



In other words, even at 0°K, the vibrator still possesses kinetic energy of 

 vibration. 



The vibrational-energy levels of diatomic molecules are more complex 

 than those of a simple harmonic vibrator. Polyatomic vibrations are 

 still more complicated. Thus, the expressions for the energy in Equa- 

 tions 9 and 10 are oversimplified, but at low values of v, they are good 

 approximations. The more exact expressions for E permit one to calcu- 

 late dissociation energies from spectroscopic data. These values agree 



