PHYSICAL CHEMISTRY IN BRIEF

CHAP. 12: BASIC TERMS OF CHEMICAL PHYSICS [CONTENTS] 425

For the oscillations of polyatomic molecules the following relation applies

Evib=

∑

i

hνi

(

vi+

1

2

)

, (12.33)

where the summation is over all possible types of independent (so called fundamental) vibrations
of molecules.

Note:In a more accurate description of molecular vibrations we would find out that for

higher vibrational quantum numbers the energies deviate from the values of the corre-

sponding harmonic oscillations. Formula

Evib=hν

(

v+

1

2

)

−hνxe

(

v+

1

2

) 2

, (12.34)

wherexeis the anharmonic constant, is a better approximation of the energy of an an-

harmonic oscillator than (12.31).

12.2.6 Motion of electrons around the nucleus

Another important type of motion is that of electrons in the electrostatic field of the atom
nucleus. For the interaction of one single electron with a positively charged nucleus, such as
occurs in the case of an isolated hydrogen atom or a He+cation, the following relation applies
for the potential energy

Vel(r) =

1

4 π ε 0

Z e^2

r

, (12.35)

wherer is the distance between the nucleus and the electron,ε 0 is the vacuum permittivity
[see12.1.1],eis the elementary charge (e= 1. 602 × 10 −^19 C), andZeis the nucleus charge.
In the absence of an external electromagnetic field, the energy of an atom depends solely
on theprincipal quantum numbern,

Eel=−

μ e^4 Z^2

8 ε^20 h^2

1

n^2

, (12.36)

whereμis the reduced mass of the electron-nucleus system andn= 1, 2 , 3 ,...