\documentclass{seminar}
\usepackage{fancybox}

\begin{document}

%################################################################################
%################################################################################

\begin{slide*}
\pagestyle{empty}
\begin{center}
{\bf \large The Induced Charge Computation Method and its Application in Monte Carlo Simulations of Inhomogeneous Dielectric Systems}	\\
\vspace{0.6cm}
\Ovalbox{{\it \large Dezs\H{o} Boda}}\\
\vspace{0.2cm}
{University of Veszprém, Department of Physical Chemistry}\\
\vspace{0.5cm}
\ovalbox{{\it Dirk Gillespie$^{1}$, Wolfgang Nonner$^{2}$, Douglas Henderson$^{3}$, Bob Eisenberg$^{1}$}}\\
\vspace{0.2cm}
{$^{1}$Rush University Medical Center, Chicago}\\
{$^{2}$University of Miami School of Medicine, Miami}\\
{$^{3}$Brigham Young University, Provo\\
\vspace{0.5cm}
Ionic Soft Matter: Novel trends in theory and applications\\
April 17, 2004, Lviv, Ukraine
}

\end{center}
\end{slide*}

%################################################################################


%################################################################################

\pagestyle{enyem}

%################################################################################

\begin{slide*}
\slideheading{Continuum models}
\begin{description}
\item[Molecular simulations of systems modelled in full atomic detail] \textbf{might be problematic:} \hspace{0.1cm}
      \begin{itemize} 
        \item \textit{time consuming} - many particles (macroscopic sampling is not possible)
        \item \textit{unknown 3D structures} (proteins)
        \item \textit{uncertainties in estimation of potential parameters}
      \end{itemize}      
\item["Coarse-graining"] \hspace{0.3cm}
      \begin{itemize} 
        \item {\it important} particles are treated explicitely (e. g. ions)
        \item {\it the rest} is smoothed into a continuum (e. g. solvent, proteins, membrane, electrode)
        \item the various components with different polarizabilities are modelled as continuums with different dielectric coefficients
	    \item it can be described by an inhomogeneous dielectric coefficient: $ \varepsilon(\textbf{r}) $ 
      \end{itemize}      
\item[Basic task:] solving the Poisson equation
\ovalbox{$-\varepsilon_{0}\nabla\cdot\left[  \varepsilon\left(  \mathbf{r}\right) \nabla
\psi\left(  \mathbf{r}\right)  \right]  =\rho\left(  \mathbf{r}\right)$}
\end{description}
\end{slide*}

%################################################################################

\begin{slide*}
\slideheading{Macroscopic electrostatics}
\begin{description}
\item[The total charge is split into two parts:] \hspace{0.3cm}
      \begin{itemize} 
        \item source (free or explicitely treated) charges: \ovalbox{$g(\mathbf{r})=\rho(\mathbf{r})/\varepsilon_{0}$} and
	    \item polarization charges \ovalbox{$h(\mathbf{r})=\rho_{pol}(\mathbf{r})/\varepsilon_{0} $} (\textbf{to be determined})
        \item constitutive relation: $\rho_{pol}(\mathbf{r})=-\nabla\cdot\mathbf{P}(\mathbf{r})$, where
        \item polarization: $ \mathbf{P}(\mathbf{r})=-\varepsilon_{0}\chi(\mathbf{r})\nabla\psi(\mathbf{r})$
        \item dielectric susceptibility: $\chi(\mathbf{r})=\varepsilon(\textbf{r})-1$
      \end{itemize} 
\item[The corresponding Poisson equation:] 
	$ \nabla^{2}\psi(\mathbf{r})=-\frac{1}{\varepsilon_{0}}[\rho(\mathbf{r})+\rho_{pol}(\mathbf{r})] $ \hspace{0.3cm}   
\item[The potential is also split into two parts:] \hspace{0.3cm}
	\begin{itemize} 
	\item 
\Ovalbox{$ \psi(\mathbf{r}) =\psi_{e}(\mathbf{r})+\psi_{i}(\mathbf{r}) = \int_{\mathcal{D}}G(\mathbf{r}-\mathbf{r}^{\prime})g(\mathbf{r}^{\prime})d\mathbf{r}^{\prime}+
\int_{\mathcal{D}}G(\mathbf{r}-\mathbf{r}^{\prime})h(\mathbf{r}^{\prime})d\mathbf{r}^{\prime} $}
	\item the Green-function satisfies $ \nabla^{2}G(\mathbf{r}-\mathbf{r}^{\prime})=-\delta(\mathbf{r}-\mathbf{r} ^{\prime}) $ 
	\item $ G( \mathbf{r}-\mathbf{r}^{\prime}) =1/ (4\pi |\mathbf{r}-\mathbf{r} ^{\prime}| )$ for the boundary condition $ \psi(\mathbf{r}) \rightarrow 0 $ if $ \mathbf{r}\rightarrow \infty $
	\end{itemize}
\end{description}
\end{slide*}

