Winter 2013
Chem 356: Introductory Quantum Mechanics
Chapter 5 – Vibrational Motion .................................................................................................................. 65
Potential Energy Surfaces, Rotations and Vibrations ............................................................................. 65
Harmonic Oscillator ................................................................................................................................ 67
General Solution for H.O.: Operator Technique ..................................................................................... 68
Vibrational Selection Rules ..................................................................................................................... 72
Poly-atomic Molecules............................................................................................................................ 73
Beyond the harmonic oscillator approximation ..................................................................................... 74
Chapter 5 – Vibrational Motion
Potential Energy Surfaces, Rotations and Vibrations
Suppose we assume the nuclei of a molecule are fixed, then we can solve the Schrodinger equation for
the electrons
Hˆ (r1 , r2 ,....rN )  E (r1 , r2 ,...rN )
This is a complicated problem, that will be discussed later (Chapter 7 and beyond)
We would get the (ground state) energy at a particular nuclear configuration { R }
Hence, assume we can solve this
We can fit a curve through the points  V ({R})
This V ({R}) is called the potential energy surface (PES) and is a crucial concept in chemistry, eg.
Chapter 5 – Vibrational Motion
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Winter 2013
Chem 356: Introductory Quantum Mechanics
Minima on the PES we associate with different isomers, for example with the reaction
A  B  C  D . Saddle points on the PES we associate with transition states
Obtaining the minima (+ curvature) we can get thermochemical information. From TS we get into rates
of reactions (Chem254, Chem 350)
Ideally, once we have obtained PES, we should solve for the Quantum energies involving the nuclei. This
is very complicated, and today can only be done for small molecules (up to 5 atoms).
What can be done is solving for molecular rotations + vibrations in the harmonic oscillator + Rigid Rotor
approximation
The crux is to make a quadratic approximation to the PES around each minimum ( = molecular species)
V ( R)  V ( Re ) 
1
 k (R  Re ) (R  Re )
2 
Chapter 5 – Vibrational Motion
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Chem 356: Introductory Quantum Mechanics
A molecule with N atoms has 3 N degrees of freedom
- 3 overall translations
- 3 overall rotations or 2 rotations for linear molecules
(3 N  6) Vibrations , [ 3N  5 for linear molecule]
  1....3N
k :
All of the information V ( Re ) , Re and k ( Re ) is obtained from an electronic structure program (like
Gaussian, Gamess, Turbomoli…)
Once we have these, one can use ‘exact’ calculations to solve for Harmonic oscillator + Rigid Rotor
energies.
The harmonic (quadratic) potential is an approximation, but often works well, certainly for stiff
molecules/ degrees of freedom.
What we will do next is: discuss harmonic oscillator for diatomic molecules
-
Ground state of harmonic oscillator
All excited states using operator technique
Generalize to polyatomic molecules
More accurate solution for diatomic (Beyond H.O.)
Selection rules, vibrational spectroscopy
Harmonic Oscillator
In the following chapter we will go on to discuss rotations
For a general polyatomic molecule we can define H.O. Hamiltonian as
H 

2
2M 
P 2 
1
 kab ( R  Re )a ( R  Re )b
2 a ,b 1...3 N
For a diatomic this can be reduced to
1 2 d2 1 2
Hˆ  
 kx
2  dx 2 2
Here

M1 M 2
M1  M 2
x  ( R  Re )
k
is the [kg] reduced mass
is deviation [m] from equilibrium
is the force constant [Nm-1]
Chapter 5 – Vibrational Motion
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Winter 2013
Chem 356: Introductory Quantum Mechanics
In McQuarrie this is derived using classical equations of motion. I will post on the Website a general
derivation to get the H.O. form for a general polyatomic, but will not discuss in class. Here I will simply
use the form for H.
The ground state solution has the form e  x
on I will discuss the full solution.
2
/2
. Let us determine the constant  , and the energy. Later
2
d  x2 /2
e
  xe  x /2
dx
2
2
d 2  x2 /2
e
  e x /2   2 x 2 e x /2
2
dx
2
2
2
 2 d 2 1 2   x2 /2  2

