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0 D
E e;i
2 E e;i
E e;i ð
E e;i ð
R I þ
0 D
R J þ ...
R I @
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
excting a material system
from a state I to a state F is
E inc ðoÞjm FI j
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 ].
Q I jn i i @m fi
m FI ¼ m
hn f jn i
hn f j
Q I þ ...;
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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