Rigid Rotor, useful for MSc and BSc Chemistry students

1. Dr. Pradeep Samantaroy
Department of Chemistry
Rayagada Autonomous College, Rayagada
pksroy82@gmail.com; pksroy82@yahoo.in
9444078968

2. r
Prepared by Dr. Pradeep Samantaroy

3. The kinetic energy of the system:
KE = ½ Iω2
= L2/ 2I since: L= Iω
where L = Angular momentum
I = Moment of Inertia
ω= Angular velocity
I = µr2
where µ is the reduced mass = m1m2/(m1+m2)
Prepared by Dr. Pradeep Samantaroy

4. Now the Schrodinger equation is
ĤΨ = EΨ
Where Ĥ = T + V
Since the potential energy is assumed to be zero
Ĥ = T = L2/ 2I
Now the angular momentum operator L2 can be written as
2
z
L2
y
L2
x
L2L 
Prepared by Dr. Pradeep Samantaroy

5. 












y
z
z
yiPzPyL yzx 













z
x
x
ziPxPzL zxy 













x
y
y
xiPyPxL zyz 
Prepared by Dr. Pradeep Samantaroy

6. TRANSFORMATION OF CO-ORDINATES
x = rsinθcosφ y=rsinθsinφ z= rcosθ
r2 = x2 + y2 + z2
Prepared by Dr. Pradeep Samantaroy

7. 















 coscotsiniLx
















 sincotcosiLy




iLz




















2
2
2
22
sin
1
sin
sin
1



L
Prepared by Dr. Pradeep Samantaroy

8. Now putting the value of L2 in the Schrodinger equation
 E
I
L


2
2




E
I

























 2
2
2
2
sin
1
sin
sin
1
2
1

0
2
sin
1
sin
sin
1
22
2
2

























E
I

0
8
sin
1
sin
sin
1
2
2
2
2
2


























E
h
I
Prepared by Dr. Pradeep Samantaroy

9. Now, Ψ is a function of θ and φ and both are independent of each other.
Hence,
Ψ(θ,φ)=Θ(θ). Φ(φ)
Now multiplying Sin2θ in the previous equation,
0sin
8
sinsin 2
2
2
2
2


























 E
h
I
E
h
I
Assume 2
2
8
,

 
2
2
2
sinsinsin
becomesequationtheNow




















Prepared by Dr. Pradeep Samantaroy

10. 2
2
2 )(
)(sin)()(
)(
sinsin)(
ofvaluethengSubstituti






















2
2
2 )(
)(
1
sin
)(
sin
)(
sin
bytthroughoudividingNow
























2
2
2
)(
)(
1
mAssume 







22
sin
)(
sin
)(
sin
mThen 
















As the terms are separated now, they can be equated to one constant, as they
are independent of each other.
Prepared by Dr. Pradeep Samantaroy

11. Now we have two different equation to solve.
Solving individual equations and then combining will provide the
solution for rigid rotor.
θ varies from 0 to π.
φ varies from 0 to 2π.
Prepared by Dr. Pradeep Samantaroy

12. Solution to φ equation
2
2
2
)(
)(
1
haveWe m







0)(
)( 2
2
2



m


constant.ionnormalizattheisN
)(
isequationsuchosolution tgeneralThe

 im
Ne
Prepared by Dr. Pradeep Samantaroy

13. Normalization of φ equation








2
1
1d
1d
1)d()(
2
0
2
2
0
2
0
*








N
N
NeNe imim
2....1,0,m
2
1
)(
becomessolutiontheHence
 

 im
e
Applying Normalization Condition
Prepared by Dr. Pradeep Samantaroy

14. 0sin
)(
sin
)(
sin 22











mThen 





Solution to Θ equation
It will become easier to solve, if this equation can be transformed to Legendre equation
dx
d
x
dx
d
dx
d
d
dx
d
d
Let
ddxx
xThen
Assume
2
2
22
-1sin.
P(x))(
sin-sin-1
sin-1
cosx











Prepared by Dr. Pradeep Samantaroy

15. 0)(
1
)(
2
)(
)1(
becomesequationtheNow
2
2
2
2
2















xP
x
m
x
xP
x
x
xP
x 
This is similar to Legendre equation, where β = l(l+1)
l is the azimuthal quantum number.
...2,1,0for)(
1
2
1
1
becomesequationLegendretheolution totheNow
2
22



lxPFind
)(x
dx
d
l!
(x)P
(x)P
dx
d
)x((x)P
s
l
l
l
l
ll
l|m|
|m|
|m|/|m|
l
Prepared by Dr. Pradeep Samantaroy

16. Now |m| ≤ l
It can’t be greater than l, as the Legendre equation becomes zero.
Using the value of x, we can find solution to Θ equation as
|m|)!(l
)(l-|m|)!l(
N
N
(x)PN
l,m
ml
|m|
lml




2
12
constant.ionnormalizattheis
)(
,
,
(x).P
|m|)!(l
)(l-|m|)!l( |m|
l



2
12
)(
Prepared by Dr. Pradeep Samantaroy

17. lm-l&,,,where l
e(x).P
|m|)!(l
(l-|m|)!)l(
e(x).P
|m|)!(l
)(l-|m|)!l(
im|m|
l
im|m|
l








3210
.
4
12
2
1
.
2
12
)().(),(
isrotatorrigidfortionwavefunccompletetheNow





Prepared by Dr. Pradeep Samantaroy

18. number.quantumrotationalasknownis
....3210Jherew
8
)1(
pySpectroscoIn
....3210where
8
)1(
8
)1(
2
2
2
2
2
2
J
,,,
I
hJJ
E
,,,l
I
hll
E
h
IE
ll











Prepared by Dr. Pradeep Samantaroy