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<H2><A ID="SECTION00051000000000000000">
4.1 Execution time</A>
</H2>
<P>
The following is a rough estimate of the complexity of a plain
scf calculation with <TT>pw.x</TT>, for NCPP. USPP and PAW
give raise additional terms to be calculated, that may add from a
few percent
up to 30-40% to execution time. For phonon calculations, each of the
3<I>N</I><SUB>at</SUB> modes requires a time of the same order of magnitude of
self-consistent calculation in the same system (possibly times a small multiple).
For <TT>cp.x</TT>, each time step takes something in the order of
<!-- MATH
$T_h + T_{orth} + T_{sub}$
-->
<I>T</I><SUB>h</SUB> + <I>T</I><SUB>orth</SUB> + <I>T</I><SUB>sub</SUB> defined below.
<P>
The time required for the self-consistent solution at fixed ionic
positions, <I>T</I><SUB>scf</SUB> , is:
<P><!-- MATH
\begin{displaymath}
T_{scf} = N_{iter} T_{iter} + T_{init}
\end{displaymath}
-->
</P>
<DIV ALIGN="CENTER">
<I>T</I><SUB>scf</SUB> = <I>N</I><SUB>iter</SUB><I>T</I><SUB>iter</SUB> + <I>T</I><SUB>init</SUB>
</DIV><P></P>
where <I>N</I><SUB>iter</SUB> = number of self-consistency iterations (<TT>niter</TT>),
<I>T</I><SUB>iter</SUB> =
time for a single iteration, <I>T</I><SUB>init</SUB> = initialization time
(usually much smaller than the first term).
<P>
The time required for a single self-consistency iteration <I>T</I><SUB>iter</SUB> is:
<P><!-- MATH
\begin{displaymath}
T_{iter} = N_k T_{diag} +T_{rho} + T_{scf}
\end{displaymath}
-->
</P>
<DIV ALIGN="CENTER">
<I>T</I><SUB>iter</SUB> = <I>N</I><SUB>k</SUB><I>T</I><SUB>diag</SUB> + <I>T</I><SUB>rho</SUB> + <I>T</I><SUB>scf</SUB>
</DIV><P></P>
where <I>N</I><SUB>k</SUB> = number of k-points, <I>T</I><SUB>diag</SUB> = time per
Hamiltonian iterative diagonalization, <I>T</I><SUB>rho</SUB> = time for charge density
calculation, <I>T</I><SUB>scf</SUB> = time for Hartree and XC potential
calculation.
<P>
The time for a Hamiltonian iterative diagonalization <I>T</I><SUB>diag</SUB> is:
<P><!-- MATH
\begin{displaymath}
T_{diag} = N_h T_h + T_{orth} + T_{sub}
\end{displaymath}
-->
</P>
<DIV ALIGN="CENTER">
<I>T</I><SUB>diag</SUB> = <I>N</I><SUB>h</SUB><I>T</I><SUB>h</SUB> + <I>T</I><SUB>orth</SUB> + <I>T</I><SUB>sub</SUB>
</DIV><P></P>
where <I>N</I><SUB>h</SUB> = number of <I>Hψ</I> products needed by iterative diagonalization,
<I>T</I><SUB>h</SUB> = time per <I>Hψ</I> product, <I>T</I><SUB>orth</SUB> = CPU time for
orthonormalization, <I>T</I><SUB>sub</SUB> = CPU time for subspace diagonalization.
<P>
The time <I>T</I><SUB>h</SUB> required for a <I>Hψ</I> product is
<P><!-- MATH
\begin{displaymath}
T_h = a_1 M N + a_2 M N_1 N_2 N_3 log(N_1 N_2 N_3 ) + a_3 M P N.
\end{displaymath}
-->
</P>
<DIV ALIGN="CENTER">
<I>T</I><SUB>h</SUB> = <I>a</I><SUB>1</SUB><I>MN</I> + <I>a</I><SUB>2</SUB><I>MN</I><SUB>1</SUB><I>N</I><SUB>2</SUB><I>N</I><SUB>3</SUB><I>log</I>(<I>N</I><SUB>1</SUB><I>N</I><SUB>2</SUB><I>N</I><SUB>3</SUB>) + <I>a</I><SUB>3</SUB><I>MPN</I>.
</DIV><P></P>
The first term comes from the kinetic term and is usually much smaller
than the others. The second and third terms come respectively from local
and nonlocal potential. <!-- MATH
$a_1, a_2, a_3$
-->
<I>a</I><SUB>1</SUB>, <I>a</I><SUB>2</SUB>, <I>a</I><SUB>3</SUB> are prefactors (i.e.
small numbers <!-- MATH
${\cal O}(1)$
-->
<IMG STYLE="height: 1.69ex; vertical-align: -0.10ex; " SRC="img3.png"
ALT="$\cal {O}$">(1)), <I>M</I> = number of valence
bands (<TT>nbnd</TT>), <I>N</I> = number of PW (basis set dimension: <TT>npw</TT>), <!-- MATH
$N_1, N_2, N_3$
-->
<I>N</I><SUB>1</SUB>, <I>N</I><SUB>2</SUB>, <I>N</I><SUB>3</SUB> =
dimensions of the FFT grid for wavefunctions (<TT>nr1s</TT>, <TT>nr2s</TT>,
<TT>nr3s</TT>; <!-- MATH
$N_1 N_2 N_3 \sim 8N$
-->
<I>N</I><SUB>1</SUB><I>N</I><SUB>2</SUB><I>N</I><SUB>3</SUB>∼8<I>N</I> ),
P = number of pseudopotential projectors, summed on all atoms, on all values of the
angular momentum <I>l</I>, and <!-- MATH
$m = 1, . . . , 2l + 1$
-->
<I>m</I> = 1,..., 2<I>l</I> + 1.
