Biology Reference

In-Depth Information

0
D

X

X

@

E
e;i

@

@

2
E
e;i

E
e;i
ð

R

Þ¼

E
e;i
ð

0

Þþ

R
I
þ

0
D

R
I
D

R
J
þ ...

R
I

@

R
I
@

R
J

I

I;J

This expansion could be evaluated in principle on every potential energy surface

E
e,i
obtaining the vibrational properties of the excited states. However, care must be

taken in using DFT for excited states since it is rigorously a ground state theory.

The dynamical matrix can be evaluated either by numerical difference applying

small finite displacements, or directly within the perturbative approach [
27
].

The evaluation of the vibrational frequencies of the system is usually the

preliminary step for the calculation of the vibrational spectra. As will be discussed

through explicit examples in the next section, the DFT approach leads to systematic

errors, especially in the high-frequency range, which are up to the order of 5-10%,

stemming from the approximation introduced by the energy functionals, by the

frozen core approximation, and by the finite basis set expansion. However, the

trends of variation of frequencies with respect to external environment variations

are generally accurately reproduced.

3.2 Calculation of the Vibrational Spectra

The production of a vibrational spectrum involves the interaction of an external

field (typically the electromagnetic field) with the vibrational modes of the system.

Once this interaction occurs, one might simply look at the variation of the probe

field after the interaction, as in the IR absorption experiments, or analyze the result

of the scattering process, as in the Raman experiments.

The absorption rate
W
I!F
of a radiation of frequency

o

excting a material system

from a state
I
to a state
F
is

E
inc
ðoÞjm
FI
j

2

W
I!F
/

;

where
E
inc
is the electric field of the incident light and

the electric dipole. After

the BO decoupling, one can treat the nuclear part quantum mechanically within the

harmonic approximation [
28
].

m

X

Q
I
jn
i
i
@m
fi

@

fi

m
FI
¼ m

hn
f
jn
i
iþ

hn
f
j

Q
I
þ ...;

I

where

o
I
and for the
i
th

electron state, and
Q
I
the corresponding vibrational coordinate. In the case of the IR

spectroscopy, however, the energy of the incident radiation is not sufficient to

electronically excite the system, thus

n
i
are the nuclear vibrational states of the mode at frequency

n
i
are both vibrational states of the

ground electronic state. Consequently, the first term of the sum gives a negligible

contribution due to orthonormality of

n
f
and

n
f
and

n
i
. In addition, given the selection

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