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<p><font size="6" color="white"><b>Theory of Cluster Dynamics</b></font><font
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size="5"><br />
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</font><font size="6"> </font><font size="5">The Toulouse -
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Erlangen Collaboration</font></p>
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</div>
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<div id="content"> <a name="oben"> </a>
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<div style="margin:15px;width:770px;border:1px solid gray;float:left;font-size:10px;"><a
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name="oben">
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</a>
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<div style="width:180px;float:left;text-align:center;font-size:12px"><a
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name="oben">
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</a><a href="formal.html">1. Theoretical developments </a> </div>
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<div style="width:200px;float:left;text-align:center;font-size:10px;">
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<a href="../analysis/detail1.html"> 2. Analysis of cluster
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dynamics </a> </div>
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<div style="width:200px;float:left;text-align:center;font-weight:900;font-size:10px;">
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<a href="../analysis/detail2.html"> 3. Clusters in strong external
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fields </a> </div>
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<div style="width:180px;float:left;text-align:center;font-weight:900;font-size:10px;">
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<a href="detailQMMM.html"> 4. Embedded clusters </a> </div>
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</div>
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<div id="WideContent">
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<div id="contentBoxWide">
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<div id="contentBoxHeader">
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<p>Time Dependent Density Functional Theory with Molecular
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Dynamics </p>
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</div>
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<div id="contentBoxContent">
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<p> </p>
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<div align="CENTER"> <font size="+2"><b> TDLDA-MD:</b></font> <br />
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<br />
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<font size="+1"><b>Time-dependent local-density approximation
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plus ionic molecular dynamics</b></font> <br />
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</div>
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<p> (<em>This is a very short summary of our formal scheme. A
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most detailed description is found in </em>[<a href="../literatur.html#own1281">303</a>].)
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</p>
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<p> The <font color="#ff0000"> electron cloud</font> is
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described by density functional theory at the level of TDLDA.
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The dynamical degrees of freedom are the set of occupied <font
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color="#ff0000">
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single-electron wavefunctions <font color="#ff0000"><!-- MATH
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$\varphi_\alpha$ --> <img src="img1.png" alt="\bgroup\color{red}$ \varphi_\alpha$\egroup"
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width="26"
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border="0"
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align="MIDDLE"
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height="33" /></font></font>.
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The <font color="#00b300"> ions</font> are treated by
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classical MD and their degrees of freedom are the <font color="#00b300">
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positions <i><b>R<sub>I</sub></b></i> and momenta <i><b>P<sub>I</sub></b></i>
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<font color="#00b300"><!-- MATH
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$({R}_I,{P}_I)$ -->
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<!-- <IMG
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WIDTH="69" HEIGHT="37" ALIGN="MIDDLE" BORDER="0" SRC="img2.png" ALT="\bgroup\color{dgreen}$ ({R}_I,{P}_I)$\egroup"></FONT></FONT>.--></font></font>.
