470 lines
13 KiB
HTML
470 lines
13 KiB
HTML
<!DOCTYPE html PUBLIC "-//W3C//DTD XHTML 1.0 Transitional//EN" "http://www.w3.org/TR/xhtml1/DTD/xhtml1-transitional.dtd">
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<title>Theory of Cluster Dynamics</title>
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<li style="margin-top:1px;border-top:1px solid #B0C4DE; "><a href="../index.html">Home</a></li>
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<li><a href="../intro.html">Introductory Overview</a></li>
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<div id="image">
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<p><font color="white" size="6"><b>Theory of Cluster Dynamics</b></font><font size="5"><br>
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</font><font size="6">
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</font><font size="5">The Toulouse - Erlangen Collaboration</font></p>
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</div>
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<div id="content">
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<a name="oben">
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<div style="margin:15px;width:770px;border:1px solid gray;float:left;font-size:10px;">
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<div style="width:220px;float:left;text-align:center;">
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<a href="../analysis/detail1.html">1. Analysis of cluster dynamics</a>
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</div>
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<div style="width:220px;float:left;text-align:center;font-size:10px;">
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<a href="../analysis/detail2.html"> 2. Clusters in external fields</a>
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</div>
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<div style="width:220px;float:left;text-align:center;font-weight:900;font-size:12px;">
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<a href="formal.html"> 3. Theoretical developments </a>
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</div>
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</div>
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</a>
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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 Dynamics </p>
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</div>
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<div id="contentBoxContent">
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<P>
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<DIV ALIGN="CENTER">
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<FONT SIZE="+2"><B> TDLDA-MD:</B></FONT>
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<BR>
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<BR><FONT SIZE="+1"><B>Time-dependent local-density approximation
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plus ionic molecular dynamics</B></FONT>
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<BR>
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</DIV>
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<P>
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(<EM>This is a very short summary of our formal scheme. A most
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detailed description is found in </EM>[<a href="../literatur.html#own1281">303</a>].)
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<P>
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The
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<FONT COLOR="#ff0000"> electron cloud</FONT> is described by density functional theory at
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the level of TDLDA. The dynamical degrees of freedom are the set of
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occupied
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<FONT COLOR="#ff0000"> single-electron wavefunctions
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<FONT COLOR="#ff0000"><!-- MATH
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$\varphi_\alpha$
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-->
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<IMG
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WIDTH="26" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
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SRC="img1.png"
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ALT="\bgroup\color{red}$ \varphi_\alpha$\egroup"></FONT></FONT>. The
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<FONT COLOR="#00b300"> ions</FONT> are treated by classical MD and their degrees of freedom are
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the
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<FONT COLOR="#00b300"> positions <i><b>R<sub>I</sub></b></i> and momenta <i><b>P<sub>I</sub></b></i>
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<FONT COLOR="#00b300"><!-- MATH
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$({R}_I,{P}_I)$
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-->
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<!-- <IMG
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WIDTH="69" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
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SRC="img2.png"
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ALT="\bgroup\color{dgreen}$ ({R}_I,{P}_I)$\egroup"></FONT></FONT>.--></FONT></FONT>. The starting
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point is the total energy given by:
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<BR>
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<DIV ALIGN="CENTER">
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<!-- MATH
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\begin{eqnarray*}
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E_{\rm total}
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&=&
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{\color{red}
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E_{\rm kin}(\{\varphi_\alpha\})
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+
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E_{\rm C}(\rho)
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+
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E_{\rm xc}^{\rm (LDA)}(\rho_\uparrow,\rho_\downarrow)
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}
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+
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E_{\rm el,ion}({\color{red} \rho},{\color{dgreen} \{{R}_I\}})
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+
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{\color{dgreen} E_{\rm ion}(\{{R}_I,{P}_I\})}
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+
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E_{\rm ext}({\color{red} \rho},{\color{dgreen} {R}_I},t)
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\quad.
