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Reciprocal space summation for Coulomb interactions

A point charge $Z$ is spreaded as $\rho(r)$ :

\begin{displaymath}
\rho(r)=Z ~(\frac{1}{2 \sigma^2_c \pi })^{3/2}exp( - \frac{r^2}{2 \sigma^2_c})
\end{displaymath}

and in reciprocal space

\begin{displaymath}
\rho(k)=Z ~exp( - \frac{k^2 \sigma^2_c }{2 })
\end{displaymath}

The dynamical-matrix block between atom $\eta$ and atom $\xi$ is given by :

\begin{displaymath}
Z_{\eta} Z_{\xi} \sum_l exp(i \vec{Q} . (r_{\xi l} -r_{\eta...
...partial_{\eta 0 x} \partial_{\xi l y} V(r_{\xi l}-r_{\eta 0} )
\end{displaymath}

and reciprocal space

\begin{displaymath}
\frac{Z_{\eta} Z_{\xi}}{(2 \pi)^3} 4 \pi \int d k^3 \frac{e...
...{k^2}
\sum_l exp(i \vec{Q} . (R_l + r_{\xi 0} -r_{\eta 0} ))
\end{displaymath}

where $R_l$ is a lattice vector. The above expression can be simplified by the help of the following identity :

\begin{displaymath}
\sum_{R_i} exp(i K.R_i) = \frac{(2 \pi)^3}{V} \sum_G \delta(K-G)
\end{displaymath}

where V is the volume of the lattice unit cell and $G$ runs over the reciprocal lattice. Finally the matrix block contributing to $Z ~C ~Z$ of equation(15)is given by :


\begin{displaymath}
\frac{Z_{\eta} Z_{\xi} 4 \pi }{(2 \pi)^3} \sum_G \frac{
exp...
...G+Q)_x (G+Q)_y exp(iG.( r_{\xi 0} -r_{\eta 0} ))
}{
(G+Q)^2
}
\end{displaymath} (16)

while the self term ($C_0$ of equation(15)) for atom $\eta$ is obtained by setting $Q=0$ and $Z_{\eta}=1$ in the above expression. In calculating the term given by formula (16) we are actually doing an error : as we sum over the whole lattice we are considering also a spurious affect coming from the interaction of the charge with itself. In a case of rigid ions the self term $ Z C_0$ simply subtracts to $Z ~C ~Z$ and the spurious terms in $C_0$ and in $C$ would delete each other. However in the case of polarizable shells, as can be seen in equation (15), $C_0$ and $C$ have a life on their own, being multiplied sometimes by $Z$ and some other time by $Y$. The spurious term has thus to be removed from $C$ and $C_0 ~Z^{-1}$. That can be obtained removing to the diagonal blocks of $C$ and $C_0 ~Z^{-1}$, for each atom $\eta$ the following quantity :

\begin{displaymath}
\frac{4 \pi}{3 \sigma_c^3} \sqrt{\frac{1}{(2 \pi)^3}}
\end{displaymath}

That corresponds to the derivative of the electric field at the center of a $3D$ gaussian distribution of charge.

The following subsection explains how the difference between a point charge and a gaussian one is recovered


next up previous contents
Next: Real-space summation of the Up: Treatment of the Coulomb Previous: Treatment of the Coulomb   Contents
Alessandro Mirone 2003-11-17