Vibrational Spectroscopy of Fluorescent Proteins: A Tool to Investigate the Structure of the Chromophore and Its Environment - Fluorescent Proteins: From Understanding to Design - page 142

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