Download Physical Chemistry 2nd Edition

Chapter 18
A Quantum Mechanical Model for the Vibration
and Rotation of Molecules
Physical Chemistry 2nd Edition
Thomas Engel, Philip Reid
Objectives
• Solving Schrödinger Equation
• Introducing Angular Momentum
• Introducing Spherical Harmonic Functions
Chapter 18: A Quantum Mechanical Model for the Vibration and Rotation of Molecules
Physical Chemistry 2nd Edition
© 2010 Pearson Education South Asia Pte Ltd
Outline
1. Solving the Schrödinger Equation for the
Quantum Mechanical Harmonic Oscillator
2. Solving the Schrödinger Equation for
Rotation in Two Dimensions
3. Solving the Schrödinger Equation for
Rotation in Three Dimensions
4. The Quantization of Angular Momentum
5. The Spherical Harmonic Functions
Chapter 18: A Quantum Mechanical Model for the Vibration and Rotation of Molecules
Physical Chemistry 2nd Edition
© 2010 Pearson Education South Asia Pte Ltd
18.1 Solving the Schrödinger Equation for the Quantum Mechanical
Harmonic Oscillator
•
•
•
Translational motion in various potentials is
described in the context of wave-particle
duality.
In applying quantum mechanics to molecules,
there are 2 motions for molecules to undergo:
vibration and rotation.
For vibration, the harmonic potential is
1 2
V  x   kx
2
where k = force constant
Chapter 18: A Quantum Mechanical Model for the Vibration and Rotation of Molecules
Physical Chemistry 2nd Edition
© 2010 Pearson Education South Asia Pte Ltd
18.1 Solving the Schrödinger Equation for the Quantum Mechanical
Harmonic Oscillator
•
The normalized wave functions are
1
2

2  x / 2
 x   An H n  x e
, and n  0,1,2,...


•
18.1 The Classical Harmonic Oscillator
Chapter 18: A Quantum Mechanical Model for the Vibration and Rotation of Molecules
Physical Chemistry 2nd Edition
© 2010 Pearson Education South Asia Pte Ltd
Example 18.1
Show that the function e
satisfies the
Schrödinger equation for the quantum harmonic
oscillator. What conditions does this place on ?
What is E?
 x 2
Chapter 18: A Quantum Mechanical Model for the Vibration and Rotation of Molecules
Physical Chemistry 2nd Edition
© 2010 Pearson Education South Asia Pte Ltd
Solution
We have
h 2 d 2 n ( x)

 V x  n x   En n  x 
2
2 dx
2
2
   V xe    h d  2xe   1 kx e 
2
dx
2
 x 2
h d e

2 dx 2
 x
2
 x 2
2
2



 x 2

 
h2
h2
1 2  x 2
 x 2
2 2  x 2

 2xe

 4 x e
 kx e
2
2
2
Chapter 18: A Quantum Mechanical Model for the Vibration and Rotation of Molecules
Physical Chemistry 2nd Edition
© 2010 Pearson Education South Asia Pte Ltd
Solution
The function is an eigenfunction of the total
energy operator only if the last two terms
cancel:
Hˆ totale x 
2
h2 

e
 x 2
1 k
if   
4 h2
2
Finally,
E
h2


h2

1 k h k

2
4h
2 
Chapter 18: A Quantum Mechanical Model for the Vibration and Rotation of Molecules
Physical Chemistry 2nd Edition
© 2010 Pearson Education South Asia Pte Ltd
18.1 Solving the Schrödinger Equation for the Quantum Mechanical
Harmonic Oscillator
•
Hermite polynomials states that
   1/ 2 x 2
 0 x     e
 
1/ 4
1/ 4
 4 

  
3
 1 x   
 
 2 x    
 4 
1/ 4

 3  x   
 9
3
1/ 4



xe1/ 2 x
2x
2x
2
3
2

 1 e 1/ 2 x

 3x e
2
 1 / 2 x 2
Chapter 18: A Quantum Mechanical Model for the Vibration and Rotation of Molecules
Physical Chemistry 2nd Edition
© 2010 Pearson Education South Asia Pte Ltd
18.1 Solving the Schrödinger Equation for the Quantum Mechanical
Harmonic Oscillator
•
The amplitude of the wave functions
approaches zero for large x values only when
k
1
1

En  h
 n    hv n   with n  1,2,3,...

