6.2 Diatomic Molecules
265
Fig. 6.2 Coordinate system
for describing vibrational
motion of a diatomic
molecule whose constituent
atoms, of masses m 1 and m 2 ,
lie along the z-axis
V (R) =
1
2 k(z 2 − z 1 − R e )
2 .
(6.2.16)
If we now utilize Newton’s Second law in the forms
F i = m i a i = m i
d 2 R i
dt 2 ;
F i = −
∂V
∂R i
R j =R i
,
and apply them to our diatomic system, we obtain the two equations
m 1
d 2 z 1
dt 2 = −
∂V (z 1 , z 2 )
∂z 1
= k(z 2 − z 1 − R e ) ,
m 2
d 2 z 2
dt 2 = −
∂V (z 1 , z 2 )
∂z 2
= −k(z 2 − z 1 − R e ) .
Upon multiplying the first of these two equations by m 2 , the second by m 1 , then
subtracting the first result from the second, we see that
m 1 m 2
d 2 (z 2 − z 1 − R e )
dt 2
= −(m 1 + m 2 )k(z 2 − z 1 − R e )
or
d 2 (z 2 − z 1 − R e )
dt 2
= −
1
m 1
+
1
m 2
k(z 2 − z 1 − R e ) .
Upon introducing the reduced mass m r through the defining relation
1
m r
≡
1
m 1
+
1
m 2
,
(6.2.17)
and defining the displacement from equilibrium, ξ , via ξ ≡ z 2 − z 1 − R e , we see
that our result can be rewritten in the traditional manner in the form of a simple
second-order ordinary differential equation, namely,
d 2 ξ
dt 2 = −
k
m r
ξ ≡ −ω
2
osc ξ .
(6.2.18)