%################################################################################

\begin{slide*}
\slideheading{Variational formalism}
\begin{description}
\item[Solution of the Poisson equation minimizes the functional] \hspace{0.3cm}
\begin{itemize}
	\item \doublebox{$ I\left[  \psi\right]  =\frac{1}{2}\int_{\mathcal{D}}\nabla\psi\cdot\nabla\psi
d\mathbf{r}-\int_{\mathcal{D}}\psi\left[  g+\frac{1}{2}\nabla\cdot\left(
\chi\nabla\psi\right)  \right]  d\mathbf{r} $}
	\item In minimum the value $-\varepsilon_{0}I[\psi]$ is the electrostatic energy
 	\item Substituting the Green function form of $ \psi(\mathbf{r})$, the functional can be expressed in terms of the charges: $ I[g ,h] $
	\item At a fixed $g(\mathbf{r})$, we have to find $ h(\mathbf{r}) $ that minimizes $ I[g ,h] $.
	\item This is equivalent to minimizing the $ h $-dependent part of the functional $ I_{2}\left[  h\right]  $ for a fixed $ g $.
\end{itemize}
	\item [So, the task is to solve the extremum condition]
\doublebox{$\frac{\displaystyle \delta I_{2}[h]}{\displaystyle \delta h(\mathbf{r})}=0$}
\end{description}
\begin{itemize}
\item \begin{small}\underline{R. Allen, J.-P. Hansen, and S. Melchionna, \textit{Phys. Chem. Chem. Phys.} \textbf{3}, 4177 (2001).}\end{small}
\item Allen \textit{et al.} solved it with the \textit{steepest descent} method.
\end{itemize}
\end{slide*}

%################################################################################

\begin{slide*}
\slideheading{Solution of the extremum condition}
\begin{center}
\doublebox{$\frac{\displaystyle \delta I_{2}[h]}{\displaystyle \delta h(\mathbf{r})}=0$}
\end{center}
We present a different solution resulting in a matrix equation.
\textbf{The following integral equation can be obtained from the extremum condition:}
\begin{center}
\shadowbox{
$
\begin{array}{rl}
&  h(\mathbf{r})\varepsilon(\mathbf{r})-\int_{\mathcal{D}}h(\mathbf{r}%
^{\prime})\nabla_{\mathbf{r}}\varepsilon(\mathbf{r})\cdot\nabla_{\mathbf{r}%
}G(\mathbf{r}-\mathbf{r}^{\prime})d\mathbf{r}^{\prime}\\
&  =\nabla_{\mathbf{r}}\varepsilon(\mathbf{r})\cdot\nabla_{\mathbf{r}}\psi
_{e}(\mathbf{r})-\frac{1}{\varepsilon_{0}}\left[  \varepsilon(\mathbf{r}%
)-1\right]  \rho(\mathbf{r})
\end{array}
$
}
\end{center}
\begin{itemize} 
\item This equation should be discretized and solved for $ h $.
\item It is \textbf{general} and valid for arbitrary $ \varepsilon(\mathbf{r}) $ and $ \rho(\mathbf{r}) $.
\item We can make our life easier by making some assumptions for $ \varepsilon(\mathbf{r}) $ and $ \rho(\mathbf{r}) $.
\end{itemize} 
%\ovalbox{$ \psi_{e}(\textbf{r})=? $}
\begin{small}\underline{D. Boda, D. Gillespie, W. Nonner, D. Henderson, B. Eisenberg, \textit{Phys. Rev. E} in press (2004).}\end{small}
\end{slide*}

