<!DOCTYPE html PUBLIC "-//W3C//DTD XHTML 1.0 Transitional//EN" "http://www.w3.org/TR/xhtml1/DTD/xhtml1-transitional.dtd"><htmlxmlns="http://www.w3.org/1999/xhtml"><head><metahttp-equiv="content-type"content="application/xhtml+xml; charset=iso-8859-1"/><title>Theory of Cluster Dynamics</title><linkhref="../style.css"rel="stylesheet"type="text/css"/></head><body><divid="container"><divid="header"><divid="menu"><divid="navMenu"><ul><listyle="margin-top:1px;border-top:1px solid #B0C4DE; "><ahref="../index.html">Home</a></li><li><ahref="../intro.html">Introductory Overview</a></li><li><ahref="../research.html">Scientific Information</a></li><li><ahref="../staff.html">Staff</a></li><li><ahref="../publications.html">Publications/Talks</a></li><li><ahref="../contact.html">Contact</a></li></ul></div></div><divid="image"><p><fontsize="6"color="white"><b>Theory of Cluster Dynamics</b></font><fontsize="5"><br/></font><fontsize="6"></font><fontsize="5">The Toulouse -
Erlangen Collaboration</font></p></div><divid="content"><aname="oben"></a><divstyle="margin:15px;width:770px;border:1px solid gray;float:left;font-size:10px;"><aname="oben"></a><divstyle="width:180px;float:left;text-align:center;font-size:12px"><aname="oben"></a><ahref="formal.html">1. Theoretical developments </a></div><divstyle="width:200px;float:left;text-align:center;font-size:10px;"><ahref="../analysis/detail1.html"> 2. Analysis of cluster
dynamics </a></div><divstyle="width:200px;float:left;text-align:center;font-weight:900;font-size:10px;"><ahref="../analysis/detail2.html"> 3. Clusters in strong external
fields </a></div><divstyle="width:180px;float:left;text-align:center;font-weight:900;font-size:10px;"><ahref="detailQMMM.html"> 4. Embedded clusters </a></div></div><divid="WideContent"><divid="contentBoxWide"><divid="contentBoxHeader"><p>Time Dependent Density Functional Theory with Molecular
Dynamics </p></div><divid="contentBoxContent"><p></p><divalign="CENTER"><fontsize="+2"><b> TDLDA-MD:</b></font><br/><br/><fontsize="+1"><b>Time-dependent local-density approximation
plus ionic molecular dynamics</b></font><br/></div><p> (<em>This is a very short summary of our formal scheme. A
most detailed description is found in </em>[<ahref="../literatur.html#own1281">303</a>].)
</p><p> The <fontcolor="#ff0000"> electron cloud</font> is
described by density functional theory at the level of TDLDA.
The dynamical degrees of freedom are the set of occupied <fontcolor="#ff0000">
single-electron wavefunctions <fontcolor="#ff0000"><!-- MATH
$\varphi_\alpha$ --><imgsrc="img1.png"alt="\bgroup\color{red}$ \varphi_\alpha$\egroup"width="26"border="0"align="MIDDLE"height="33"/></font></font>.
The <fontcolor="#00b300"> ions</font> are treated by
classical MD and their degrees of freedom are the <fontcolor="#00b300">
positions <i><b>R<sub>I</sub></b></i> and momenta <i><b>P<sub>I</sub></b></i><fontcolor="#00b300"><!-- MATH
$({R}_I,{P}_I)$ --><!-- <IMG
WIDTH="69" HEIGHT="37" ALIGN="MIDDLE" BORDER="0" SRC="img2.png" ALT="\bgroup\color{dgreen}$ ({R}_I,{P}_I)$\egroup"></FONT></FONT>.--></font></font>.