1


kx
e



 2 x 2  kx 2  e x /2  E0 e x /2
 2 2 2

 2
2
2
dx




E0 
2
2

1
1 2 2
k
0
2
2 
E0 

k
2
2
k


1
2

;


1
k
1

2


k

General Solution for H.O.: Operator Technique (see appendix 5 in McQuarrie)
 2 d2 1 2
Hˆ 
 kx
2m dx 2 2
Solution e  x
2
/2
   /   k /
Define
qx 
e  x
2
/2

e q
2
/2
1
1
2
q
k  x


1

x
k 



1
2
q
1
 q   1
2 


k 
x x q 
 q
Chapter 5 – Vibrational Motion
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Winter 2013
Chem 356: Introductory Quantum Mechanics
 2
Hˆ 
2
 k 2
q
2

1
k
q2
2
k
k  1 2 1 2 
 q 

  2 q 2 2 

 1 d2 1 2 
 
 q 
2
2 
 2 dq
‘ q ’ are called dimensionless coordinates (Check that q 
1
 k x is indeed dimensionless!)
Commutation Relation:
 d 
 q, dq   1


 d
d 
 q  q  f (q )   f (q )
 dq dq 
Define new operators:
1 
d 
bˆ  
q  
dq 
2
Then
b b 

d 
d 
 q   q  
2 
dq 
dq 
d2  d 
2
q

 q,


2 
dq 2  dq  
1
 Hˆ 

2
1

Hˆ    bˆ  b  
2



1 
d 
bˆ 
q  
dq 
2

Commutation Relations, between b operators:
bˆ, bˆ   bˆ  , bˆ   0
  



bˆ, bˆ   1  q  d  ,  q  d  

 2
dq  
dq  

Hence
 d   d   d d 
1
  [ q , q ]   , q    q,    ,  
2
 dq   dq   dq dq  
1
  1  (1)   1
2
bˆ, bˆ   1
bˆ , bˆ   1




Chapter 5 – Vibrational Motion
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Winter 2013
Chem 356: Introductory Quantum Mechanics
Using this form of the Hamiltonian and the commutation relations we can derive the eigenvalues and
eigenfunctions of H.O.!!
a) If  n (q ) is eigenfunction with eigenvalue En then
i)
bˆ n (q) is eigenfunction with En1  En  
ii)
bˆ n (q) is eigenfunction with En1  En  
i)
1

Hˆ bˆ  n (q )     bˆ  bˆ   bˆ  n (q )
2

Proof:



1 

  bˆ bˆ, bˆ   bˆ  bˆ  bˆ    n (q)
2 


 bˆ   Hˆ  (q)


n
 ( En  )bˆ n (q)
ii)


1

Hˆ bˆ n (q )     bˆ  bˆ   bˆ n (q )
2


ˆ ˆ   1  bˆ (q )
   bˆ  , bˆ   bb
 n
2


1 

   1  bˆ  bˆ  bˆ  bˆ    n (q)
2 


 bˆ    Hˆ  n (q)   En    bˆ n (q)


b̂  and b̂ are called the raising and lowering operators, or ladder operators
Chapter 5 – Vibrational Motion
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Winter 2013
Chem 356: Introductory Quantum Mechanics
What about the ground state?
bˆ n (q)   E0    bˆ n (q)
Still true, but E0   < E0 !!
Only way out:
bˆ 0 (q)  0
1 
d 
 q   0 (q)  0
dq 
2
Differential equation with solution  0 ( q )  e
1
 q2
2
1
1
 q2