%################################################################################

\begin{slide*}
\slideheading{Point charges as source charges}
\begin{itemize} 
\item Suppose that the source charges are point charges: \ovalbox{$ \ovalbox{$\rho(\mathbf{r})=\sum_{k}z_{k}e\delta(\mathbf{r}-\mathbf{r}_{k})	 $} $}
\item Polarization charges of magnitude \ovalbox{$ -z_{k}e\frac{\varepsilon(\mathbf{r}_{k})-1}{\varepsilon(\mathbf{r}_{k})} \delta(\mathbf{r}-\mathbf{r}_{k}) $} are induced around the point charges (localized on them).
\item The electric potential produced by the point charges \textbf{and} the polarization charges induced on them: 
\ovalbox{$\psi_{e}(\mathbf{r})=\frac{\displaystyle e}{\displaystyle 4\pi\varepsilon_{0}}\sum_{k}\frac{\displaystyle z_{k}} {\displaystyle \varepsilon(\mathbf{r}_{k})|\mathbf{r}-\mathbf{r}_{k}|}$}\\ 
where $ \varepsilon(\mathbf{r}_{k}) $ is the dielectric coefficient at the position of the $ k $th source charge.
\item From now on, $ h(\textbf{r}) $  denotes \textbf{only} the polarization charges induced on the dielectric interfaces. In this case, the basic equation reduces to :\\

\shadowbox{
$
h(\mathbf{r})\varepsilon(\mathbf{r})-\int_{\mathcal{D}}h(\mathbf{r}^{\prime
})\nabla_{\mathbf{r}}\varepsilon(\mathbf{r})\cdot\nabla_{\mathbf{r}%
}G(\mathbf{r}-\mathbf{r}^{\prime})d\mathbf{r}^{\prime}=\nabla_{\mathbf{r}%
}\varepsilon(\mathbf{r})\cdot\nabla_{\mathbf{r}}\psi_{e}(\mathbf{r})$
}
\item This is valid for arbitrary $ \varepsilon(\mathbf{r}) $.
\end{itemize}
\end{slide*}

%################################################################################

\begin{slide*}
\slideheading{Sharp dielectric interfaces}
%\begin{center}
%$ \Longrightarrow $
%\ovalbox{$
%h(\mathbf{r})\varepsilon(\mathbf{r})-\int_{\mathcal{D}}h(\mathbf{r}^{\prime
%})\nabla_{\mathbf{r}}\varepsilon(\mathbf{r})\cdot\nabla_{\mathbf{r}%
%}G(\mathbf{r}-\mathbf{r}^{\prime})d\mathbf{r}^{\prime}=\nabla_{\mathbf{r}%
%}\varepsilon(\mathbf{r})\cdot\nabla_{\mathbf{r}}\psi_{e}(\mathbf{r})$
%}
%\end{center}
\begin{itemize} 
\item When the dielectric is piecewise uniform, $ \varepsilon(\mathbf{r}) $ jumps from one value to another along a boundary.
\item In this case, the volume integrals over $ \mathcal{D} $ become surface integrals over the dielectric boundary surfaces $ \mathcal{B} $.
\item The basic equation then reduces to:
\begin{center}
\shadowbox{$h(\mathbf{s})\varepsilon(\mathbf{s})-\Delta\varepsilon(\mathbf{s}%
)\int_{\mathcal{B}}h(\mathbf{s}^{\prime})\nabla_{\mathbf{s}}G(\mathbf{s}%
-\mathbf{s}^{\prime})\cdot\mathbf{n}(\mathbf{s})d\mathbf{s}^{\prime}%
=\Delta\varepsilon(\mathbf{s})\nabla\psi_{e}(\mathbf{s})\cdot\mathbf{n}%
(\mathbf{s})$}
\end{center}
where
\item $ h(\mathbf{s}) $ is now surface charge distribution
\item $ \varepsilon(\mathbf{s}) $ is the mean average of the dielectric constants on the two sides of the surface
\item $ \Delta\varepsilon(\mathbf{s}) $ is their difference, and 
\item $ \mathbf{n}(\mathbf{s})	 $ is a normal vector of the surface


