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<TITLE>8. Effective Core Potentials</TITLE>
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<B> Next:</B> <A NAME="tex2html1123"
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HREF="node11.html">9. Relativistic All-electron Approximations</A>
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<B> Previous:</B> <A NAME="tex2html1113"
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HREF="node9.html">7. Basis sets</A>
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HREF="node2.html">Contents</A></B>
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<A NAME="CHILD_LINKS"><STRONG>Subsections</STRONG></A>
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<UL>
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<LI><A NAME="tex2html1124"
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HREF="node10.html#SECTION001010000000000000000">8.1 Scalar ECPs</A>
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<LI><A NAME="tex2html1125"
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HREF="node10.html#SECTION001020000000000000000">8.2 Spin-orbit ECPs</A>
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</UL>
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<!--End of Table of Child-Links-->
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<HR>
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<H1><A NAME="SECTION001000000000000000000">
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8. Effective Core Potentials</A>
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</H1>
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<A NAME="sec:ecp"></A>
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Effective core potentials (ECPs) are a useful means of replacing the core
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electrons in a calculation with an effective potential, thereby eliminating
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the need for the core basis functions, which usually require a large set of
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Gaussians to describe them. In addition to replacing the core, they may be
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used to represent relativistic effects, which are largely confined to the
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core. In this context, both the scalar (spin-free) relativistic effects and
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spin-orbit (spin-dependent) relativistic effects may be included in
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effective potentials. NWChem has the facility to use both, and these are
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described in the next two sections.
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<P>
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A brief recapitulation of the development of RECPs is given here, following
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Pacios and Christiansen<A NAME="tex2html15"
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HREF="footnode.html#foot2077"><SUP>8.1</SUP></A>. The process can be viewed as starting
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from an atomic Dirac-Hartree-Fock calculation, done in <I>jj</I> coupling,
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and producing relativistic effective potentials (REPs) for each <IMG
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WIDTH="9" HEIGHT="15" ALIGN="BOTTOM" BORDER="0"
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SRC="img65.gif"
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ALT="$l$"> and
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<IMG
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WIDTH="12" HEIGHT="28" ALIGN="MIDDLE" BORDER="0"
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SRC="img66.gif"
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ALT="$j$"> value, <!-- MATH
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$U^{\rm REP}_{lj}$
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-->
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<IMG
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WIDTH="44" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
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SRC="img67.gif"
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ALT="$U^{\rm REP}_{lj}$">. From these, a local potential is
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extracted, which for example contains the Coulomb potential of the core
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electrons balanced by the part of the nuclear attraction which cancels the
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core electron charge. The residue is expressed in a semi-local form,
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<BR>
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<DIV ALIGN="RIGHT">
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<!-- MATH
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\begin{equation}
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U^{\rm REP} = U^{\rm REP}_{LJ}(r) + \sum_{l=0}^{L-1}
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\sum_{j=|l-1/2}^{l+1/2} \left[ U^{\rm REP}_{lj}(r) -
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U^{\rm REP}_{LJ}(r) \right] \sum_m | lj m \rangle \langle lj m |
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\end{equation}
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-->
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<TABLE WIDTH="100%" ALIGN="CENTER">
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<TR VALIGN="MIDDLE"><TD NOWRAP><IMG
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WIDTH="476" HEIGHT="102" BORDER="0"
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SRC="img68.gif"
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ALT="\begin{displaymath}
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U^{\rm REP} = U^{\rm REP}_{LJ}(r) + \sum_{l=0}^{L-1}
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\sum_{j...
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...}_{LJ}(r) \right] \sum_m \vert lj m \rangle \langle lj m \vert
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\end{displaymath}"></TD>
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<TD WIDTH=10 ALIGN="RIGHT">
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(8.1)</TD></TR>
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</TABLE>
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<BR CLEAR="ALL"></DIV><P></P>
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where <IMG
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WIDTH="15" HEIGHT="14" ALIGN="BOTTOM" BORDER="0"
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SRC="img63.gif"
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ALT="$L$"> is one larger than the maximum angular momentum in the atom.