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The starting point is the total energy given by: <br />
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</p>
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<div align="CENTER">
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<!-- MATH
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\begin{eqnarray*}E_{\rm total} &=& {\color{red} E_{\rm kin}(\{\varphi_\alpha\}) + E_{\rm C}(\rho) + E_{\rm xc}^{\rm (LDA)}(\rho_\uparrow,\rho_\downarrow) } + E_{\rm el,ion}({\color{red} \rho},{\color{dgreen} \{{R}_I\}}) + {\color{dgreen} E_{\rm ion}(\{{R}_I,{P}_I\})} + E_{\rm ext}({\color{red} \rho},{\color{dgreen} {R}_I},t) \quad.\end{eqnarray*} -->
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<table width="100%" cellpadding="0" align="CENTER">
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<tbody>
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<tr valign="MIDDLE">
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<td nowrap="nowrap" align="RIGHT"><img src="img3.png" alt="$\displaystyle E_{\rm total}$"
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width="47"
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border="0"
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align="MIDDLE"
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height="35" /></td>
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<td nowrap="nowrap" width="10" align="CENTER"><img src="img4.png"
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alt="$\displaystyle =$"
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width="19"
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border="0"
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align="MIDDLE"
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height="33" /></td>
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<td nowrap="nowrap" align="LEFT"><img src="img5.png" alt="$\displaystyle {\color{red} E_{\rm kin}(\{\varphi_\alpha\})+ E_{\rm C}(\rho)+ ......}_I,{P}_I\})} +E_{\rm ext}({\color{red} \rho},{\color{dgreen} {R}_I},t) \quad.$"
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width="690"
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border="0"
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align="MIDDLE"
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height="43" /></td>
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<td width="10" align="RIGHT"> </td>
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</tr>
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</tbody>
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</table>
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</div>
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<br clear="ALL" />
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<p> The electronic kinetic energy <font color="#00b300"><!-- MATH
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${\color{red} E_{\rm kin}}$ --> <img src="img6.png" alt="\bgroup\color{dgreen}$ {\color{red} E_{\rm kin}}$\egroup"
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width="37"
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border="0"
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align="MIDDLE"
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height="35" /></font>
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employs the single-electron wavefunctions <font color="#00b300"><!-- MATH
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${\color{red} \varphi_\alpha}$ --> <img src="img7.png" alt="\bgroup\color{dgreen}$ {\color{red} \varphi_\alpha}$\egroup"
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width="26"
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border="0"
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align="MIDDLE"
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height="33" /></font>
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which maintains the quantum mechanical shell effects. All
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other electronic energies refer only to the local
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spin-densities or total density
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<!-- MATH
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${\color{red} \rho=\rho_\uparrow+\rho_\downarrow}$ --> <img src="img8.png" alt="\bgroup\color{dgreen}$ {\color{red} \rho=\rho_\uparrow+\rho_\downarrow}$\egroup"
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width="92"
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border="0"
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align="MIDDLE"
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height="33" />;
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the Coulomb energy <font color="#00b300"><!-- MATH
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${\color{red} E_{\rm C}}$ --> <img src="img9.png" alt="\bgroup\color{dgreen}$ {\color{red} E_{\rm C}}$\egroup"
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width="29"
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border="0"
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align="MIDDLE"
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height="35" /></font>
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naturally, and the exchange-correlation energy <font color="#00b300"><!-- MATH
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${\color{red} E_{\rm xc}}$ --> <img src="img10.png" alt="\bgroup\color{dgreen}$ {\color{red} E_{\rm xc}}$\egroup"
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width="32"
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border="0"
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align="MIDDLE"
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height="35" /></font>
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by virtue of the LDA (often augmented by a self-interaction
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correction (SIC) <a href="../literatur.html#own1252">[277]</a>).
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The electron-ion coupling <font color="#00b300"><!-- MATH
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$E_{\rm el,ion}$ --> <img src="img11.png" alt="\bgroup\color{dgreen}$ E_{\rm el,ion}$\egroup"
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width="51"
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border="0"
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align="MIDDLE"
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height="35" /></font>
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is realized by pseudo-potentials, mostly soft local ones <a href="../literatur.html#own1216">[249]</a>.
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The ionic part <font color="#00b300"><!-- MATH
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${\color{dgreen} E_{\rm ion}}$ --> <img src="img12.png" alt="\bgroup\color{dgreen}$ {\color{dgreen} E_{\rm ion}}$\egroup"
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width="37"
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border="0"
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align="MIDDLE"
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height="35" /></font>
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is composed of Coulomb interaction and kinetic energy.
