Methods Sensitive to
Free Radical Structure
• Resonance Raman
• Electron-Spin Resonance (ESR) or
Electron Paramagnetic Resonance (EPR)
Motivation
• Absorption spectra of free radical and
excited states are generally broad and
featureless
• Conductivity is not species specific
• Conductivity is additive with respect to
ionic content of the cell
Specific Vibrations?
• Now have vibrational spectroscopy in laser
flash photolysis, usually in organic
solvents
• Water is a good filter of infrared and
masks vibrational features of free radicals
• Raman is weak, second-order effect
• What about Resonance Enhanced
Raman?
G.N.R. Tripathi
LIGHT SCATTERING
Medium
Ei
Incident light 
0
Rayleigh
+
-
Emergent light
s Scattered light
s = 0
s = 0  mn Raman
Pi
= αij Ej
P = Induced electric dipole moment
E = Electric field of the electromagnetic
radiation
αij = Elements of polarizability tensor
ENHANCEMENT OF RAMAN SCATTERING
(via αij )
e
em
0
n
mn
m
Probability
Imn = Const. I0 (0  mn)4  I(  ) mn I2

Amplitude
(  ) mn = (1/h)  MmeMen / (em  0 + i e)
e
+ non-resonant terms
G.N.R. Tripathi
G.N.R. Tripathi
RESONANCE RAMAN
em >> 0
em - 0
~0
Normal Raman
Resonance Raman
|(  ) mn 2 = Const. × (MmeMen)2 /  2
Enhancement up to 107-108
Pulse radiolysis time-resolved resonance Raman
Identification, structure, reactivity and
reaction mechanism of short-lived radicals and
excited electronic states in condensed media
Relevance: Theoretical chemistry, chemical
dynamics, biochemistry, ,paper and pulp-industry, etc.
[Ru(bpy)2dppz]2+ bound to DNA
2,2’- bipyridyl
dppz = dipyridophenazine
NO
DNA present
DNA present
1000
1200
1300
1400
Raman shift (cm-1)
http://www.lot-oriel.com/site/site_down/cc_appexraman_deen.pdf
Two-slit experiment
Selection Rules
for the Amplitudes of Transitions
Franck-Condon
Factor
Electronic Transition
Elements
(Dipole allowed)
For Resonance enhancement both
must be non-zero
http://www.personal.dundee.ac.uk/~tjdines/Raman/RR3.HTM
Relationship to Radiationless
Transitions and Absorption
dP(nm)/dt = (42/h) |Vmn|2 FC (Em)
This is a probability. Quantum mechanics usually calculates amplitudes
which are “roughly the square root” (being careful about complex numbers)
Taking the square root, shows that the amplitudes for
radiationless transitions are first-order in the interaction V
Likewise, simple absorption and spontaneous emission are first-order
processes with regard to an interaction Vrad
Connection to Wavefunctions

 ( xb , t b )   K (b, a) f (a)dxa
-
So we can use the path integral to see how one
non-stationary state (f) at time ta propagates into another
 at time tb
In terms of the stationary states of the system

 i

K (b, a)   n ( xb )n* ( xa ) exp  En (t b  t a ), for t b  t a
 

n 1
K (b, a)  0, for t b  ta
Expansion of part of exponential
for small potentials
2
2
i tb
1  i   tb
 i tb

exp   V ( x, t )dt   1   V ( x, t )dt     V ( x, t )dt  

 ta
2!     ta
  ta

Putting this back into the Amplitude Kv(b,a) gives a
perturbation expansion
of the
path integral
K V (b, a)  K 0 (b, a)  K (1) (b, a)  K ( 2 ) (b, a)  
Interpretation of First Term

 i tb m 2  
K 0 (b, a)    exp  
x dt  Dx (t )
a
  ta 2


b
Represents propagation of a free particle from (xa,ta) to (xb,tb)
with no scattering by the potential
b
V
a
Second Term
i b
 i t b m 2   tb
K (b, a)     exp  
x dt   V x( s), s ds Dx (t )
 a
  ta 2
  ta
(1)
Interpretation of Second Term
i tb
K (b, a)   
 ta
(1)
b
tb
t
tc
ta

 i tb m 2  
a  exp   ta 2 x dt  V x(s), sDx (t ) ds
b
c
a
x
Particle moves from a to c as a free particle.
At c it is scattered by V[x(s),s] = Vc.
After it moves as a free particle to b.
The amplitude is then integrated over xc,
namely over all paths.


 i tb m 2  
a  exp   ta 2 x dt  V x(s), sDx (t )  - K 0 (b, c)V ( xc , tc ) K 0 (c, a)dxc
b
Physical Meaning of 2nd Term
i tb 
K (b, a )     K 0 (b, c)V ( xc , t c ) K 0 (c, a )dxc dt c
 t a -
(1)
Represents propagation of a particle from (xa,ta) to (xb,tb)
that may be scattered once by the potential at (xc,tc)
b
V
c
a
Interpretation of Third Term
K
( 2)
tb
1 b
 i tb m 2   tb
(b, a)  
x dt    V x( s), s ds  V x( s), sds Dx (t )
 exp  

a
t
ta
2 
 a 2
  ta
Represents propagation of a particle from (xa,ta) to (xb,tb)
that may be scattered twice by the potential,
once at (x(s),s) and once at (x(s),s)
b
V
a
Selection Rules (A-term)
A-term: Condon approximation - the transition polarizability is controlled by
the pure electronic transition moment and vibrational overlap
integrals
The A-term is non-zero if two conditions are fulfilled:
(i) The transition dipole moments [m]ge0 and [m]eg0 are both non-zero.
(ii) The products of the vibrational overlap integrals, i.e. Franck-Condon
factors, <ng|e><e|mg> are non-zero for at least some values of
the excited state vibrational quantum number .
Consideration of
Franck-Condon
Factors
<ne|g> = 0
orthogonal
<ne|g>  0
Totally
Symmetric
Vibrational
Mode
http://www.personal.dundee.ac.uk/~tjdines/Raman/RR4.HTM
<ne|g>  0
Non-symmetric
Or Symmetric
<ne|g>  0
Totally
Symmetric
Vibrational
Mode
Why must these modes totally
symmetric vibrations?
He(Q) = Qg + DQ + (k/2)Q2
Hg(Q) = Qg + (k/2)Q2
All terms in the Hamiltonian must be totally symmetric,
Therefore, the displacement DQ must also be totally symmetric
G.N.R. Tripathi
G.N.R. Tripathi
G.N.R. Tripathi