2
2

•
The frequency of oscillation is given by
v
•
1
2
k

18.2 Energy Levels and Eigenfunctions for
the Harmonic Oscillator
Chapter 18: A Quantum Mechanical Model for the Vibration and Rotation of Molecules
Physical Chemistry 2nd Edition
© 2010 Pearson Education South Asia Pte Ltd
Example 18.2
 x   4 3 /   4 xe1/ 2 x
a. Is
an eigenfunction of the kinetic
energy operator? Is it an eigenfunction of the
potential energy operator?
b. What are the average values of the kinetic and
potential energies for a quantum mechanical
oscillator in this state?
1
2
Chapter 18: A Quantum Mechanical Model for the Vibration and Rotation of Molecules
Physical Chemistry 2nd Edition
© 2010 Pearson Education South Asia Pte Ltd
Solution
a. Neither the potential energy operator nor the
kinetic energy operator commutes with the total
energy operator. Therefore, because
 x   4 /   xe   is an eigenfunction of the total
energy operator, it is not an eigenfunction of
the potential or kinetic energy operators.
3
1
4
 1/ 2 x 2
Chapter 18: A Quantum Mechanical Model for the Vibration and Rotation of Molecules
Physical Chemistry 2nd Edition
© 2010 Pearson Education South Asia Pte Ltd
Solution
b. The fourth postulate states how the average
value of an observable can be calculated.
Because
then
2
2
h
d
Eˆ potential( x)  V ( x) and Eˆ kinetic ( x)  
2 dx 2
E potential   1* ( x)V ( x) 1 ( x)dx

 4
  

 
3
1  4
 k 
2  
1/ 4
2 1

 4
 xe1/ 2 x  kx2 
2
 

3
1/ 2 



4 x
x
 e

2
 4
dx  k 
 
3
3
1/ 4






xe(1/ 2)x dx
2
1
2 
4 x
x
 e dx

Chapter 18: A Quantum Mechanical Model for the Vibration and Rotation of Molecules
Physical Chemistry 2nd Edition
© 2010 Pearson Education South Asia Pte Ltd
2
Solution
The limits can be changed as indicated in the last
integral because the integrand is an even function
of x. To obtain the solution, the following standard
integral is used:

1 3  5    2n  1 
2 n x
x
e
2
dx 
2n1 a n1
0
a
The calculated values for the average potential and
kinetic energy are
E potential
1  4
 k 
2  
3
1/ 2



  3
3k 3
k

 2 
 h
 a  4
4 4 


Chapter 18: A Quantum Mechanical Model for the Vibration and Rotation of Molecules
Physical Chemistry 2nd Edition
© 2010 Pearson Education South Asia Pte Ltd
Solution
Thus
Ekinetic
 h d2 
  x dx
   x  
2  1
 2  dx 
*
1
1/ 4
1/ 4

3
2
2
 4 3 
1/ 2 x 2  h d  4 


 
 xe
  
2 
 
 2  dx   
 
1/ 2 
h 2  4 3 



2   
h  4


2  
2
3
2



1/ 2

x 4  3x 2 e x dx
2

1/ 2 
h  4 3 



2   
2
 
xe1/ 2 x dx
 
2

x  3x e
4
2
x 2 dx
0
 2  3 
  1 

 

3



2 


2


4




 
k
3 h 2 3
 h


4
4 
Chapter 18: A Quantum Mechanical Model for the Vibration and Rotation of Molecules
Physical Chemistry 2nd Edition
© 2010 Pearson Education South Asia Pte Ltd
2
Solution
In general, we find that for the nth state,
Ekinetic,n  E potential,n 
h k
1
n