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\end{eqnarray*}
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-->
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<TABLE CELLPADDING="0" ALIGN="CENTER" WIDTH="100%">
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<TR VALIGN="MIDDLE"><TD NOWRAP ALIGN="RIGHT"><IMG
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WIDTH="47" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
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SRC="img3.png"
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ALT="$\displaystyle E_{\rm total}$"></TD>
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<TD WIDTH="10" ALIGN="CENTER" NOWRAP><IMG
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WIDTH="19" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
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SRC="img4.png"
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ALT="$\displaystyle =$"></TD>
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<TD ALIGN="LEFT" NOWRAP><IMG
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WIDTH="690" HEIGHT="43" ALIGN="MIDDLE" BORDER="0"
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SRC="img5.png"
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ALT="$\displaystyle {\color{red}
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E_{\rm kin}(\{\varphi_\alpha\})
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+
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E_{\rm C}(\rho)
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+
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...
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...}_I,{P}_I\})}
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+
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E_{\rm ext}({\color{red} \rho},{\color{dgreen} {R}_I},t)
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\quad.$"></TD>
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<TD WIDTH=10 ALIGN="RIGHT">
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</TD></TR>
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</TABLE></DIV>
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<BR CLEAR="ALL">
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<P>
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The electronic kinetic energy
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<FONT COLOR="#00b300"><!-- MATH
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${\color{red} E_{\rm kin}}$
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-->
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<IMG
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WIDTH="37" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
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SRC="img6.png"
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ALT="\bgroup\color{dgreen}$ {\color{red} E_{\rm kin}}$\egroup"></FONT> employs the
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single-electron wavefunctions
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<FONT COLOR="#00b300"><!-- MATH
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${\color{red} \varphi_\alpha}$
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-->
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<IMG
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WIDTH="26" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
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SRC="img7.png"
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ALT="\bgroup\color{dgreen}$ {\color{red} \varphi_\alpha}$\egroup"></FONT> which maintains
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the quantum mechanical shell effects. All other electronic energies
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refer only to the local spin-densities or total density
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<!-- MATH
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${\color{red} \rho=\rho_\uparrow+\rho_\downarrow}$
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-->
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<IMG
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WIDTH="92" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
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SRC="img8.png"
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ALT="\bgroup\color{dgreen}$ {\color{red} \rho=\rho_\uparrow+\rho_\downarrow}$\egroup">; the Coulomb energy
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<FONT COLOR="#00b300"><!-- MATH
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${\color{red} E_{\rm C}}$
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-->
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<IMG
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WIDTH="29" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
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SRC="img9.png"
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ALT="\bgroup\color{dgreen}$ {\color{red} E_{\rm C}}$\egroup"></FONT> naturally, and the exchange-correlation energy
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<FONT COLOR="#00b300"><!-- MATH
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${\color{red} E_{\rm xc}}$
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-->
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<IMG
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WIDTH="32" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
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SRC="img10.png"
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ALT="\bgroup\color{dgreen}$ {\color{red} E_{\rm xc}}$\egroup"></FONT> by virtue of the LDA (often augmented by a
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self-interaction correction (SIC) <a href="../literatur.html#own1252">[277]</a>). The electron-ion coupling
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<FONT COLOR="#00b300"><!-- MATH
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$E_{\rm el,ion}$
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-->
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<IMG
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WIDTH="51" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
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SRC="img11.png"
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ALT="\bgroup\color{dgreen}$ E_{\rm el,ion}$\egroup"></FONT> is realized by pseudo-potentials, mostly soft local
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ones <a href="../literatur.html#own1216">[249]</a>. The ionic part
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<FONT COLOR="#00b300"><!-- MATH
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${\color{dgreen} E_{\rm ion}}$
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-->
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<IMG
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WIDTH="37" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
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SRC="img12.png"
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ALT="\bgroup\color{dgreen}$ {\color{dgreen} E_{\rm ion}}$\egroup"></FONT> is composed of Coulomb
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interaction and kinetic energy. Excitation mechanisms (laser, ionic
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collisions) are described in
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<FONT COLOR="#00b300"><!-- MATH
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$E_{\rm ext}$
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-->
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<IMG
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WIDTH="37" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
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SRC="img13.png"
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ALT="\bgroup\color{dgreen}$ E_{\rm ext}$\egroup"></FONT> as external time-dependent
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potentials.
