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various GFP mutants is shown in Fig. 2b (see Sect. 4.4 ). By comparing the time-
resolved spectra collected with and without the actinic pump, it is possible to
analyze the vibrational differences of the molecule in the excited state and therefore
to explore its induced structural changes.
3 Theory: The Density Functional Theory Framework
The accurate evaluation of vibrational frequencies and spectra requires the use of
quantum chemistry (QM) modeling techniques, treating explicitly the electronic
system. One in principle should solve the complete electrons-nuclei Schr
C I ¼
H mol C I ¼ T K þ T e þ V nuc ; nuc þ V nuc ;e þ V e;e
E I C I ;
is the total wavefunction depending on the complete set of nuclear and
electronic coordinates {R , r} and the operators in [] are the kinetic energy of nuclei,
the kinetic energy of electrons, and the electrostatic potential energy of nuclei,
of electrons-nuclei and of electrons interactions, respectively. In order to simplify
the problem, basically all the solving techniques rely on the Born-Oppenheimer
(BO) approximation, which consists in separating the total wavefunction in the
product of nuclear (
C e,i ) wave functions (the last one having the
coordinates of the nuclei R as parameters)
w a ) and electronic (
; R
C I ð
Þ¼w a ð
ÞC e;i ð
under the assumption that the electron dynamics are much faster than the nuclear
ones. Using this form of the wavefunction, the total Hamiltonian separates, produc-
ing two Schrodinger equations
w a ð
T K þ
E e;i ð
E tot w a ð
C e;i ð
T e þ V nuc ; nuc þ V nuc ;e þ V e;e
; R
; R
E e; i ð
ÞC e;i ð
for nuclei and electrons, respectively. The second line is the Schrodinger equation
of an electron system at fixed nuclear configuration R and its eigenvalues depend
Fig. 3 (continued) transitions ( blue line ); in (d) the gain spectrum, i.e., the ratio of the probe
continua with Raman pump pulse on and off, is depicted as a blue line .(e, f): Diagrams depicting
the femtosecond-stimulated Raman scattering (FSRS) technique: (e) energy level diagram
showing the electronic and vibrational resonances; (f) timing diagram for the actinic ( red ),
Raman pump ( green ), and Raman-continuum probe ( blue ) pulses. Panels (c-f) reproduced with
permission from [ 17 ], coyright 2010 John Wiley and Sons
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