The starting point is the total energy given by: <br/></p><divalign="CENTER"><!-- MATH
\begin{eqnarray*}E_{\rm total} &=& {\color{red} E_{\rm kin}(\{\varphi_\alpha\}) + E_{\rm C}(\rho) + E_{\rm xc}^{\rm (LDA)}(\rho_\uparrow,\rho_\downarrow) } + E_{\rm el,ion}({\color{red} \rho},{\color{dgreen} \{{R}_I\}}) + {\color{dgreen} E_{\rm ion}(\{{R}_I,{P}_I\})} + E_{\rm ext}({\color{red} \rho},{\color{dgreen} {R}_I},t) \quad.\end{eqnarray*} --><tablewidth="100%"cellpadding="0"align="CENTER"><tbody><trvalign="MIDDLE"><tdnowrap="nowrap"align="RIGHT"><imgsrc="img3.png"alt="$\displaystyle E_{\rm total}$"width="47"border="0"align="MIDDLE"height="35"/></td><tdnowrap="nowrap"width="10"align="CENTER"><imgsrc="img4.png"alt="$\displaystyle =$"width="19"border="0"align="MIDDLE"height="33"/></td><tdnowrap="nowrap"align="LEFT"><imgsrc="img5.png"alt="$\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.$"width="690"border="0"align="MIDDLE"height="43"/></td><tdwidth="10"align="RIGHT"></td></tr></tbody></table></div><brclear="ALL"/><p> The electronic kinetic energy <fontcolor="#00b300"><!-- MATH
${\color{red} E_{\rm kin}}$ --><imgsrc="img6.png"alt="\bgroup\color{dgreen}$ {\color{red} E_{\rm kin}}$\egroup"width="37"border="0"align="MIDDLE"height="35"/></font>
employs the single-electron wavefunctions <fontcolor="#00b300"><!-- MATH
${\color{red} \varphi_\alpha}$ --><imgsrc="img7.png"alt="\bgroup\color{dgreen}$ {\color{red} \varphi_\alpha}$\egroup"width="26"border="0"align="MIDDLE"height="33"/></font>
which maintains the quantum mechanical shell effects. All
other electronic energies refer only to the local
spin-densities or total density
<!-- MATH
${\color{red} \rho=\rho_\uparrow+\rho_\downarrow}$ --><imgsrc="img8.png"alt="\bgroup\color{dgreen}${\color{red}\rho=\rho_\uparrow+\rh
The electron-ion coupling <fontcolor="#00b300"><!-- MATH
$E_{\rm el,ion}$ --><imgsrc="img11.png"alt="\bgroup\color{dgreen}$ E_{\rm el,ion}$\egroup"width="51"border="0"align="MIDDLE"height="35"/></font>
is realized by pseudo-potentials, mostly soft local ones <ahref="../literatur.html#own1216">[249]</a>.
The ionic part <fontcolor="#00b300"><!-- MATH
${\color{dgreen} E_{\rm ion}}$ --><imgsrc="img12.png"alt="\bgroup\color{dgreen}$ {\color{dgreen} E_{\rm ion}}$\egroup"width="37"border="0"align="MIDDLE"height="35"/></font>
is composed of Coulomb interaction and kinetic energy.
Excitation mechanisms (laser, ionic collisions) are described
in <fontcolor="#00b300"><!-- MATH
$E_{\rm ext}$ --><imgsrc="img13.png"alt="\bgroup\color{dgreen}$ E_{\rm ext}$\egroup"width="37"border="0"align="MIDDLE"height="35"/></font>
as external time-dependent potentials. </p><p> The coupled equations of motion are obtained in standard
manner by variation. They read
<!-- MATH
\begin{displaymath}{\color{red} \imath\partial_t\varphi_\alpha = \Big(\frac{\hat{p}^2}{2m} + \frac{\delta E_{\rm total}}{\delta\rho_{\sigma_\alpha}}\Big) \varphi_\alpha } \qquad,\qquad {\color{dgreen} \partial_t{R}_I = \frac{{P}_I}{M_I} \quad,\quad \partial_t{P}_I = -\nabla_{{R}_I}E_{\rm total}} \quad.\end{displaymath} --></p><p></p><divalign="CENTER"><imgsrc="img14.png"alt="\bgroup\color{dgreen}$\displaystyle {\color{red} \imath\partial_t\varphi_\alpha......M_I} \quad,\quad\partial_t{P}_I =-\nabla_{{R}_I}E_{\rm total}} \quad.$\egroup"width="618"border="0"align="MIDDLE"height="65"/></div><p> where <fontcolor="#00b300"><!-- MATH
${\color{red} \sigma_\alpha}$ --><imgsrc="img15.png"alt="\bgroup\color{dgreen}$ {\color{red} \sigma_\alpha}$\egroup"width="24"border="0"align="MIDDLE"height="33"/></font>
is the spin orientation of the state <fontcolor="#00b300"><!-- MATH
${\color{red} \alpha}$ --><imgsrc="img16.png"alt="\bgroup\color{dgreen}$ {\color{red} \alpha}$\egroup"width="16"border="0"align="BOTTOM"height="19"/></font>.
The equations imply a non-adiabatic coupling which goes beyond
usual Born-Oppenheimer approach. Non-adiabatic effects become
crucial in cluster dynamics induced by strong fields. The
numerical solution involves the representation of the
wavefunctions on a spatial grid, time-splitting for the
electronic propagation and the Verlet algorithm for MD, for
details see [<ahref="../literatur.html#own1230">254</a>]. The
obtained wavefunctions, densities, and ionic coordinates allow
to compute a wide variety of observables,
<!-- at the side of the electrons -->e.g. <fontcolor="#ff0000">
optical absorption spectra</font> [<ahref="../literatur.html#own1155">9</a>],
or <fontcolor="#ff0000"> ionization</font> [<ahref="../literatur.html#own1186">208</a>]
for electronic degrees of freedom. The <fontcolor="#00b300">
ionic configurations</font> can be measured indirectly
through optical response and its dynamics with various pump
and probe scenarios [<ahref="../literatur.html#own1246">290</a>].
