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conditions the variation of the polarizability with respect to the normal coordinates
is mainly due to the denominator, and neglects the variation of the dipoles. In these
conditions, one can write [
30
]:
m
0
r
m
r
0
m
0
r
m
r
0
@
2
@
E
r
G
'
h
o
@
Q
I
E
r
E
0
=
2
i
@
Q
I
1
h
o
ð
E
r
E
0
1
=
2
i
GÞ
(the derivative of
E
0
vanishes if the system is in the equilibrium geometry). The
appearance of the excited state energy can be explained in the following way:
in resonance conditions, the system is actually excited and stays for some time on
the excited state, where the atoms acquire velocities according to the forces, which
coincide with the derivative
Q
I
. Then it goes back to the ground state and
moves according to the acquired velocities. Thus, the modes whose displacement
has a large projection onto the excited state force are selected. Although the very
on-resonance conditions are usually avoided due to the dominance of other pheno-
mena, such as fluorescence, this picture is still a good representation also for the
near-resonance conditions.
As in the case of the polarizability and dipole moment derivatives, one could
calculate the derivative of the
E
r
with respect to
Q
directly or numerically. However
in this case there is an alternative approach that consists in simulating the whole
process of excitation and subsequent dynamical evolution of the system. This
approach is general and can be applied to any kind of external perturbation and
has the advantage of going beyond the linear response approximation. The algo-
rithm is the following: starting from the relaxed system, (a) induce changes due to
the external perturbation, (b) simulate the dynamics of the system for a sufficiently
long time after perturbation, and (c) analyze the trajectory to extract the vibrational
spectrum.
Step (a) depends on the kind of process one is considering and generally
produces a set of nuclear displacements or equivalently a set of atomic velocities.
In the case of resonance Raman, these are obtained by performing a few steps of
excited state dynamics [
31
]. This and the subsequent point (b) require an algorithm
to simulate the dynamics of the system. Within the DFT frame one can use the
Born-Oppehneimer (BO) dynamics or the Car-Parrinello dynamics (CP) [
32
].
Within the BO scheme, one simply evolves the nuclei with Newtonian dynamics
using the forces evaluated within the DFT approach as derivatives of the
E
r
(
R
)
with respect to
R.
The classical dynamics of the nuclear system is integrated
numerically, and for each step of the nuclear system the KS equation for the elec-
tron system must be solved, involving a full electronic wavefunction optimization.
In this way, a line on the exact BO surface
E
r
(
R
) is followed.
The CP approach is an approximate way to solve the BO dynamics. A lagrangian
system including both the nuclear coordinates and the electronic coordinates is
defined. The electronic quantum coordinates are transformed in classical functional
coordinates, the KS orbitals
∂
E
r
/
∂
'
i
, and a fictitious classical mass
m
is associated
with them:
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