CHAP. 12: BASIC TERMS OF CHEMICAL PHYSICS [CONTENTS] 424
wherel= 0, 1 ,.. ., is the rotation quantum number, often denoted by symbolJ. The energy
values forl >0 correspond to several eigenfunctions whose form depends on another quantum
numberm(m=−l,−l+ 1,... ,+l). The energy levels of the rigid rotor are thus degenerate
and the degree of degeneracygis
g= 2l+ 1. (12.29)
The general closed solution of the Schr ̈odinger equation for the rotation of more complex
molecules does not exist and can be obtained only approximately.
12.2.5 Vibration
Vibration is the last type of molecular motion to be described. The potential energy of the
simplest two-atom molecule can be approximately written as theharmonic oscillator
Vvib=1
2
κ(x−xeq)^2 , (12.30)wherexis the distance of atoms andxeqthe equilibrium distance corresponding to the minimum
potential energy. The proportionality constantκis theforce constantof vibration which is
the measure of the force with which the molecule is kept in the equilibrium position. The higher
the value ofκthe higher is the force returning the particle to the equilibrium position. The
eigenvalues of the vibrational energy obtained from the solution of the Schr ̈odinger equation
(12.19) are
Evib=hν(
v+1
2
)
, (12.31)wherevis the vibrational quantum number andνis the characteristic frequency of vibration
given by the relation
ν=1
2 π√
κ
μ, (12.32)
andμis again the reduced mass (12.27).
The vibrational quantum number acquires the valuesv= 0, 1 , 2 ,.. .The lowest energy level
1
2 hνcorresponding tov= 0 is usually referred to as thezero point energy. The degree of
degeneracy of individual levels isg= 1.