d   2 q2
 ( q  q )e 2  0
q 
e
dq


Putting it all together
1
 1  4  q2
normalized
 0 (q)    e 2
 
1 ˆ n
normalized
 n (q) 
b  0 (q)
n!
1
 
1 
d 
bˆ  
q  
dq 
2
1

n  0,1, 2,3....
En   n   
2

First couple of unnormalized functions in terms of q :
 0 (q)  e
1
 q2
2

 1 (q)   q 


 2 (q)   q 


 3 (q)   q 

2
d   2 q2
 2qe  q /2
e
dq 
1
2
d   q2 /2
  2q 2  1 e q /2
 qe
dq 
2
d 
2
 q 2 /2
  4q3  6q  e q /2
  2q  1 qe
dq 
Etc.
We can obtain all eigenfunctions by differentiation
 n ( x)  substitute x  q 
+ normalize
Chapter 5 – Vibrational Motion
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The wave functions take the form H n ( q )e
1
 q2
2
H n (q) are Hermite polynomials, they are either odd or even functions. (Each
polynomial contains only odd or only even terms)
Vibrational Selection Rules
Later we will discuss more generally the radiation process. For now the transition dipole
moment is used to define the strength of a spectroscopic translation

n
*( x) ( x) m ( x)dx
mn
 d 
 x  .......
 dx  x0
 ( x )  0 ( x )  
d
is the change in dipole with changing x
dx
0 is the dipole at equilibrium distance,
(internuclear-distance)
Eg.
d
0
dx
d
0
dx
d
0
dx
N2 :
HF:
CO:

n
large
small
*( x) ( x) m ( x)dx
 0   n * ( x) m ( x)dx  0 (if n  m )
d
dx 
d

dx
d

dx

 n * ( x) x m ( x)dx

2



2
n
n
* (q)q m (q)dq


* (q) bˆ  bˆ  m (q)dq
bˆ( m ) ~  m1
bˆ ( m ) ~  m1
For Harmonic Oscillator we can only get allowed transitions if n  1 or n  1
E   
Chapter 5 – Vibrational Motion
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Winter 2013
Chem 356: Introductory Quantum Mechanics
Poly-atomic Molecules
 2 2 1
Hˆ  
  k  ( R  Re ) ( R  Re ) 
2
2  ,

By some manipulation (see notes on webpage) this can be written as a sum of H.O.
Hamiltonians…….tedious derivations
 1 d2 1 2 
Hˆ   i 
 qi 
2
2 
i
 2 dqi
1

  i bˆi  bˆi  
2

i
The coordinates q : are linear combinations of atomic displacement vectors
3 N coordinate  3 translation, 3 rotation, (3 N -6) vibration
For linear molecule, rotation around axis is not a displacement
 2 rotations, and (3 N -5) vibrations
Eg. Water
Normal modes : display symmetry of molecule
The eigenfunctions are simply products of 1 dimensional H.O. functions
 k (q1 )
 l (q2 )  m (q3 )
1
1

 1

Ek ,l ,m   k   1   l   2   m   3
2
2
2



Very simple solution. One needs to diagonalize the “mass weighted hessian”
1
M
k
1
M
3N 3N Matrix
Reasonable approximations to all vibrational levels
Statistical Mechanics
Chapter 5 – Vibrational Motion
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Winter 2013
Chem 356: Introductory Quantum Mechanics
Beyond the harmonic oscillator approximation
The true potential is not quadratic. For large molecules this is not so easy to correct (people would use
low order perturbation theory, based on a quartic force field)
For smalls molecules, in particular diatomics, one can solve for the full vibrational problem
The exact energies are not equidistant. A better approximation is
2
1
1


E (n)  e  n    e xe  n  
2
2


The energy levels are usually called g (V )
There are a finite number of bound levels
Another effect is that transition moments can be non-zero even when n  1 . This leads to the
observation of overtones.
An often used form for the potential for a diatomic is the Morse potential
V ( x)  De 1  e  x 
V
x
0
2
x0
equilibrium geometry
x 0
1V
De  2  k
2
2 x
2
 
k
De
De can be used from experiment ( D0 measurable)
This is still an approximate potential
Today: Potential energy curves can be calculated using electronic structure programs.
Chapter 5 – Vibrational Motion
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