</p><p></p><p> Often, we use a <fontcolor="#ff0000"> semi-classical
description for the electronic dynamics</font> at the level
of Vlasov-LDA, particularly for energetic processes and/or
large clusters. Instead of the <fontcolor="#ff0000">
wavefunctions</font>, the key ingredient becomes here the <fontcolor="#ff0000">
one-electron phase-space distribution <fontcolor="#ff0000"><!-- MATH
$f({r},{p},t)$ --><imgsrc="img17.png"alt="\bgroup\color{red}$ f({r},{p},t)$\egroup"width="71"border="0"align="MIDDLE"height="37"/></font></font>.
The quantum-mechanical propagation for the electrons is
replaced by the Vlasov equation
<!-- MATH
\begin{displaymath}{\color{red} \partial_t f = \frac{{p}}{m}\nabla_{r}f - \Big( \nabla_{r}\frac{\delta E_{\rm total}}{\delta\rho_{\sigma_\alpha}} \Big) \nabla_{p}f }\end{displaymath} --></p><p></p><divalign="CENTER"><imgsrc="img18.png"alt="\bgroup\color{red}$\displaystyle {\color{red} \partial_t f= \frac{{p}}{m}\nabl......{\delta E_{\rm total}}{\delta\rho_{\sigma_\alpha}} \Big)\nabla_{p}f }$\egroup"width="270"border="0"align="MIDDLE"height="61"/></div><p></p><p> again non-adiabatically coupled to ionic motion as above.
Note that formally the same Kohn-Sham potential <fontcolor="#ff0000"><!-- MATH
${\color{red} {\delta E_{\rm total}}\big/{\delta\rho_{\sigma_\alpha}}}$ --><imgsrc="img19.png"alt="\bgroup\color{red}$ {\color{red} {\delta E_{\rm total}}\big/{\delta\rho_{\sigma_\alpha}}}$\egroup"width="102"border="0"align="MIDDLE"height="41"/></font>
is employed. For a derivation and justification from TDLDA see
[<ahref="../literatur.html#own1163">182</a>]. The Vlasov-LDA
equation is solved with the test-particle method where the
distribution function <fontcolor="#ff0000"><!-- MATH
${\color{red} f}$ --><imgsrc="img20.png"alt="\bgroup\color{red}$ {\color{red} f}$\egroup"width="16"border="0"align="MIDDLE"height="35"/></font>
is represented as a sum of Gaussian test-particles which are
propagated again by the Verlet algorithm [<ahref="../literatur.html#own1248">273</a>].
</p><p> The semi-classical description makes it feasible to include
dynamical correlations from electron-electron collisions. This
is achieved by adding an Uehling-Uhlenbeck collision term
leading to
<!-- MATH
\begin{displaymath}{\color{red} \partial_t f = \frac{{p}}{m}\nabla_{r}f - \Big( \nabla_{r}\frac{\delta E_{\rm total}}{\delta\rho_{\sigma_\alpha}} \Big) \nabla_{p}f + I_{\rm UU}(f) } \quad.\end{displaymath} --></p><p></p><divalign="CENTER"><imgsrc="img21.png"alt="\bgroup\color{red}$\displaystyle {\color{red} \partial_t f= \frac{{p}}{m}\nabl......\delta\rho_{\sigma_\alpha}} \Big)\nabla_{p}f +I_{\rm UU}(f) }\quad. $\egroup"width="372"border="0"align="MIDDLE"height="61"/></div><p> The collision term <fontcolor="#ff0000"><!-- MATH
${\color{red} I_{\rm UU}}$ --><imgsrc="img22.png"alt="\bgroup\color{red}$ {\color{red} I_{\rm UU}}$\egroup"width="34"border="0"align="MIDDLE"height="35"/></font>
is a non-linear functional of the distribution function <fontcolor="#ff0000"><!-- MATH
${\color{red} f}$ --><imgsrc="img20.png"alt="\bgroup\color{red}$ {\color{red} f}$\egroup"width="16"border="0"align="MIDDLE"height="35"/></font>.
It contains terms up to third power in <fontcolor="#ff0000"><!-- MATH
${\color{red} f}$ --><imgsrc="img20.png"alt="\bgroup\color{red}$ {\color{red} f}$\egroup"width="16"border="0"align="MIDDLE"height="35"/></font>.
It is constructed from local and instantaneous collisions
which obey energy conservation, momentum conservation, and the
Pauli principle [<ahref="../literatur.html#own1248">273</a>].
The resulting equation is called the Vlasov-Uehling-Uhlenbeck
approach (VUU). </p><p></p><center><tablewidth="70%"><tbody><tr><tdalign="right"><ahref="#top">Back to top </a></td></tr></tbody></table></center><!-- <HR> --><!--Table of Child-Links--><aname="CHILD_LINKS"></a><!--End of Table of Child-Links--><!-- <HR>
<ADDRESS>Paul-Gerhard Reinhard2006-03-18</ADDRESS> --><!-- START CONTENT HERE --></div></div></div></div><divid="footer"><p></p></div></div></div></body></html>
