Vibrational Spectroscopy of Fluorescent Proteins: A Tool to Investigate the Structure of the Chromophore and Its Environment - Fluorescent Proteins: From Understanding to Design - page 144

X

Þ¼ @

E KS
@

ij L ij @

M I R I ð t

R I þ R I hf i jf j i @

X
j L ij jf j i:

Þ¼ d

E KS
dhf i j þ

mf i ð t

This allows evolving the electronic and nuclear system at the same time: after
the full electronic optimization at the starting nuclear step, there is no need to re-
optimize the electronic system, which is simply evolved with a single electronic
calculation for each nuclear step. Since the electronic part is the bottleneck of the
calculation, this generally allows a save in computational cost. The electron system
dynamics is driven by the value of the fictitious electron mass. The exact BO
approximation is recovered if 0, but this would need an infinitesimally small

timestep for the evolution of the whole system. Exact BO or CP can be used for the
step (a) and to generate a long trajectory for step (b).
Once a dynamical trajectory of the system is available, the vibrational spectrum
can be obtained by the Fourier transform (FT) of the velocity self-correlation
function [point (c)] [ 33 ]:

m ¼

S ðoÞ¼ FT ½ c ðtÞ ¼ FT ½h v ð t Þ v ð t þ tÞi:

S ( ) will have resonances (peaks) at the frequencies corresponding to the vibra-

tional modes of the system, whose height will be larger for the modes that have
larger superposition with the starting displacement generated by the perturbation,
i.e., the spectrum will reflect the external perturbation. The width of the peaks
includes several effects. One is the necessarily finite length of the simulation, which
implies a numerical enlargement of the peaks in the Fourier space. This adds up to
the physical broadening due to anharmocity that implies also interactions between
the modes and displacement of the frequencies from those evaluated within the
harmonic approach. In addition if the dynamics is evaluated within the CP scheme
an additional softening of the frequencies of about 1-3% is due to the fictitious
electron mass

o

that has the effect of slowing down the dynamics [ 34 ].
A schematic representation of the IR, Raman, and Resonance Raman processes
is reported in Fig. 1c , together with the corresponding spectra of the same mole-
cule, i.e., the GFP model chromophore HDBI. The difference between IR and
Raman spectra is apparent: the selection rules for the two processes are very
different, thus the Raman active modes in this molecule are much less than the IR
active modes. The difference between Raman and Resonance Raman spectra
are less evident in this kind of molecule. They are more evident, e.g., when the
chromophore is in the protein, where the ResonanceRamantechniqueisableto
select the modes of the chromophore, related to the electronic excitation, from
those of the protein.

m