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<H2><A NAME="SECTION00053000000000000000">
4.3 Format of arrays containing charge density, potential, etc.</A>
</H2>

<P>
The index of arrays used to store functions defined on 3D meshes is
actually a shorthand for three indices, following the FORTRAN convention 
("leftmost index runs faster"). An example will explain this better. 
Suppose you have a 3D array <TT>psi(nr1x,nr2x,nr3x)</TT>. FORTRAN 
compilers store this array sequentially  in the computer RAM in the following way:
<PRE>
        psi(   1,   1,   1)
        psi(   2,   1,   1)
        ...
        psi(nr1x,   1,   1)
        psi(   1,   2,   1)
        psi(   2,   2,   1)
        ...
        psi(nr1x,   2,   1)
        ...
        ...
        psi(nr1x,nr2x,   1)
        ...
        psi(nr1x,nr2x,nr3x)
etc
</PRE>
Let <TT>ind</TT> be the position of the <TT>(i,j,k)</TT> element in the above list: 
the following relation
<PRE>
        ind = i + (j - 1) * nr1x + (k - 1) *  nr2x * nr1x
</PRE>
holds. This should clarify the relation between 1D and 3D indexing. In real
space, the <TT>(i,j,k)</TT> point of the FFT grid with dimensions 
<TT>nr1</TT> (<IMG
 WIDTH="19" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="img9.png"
 ALT="$ \le$"><TT>nr1x</TT>), 
<TT>nr2</TT>  (<IMG
 WIDTH="19" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="img9.png"
 ALT="$ \le$"><TT>nr2x</TT>), , <TT>nr3</TT> (<IMG
 WIDTH="19" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="img9.png"
 ALT="$ \le$"><TT>nr3x</TT>), is
<P><!-- MATH
 \begin{displaymath}
r_{ijk}=\frac{i-1}{nr1} \tau_1  +  \frac{j-1}{nr2} \tau_2 +
\frac{k-1}{nr3} \tau_3
\end{displaymath}
 -->
</P>
<DIV ALIGN="CENTER">
<I>r</I><SUB>ijk</SUB> = <IMG
 WIDTH="48" HEIGHT="61" ALIGN="MIDDLE" BORDER="0"
 SRC="img10.png"
 ALT="$\displaystyle {\frac{{i-1}}{{nr1}}}$"><IMG
 WIDTH="20" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="img11.png"
 ALT="$\displaystyle \tau_{1}^{}$"> + <IMG
 WIDTH="50" HEIGHT="61" ALIGN="MIDDLE" BORDER="0"
 SRC="img12.png"
 ALT="$\displaystyle {\frac{{j-1}}{{nr2}}}$"><IMG
 WIDTH="20" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="img13.png"
 ALT="$\displaystyle \tau_{2}^{}$"> + <IMG
 WIDTH="52" HEIGHT="61" ALIGN="MIDDLE" BORDER="0"
 SRC="img14.png"
 ALT="$\displaystyle {\frac{{k-1}}{{nr3}}}$"><IMG
 WIDTH="20" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="img15.png"
 ALT="$\displaystyle \tau_{3}^{}$">
</DIV><P></P>
where the <IMG
 WIDTH="18" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="img16.png"
 ALT="$ \tau_{i}^{}$"> are the basis vectors of the Bravais lattice. 
The latter are stored row-wise in the <TT>at</TT> array:
<IMG
 WIDTH="20" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="img17.png"
 ALT="$ \tau_{1}^{}$"> = <TT>at(:, 1)</TT>, 
<IMG
 WIDTH="20" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="img18.png"
 ALT="$ \tau_{2}^{}$"> = <TT>at(:, 2)</TT>, 
<IMG
 WIDTH="20" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="img19.png"
 ALT="$ \tau_{3}^{}$"> = <TT>at(:, 3)</TT>.

<P>
The distinction between the dimensions of the FFT grid,
<TT>(nr1,nr2,nr3)</TT> and the physical dimensions of the array,
<TT>(nr1x,nr2x,nr3x)</TT> is done only because it is computationally
convenient in some cases that the two sets are not the same.
In particular, it is often convenient to have <TT>nrx1</TT>=<TT>nr1</TT>+1
to reduce memory conflicts.

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