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<P>
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The coupled equations of motion are obtained in standard manner by
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variation. They read
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<!-- MATH
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\begin{displaymath}
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{\color{red}
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\imath\partial_t\varphi_\alpha
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=
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\Big(\frac{\hat{p}^2}{2m}
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+
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\frac{\delta E_{\rm total}}{\delta\rho_{\sigma_\alpha}}\Big)
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\varphi_\alpha
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}
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\qquad,\qquad
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{\color{dgreen} \partial_t{R}_I
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=
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\frac{{P}_I}{M_I}
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\quad,\quad
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\partial_t{P}_I
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=
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-\nabla_{{R}_I}E_{\rm total}}
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\quad.
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\end{displaymath}
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-->
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<P></P>
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<DIV ALIGN="CENTER">
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<IMG
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WIDTH="618" HEIGHT="65" ALIGN="MIDDLE" BORDER="0"
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SRC="img14.png"
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ALT="\bgroup\color{dgreen}$\displaystyle {\color{red}
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\imath\partial_t\varphi_\alpha...
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...M_I}
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\quad,\quad
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\partial_t{P}_I
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=
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-\nabla_{{R}_I}E_{\rm total}}
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\quad.
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$\egroup">
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</DIV><P>
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where
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<FONT COLOR="#00b300"><!-- MATH
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${\color{red} \sigma_\alpha}$
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-->
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<IMG
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WIDTH="24" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
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SRC="img15.png"
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ALT="\bgroup\color{dgreen}$ {\color{red} \sigma_\alpha}$\egroup"></FONT> is the spin orientation of the state
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<FONT COLOR="#00b300"><!-- MATH
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${\color{red} \alpha}$
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-->
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<IMG
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WIDTH="16" HEIGHT="19" ALIGN="BOTTOM" BORDER="0"
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SRC="img16.png"
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ALT="\bgroup\color{dgreen}$ {\color{red} \alpha}$\egroup"></FONT>. The equations imply a non-adiabatic coupling which
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goes beyond usual Born-Oppenheimer approach. Non-adiabatic effects
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become crucial in cluster dynamics induced by strong fields. The
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numerical solution involves the representation of the wavefunctions on
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a spatial grid, time-splitting for the electronic propagation and the
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Verlet algorithm for MD, for details see [<a href="../literatur.html#own1230">254</a>]. The obtained
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wavefunctions, densities, and ionic coordinates allow to compute a
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wide variety of observables, <!-- at the side of the electrons -->e.g.
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<FONT COLOR="#ff0000"> optical absorption spectra</FONT> [<a href="../literatur.html#own1155">9</a>],
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<FONT COLOR="#ff0000"> angular distributions</FONT>
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[<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
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<FONT COLOR="#ff0000"> ionization</FONT> [<a href="../literatur.html#own1186">208</a>] for electronic degrees of freedom.
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The
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<FONT COLOR="#00b300"> ionic configurations</FONT> can be measured indirectly through optical
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response and its dynamics with various pump and probe scenarios
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[<a href="../literatur.html#own1246">290</a>].
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<P></P>
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<P>
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Often, we use a
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<FONT COLOR="#ff0000"> semi-classical description for the electronic
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dynamics</FONT> at the level of Vlasov-LDA, particularly for energetic
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processes and/or large clusters. Instead of the
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<FONT COLOR="#ff0000"> wavefunctions</FONT>,
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the key ingredient becomes here the
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<FONT COLOR="#ff0000"> one-electron phase-space
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distribution
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<FONT COLOR="#ff0000"><!-- MATH
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$f({r},{p},t)$
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-->
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<IMG
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WIDTH="71" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
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SRC="img17.png"
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ALT="\bgroup\color{red}$ f({r},{p},t)$\egroup"></FONT></FONT>. The quantum-mechanical
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propagation for the electrons is replaced by the Vlasov equation
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<!-- MATH
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\begin{displaymath}
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{\color{red}
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\partial_t f
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=
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\frac{{p}}{m}\nabla_{r}f
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-
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\Big(
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\nabla_{r}\frac{\delta E_{\rm total}}{\delta\rho_{\sigma_\alpha}}
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\Big)
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\nabla_{p}f
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}
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\end{displaymath}
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-->
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<P></P>
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<DIV ALIGN="CENTER">
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<IMG
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WIDTH="270" HEIGHT="61" ALIGN="MIDDLE" BORDER="0"
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SRC="img18.png"
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ALT="\bgroup\color{red}$\displaystyle {\color{red}
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\partial_t f
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=
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\frac{{p}}{m}\nabl...