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Excitation mechanisms (laser, ionic collisions) are described
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in <font color="#00b300"><!-- MATH
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$E_{\rm ext}$ --> <img src="img13.png" alt="\bgroup\color{dgreen}$ E_{\rm ext}$\egroup"
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width="37"
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border="0"
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align="MIDDLE"
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height="35" /></font>
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as external time-dependent potentials. </p>
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<p> The coupled equations of motion are obtained in standard
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manner by variation. They read
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<!-- MATH
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\begin{displaymath}{\color{red} \imath\partial_t\varphi_\alpha = \Big(\frac{\hat{p}^2}{2m} + \frac{\delta E_{\rm total}}{\delta\rho_{\sigma_\alpha}}\Big) \varphi_\alpha } \qquad,\qquad {\color{dgreen} \partial_t{R}_I = \frac{{P}_I}{M_I} \quad,\quad \partial_t{P}_I = -\nabla_{{R}_I}E_{\rm total}} \quad.\end{displaymath} -->
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</p>
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<p></p>
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<div align="CENTER"> <img src="img14.png" alt="\bgroup\color{dgreen}$\displaystyle {\color{red} \imath\partial_t\varphi_\alpha......M_I} \quad,\quad\partial_t{P}_I =-\nabla_{{R}_I}E_{\rm total}} \quad.$\egroup"
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width="618"
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border="0"
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align="MIDDLE"
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height="65" />
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</div>
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<p> where <font color="#00b300"><!-- MATH
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${\color{red} \sigma_\alpha}$ --> <img src="img15.png" alt="\bgroup\color{dgreen}$ {\color{red} \sigma_\alpha}$\egroup"
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width="24"
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border="0"
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align="MIDDLE"
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height="33" /></font>
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is the spin orientation of the state <font color="#00b300"><!-- MATH
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${\color{red} \alpha}$ --> <img src="img16.png" alt="\bgroup\color{dgreen}$ {\color{red} \alpha}$\egroup"
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width="16"
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border="0"
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align="BOTTOM"
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height="19" /></font>.
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The equations imply a non-adiabatic coupling which goes beyond
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usual Born-Oppenheimer approach. Non-adiabatic effects become
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crucial in cluster dynamics induced by strong fields. The
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numerical solution involves the representation of the
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wavefunctions on a spatial grid, time-splitting for the
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electronic propagation and the Verlet algorithm for MD, for
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details see [<a href="../literatur.html#own1230">254</a>]. The
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obtained wavefunctions, densities, and ionic coordinates allow
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to compute a wide variety of observables,
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<!-- at the side of the electrons -->e.g. <font color="#ff0000">
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optical absorption spectra</font> [<a href="../literatur.html#own1155">9</a>],
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<font color="#ff0000"> angular distributions</font> [<a href="../literatur.html#own1288">313</a>],
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<font color="#ff0000"> emission spectra</font> [<a href="../literatur.html#own1285">304</a>],
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or <font color="#ff0000"> ionization</font> [<a href="../literatur.html#own1186">208</a>]
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for electronic degrees of freedom. The <font color="#00b300">
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ionic configurations</font> can be measured indirectly
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through optical response and its dynamics with various pump
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and probe scenarios [<a href="../literatur.html#own1246">290</a>].
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</p>
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<p></p>
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<p> Often, we use a <font color="#ff0000"> semi-classical
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description for the electronic dynamics</font> at the level
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of Vlasov-LDA, particularly for energetic processes and/or
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large clusters. Instead of the <font color="#ff0000">
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wavefunctions</font>, the key ingredient becomes here the <font
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color="#ff0000">
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one-electron phase-space distribution <font color="#ff0000"><!-- MATH
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$f({r},{p},t)$ --> <img src="img17.png" alt="\bgroup\color{red}$ f({r},{p},t)$\egroup"
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width="71"
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border="0"
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align="MIDDLE"
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height="37" /></font></font>.
