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\def\d{{\bf d}}
\def\E{\textsf{E}}
\def\R{{\bf R}}
\def\Z{{Z^*}}
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\section{Definitions}

Phonon eigenvectors at ${\bf q}=0$ (or normal modes in finite systems)
are denoted by $U_s^\alpha(\nu)$ for atom $s$ and Cartesian component 
$\alpha$. They are orthonormalized as follows:
\begin{equation}
\sum_{st,\alpha\beta} U_s^\alpha(\mu) M_s\delta_{st}\delta_{\alpha\beta} 
                      U_t^\beta(\nu) = \delta_{\mu\nu}
\end{equation}
where $M_s$ is the mass of atom $s$.

One introduces the normal mode coordinate $q_\nu$ by defining
the actual atomic displacements $u_s^\alpha$ as
\begin{equation}
u_s^\alpha = \sum_{\nu} q_\nu U_s^\alpha(\nu).
\end{equation}
Derivatives wrt $q_\nu$ are then given by
\begin{equation}
{\partial A\over\partial q_\nu} = \sum_{s,\alpha} 
{\partial A\over\partial R_s^\alpha} U_s^\alpha(\nu)
\end{equation}
where $\R_s$ is the position of the $s-$th atom and
the quantity $A$ is a function of atomic positions:
$A \equiv A(\R_1,..,\R_n)$.

\section{IR cross section}

The IR cross section, $I_\nu^{IR}$, for normal mode $\nu$
in a gas is given by
\begin{equation}
I_\nu^{IR} = {{\cal N}\pi\over 3 c} \left| 
                 {\partial \d \over \partial q_\nu}\right|^2
\end{equation}
where ${\cal N}$ is the particle density, $c$ is the speed of
light, $\d$ is the electric dipole of the system. In a 
condensed-matter system, ${\cal N}=1/\Omega$, where $\Omega$ 
is the volume of the unit cell, and $\d$ is the electric dipole
per unit cell. The quantity 
$f^{IR} = |\partial \d / \partial q_\nu|^2 $ is usually referred to 
as the ``absolute IR activity''. In terms of effective charges 
$\Z$:
\begin{equation} 
f^{IR} = 
e^2\sum_\alpha\left|\sum_{s\beta}\Z^{\alpha\beta}_s U_s^{\beta}\right|^2. 
\end{equation}

\section{Raman cross section}

The nonresonant Raman cross section is written in terms
of the Raman tensor $r_{\alpha\beta}(\nu)$:
\begin{equation}
r_{\alpha\beta}(\nu) = {\partial \chi_{\alpha\beta}\over \partial q_\nu}
                     = \sum_{s\gamma }{\partial \chi_{\alpha\beta}\over
                     \partial R_s^\gamma} U_s^\gamma(\nu),
\end{equation}
where $\chi_{\alpha\beta}$ is the electronic polarizability of the system:
$\chi_{\alpha\beta} = (\epsilon^\infty_{\alpha\beta}-1)\Omega/4\pi$ in
terms of the electronic dielectric tensor $\epsilon^\infty$. 
The derivative of $\chi$ is a third-order derivative of the energy $E$:
\begin{equation}
P^{Ram}_{\alpha\beta,s\gamma}
= {\partial^3 E \over \partial\E_\alpha \partial\E_\beta \partial R_s^\gamma}
= {\partial \chi_{\alpha\beta}\over\partial R_s^\gamma},
\end{equation}
where $\E$ is the electric field.
For a typical experimental setup: incident and outgoing signal along
orthogonal directions, plane-polarized incident beam, what is measured
(the Raman activity $I^{Ram}$) is given by
\begin{equation}
I^{Ram} = 45 a^2 + 7 c^2,
\end{equation}
where
\begin{equation}
a = {1\over 3} (r_{11}+r_{22}+r_{33}),
c^2={1\over 2} \left[ (r_{11}-r_{22})^2 +(r_{11}-r_{33})^2+(r_{22}-r_{33})^2
                      +6 (r_{12}^2 +r_{13}^2+r_{23}^2)\right]
\end{equation}
and it is understood that all quantities refer to mode $\nu$. For degenerate
modes one has to sum over different modes. The depolarization ratio $\rho$
-- the ratio between the intensity perpendicular and parallel to the incident
polarization -- varies from 0 to $3/4$, vanishes for totally symmetric 
modes, and is given by
\begin{equation}
\rho = {3c^2\over 45 a^2 + 4 c^2}.
\end{equation}

\section{Clausius-Mossotti formula}

For molecular systems, $\chi$ should be replaced by the molecular 
polarizability $\tilde\alpha$. This can be estimated from a supercell
calculation using a Clausius-Mossotti approach. For an isotropic
system:
\begin{equation}
\tilde\alpha = {3\Omega\over 4\pi}\left({\epsilon-1\over\epsilon+2}\right)
             = {3\chi \over\epsilon+2}
\end{equation}
and
\begin{equation}
{\partial \tilde\alpha\over R_s^\gamma} = 
   {3\Omega\over 4\pi}{\partial\epsilon\over\partial R_s^\gamma}
   {3\over\left(\epsilon+2\right)^2} = 
   {\partial \chi\over\partial  R_s^\gamma}
   \left({3\over\epsilon+2}\right)^2
\end{equation}
For weakly anisotropic system, one may replace the factor 
$\epsilon+2$ with $\mbox{Tr}\epsilon/3+2$.

\section{Units}

Absolute IR activities are typically given in units of
(Debye/\AA)$^2$amu$^{-1}$ (1 Debye/\AA =10$^{-10}$ esu;
1 amu =$1.660538\times 10^{-27}$ Kg). 
Other frequently encountered unit are km/mol and cm$^{-2}$ atm$^{-1}$:
1 (Debye/\AA)$^2$amu$^{-1}$ = 42.255 km/mol = 
171.65 cm$^{-2}$ atm$^{-1}$ at 0 C and 1 atm.
Third-order derivatives $P^{Ram}$ are typically given in units of \AA$^2$,
Raman activities $I^{Ram}$ in units of \AA$^4$amu$^{-1}$. 

In the code, everything is in atomic Rydberg units (aRu):
$e^2=2, m=1/2, \hbar=1$. Conversion factors : 
e$^2$ = 2 aRu = $4.80324^2 \times 10^{-20}$ esu$^2 = 
4.80324^2$ (Debye/\AA)$^2$ \\
1 aRu mass unit = 2 electron mass = 1.821876376$\times 10^{-30}$ Kg
= 0.00109716 amu; \\
1 aRu length unit = 1 bohr radius = 0.529177 \AA. Note that
$\Z$ is adimensional.

The conversion factors are:
\begin{itemize}
\item 1 aRu = 10514.0155 (Debye/\AA)$^2$amu$^{-1}$ for $f^{IR}$;
\item 1 aRu = 0.2800283 \AA$^2$ for $P^{Ram}$;
\item 1 aRu = 71.47166 \AA$^4$ amu$^{-1}$ for $I^{Ram}$.
\end{itemize}

\section{Bibliography}

D. Porezag and M. R. Pederson, Phys. Rev. {\bf B} 54, 7830 (1996).\\
P. Umari, X. Gonze, and A. Pasquarello, Phys. Rev. {\bf B} 69, 235102 (2004).

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