\end{itemize}
\end{slide*}

%################################################################################

\begin{slide*}
\slideheading{The matrix equation}
%\begin{center}
%$ \Longrightarrow $\ovalbox{
%$h(\mathbf{s})\varepsilon(\mathbf{s})-\Delta\varepsilon(\mathbf{s}%
%)\int_{\mathcal{B}}h(\mathbf{s}^{\prime})\nabla_{\mathbf{s}}G(\mathbf{s}%
%-\mathbf{s}^{\prime})\cdot\mathbf{n}(\mathbf{s})d\mathbf{s}^{\prime}%
%=\Delta\varepsilon(\mathbf{s})\nabla\psi_{e}(\mathbf{s})\cdot\mathbf{n}%
%(\mathbf{s})$}
%\end{center}
The previous equation is valid for any discrete value of $ \mathbf{s}_{\alpha} $. Use the notations
\begin{itemize} 
\item normal vector: $\mathbf{n}_{\alpha}=\mathbf{n}(\mathbf{s}_{\alpha})	$
\item mean dielectric constant: $\varepsilon_{\alpha}=\varepsilon(\mathbf{s}_{\alpha})	$
\item jump in the dielectric constant: $\Delta\varepsilon_{\alpha}=\Delta\varepsilon(\mathbf{s}_{\alpha})$
\end{itemize}
The surface $ \mathcal{B} $ is discretized into surface elements. Let us suppose that the induced surface charge is constant on surface element $ \beta $ denoted by $ h_{\beta} $. Then our equation becomes
\begin{center}
\shadowbox{$\sum_{\beta}  h_{\beta} \left[ \varepsilon_{\beta} \delta_{\alpha\beta} - \Delta\varepsilon_{\alpha}  \int_{a_{\beta}}  \nabla_{\textbf{s}_{\alpha}} G(\textbf{s}_{\alpha}-\textbf{s}_{\beta}) \cdot \textbf{n}_{\alpha} d\textbf{s}_{\beta} \right]
= \Delta\varepsilon_{\alpha} \nabla \psi_{e} (\textbf{s}_{\alpha}) \cdot \textbf{n}_{\alpha} .
\label{surface2}$}
\end{center}
where the integral expresses the polarization of the surface element $ \beta $ by the induced charge at $ \mathbf{s}_{\alpha} $, and vice versa. The integral is taken for the $ \beta $th surface element.

%The surface integral becomes a sum, and we obtain:\\
%\begin{center}
%\shadowbox{$\sum_{\beta}h_{\beta}\left(  \delta_{\alpha\beta}\varepsilon_{\beta}%
%-\Delta\varepsilon_{\alpha}\frac{\displaystyle \partial G_{\alpha\beta}}{\displaystyle \partial n_{\alpha
%}}a_{\beta}\right)  =\Delta\varepsilon_{\alpha}\frac{\displaystyle \partial\psi_{\alpha}%
%}{\displaystyle \partial n_{\alpha}}\label{h_eq}$}
%\end{center}

\end{slide*}

%\begin{itemize} 
%\item vector of center of mass: $ \mathbf{s}_{\beta} $
%\item area: $a_{\beta}$
%\end{itemize}