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The quantum-mechanical propagation for the electrons is
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replaced by the Vlasov equation
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<!-- MATH
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\begin{displaymath}{\color{red} \partial_t f = \frac{{p}}{m}\nabla_{r}f - \Big( \nabla_{r}\frac{\delta E_{\rm total}}{\delta\rho_{\sigma_\alpha}} \Big) \nabla_{p}f }\end{displaymath} -->
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</p>
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<p></p>
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<div align="CENTER"> <img src="img18.png" alt="\bgroup\color{red}$\displaystyle {\color{red} \partial_t f= \frac{{p}}{m}\nabl......{\delta E_{\rm total}}{\delta\rho_{\sigma_\alpha}} \Big)\nabla_{p}f }$\egroup"
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width="270"
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border="0"
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align="MIDDLE"
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height="61" />
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</div>
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<p></p>
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<p> again non-adiabatically coupled to ionic motion as above.
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Note that formally the same Kohn-Sham potential <font color="#ff0000"><!-- MATH
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${\color{red} {\delta E_{\rm total}}\big/{\delta\rho_{\sigma_\alpha}}}$ --> <img
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src="img19.png"
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alt="\bgroup\color{red}$ {\color{red} {\delta E_{\rm total}}\big/{\delta\rho_{\sigma_\alpha}}}$\egroup"
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width="102"
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border="0"
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align="MIDDLE"
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height="41" /></font>
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is employed. For a derivation and justification from TDLDA see
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[<a href="../literatur.html#own1163">182</a>]. The Vlasov-LDA
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equation is solved with the test-particle method where the
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distribution function <font color="#ff0000"><!-- MATH
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${\color{red} f}$ --> <img src="img20.png" alt="\bgroup\color{red}$ {\color{red} f}$\egroup"
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width="16"
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border="0"
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align="MIDDLE"
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height="35" /></font>
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is represented as a sum of Gaussian test-particles which are
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propagated again by the Verlet algorithm [<a href="../literatur.html#own1248">273</a>].
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</p>
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<p> The semi-classical description makes it feasible to include
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dynamical correlations from electron-electron collisions. This
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is achieved by adding an Uehling-Uhlenbeck collision term
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leading to
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<!-- MATH
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\begin{displaymath}{\color{red} \partial_t f = \frac{{p}}{m}\nabla_{r}f - \Big( \nabla_{r}\frac{\delta E_{\rm total}}{\delta\rho_{\sigma_\alpha}} \Big) \nabla_{p}f + I_{\rm UU}(f) } \quad.\end{displaymath} -->
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</p>
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<p></p>
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<div align="CENTER"> <img src="img21.png" alt="\bgroup\color{red}$\displaystyle {\color{red} \partial_t f= \frac{{p}}{m}\nabl......\delta\rho_{\sigma_\alpha}} \Big)\nabla_{p}f +I_{\rm UU}(f) }\quad. $\egroup"
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width="372"
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border="0"
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align="MIDDLE"
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height="61" />
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</div>
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<p> The collision term <font color="#ff0000"><!-- MATH
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${\color{red} I_{\rm UU}}$ --> <img src="img22.png" alt="\bgroup\color{red}$ {\color{red} I_{\rm UU}}$\egroup"
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width="34"
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border="0"
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align="MIDDLE"
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height="35" /></font>
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is a non-linear functional of the distribution function <font
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color="#ff0000"><!-- MATH
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${\color{red} f}$ --> <img src="img20.png" alt="\bgroup\color{red}$ {\color{red} f}$\egroup"
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width="16"
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border="0"
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align="MIDDLE"
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height="35" /></font>.
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It contains terms up to third power in <font color="#ff0000"><!-- MATH
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${\color{red} f}$ --> <img src="img20.png" alt="\bgroup\color{red}$ {\color{red} f}$\egroup"
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width="16"
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border="0"
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align="MIDDLE"
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height="35" /></font>.
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It is constructed from local and instantaneous collisions
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which obey energy conservation, momentum conservation, and the
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Pauli principle [<a href="../literatur.html#own1248">273</a>].
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The resulting equation is called the Vlasov-Uehling-Uhlenbeck
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approach (VUU). </p>
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<p> </p>
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