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The scalar potential is obtained by averaging the REPs for each <IMG
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WIDTH="12" HEIGHT="28" ALIGN="MIDDLE" BORDER="0"
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SRC="img66.gif"
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ALT="$j$"> for a
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given <IMG
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WIDTH="9" HEIGHT="15" ALIGN="BOTTOM" BORDER="0"
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SRC="img65.gif"
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ALT="$l$"> to give an averaged relativistic effective potential, or AREP,
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<BR>
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<DIV ALIGN="RIGHT">
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<!-- MATH
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\begin{equation}
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U^{\rm AREP}_l(r) = \frac{1}{2l+1} \left[ lU^{\rm REP}_{l-1/2}(r)
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+ (l+1) U^{\rm REP}_{l+1/2}(r) \right].
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\end{equation}
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-->
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<TABLE WIDTH="100%" ALIGN="CENTER">
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<TR VALIGN="MIDDLE"><TD NOWRAP><IMG
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WIDTH="357" HEIGHT="80" BORDER="0"
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SRC="img69.gif"
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ALT="\begin{displaymath}
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U^{\rm AREP}_l(r) = \frac{1}{2l+1} \left[ lU^{\rm REP}_{l-1/2}(r)
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+ (l+1) U^{\rm REP}_{l+1/2}(r) \right].
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\end{displaymath}"></TD>
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<TD WIDTH=10 ALIGN="RIGHT">
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(8.2)</TD></TR>
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</TABLE>
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<BR CLEAR="ALL"></DIV><P></P>
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These are summed into the full potential.
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<P>
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The spin-orbit potential is obtained from the difference between the REPs
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for the two <IMG
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WIDTH="12" HEIGHT="28" ALIGN="MIDDLE" BORDER="0"
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SRC="img66.gif"
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ALT="$j$"> values for a given l, and may be
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represented in terms of an effective spin-orbit operator,
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<BR>
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<DIV ALIGN="RIGHT">
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<!-- MATH
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\begin{equation}
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H^{\rm SO} = {\bf s} \cdot \sum_{l=1}^{L-1} \frac{2}{2l+1}
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\Delta U^{\rm REP}_{l} \sum_{mm'}
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| lm \rangle \langle lm | \hat l| lm' \rangle \langle lm' |.
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\end{equation}
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-->
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<TABLE WIDTH="100%" ALIGN="CENTER">
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<TR VALIGN="MIDDLE"><TD NOWRAP><IMG
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WIDTH="366" HEIGHT="96" BORDER="0"
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SRC="img70.gif"
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ALT="\begin{displaymath}
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H^{\rm SO} = {\bf s} \cdot \sum_{l=1}^{L-1} \frac{2}{2l+1}
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...
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...le \langle lm \vert \hat l\vert lm' \rangle \langle lm' \vert.
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\end{displaymath}"></TD>
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<TD WIDTH=10 ALIGN="RIGHT">
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(8.3)</TD></TR>
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</TABLE>
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<BR CLEAR="ALL"></DIV><P></P>
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where
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<BR>
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<DIV ALIGN="RIGHT">
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<!-- MATH
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\begin{equation}
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\Delta U^{\rm REP}_{l} = U^{\rm REP}_{l+1/2}(r)
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- U^{\rm REP}_{l-1/2}(r).
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\end{equation}
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-->
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<TABLE WIDTH="100%" ALIGN="CENTER">
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<TR VALIGN="MIDDLE"><TD NOWRAP><IMG
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WIDTH="227" HEIGHT="68" BORDER="0"
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SRC="img71.gif"
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ALT="\begin{displaymath}
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\Delta U^{\rm REP}_{l} = U^{\rm REP}_{l+1/2}(r)
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- U^{\rm REP}_{l-1/2}(r).
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\end{displaymath}"></TD>
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<TD WIDTH=10 ALIGN="RIGHT">
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(8.4)</TD></TR>
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</TABLE>
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<BR CLEAR="ALL"></DIV><P></P>
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The spin-orbit integrals generated by NWChem are the integrals over the sum,
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including the factor of <IMG
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WIDTH="73" HEIGHT="31" ALIGN="MIDDLE" BORDER="0"
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SRC="img72.gif"
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ALT="$2/(2l+1)$">, so that they may be treated as an
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effective spin-orbit operator without further factors introduced.