<P>
The time <I>T</I><SUB>orth</SUB> required by orthonormalization is
<P><!-- MATH
\begin{displaymath}
T_{orth} = b_1 N M_x^2
\end{displaymath}
-->
</P>
<DIV ALIGN="CENTER">
<I>T</I><SUB>orth</SUB> = <I>b</I><SUB>1</SUB><I>NM</I><SUB>x</SUB><SUP>2</SUP>
</DIV><P></P>
and the time <I>T</I><SUB>sub</SUB> required by subspace diagonalization is
<P><!-- MATH
\begin{displaymath}
T_{sub} = b_2 M_x^3
\end{displaymath}
-->
</P>
<DIV ALIGN="CENTER">
<I>T</I><SUB>sub</SUB> = <I>b</I><SUB>2</SUB><I>M</I><SUB>x</SUB><SUP>3</SUP>
</DIV><P></P>
where <I>b</I><SUB>1</SUB> and <I>b</I><SUB>2</SUB> are prefactors, <I>M</I><SUB>x</SUB> = number of trial wavefunctions
(this will vary between <I>M</I> and 2÷4<I>M</I>, depending on the algorithm).
<P>
The time <I>T</I><SUB>rho</SUB> for the calculation of charge density from wavefunctions is
<P><!-- MATH
\begin{displaymath}
T_{rho} = c_1 M N_{r1} N_{r2}N_{r3} log(N_{r1} N_{r2} N_{r3}) +
c_2 M N_{r1} N_{r2} N_{r3} + T_{us}
\end{displaymath}
-->
</P>
<DIV ALIGN="CENTER">
<I>T</I><SUB>rho</SUB> = <I>c</I><SUB>1</SUB><I>MN</I><SUB>r1</SUB><I>N</I><SUB>r2</SUB><I>N</I><SUB>r3</SUB><I>log</I>(<I>N</I><SUB>r1</SUB><I>N</I><SUB>r2</SUB><I>N</I><SUB>r3</SUB>) + <I>c</I><SUB>2</SUB><I>MN</I><SUB>r1</SUB><I>N</I><SUB>r2</SUB><I>N</I><SUB>r3</SUB> + <I>T</I><SUB>us</SUB>
</DIV><P></P>
where <!-- MATH
$c_1, c_2, c_3$
-->
<I>c</I><SUB>1</SUB>, <I>c</I><SUB>2</SUB>, <I>c</I><SUB>3</SUB> are prefactors, <!-- MATH
$N_{r1}, N_{r2}, N_{r3}$
-->
<I>N</I><SUB>r1</SUB>, <I>N</I><SUB>r2</SUB>, <I>N</I><SUB>r3</SUB> =
dimensions of the FFT grid for charge density (<TT>nr1</TT>,
<TT>nr2</TT>, <TT>nr3</TT>; <!-- MATH
$N_{r1} N_{r2} N_{r3} \sim 8N_g$
-->
<I>N</I><SUB>r1</SUB><I>N</I><SUB>r2</SUB><I>N</I><SUB>r3</SUB>∼8<I>N</I><SUB>g</SUB>,
where <I>N</I><SUB>g</SUB> = number of G-vectors for the charge density,
<TT>ngm</TT>), and
<I>T</I><SUB>us</SUB> = time required by PAW/USPPs contribution (if any).
Note that for NCPPs the FFT grids for charge and
wavefunctions are the same.
<P>
The time <I>T</I><SUB>scf</SUB> for calculation of potential from charge density is
<P><!-- MATH
\begin{displaymath}
T_{scf} = d_2 N_{r1} N_{r2} N_{r3} + d_3 N_{r1} N_{r2} N_{r3}
log(N_{r1} N_{r2} N_{r3} )
\end{displaymath}
-->
</P>
<DIV ALIGN="CENTER">
<I>T</I><SUB>scf</SUB> = <I>d</I><SUB>2</SUB><I>N</I><SUB>r1</SUB><I>N</I><SUB>r2</SUB><I>N</I><SUB>r3</SUB> + <I>d</I><SUB>3</SUB><I>N</I><SUB>r1</SUB><I>N</I><SUB>r2</SUB><I>N</I><SUB>r3</SUB><I>log</I>(<I>N</I><SUB>r1</SUB><I>N</I><SUB>r2</SUB><I>N</I><SUB>r3</SUB>)
</DIV><P></P>
where <I>d</I><SUB>1</SUB>, <I>d</I><SUB>2</SUB> are prefactors.
<P>
For hybrid DFTs, the dominant term is by far the calculation of the
nonlocal (<I>V</I><SUB>x</SUB><I>ψ</I>) product, taking as much as
<P><!-- MATH
\begin{displaymath}
T_{exx} = e N_k N_q M^2 N_1 N_2N_3 log(N_1 N_2 N_3)
\end{displaymath}
-->
</P>
<DIV ALIGN="CENTER">
<I>T</I><SUB>exx</SUB> = <I>eN</I><SUB>k</SUB><I>N</I><SUB>q</SUB><I>M</I><SUP>2</SUP><I>N</I><SUB>1</SUB><I>N</I><SUB>2</SUB><I>N</I><SUB>3</SUB><I>log</I>(<I>N</I><SUB>1</SUB><I>N</I><SUB>2</SUB><I>N</I><SUB>3</SUB>)
</DIV><P></P>
where <I>N</I><SUB>q</SUB> is the number of points in the <I>k</I> + <I>q</I> grid, determined by
options <TT>nqx1,nqx2,nqx3</TT>, <I>e</I> is a prefactor.
<P>
The above estimates are for serial execution. In parallel execution,
each contribution may scale in a different manner with the number of processors (see below).
<P>
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