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Equation Section 23University of Washington
Department of Chemistry
Chemistry 553
Fall Quarter 2003
Lecture 23: Correlation Functions and Spectral Line Shapes
12/04/02
Text Reading: Ch 21,22
A. Introduction

Spectral line shapes and autocorrelation functions are related by a Fourier
transform. At the heart of this relationship is the fact that energy absorbed from a
weak field is dissipated by fluctuations that are characteristic of the system at
equilibrium. In the following analysis we assume that A=B=u, where u is the
electric or magnetic dipole moment.
B. The Dielectric Constant

We will be considering the interaction of radiation fields with electric dipoles. So
let’s consider some properties of polarizeable materials. The simplest model is a
parallel plate capacitor. Assume the plates are separated by a distance d and have
area A. The normal vector to the plates is in the z direction. The electric field
between the plates of the capacitor is

E0  z
(23.1)
0
where =q/A is the surface charge density of the plates.The constant 0 is the free
A
space permittivity. The capacitance is C0  0 where it is assumed that the space
d
between the plate is vacuum. If the space between the capacitor plates is filled by a
A
dielectric, the capacitance is C 
where  is the permittivity of the dielectric. The
d

C
E

relative permittivity is defined as  r  
, where E0 and E are the fields
 0 C0 E0
that exist between the plates where the space is evacuated (0) or filled with dielectric.
 The electric polarization P of the material is related to the electric field E by
(23.2)
P   0  e   E

The subscript e indicates the electric susceptibility. The complex, frequencydependent electric susceptibility is related to the relative permittivity by
(23.3)
 r    1  e  
Therefore the permittivity, also called the dielectric constant, is also a complex
number i.e.  r     r    i r   . The relationship of the in-phase and out-ofphase components of the permittivity are related to the response function by


0
0
   1      dt   t  cos  t and        dt   t  sin  t
(23.4)

Now the expression for the rate of energy absorption from the field is…
U
 r     0
U
1
where U   0 E02 and  0 is the vacuum permittivity. Then
2
U
 
 r     0        E02   U  D
U
2 
see Lecture 21.

 BA 1


2
1

1  e

i Q
1  e 
 



dteitBA  t  

e
  m
    nm  Amn Bnm
m,n

1
 i t
e 
 dte 
2 i Q 
m,n
1  e 

 
i
m
1  e
 e   m


Q
m,n 
  mn
e
 i mn t
Amn Bnm

    nm   mn  nm

 e  m


Q
m,n 

2
    nm   mn
i

(23.7)
wher A=B=u which is the electric dipole moment.
 The r.hs. of (23.7) is pure imaginary so…
i 1  e     e   m
 i   i 








Q
m,n 

    nm   mn

2
(23.8)
 e  m 
2
   


    nm   mn
Q 
m,n 
Multiply (23.8) by N/V to make it per unit volume…and divide by the free space
permittivity to make the l.h.s. a relative quantity…
 1  e 

(23.6)
We now calculate the complex susceptibility using linear response theory, see
Lecture 22…
BA   

(23.5)


C. The Spectral Line Shape


By definition the spectral line shape is:
3  0   
I   
  pi f u i i u f   f ,i   
 1  e     VN  i , f
Note the definition of the delta function

1
1
   
dt eit and so   f ,i    

2 
2

 dt e



i  f , i  t
(23.9)
(23.10)

Note the factor of three in (23.9) is introduced to indicate an average transition
moment. In all calculations up to here the direction assumed is x. Then
2
 mn x  13  mn   nm . Combining (23.9) and (23.10)
I   
3  0   
 1  e 

 
N
V

1
  pi f u i i u f 
i, f

 2
  E f  Ei
 

exp
it


 
  
 

 (23.11)

where  f ,i  E f  Ei

(23.11) may be further reduced using standard methods…

1
iE t
I   
dt pi f e f ueiEit i i u f e i t


2 i , f 


1
2
1
2

  dt p
i
f eiEHt ue iEHt i i u f e it
i , f 

  dt p
i
(23.12)
f u  t  i e i t
iu f
i , f 
whereu  t   eiEHt ue iEHt

We then use the closure property
k
k  1 to collapse the double
k
summation to a single summation

1
I   
  dt pi i u f
2 i , f 


1
2
1
2

  dt p
i
i

f u  t  i e  i t
i u  u  t  i e  i t

 dt
(23.13)
u  u  t  e  i t 

1
2

 dt C  t  e
 i t

where C  t   u  u  t    pi i u  u  t  i
i
D. Summary


(23.13) shows that the spectral lineshape and the correlation function C(t) are
related by a Fourier transform. (23.13) can be inverted


1
 i t
I   
dt
C
t
e

C
t

(23.14)

   d I   eit

2 

(23.19) and (23.20) can be applied to a number of spectroscopies by making
the appropriate substitution for the operator in the correlation function
expression. For example
o Microwave Spectroscopy: u=u0, the permanent dipole moment.
 u 
o Infrared: u  
 Q , where Q is a normal coordinate
 Q 0
o Rayleigh Scattering: u  uind  eˆi    eˆs where  is the polarizability
tensor, and eˆi , s are the unit vectors in the direction of the incident and
scattered radiation.
  
o Raman Scattering: u  uind  eˆi     eˆs where    

 Q 0
o Magnetic Resonance: correlation functions in magnetic resonance
involve the magnetic dipole moment of the particle. Relaxation rates in
magnetic resonance involve spectral densities that are Fourier
transforms of correlation functions.