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...{\delta E_{\rm total}}{\delta\rho_{\sigma_\alpha}}
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\Big)
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\nabla_{p}f
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}
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$\egroup">
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</DIV><P></P>
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<P>
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again non-adiabatically coupled to ionic motion as above.
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Note that formally the same Kohn-Sham potential
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<FONT COLOR="#ff0000"><!-- MATH
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${\color{red} {\delta E_{\rm total}}\big/{\delta\rho_{\sigma_\alpha}}}$
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-->
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<IMG
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WIDTH="102" HEIGHT="41" ALIGN="MIDDLE" BORDER="0"
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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}}}$\egroup"></FONT>
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is employed. For a derivation and justification from TDLDA see
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[<a href="../literatur.html#own1163">182</a>]. The Vlasov-LDA equation is solved with the
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test-particle method where the distribution function
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<FONT COLOR="#ff0000"><!-- MATH
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${\color{red} f}$
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-->
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<IMG
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WIDTH="16" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
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SRC="img20.png"
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ALT="\bgroup\color{red}$ {\color{red} f}$\egroup"></FONT> is
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represented as a sum of Gaussian test-particles which are propagated
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again by the Verlet algorithm [<a href="../literatur.html#own1248">273</a>].
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<P>
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The semi-classical description makes it feasible to include dynamical
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correlations from electron-electron collisions. This is achieved by
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adding an Ühling-Uhlenbeck collision term leading to
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<!-- MATH
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\begin{displaymath}
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{\color{red}
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\partial_t f
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=
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\frac{{p}}{m}\nabla_{r}f
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-
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\Big(
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\nabla_{r}\frac{\delta E_{\rm total}}{\delta\rho_{\sigma_\alpha}}
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\Big)
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\nabla_{p}f
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+
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I_{\rm UU}(f)
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}
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\quad.
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\end{displaymath}
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-->
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<P></P>
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<DIV ALIGN="CENTER">
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<IMG
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WIDTH="372" HEIGHT="61" ALIGN="MIDDLE" BORDER="0"
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SRC="img21.png"
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ALT="\bgroup\color{red}$\displaystyle {\color{red}
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\partial_t f
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=
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\frac{{p}}{m}\nabl...
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...\delta\rho_{\sigma_\alpha}}
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\Big)
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\nabla_{p}f
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+
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I_{\rm UU}(f)
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}
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\quad.
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$\egroup">
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</DIV><P>
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The collision term
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<FONT COLOR="#ff0000"><!-- MATH
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${\color{red} I_{\rm UU}}$
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-->
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<IMG
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WIDTH="34" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
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SRC="img22.png"
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ALT="\bgroup\color{red}$ {\color{red} I_{\rm UU}}$\egroup"></FONT> is a non-linear functional of
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the distribution function
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<FONT COLOR="#ff0000"><!-- MATH
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${\color{red} f}$
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-->
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<IMG
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WIDTH="16" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
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SRC="img20.png"
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ALT="\bgroup\color{red}$ {\color{red} f}$\egroup"></FONT>. It contains terms up to third
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power in
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<FONT COLOR="#ff0000"><!-- MATH
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${\color{red} f}$
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-->
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<IMG
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WIDTH="16" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
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SRC="img20.png"
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ALT="\bgroup\color{red}$ {\color{red} f}$\egroup"></FONT>. It is constructed from local and instantaneous
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collisions which obey energy conservation, momentum conservation, and
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the Pauli principle [<a href="../literatur.html#own1248">273</a>]. The resulting equation is called the
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Vlasov-Ühling-Uhlenbeck approach (VUU).
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<P>
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<center>
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<table width="70%">
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<tr>
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<td align="right">
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<a href="#top">Back to top </a>
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<!-- <HR>
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<ADDRESS>
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Paul-Gerhard Reinhard
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2006-03-18
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</ADDRESS> -->
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