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<P>
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The effective potentials, both scalar and spin-orbit, are fitted to
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Gaussians with the form
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<BR><P></P>
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<DIV>
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<!-- MATH
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\begin{displaymath}
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r^2U_l(r) = \sum_{k} A_{lk} r^{n_{lk}} e^{-B_{lk}r^{2}}
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\end{displaymath}
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-->
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<IMG
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WIDTH="198" HEIGHT="84" BORDER="0"
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SRC="img73.gif"
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ALT="\begin{displaymath}
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r^2U_l(r) = \sum_{k} A_{lk} r^{n_{lk}} e^{-B_{lk}r^{2}}
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\end{displaymath}">
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</DIV>
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<BR CLEAR="ALL">
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<P></P>
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where <IMG
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WIDTH="28" HEIGHT="29" ALIGN="MIDDLE" BORDER="0"
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SRC="img74.gif"
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ALT="$A_{lk}$"> is the contraction coefficient, <IMG
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WIDTH="26" HEIGHT="28" ALIGN="MIDDLE" BORDER="0"
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SRC="img75.gif"
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ALT="$n_{lk}$"> is the
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exponent of the ``r'' term (r-exponent), and <IMG
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WIDTH="28" HEIGHT="29" ALIGN="MIDDLE" BORDER="0"
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SRC="img76.gif"
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ALT="$B_{lk}$"> is the Gaussian
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exponent. The <IMG
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WIDTH="26" HEIGHT="28" ALIGN="MIDDLE" BORDER="0"
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SRC="img75.gif"
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ALT="$n_{lk}$"> is shifted by 2, in accordance with most of the ECP
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literature and implementations, i.e., an <IMG
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WIDTH="55" HEIGHT="28" ALIGN="MIDDLE" BORDER="0"
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SRC="img77.gif"
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ALT="$n_{lk} = 0$"> implies
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<IMG
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WIDTH="29" HEIGHT="17" ALIGN="BOTTOM" BORDER="0"
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SRC="img78.gif"
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ALT="$r^{-2}$">. The current implementation allows <IMG
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WIDTH="26" HEIGHT="28" ALIGN="MIDDLE" BORDER="0"
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SRC="img75.gif"
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ALT="$n_{lk}$"> values
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of only 0, 1, or 2.
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<P>
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<H1><A NAME="SECTION001010000000000000000"></A>
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<A NAME="sec:scalar_ecp"></A>
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<BR>
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8.1 Scalar ECPs
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</H1>
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<P>
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The optional directive <code>ECP</code> allows the user to describe an effective core
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potential (ECP) in terms of contracted Gaussian functions as given above.
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Potentials using these functions must be specified explicitly by user input
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in the <code>ECP</code> directive. This directive has essentially the same form
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and properties as the standard <code>BASIS</code> directive, except for essential
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differences required for ECPs. Because of this, the ECP is treated
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internally as a basis set. The form of the input for the
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<code>ECP</code> directive is as follows:
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<P>
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<PRE>
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ECP [<string name default "ecp basis">] \
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[print || noprint default print]
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<string tag> library [<string tag_in_lib>] \
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<string standard_set> [file <filename>] \
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[except <string tag list>]
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<string tag> [nelec] <integer number_of_electrons_replaced>
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...
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<string tag> <string shell_type>
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<real r-exponent> <real Gaussian-exponent> <real list_of_coefficients>
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...
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END
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</PRE>
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<P>
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ECPs are automatically segmented, even if general contractions are input.
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The projection operators defined in an ECP are spherical by default, so
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there is no need to include the <code>CARTESIAN</code> or <code>SPHERICAL</code> keyword
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as there is for a standard basis set. ECPs are associated with centers in
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geometries through tags or names of centers. These tags must match in the
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same manner as for basis sets the tags in a <code>GEOMETRY</code> and
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<code>ECP</code> directives, and are limited to sixteen (16) characters.
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Each center with the same tag will have the same ECP. By default, the
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input module prints each ECP that it encounters. The <code>NOPRINT</code>
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option can be used to disable printing. There can be only one active
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ECP, even though several may exist in the input deck. The ECP modules
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load ``ecp basis'' inputs along with any ``ao basis'' inputs present.
