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rules for eigenstates of the harmonic oscillator, the second term is nonvanishing

only when

n
i
differ by one, implying that the system has adsorbed an

amount of energy corresponding to

n
f
and

o
I
. The intensity of the spectral lines is pro-

portional to the square of the derivative of the dipole with respect to the vibrational

coordinate.

The Raman process is the inelastic scattering of light by an interaction with

the vibrations of the system. Consequently, the Raman amplitudes are propor-

tional to the response function of the system to an external time dependent electric

field, i.e., the dynamical electron polarizability:

h

"

#

r

n
I
þ

X

m
0
r
m
r
0

m
0
r
m
r
0

1

2
h
o
I

@

a
a
Stokes
¼

G
þ

;

h
o

h
o

h
o
I
þ

@

Q
I

E
r

E
0

1

=

2
i

E
r

E
0
þ

1

=

2
i

G

I;r

where

o

is the frequency of the incident radiation,

o
I
is the frequency of the

I
th vibrational mode and
Q
I
its eigenvector.

m
0
r
is the electric dipole evaluated

between the ground and the excited state
r
over which the sum is made and

a

b

,

are

Cartesian coordinates;

1

h

n
I
þ
1
i¼

e
h
o
I
=KT
;

1

where
T
is temperature and
K
is the Boltzmann constant. If the Raman signal is

collected at 90
with respect to the direction of the incoming light, the spectral

intensity is:

4

Þ¼
ðo o
I
Þ

E
0
E
inc
ða

2

yy

2

zy

I

ðp=

þ a

Þ:

2

32
p

2
c
3

The calculation of IR and Raman spectra involves the evaluation of the deriva-

tive of some operators (the dipole moment and the electronic polarizability, respec-

tively) with respect to a normal coordinate. Again, these are evaluated using the

finite difference method and the linear response theory [
29
]. We observe that the

DFT - a ground state theory - gives results in good quantitative agreement with

experimental intensities, in spite of the fact that the calculation of the polarizability

involves a sum over the electronic excited states.

The above formula for the electronic polarizability involves an explicit depen-

dence of the sum terms on the incident radiation frequency,

. When the Raman

experiment is performed in far-from-resonance conditions, i.e., with an incident

frequency far from any electronic excitation of the system, this dependence can be

neglected with a great save of computational cost. Conversely, the full calculation

is necessary in the so called preresonance conditions. However, in the very near

resonance conditions a different approximation can be considered. If the photon

energy or the incident light is very close to an electronic excitation energy, one

single term of the sum over
r
is dominant. Additionally, one assumes that in these

o

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