%################################################################################

\begin{slide*}
\slideheading{The matrix equation}
%\begin{center}
%$ \Longrightarrow $\ovalbox{$\sum_{\beta}h_{\beta}\left(  \delta_{\alpha\beta}\varepsilon_{\beta}%
%-\Delta\varepsilon_{\alpha}\frac{\displaystyle \partial G_{\alpha\beta}}{\displaystyle \partial n_{\alpha
%}}a_{\beta}\right)  =\Delta\varepsilon_{\alpha}\frac{\displaystyle \partial\psi_{\alpha}%
%}{\displaystyle \partial n_{\alpha}}\label{h_eq}$}
%\end{center}
If $\mathbf{s}_{\alpha}$ is the center of the $ \alpha $th surface element, the previous equation can be written in the matrix form: \ovalbox{${\Huge A\mathbf{h}=\mathbf{c}} $}, where
\begin{itemize}
\item the \ovalbox{$A_{\alpha\beta}=\varepsilon_{\beta} \delta_{\alpha\beta} - \Delta\varepsilon_{\alpha}  \int_{a_{\beta}}  \nabla_{\textbf{s}_{\alpha}} G(\textbf{s}_{\alpha}-\textbf{s}_{\beta}) \cdot \textbf{n}_{\alpha} d\textbf{s}_{\beta}$} matrix depends only on the geometry of the dielectric
\item $ \mathbf{h} $ is the vector of induced charges 
\item \ovalbox{$c_{\alpha}=\Delta\varepsilon_{\alpha} \nabla \psi_{e} (\textbf{s}_{\alpha}) \cdot \textbf{n}_{\alpha}$} depends on the source charges
\item If the geometry does not change in a simulation, the matrix need be inverted only once at the beginning of the simulation. 
\item Moving the source charges (the ions) changes the vector $ \mathbf{c} $, from which the induced charges are calculated by the matrix-vector multiplication $ \mathbf{h}=A^{-1}\mathbf{c}  $.
\end{itemize}

\end{slide*}

%################################################################################

\begin{slide*}
\slideheading{Calculation of the integral}

Regarding the calculation of the integral $ I_{\alpha\beta}=\int_{a_{\beta}}  \nabla_{\textbf{s}_{\alpha}} G(\textbf{s}_{\alpha}-\textbf{s}_{\beta}) \cdot \textbf{n}_{\alpha} d\textbf{s}_{\beta}$ there are two approaches 
\begin{enumerate}
\item Treating the surface charge $ h_{\beta} $ as a \textbf{point charge }of magnitude $ h_{\beta}a_{\beta} $ at the center of the $ \beta $th surface element of area $ a_{\beta} $
\begin{center}
\ovalbox{$I_{\alpha\beta} = \nabla_{\textbf{s}_{\alpha}} G(\textbf{s}_{\alpha}-\textbf{s}_{\beta}) \cdot \textbf{n}_{\alpha} a_{\beta}$}
\end{center}
for $ \beta \neq \alpha $ and $ 0 $ otherwise [\underline{Boda \textit{et al.}, \textit{Phys. Rev. E}, in press}].
This matrix has been developed before from the boundary condition $ D_{1n}=D_{2n} $, $ E_{1t}=E_{2t} $. [\underline{Hoshi \textit{et al.} JCP, 1987; Lu and Green, PCPS, 1997}]
\item Treating  $ h_{\beta} $ as a constant \textbf{surface charge} over the $ \beta $th surface element, and calculating $ I_{\alpha\beta} $ numerically after an appropriate parametrization of the surface [\underline{Nonner and Gillespie, \textit{Biophys. J.} in preparation}]. This approach is especially important if the surface is curved, therefore, we use the name \textit{curvature correction}.