2 
2
Chapter 18: A Quantum Mechanical Model for the Vibration and Rotation of Molecules
Physical Chemistry 2nd Edition
© 2010 Pearson Education South Asia Pte Ltd
18.1 Solving the Schrödinger Equation for the Quantum Mechanical
Harmonic Oscillator
•
18.3 Probability of Finding the Oscillator
in the Classically Forbidden Region
Chapter 18: A Quantum Mechanical Model for the Vibration and Rotation of Molecules
Physical Chemistry 2nd Edition
© 2010 Pearson Education South Asia Pte Ltd
18.2 Solving the Schrödinger Equation for Rotation in Two
Dimensions
•
Consider rotation, the total energy operator can
be written as a sum of individual operators for
the molecule:
Hˆ total  Hˆ transrcm   Hˆ vib  internal   Hˆ rot cm , cm 
•
Also the system wave function is a product of
the eigenfunctions for the three degrees of
freedom:
 total   trans rcm  vib  internal  rot  cm , cm 
Chapter 18: A Quantum Mechanical Model for the Vibration and Rotation of Molecules
Physical Chemistry 2nd Edition
© 2010 Pearson Education South Asia Pte Ltd
Example 18.3
The bond length for H19F is 91.68 × 10-12 m.
Where does the axis of rotation intersect the
molecular axis?
Chapter 18: A Quantum Mechanical Model for the Vibration and Rotation of Molecules
Physical Chemistry 2nd Edition
© 2010 Pearson Education South Asia Pte Ltd
Solution
If xH and xF are the distances from the axis of
rotation to the H and F atoms, respectively, we can
12
x

x

91
.
68

10
write H F
and xHmH=xFmF.
Substituting mF=19.00 amu and mH=1.008 amu,
we find that xF=4.58×10-12 m and
xH=87.10 × 10-12 m. The axis of rotation is very
close to the F atom. This is even more pronounced
for HI or HCl.
Chapter 18: A Quantum Mechanical Model for the Vibration and Rotation of Molecules
Physical Chemistry 2nd Edition
© 2010 Pearson Education South Asia Pte Ltd
18.2 Solving the Schrödinger Equation for Rotation in Two
Dimensions
•
The eigenfunction
angle Ф.
 
depends only on the
    A eiml and     A eiml
•
The solutions above correspond to clockwise
and counterclockwise rotation.
Chapter 18: A Quantum Mechanical Model for the Vibration and Rotation of Molecules
Physical Chemistry 2nd Edition
© 2010 Pearson Education South Asia Pte Ltd
Example 18.4
Determine the normalization constant A in
Chapter 18: A Quantum Mechanical Model for the Vibration and Rotation of Molecules
Physical Chemistry 2nd Edition
© 2010 Pearson Education South Asia Pte Ltd
Solution
The variable  can take on values between 0 and
2π. The following result is obtained:
2
*

 ml   ml  d  1
0
2
A   e
2

iml iml
e
d  A 
0
A 
2
2
 d  1
0
1
2
Convince yourself that
A has the same value.
Chapter 18: A Quantum Mechanical Model for the Vibration and Rotation of Molecules
Physical Chemistry 2nd Edition
© 2010 Pearson Education South Asia Pte Ltd
18.2 Solving the Schrödinger Equation for Rotation in Two
Dimensions
•
The energy-level spectrum is discrete and is
given by
h 2 ml2 h 2 ml2
Eml 

for ml  0,1,2,3..
2
2r0
2l
•
where ml = quantum number
We say that the energy levels with
twofold degenerate.
Chapter 18: A Quantum Mechanical Model for the Vibration and Rotation of Molecules
Physical Chemistry 2nd Edition
© 2010 Pearson Education South Asia Pte Ltd
are
18.2 Solving the Schrödinger Equation for Rotation in Two
Dimensions
•
•
•
For rotation in the x-y plane, the angular
momentum vector lies on the z axis.
The angular momentum operator in these
coordinates takes the simple form iˆz  ih   
Applying this operator
to an eigenfunction,
iml
ml h iml

ih
de
ˆl    

e  ml h   
z

2 d
2
Chapter 18: A Quantum Mechanical Model for the Vibration and Rotation of Molecules
Physical Chemistry 2nd Edition
© 2010 Pearson Education South Asia Pte Ltd
18.3 Solving the Schrödinger Equation for Rotation in Three
Dimensions
•
For molecule rotating in two dimensions:
     A eiml and      A e iml , for ml  0,1,2,3,...
•
To make sure Y(θ,Ф) are single-valued functions
and amplitude remains finite, the following
conditions must be met.
  l l  1, for l  0,1,2,3,... and
ml  l ,l  1,l  2,...,0,..., l  2, l  1, l
Chapter 18: A Quantum Mechanical Model for the Vibration and Rotation of Molecules
Physical Chemistry 2nd Edition
© 2010 Pearson Education South Asia Pte Ltd
18.3 Solving the Schrödinger Equation for Rotation in Three
Dimensions
•
Both l and ml must be integers and the
spherical harmonic functions are written in the
form
Y  ,    Yl ml  ,    lml  ml  
•
The quantum number l is associated with the
total energy observable,
h2
El  l l  1, for l  0,1,2,3,...
2I
Chapter 18: A Quantum Mechanical Model for the Vibration and Rotation of Molecules
Physical Chemistry 2nd Edition
© 2010 Pearson Education South Asia Pte Ltd
18.4 The Quantization of Angular Momentum
•
•
The spherical harmonic functions,
are
eigenfunctions of the total energy operator for
a molecule that rotates freely in three
dimensions.
The eigenvalue equation for the operator lˆ2 can
be written as
lˆ2Yl ml  ,   h2l l  1Yl ml  , 
Chapter 18: A Quantum Mechanical Model for the Vibration and Rotation of Molecules
Physical Chemistry 2nd Edition
© 2010 Pearson Education South Asia Pte Ltd
18.4 The Quantization of Angular Momentum
•
The operators lˆx , lˆy and lˆz have the following
form in Cartesian coordinates:
 