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ECPs may be used in both energy and gradient calculations.
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<P>
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ECPs are named in the same fashion as geometries or regular basis
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sets, with the default name being <code>"ecp basis"</code>. It should be
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clear from the above discussion on geometries and database entries how
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indirection is supported. All directives that are in common with the
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standard Gaussian basis set input have the same function and syntax.
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<P>
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As for regular basis sets, ECPs may be obtained from the standard library.
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The names of the sets of ECPs available in the standard
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library (their coverage is described in Appendix <A HREF="node41.html#sec:knownbasis">A</A>) are
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<UL>
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<LI><code>"Hay-Wadt MB (n+1) ECP"</code>
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</LI>
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<LI><code>"Hay-Wadt VDZ (n+1) ECP"</code>
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</LI>
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<LI><code>"LANL2DZ ECP"</code>
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</LI>
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<LI><code>"SBKJC VDZ ECP"</code>
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</LI>
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<LI><code>"Stuttgart RLC ECP"</code>
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</LI>
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<LI><code>"Stuttgart RSC ECP"</code>
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</LI>
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<LI><code>"CRENBL ECP"</code>
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</LI>
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<LI><code>"CRENBS ECP"</code>
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</LI>
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</UL>
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<P>
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The keyword <code>nelec</code> allows the user to specify the number of core
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electrons replaced by the ECP. Additional input lines define the
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specific coefficients and exponents. The variable <code><shell_type></code>
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is used to specify the components of the ECP. The keyword <code>ul</code>
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entered for <code><shell_type></code> denotes the local part of the ECP.
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This is equivalent to the highest angular momentum functions specified
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in the literature for most ECPs. The standard entries (<code>s, p, d</code>,
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etc.) for <code>shell_type</code> specify the angular momentum projector
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onto the local function. The shell type label of <code>s</code> indicates
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the <code>ul-s</code> projector input, <code>p</code> indicates the <code>ul-p</code>,
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etc.
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<P>
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For example, the Christiansen, Ross and Ermler ARECPs are available in
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the standard basis set libary named <code>{crenbl_ecp}</code>. To perform a
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calculation on uranyl (UO<IMG
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WIDTH="21" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
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SRC="img79.gif"
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ALT="$_2^{2+}$">) with all-electron oxygen
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(aug-cc-pvdz basis), and uranium with an ARECP and using the
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corresponding basis the following input can be used
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<PRE>
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geometry
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U 0 0 0
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O 0 0 1.65
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O 0 0 -1.65
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end
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basis
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U library crenbl_ecp
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O library aug-cc-pvdz
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end
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ecp
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U library crenbl_ecp
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end
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</PRE>
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<P>
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The following is an example of explicit input of an ECP for H<IMG
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WIDTH="11" HEIGHT="28" ALIGN="MIDDLE" BORDER="0"
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SRC="img5.gif"
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ALT="$_2$">CO.
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It defines an ECP for the carbon and oxygen atoms in the molecule.
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<P>
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<PRE>
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ecp
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C nelec 2 # ecp replaces 2 electrons on C
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C ul # d
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1 80.0000000 -1.60000000
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1 30.0000000 -0.40000000
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2 0.5498205 -0.03990210
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C s # s - d
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0 0.7374760 0.63810832
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0 135.2354832 11.00916230
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2 8.5605569 20.13797020
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C p # p - d
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2 10.6863587 -3.24684280
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2 23.4979897 0.78505765
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O nelec 2 # ecp replaces 2 electrons on O
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O ul # d
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1 80.0000000 -1.60000000
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1 30.0000000 -0.40000000
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2 1.0953760 -0.06623814
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O s # s - d
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0 0.9212952 0.39552179
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0 28.6481971 2.51654843
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2 9.3033500 17.04478500
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O p # p - s
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2 52.3427019 27.97790770
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2 30.7220233 -16.49630500
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end
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</PRE>
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<P>
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|
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<H1><A NAME="SECTION001020000000000000000"></A>
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|
<A NAME="sec:spinorb_ecp"></A>
|
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<BR>
|
|
8.2 Spin-orbit ECPs
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</H1>
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<P>
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The Spin-orbit ECPs can be used with the Density Functional Approach, but
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one has to run the calculations without symmetry. Note: when a Hartree-Fock
|
|
method is specified the spin-orbit input will be ignored.
