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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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