 
lˆx  ih y  z 
y 
 x

 
lˆy  ih z  x 
z 
 x
 

lˆz  ih x  y 
x 
 y
Chapter 18: A Quantum Mechanical Model for the Vibration and Rotation of Molecules
Physical Chemistry 2nd Edition
© 2010 Pearson Education South Asia Pte Ltd
18.4 The Quantization of Angular Momentum
•
The operators have the following form in
spherical coordinates:
ˆl  ih   sin    cot  cos   
x


 

ˆl  ih   cos    cot  sin   
y


 

ˆl  ih   
z
  
 
Chapter 18: A Quantum Mechanical Model for the Vibration and Rotation of Molecules
Physical Chemistry 2nd Edition
© 2010 Pearson Education South Asia Pte Ltd
18.4 The Quantization of Angular Momentum
•
For the operators in Cartesian coordinates, the
commutators relating the operators
are given by
lˆ , lˆ   ihlˆ
lˆ , lˆ   ihlˆ
lˆ , lˆ   ihlˆ
•
x
y
z
y
z
x
z
x
y
Thus the spherical harmonics is as
ˆl Y ml  ,      ih   1 e iml    m h  ,
z l
l




 2


for ml  0,1,2,3,...,l


Chapter 18: A Quantum Mechanical Model for the Vibration and Rotation of Molecules
Physical Chemistry 2nd Edition
© 2010 Pearson Education South Asia Pte Ltd
18.5 The Spherical Harmonic Functions
•
For spherical harmonic functions, these are the
first few values of l and ml:
Y00  ,   
1
4 1/ 2
1/ 2
 3 
Y  ,    


4


cos 
0
1
1/ 2
 5 
Y  ,    

 16 
sin e i
1
1
1/ 2
 5 
Y  ,    

 16 
0
2
3 cos
2
  1
1/ 2
 5 
Y  ,     
 8 
sin  cos e i
1
2
1/ 2
 15 
Y  ,    

 32 
1
2
sin 2 e i
Chapter 18: A Quantum Mechanical Model for the Vibration and Rotation of Molecules
Physical Chemistry 2nd Edition
© 2010 Pearson Education South Asia Pte Ltd
18.5 The Spherical Harmonic Functions
•
The functions which form an orthonormal set
are given in the following equations:




1 1
3
1
px 
Y1  Y1 
sin  cos 
4
2
1
3
1
1
py 
Y1  Y1 
sin  sin 
4
2i
3
pz  Y 
cos 
4
0
1
d z 2  Y10 


5
3 cos 2   1
16




d xz 
1 1
15
Y2  Y21 
sin  cos  cos 
4
2
d yz 
1
15
Y21  Y21 
sin  cos  sin 
4
2i
d x2  y2 
d xy 


1
15
Y22  Y2 2 
sin 2  cos 2
a 6
2i


1
15
Y22  Y2 2 
sin 2  sin 2
16
2i
Chapter 18: A Quantum Mechanical Model for the Vibration and Rotation of Molecules
Physical Chemistry 2nd Edition
© 2010 Pearson Education South Asia Pte Ltd
18.5 The Spherical Harmonic Functions
•
3D perspective plots of the p and d linear
combinations of the spherical harmonics.
Chapter 18: A Quantum Mechanical Model for the Vibration and Rotation of Molecules
Physical Chemistry 2nd Edition
© 2010 Pearson Education South Asia Pte Ltd