lattice. 
For all other temperatures, the indirect band gaps decrease with increasing 
temperature and are smaller than that of the the static lattice. 
In the case of the direct band gap, the zero-point band gaps are larger than 
that of the static lattice. For all other temperatures, the direct 
band gaps decrease in comparison with the zero-point vibration gap but 
are higher than that of the the static lattice up to temperature 
of 800 K. 


\begin{table}[!h]
\begin{tabular}{llccll} \hline
PP & \multicolumn{3}{c} {Parameters} & \multicolumn{2}{c}{Band gap} \\ \cline{2-4}
Name &  ~Valence & ~r$_c$ & ~~~E$_{xc}$ & \multicolumn{2}{c}{Finite T trend}\\ \cline{5-6}
     &  ~electrons   & ~(a.u.)  &   & Indirect & Direct \\ \hline 
Si$_1$  & 3s$^2$ 3p$^2$            & 1.7 & PBE &  Inc-Dec  & Inc-Dec \\
Si$_2$  & 3s$^2$ 3p$^{1.5}$        & 1.7 & PBE &  Inc-Dec  & Inc-Dec \\
Si$_3$  & 3s$^2$ 3p$^1$            & 1.7 & PBE &  Inc-Dec  & Inc-Dec \\
Si$_4$  & 3s$^2$ 3p$^2$ 3d$^0$     & 1.7 & PBE &  Inc-Dec  & Inc-Dec \\
Si$_5$  & 3s$^2$ 3p$^2$            & 2.0 & PBE &  Decrease & Inc-Dec \\
Si$_6$  & 3s$^2$ 3p$^{1.5}$        & 2.0 & PBE &  Decrease & Inc-Dec \\
Si$_7$  & 3s$^2$ 3p$^1$            & 2.0 & PBE &  Decrease & Inc-Dec \\
Si$_8$  & 3s$^2$ 3p$^2$ 3d$^0$     & 2.0 & PBE &  Decrease & Decrease \\
Si$_9$  & 3s$^2$ 3p$^{1.5}$ 3d$^0$ & 2.0 & PBE &  Decrease & Decrease \\
Si$_{10}$ & 3s$^2$ 3p$^1$ 3d$^0$   & 2.0 & PBE &  Decrease & Decrease \\
Si$_{pbe-gipaw}$ & 3s$^2$ 3p$^1$ 3d$^0$   & 2.2 & PBE & Decrease & Decrease \\
Si$_{12}$ & 3s$^2$ 3p$^1$ 3d$^0$   & 2.2 & PBE &  Decrease & Decrease \\
Si$_{13}$ & 3s$^2$ 3p$^2$ 3d$^0$   & 1.8 & PBE &  Decrease & Decrease \\
Si$_{14}$ & 3s$^2$ 3p$^2$ 3d$^0$   & 1.9 & PBE &  Decrease & Decrease \\
Si$_{15}$ & 3s$^2$ 3p$^2$ 3d$^0$   & 2.1 & PBE &  Decrease & Decrease \\
Si$_{16}$ & 3s$^2$ 3p$^2$ 3d$^0$   & 2.2 & PBE &  Decrease & Decrease \\
         \hline \hline

\end{tabular}
\caption{The trend in the indirect and direct band gap values with 
temperature increased up to 1000 K for the different silicon Troullier-Martins 
\textit{ab initio} pseudopotentials generated with PBE 
exchange correlation functional to study finite temperature 
behavior.}
\label{ppsi-g}
\end{table}


For the pseudopotentials Si$_5$-Si$_7$, that were generated with a  
cutoff radius of 2.0 a.u, the zero-point indirect band gap shows the correct 
behaviour i.e. smaller than the indirect band gap of the static lattice. 
The trend in the direct band gaps however is the same as that of 
Si$_1$-Si$_4$ pseudopotentials up to a temperature of 600 K. Thus, the
pseudopotentials Si$_1$-Si$_7$ do not exhibit temperature transferability.


\begin{figure}
\includegraphics[scale=0.46]{RCutVerZPRSi.pdf}
\caption{Variation of zero-point band-gap shifts with cut-off radius for silicon} \label{SiliconRcutPlot}
\end{figure}


Table~\ref{ppsi-g} shows that the pseudopotentials Si$_8$-Si$_{10}$ differ from 
pseudopotentials Si$_5$-Si$_7$ in that the unbound 3d$^0$ state was 
incorporated in the generation condition. These pseudopotentials show the 
correct indirect and direct band gap behavior and are hence, temperature 
transferable. This leads to the conclusion that incorporation of the unbound 
3d$^0$ state in the pseudopotential generation configuration is essential 
for temperature transferability of Si pseudopotentials. 
In addition, the pseudopotential Si$_{12}$ was generated with the same 
conditions as Si$_{pbe-gipaw}$ except that it did not 
have the gipaw construction. Both Si$_{12}$ and Si$_{pbe-gipaw}$ (Table ~\ref{si-gipaw-bg})
 give virtually identical indirect and direct band gap 
shifts at all temperatures. This suggests that the gipaw construction does 
not affect temperature transferability.

To examine the dependence of the band gap shifts on the cut-off radius, 
pseudopotentials Si$_{13}$ - Si$_{16}$ were generated in the
3s$^2$ 3p$^2$ 3d$^0$  configuration with the cut-off radius 
varying from 1.8 a.u to 2.2 a.u. Figure~\ref{SiliconRcutPlot} plots the 
zero-point indirect and direct band gap shifts as a function of the cut-off 
radius.
It is again seen that the zero-point indirect band gap shift varies strongly 
with the cut-off radius. In comparison, the zero-point direct band gap shift 
varies much more slowly with the cut-off radius.

To summarize, the temperature transferability of Troullier-Martins 
pseudopotentials is dependent on the choice of the cut-off radius and the 
inclusion/exclusion of the unbound 3d$^0$ 
state in the generation configuration. The cut-off radius strongly affects 
the indirect band gap.

These observations in combination with the usual criteria for pseudopotential 
generation indicate that only in a small subset of the generation parameter 
space of Troullier-Martins pseudopotentials the new criterion of temperature 
transferability is also satisfied. These observations will also be helpful in 
generating temperature transferable pseudopotentials for other elements that 
are constituents of compound semiconductors.

The question of whether other category of pseudopotentials, Optimized, 
Ultrasoft and ONCV pseudopotentials can also be generated to exhibit
temperature transferability needs to be explored.

\subsection{Finite temperature band structure - Diamond}\label{tbs-c}

We calculated the finite temperature electronic structure of diamond (a 
wide band gap material) at various temperatures listed in Table~\ref{msdv} 
from 0 – 1000 K by using the DW corrected pseudopotential form factors. 
While the calculations were performed for all the 
pseudopotentials listed in Table~\ref{ppc-g}, we discuss here the results 
obtained for pseudopotentials that exhibit the correct band gap trends. 


\begin{table}[!h]
\begin{tabular}{lcllrc} \hline \hline
PP & a  & \multicolumn{4}{c} {Band gap} \\ \cline{3-6}
Name                &        & \multicolumn{2}{c}{Static lattice} & \multicolumn{2}{c}{Zero point shift} \\ 
                    & (a.u.) & \multicolumn{2}{c}{(eV)} & \multicolumn{2}{c}{(meV)} \\
            &        &  Indirect  &  Direct  & Indirect & Direct  \\ \hline 
C$_1$    & 6.74 & ~4.26  & 5.634 & ~102 & 140 \\
C$_2$    & 6.73 & ~4.28  & 5.635 & ~102 & 142 \\
C$_3$    & 6.71 & ~4.33  & 5.66  & ~103