\end{enumerate}

\end{slide*}

%################################################################################

\begin{slide*}
\slideheading{Calculation of the energy}

\begin{itemize} 
\item The \textbf{source charge - source charge} interaction energy:
\ovalbox{$ W_{e}=\frac{1}{2}\sum_{j}ez_{j}\psi_{e}(\mathbf{r}_{j})$}
\item The \textbf{source charge - induced charge} interaction energy:
\begin{center}
\ovalbox{$ W_{i} = \frac{e}{8\pi}  \sum_{j} z_{j} \sum_{\beta} h_{\beta} \int_{a_{\beta}} \frac{d\mathbf{s}}{|\mathbf{s}-\mathbf{r}_{j}|}$}
\end{center}
Similarly to the calculation of the integral $ I_{\alpha\beta} $, this energy can also be calculated on two levels of approximation:
\begin{enumerate}
\item Treating the induced charge as a\textbf{ point charge }of magnitude $ h_{\beta}a_{\beta} $, the integral in the above equation becomes $ \frac{a_{\beta}}{|\mathbf{s}_{\beta}-\mathbf{r}_{j}|} $.
\item Treating the induced charge as a \textbf{surface charge} with the constant value $ h_{\beta} $ over the $ \beta $th surface element, the integral should be calculated numerically. This is a \underline{time consuming step in a simulation} because the $  |\mathbf{s}-\mathbf{r}_{j}|$ distances have to be evaluated once the $ j $th ion is displaced.
\end{enumerate}

\end{itemize}

\end{slide*}


%################################################################################

\begin{slide*}
\slideheading{Ion is modelled as a hard sphere with a point charge in the center} 
\begin{itemize} 
\begin{small}\item The dielectric constant inside the hard spheres is the same as outside.
\item The ions cannot overlap with the dielectric boundaries and cannot leave their host dielectric.
\item Otherwise, new dielectric boundaries would appear - matrix $ A $ should be inverted again.\end{small}
\end{itemize}
\begin{center}
\scalebox{0.35}{\includegraphics*{ionmodel.eps}}
\end{center}
\end{slide*}

%################################################################################

\begin{slide*}
\slideheading{One flat interface: $\varepsilon_{1} | \varepsilon_{2}$ geometry - ions on the right side}
\begin{center}
Comparison of density profiles obtained from the ICC and the image charge method \\
for various resolutions of the grid. [\underline{Boda \textit{et al.} PRE in press}]\\
\vspace{0.5cm}
\scalebox{0.4}{\includegraphics*{flat1.eps}}
\scalebox{0.4}{\includegraphics*{flat2.eps}}
\end{center}
\end{slide*}
%################################################################################

\begin{slide*}
\slideheading{Two flat interfaces: $\varepsilon_{1} | \varepsilon_{2} | \varepsilon_{3}$ geometry - electrical double layer}
It is usual to assume that the inner layer has a dielectric constant different from that of the diffuse layer or the elecrode because of the need for empirical adjustments to obtain agreement with experiments.
%By choosing an appropriate value for $ \varepsilon_{2} $, the capacitance of the double layer can be fitted to experimental values.
The elecrostatic consistency of the approach should be studied.
[\underline{Boda, Gillespie, Nagy, and Henderson, JCP, submitted}]
\begin{center}
%\vspace{0.5cm}
\scalebox{0.38}{\includegraphics*{dlgeometry.eps}} \hspace{0.5cm}
\scalebox{0.34}{\includegraphics*{dlprofile.eps}}
\end{center}
\end{slide*}

%################################################################################

\begin{slide*}
\slideheading{A dielectric sphere $\varepsilon_{1}=80$ embedded in a dielectric $\varepsilon_{2}=2$}
\begin{itemize} 
\item \underline{Radius of the sphere}: $R=5$ {\AA}. 
\item \underline{Distance of a source charge $ e $ from the center of the sphere}: $d=4$ {\AA}
\item \underline{Grid}: spherical coordinates $\vartheta$ and $\phi$ evenly divided into intervals $\Delta\theta = \Delta\phi$, $ N $ is the number of surface elements.
\item \underline{Analytic solution}: series in terms of Legendre polynomials.
\end{itemize}

