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<H2><A ID="SECTION00042000000000000000">
3.2 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> (≤<TT>nr1x</TT>),
<TT>nr2</TT> (≤<TT>nr2x</TT>), , <TT>nr3</TT> (≤<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 STYLE="height: 4.78ex; vertical-align: -1.69ex; " SRC="img2.png"
ALT="$\displaystyle {\frac{{i-1}}{{nr1}}}$"><I>τ</I><SUB>1</SUB> + <IMG STYLE="height: 4.78ex; vertical-align: -1.69ex; " SRC="img3.png"
ALT="$\displaystyle {\frac{{j-1}}{{nr2}}}$"><I>τ</I><SUB>2</SUB> + <IMG STYLE="height: 4.84ex; vertical-align: -1.69ex; " SRC="img4.png"
ALT="$\displaystyle {\frac{{k-1}}{{nr3}}}$"><I>τ</I><SUB>3</SUB>
</DIV><P></P>
where the <I>τ</I><SUB>i</SUB> are the basis vectors of the Bravais lattice.
The latter are stored row-wise in the <TT>at</TT> array:
<I>τ</I><SUB>1</SUB> = <TT>at(:, 1)</TT>,
<I>τ</I><SUB>2</SUB> = <TT>at(:, 2)</TT>,
<I>τ</I><SUB>3</SUB> = <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.
<P>
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