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|
|
|
<P>
|
|
Spin-orbit ECPs are fitted in precisely the same functional form as the
|
|
scalar RECPs and have the same properties, with the exception that there is
|
|
no local potential ul, no <IMG
|
|
WIDTH="12" HEIGHT="14" ALIGN="BOTTOM" BORDER="0"
|
|
SRC="img61.gif"
|
|
ALT="$s$"> potential and no effective charge has to be
|
|
defined. Spin-orbit potentials are
|
|
specified in the same way as ECPs except that the directive <code>SO</code> is
|
|
used instead of <code>ECP</code>. Note that there currently are no spin-orbit
|
|
ECPs defined in the standard NWChem library. The <code>SO</code>
|
|
directive is as follows:
|
|
|
|
<P>
|
|
<PRE>
|
|
SO [<string name default "so basis">] \
|
|
[print || noprint default print]
|
|
|
|
<string tag> library [<string tag_in_lib>] \
|
|
<string standard_set> [file <filename>]
|
|
[except <string tag list>]
|
|
...
|
|
|
|
<string tag> <string shell_type>
|
|
<real r-exponent> <real Gaussian-exponent> <real list_of_coefficients>
|
|
...
|
|
|
|
END
|
|
</PRE>
|
|
|
|
<P>
|
|
Note: in the literature the coefficients of the spin-orbit potentials are NOT
|
|
always defined in the same manner. The NWChem code assumes that the spin-orbit
|
|
potential defined in the input is of the form:
|
|
<BR>
|
|
<DIV ALIGN="RIGHT">
|
|
|
|
<!-- MATH
|
|
\begin{equation}
|
|
\Delta U^{\rm NWChem}_{l} = \frac{2}{2l+1} \Delta U_{l}
|
|
\end{equation}
|
|
-->
|
|
<TABLE WIDTH="100%" ALIGN="CENTER">
|
|
<TR VALIGN="MIDDLE"><TD NOWRAP><IMG
|
|
WIDTH="175" HEIGHT="80" BORDER="0"
|
|
SRC="img80.gif"
|
|
ALT="\begin{displaymath}
|
|
\Delta U^{\rm NWChem}_{l} = \frac{2}{2l+1} \Delta U_{l}
|
|
\end{displaymath}"></TD>
|
|
<TD WIDTH=10 ALIGN="RIGHT">
|
|
(8.5)</TD></TR>
|
|
</TABLE>
|
|
<BR CLEAR="ALL"></DIV><P></P>
|
|
For example, in the literature the Stuttgart potentials are defined as
|
|
<IMG
|
|
WIDTH="33" HEIGHT="29" ALIGN="MIDDLE" BORDER="0"
|
|
SRC="img81.gif"
|
|
ALT="$\Delta U_{l}$"> and, hence, have to be multiplied by <IMG
|
|
WIDTH="73" HEIGHT="31" ALIGN="MIDDLE" BORDER="0"
|
|
SRC="img82.gif"
|
|
ALT="$2/(2{l}+1)$">. On the
|
|
other hand, the CRENBL potentials in the published papers are defined as
|
|
<!-- MATH
|
|
$\frac{l}{2l+1} \Delta U_{l}$
|
|
-->
|
|
<IMG
|
|
WIDTH="63" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
|
|
SRC="img83.gif"
|
|
ALT="$\frac{l}{2l+1} \Delta U_{l}$"> and, hence, have to be multiplied by
|
|
<IMG
|
|
WIDTH="25" HEIGHT="31" ALIGN="MIDDLE" BORDER="0"
|
|
SRC="img84.gif"
|
|
ALT="$2/{l}$"> (Warning: on the CRENBL website the spin-orbit potentials already
|
|
have been corrected with the <IMG
|
|
WIDTH="25" HEIGHT="31" ALIGN="MIDDLE" BORDER="0"
|
|
SRC="img84.gif"
|
|
ALT="$2/{l}$"> factor).
|
|
|
|
<P>
|
|
|
|
<P>
|
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<ADDRESS>
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Edoardo Apra
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2004-05-25
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