Biology Reference

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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

€

odinger

equation

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
;

where

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

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

R

;

r

Þ¼w
a
ð

R

ÞC
e;i
ð

r

Þ

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

R

Þ

R

Þ¼

E
tot
w
a
ð

R

Þ

C
e;i
ð

T
e
þ V
nuc
;
nuc
þ V
nuc
;e
þ V
e;e

;
R

;
R

r

Þ¼

E
e;
i
ð

R

ÞC
e;i
ð

r

Þ

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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