%Ez 32 indukált töltést jelent.\\

\begin{center}
\scalebox{0.3}{\includegraphics*{1sphere.eps}} \hspace{1cm}
\scalebox{0.3}{\includegraphics*{sphere-grid.eps}}
\end{center}	
\end{slide*}

%################################################################################

\begin{slide*}
%\slideheading{Charge inside a dielectric sphere}
%Dielectric sphere $\varepsilon_{1}=80$ embedded in a dielectric $\varepsilon_{2}=2$
\begin{center}
\scalebox{0.43}{\includegraphics*{provo1.eps}}
\end{center}
The \textit{curvature correction} is important!!! 
\end{slide*}
%################################################################################

\begin{slide*}
\slideheading{The effective force acting between two charged dielectric spheres}
The effective pair potential is defined as \ovalbox{$ \varphi(r) =W(r)-W(r\rightarrow\infty) $}. The energy $ W(r\rightarrow\infty) $ corresponds to the interaction energy between the charge and the induced charge on its own sphere. The effective force is \ovalbox{$ F(r)=-d\varphi(r)/dr $}. 
\begin{center}
\scalebox{0.25}{\includegraphics*{fig6.eps}} \hspace{0.3cm}
\scalebox{0.43}{\includegraphics*{fig7.eps}}
\end{center}
%The interaction between the spheres becomes weaker as the dielectric constant inside the spheres decreases. 
\end{slide*}

%################################################################################


\begin{slide*}
\slideheading{MC simulation for a model calcium channel}
The selectivity filter of the calcium channel is known to be lined by 4 glutamate groups (EEEE locus). They are represented by 8 half charged oxygen ions that are confined to the filter but can move freely inside. A 0.1M NaCl solution used and a few Ca$ ^{++} $ ions. \textbf{Question:} whether the Na$ ^{+} $ or the Ca$ ^{++} $ enters the filter with a higher probability?
\begin{center}
\rotatebox{0}{\scalebox{0.40}{\includegraphics*{geometry.eps}}} \hspace{0.2cm}
\frame{\scalebox{0.2}{\includegraphics*{grid.eps}}}
\end{center}
\end{slide*}

%################################################################################

\begin{slide*}
%\slideheading{Density profiles of the various ions in the channel}
%We have about 100 times more Na$ ^{+} $ ion in the bath than Ca$ ^{++} $ ion.
\begin{center}
\scalebox{0.43}{\includegraphics*{lg1.9.eps}}
\end{center}
\end{slide*}

%################################################################################

\begin{slide*}
\begin{center}
\frame{\scalebox{0.22}{\includegraphics*{CaNa.eps}}}
\end{center}
\end{slide*}

%################################################################################

\begin{slide*}
\begin{center}
\frame{\scalebox{0.22}{\includegraphics*{1Ca-b.eps}}}
\end{center}
\end{slide*}

\begin{slide*}
\slideheading{Acknowledgement}
\begin{itemize} 
\item Dr. David Busath (Brigham Young University)
\item T\'{\i}mea Nagy (University of Veszpr\'em)
\item Tibor Varga  (University of Veszpr\'em)
\end{itemize}
\end{slide*}








\end{document}


%################################################################################

\begin{slide*}
\slideheading{Conclusions}
\begin{itemize} 
\item Approach 2 for the calculation of $ I_{\alpha\beta} $ is important to the accurate determination of the energy.
\item This increases only the \textbf{fill time} of the matrix, and does not influence the \textbf{size} of the matrix.
\item The matrix is filled only once at the beginning of the simulation.
\item \underline{Conclusion:} Approach 2 in filling the matrix is recommended in MC simulations.
\item Approach 2 for the calculation of $ W_{i} $ influences the potential close to the dielectric surface.
\item This would be a time consuming step in simulations.
\item Because usually we model ions with charged hard spheres with a closest approach $ d/2 $ to the surface, this problem does not influence these simulations.
\item The interaction between the two dielectric spheres becomes weaker as the dielectric constant inside the spheres decreases.
\end{itemize}
\end{slide*}

