Transcript Anharmonicity In real molecules, highly sensitive vibrational spectroscopy can detect overtones, which are transitions originating from the n = 0 state for which Δn.

Anharmonicity
In real molecules, highly sensitive vibrational
spectroscopy can detect overtones, which are
transitions originating from the n = 0 state for
which Δn = +2, +3, …
Overtones are due to anharmonicity. A good
approximation of realistic anharmonicity is given
by the Morse potential.
V (r)  De 1 e

 (rr0 ) 2
V (r)  De 1 e

 (rr0 ) 2
V(x)  De 1 2e
x

2x
e
Put x = r – r0 and
Taylor expand:

 De 1 21 x   x  1 2x  4 x
2
1
2
2
1
2
Comparing to the harmonic oscillator
we see that k  2De
So we do De cDe




c
2
2
2
 D 
V(x)  12 kx2
 k 
 2 
    


2De 
2De 
1/ 2
1/ 2
to keep
the force constant
the same but change the
 anharmonicity
e
2
x
2
V (x)  De 1 e

x 2
use De = 40, α = 1;
then scale by c
Energy levels
Morse model

 
2
E n  2D n  1/2  n  1/2
1/ 2 
2De 

1/ 2
e
 n (x)  N n y

 n1/ 2 y / 2 2  2n1
e
D1/e 2
1
0n


2
 (2  2n 1)n!
N n  

  (2  n) 
1/ 2
a
n
L (x)
Ln
(y)
dissociated above this
y  2e
x

D1/e 2

are the generalized Laguerre polynomials


Harmonic oscillator model

 
2
MORSE
1/ 2
En
 2De n  1/2  n  1/2
1/ 2 
2De 

E nHO  2D1/e 2 n 1/2

x 2 / 2
n (x)  N n Hn ( x)e

1/ 2
k 
   2 
h 
1/ 2
 
Nn 
1/ 2  
n
2 n!  
polynomials
Hn (x) are the Hermite

1
1/ 4
Wavefunctions: harmonic oscillator
Wavefunctions: Morse oscillator
Wavefunctions: harmonic vs. Morse
Wavefunctions
Wavefunctions
Expectation value of position
Expectation value of position
Expectation value of position
Selection rules

z nn'   
*
n'
z n

For anharmonicity, can replace
the H.O. wavefunctions with
Morse wavefunctions…
d
d 2 x 2
z  0 
x 2
L
dx 0
dx 0 2!

…or can keep more terms in
the Taylor expansion of the
dipole moment
Selection rules
Correspondence principle

Pclassical(x)   2E /k  x
2

1
Where xturn is the maximum value of x
2
2E /k  x turn
Correspondence principle
Correspondence principle
Correspondence
principle