readme.tex

已经编译好的levoo程序

\end{itemize}

\section{Definitions}

Here we provide some basic definitions for a number of quantities which are
calculated in the package. 

\begin{itemize}
\item Partial electronic density for the state \( n \) is defined as:
\begin{equation}
\label{1}
\rho _{n}(\mathbf{R})=\frac{1}{N}\sum _{\mathbf{k}\in BZ}\left| \psi _{n\mathbf{k}}(\mathbf{R})\right| ^{2}=\sum _{\mathbf{k}\in SP}\omega _{\mathbf{k}}\left| \psi _{n\mathbf{k}}(\mathbf{R})\right| ^{2}
\end{equation}
 where SP means a set of (special) \textbf{k}-points (i.e. the actual \textbf{k}-point
sampling), \( \psi _{n\mathbf{k}}(\mathbf{R}) \) is the wave function and \( \omega _{\mathbf{k}} \)
is the weighting factor.
\item Partial electronic density for an \emph{island} of bands \( n\in [n_{1},n_{2}] \)
is
\begin{equation}
\label{2}
\rho _{[n_{1},n_{2}]}(\mathbf{R})=\sum ^{n_{2}}_{n=n_{1}}\rho _{n}(\mathbf{R})
\end{equation}
 
\item Total Density of States (DOS): 
\begin{equation}
\label{3}
N(\epsilon )=\frac{1}{N}\sum _{n}\sum _{\mathbf{k}\in BZ}\delta (\epsilon -\epsilon _{n\mathbf{k}})
\end{equation}
where the sum over \( n \) is run over all states (both occupied and unoccupied)
and \( \epsilon _{n\mathbf{k}} \) are Kohn-Sham eigenvalues.
\item Local DOS (projected on a sphere): 
\begin{equation}
\label{4}
N_{\mathbf{P}_{R}}(\epsilon )=\frac{1}{N}\sum _{n}\sum _{\mathbf{k}\in BZ}A_{n\mathbf{k}}\delta (\epsilon -\epsilon _{n\mathbf{k}})
\end{equation}
where \( \mathbf{P}_{R} \) is a sphere of radius \( R \) centered at point
\( \mathbf{P} \), and the weighting factor is defined as:
\begin{equation}
\label{5}
A_{n\mathbf{k}}=\int _{\mathbf{r}\in \mathbf{P}_{R}}|\Psi _{n\mathbf{k}}(\mathbf{r})|^{2}d\mathbf{r}
\end{equation}
 where integration is performed over the sphere. 
\item \textit{\emph{Angular-momenta (\( s,p,d \)) projected DOS}} is defined as above,
but with the following factors: 
\begin{equation}
\label{6}
A_{n\mathbf{k}}=\int _{\mathbf{r}\in \mathbf{P}_{R}}\, \sum _{m=-l}^{l}\left| \left\langle \psi _{n\mathbf{k}}(\mathbf{r})\right| \left. R_{nl}(r)S_{lm}(\widehat{\mathbf{r}})\right\rangle \right| ^{2}d\mathbf{r}
\end{equation}
 where \( S_{lm}(\widehat{\mathbf{r}}) \) is a real spherical function for
the momenta \( l=0,1,2 \). 
\item DOS projected on a layer (slab calculation). The factors in this case are:
\begin{equation}
\label{7}
A_{n\mathbf{k}}=\int _{\mathbf{r}\in (layer)}|\Psi _{n\mathbf{k}}(\mathbf{r})|^{2}d\mathbf{r}
\end{equation}
where the layer is thought to be perpendicular to the \( z \)-axis.
\end{itemize}
In the case of the projected DOS (LDOS) and also while analysing the density,
real space integrals are calculated numerically. Two methods are implemented
which are referred to as ``\emph{conserving}'' and ``\emph{non-conserving}''
algorithms. 

\begin{itemize}
\item \textit{\emph{The conserving}} method provides the correct charge inside a sphere
or a layer, each grid point in the cell is scanned once and only those grid
points contribute which are positioned inside the area of interest (subject
to an arbitrary lattice translation). As the region of integration increases
(e.g. the radius of the sphere) the integration volume approaches that of the
cell since equivalent points separated by a translation are ignored. 
\item In the \textit{\emph{non-conserving}} method a finer grid is constructed in
the region of interest (= a sphere or a layer) and the integrals are calculated
by summing up contributions on this finer grid (interpolation is used). This
method does not give the charge conservation for large regions going over to
adjacent cells since equivalent points will all be included. 
\item If the size of the cell is large enough with respect to the region of integration,
than both methods should give close results. 
\item The conserving method is extremely demanding and scales linearly with the size
of the system. The non-conserving method scales linearly with the size of its
own grid. However, since this grid is limited to the size of the integration
region, the time of the calculation does not depend on the size of the system
at all, so that this method is extremely fast.
\item In the conserving method the original grid is used with the directions along
the cell basic vectors \( \mathbf{a}_{i} \). That is why this method is inappropriate
for calculations of the angular momenta projected DOS for non-simple-cubic cells
or for the calculation of the dipole/quadrupole momenta of a molecule. On the
contrary, the non-conserving method has its own grid which is always cubic (along
the Cartesian axes) so that it has the atomic symmetry and is ideal for these
calculations.
\end{itemize}

\section{Program \textit{tetr}}


\subsection{Installation and general information}

The code \textbf{tetr} must be compiled only once by performing the command
\begin{verbatim}
 make -f make.tetr
\end{verbatim} in your \noun{}\texttt{\~{}/TOOLS/TETR} directory\noun{.} 

The code is user-friendly and works interactively (menu-driven). After choosing
the PW code you are working with from the very first menu, the main menu appears
which essentially reflects all the features supported at present. If you use
the first 3 options, it assumes that you are going to construct your geometry
file from scratch, so that the existing file (if any) will not be read in; all
other options, \emph{if invoked first}, require an existing geometry file. This
file is either: 

\begin{itemize}
\item \texttt{\noun{fort.15}} for \textbf{CETEP} \texttt{}
\item \texttt{\noun{{[}Seed{]}.coord}} for \textbf{CASTEP} \texttt{}
\item \texttt{CONTCAR} for \textbf{VASP}
\end{itemize}
Just to make it clear: if you start with the first 3 options, you will be kicked
back to the main menu with some geometry in the memory and you can continue
working on it using other options; in this case if there is a geometry file,
it will be ignored. Note that at present you \emph{have} to go to option 5 to
actually write the geometry and the \( \mathbf{k} \)-point files. By using
Quit (option 15) you will loose all the data.

Note that in some rather complicated situations it is easier to fill in atomic
``flesh'' into your cell solely by a proper expansion (see section \ref{Sec::extention})
of the primitive unit cell (UC) of some reference system (e.g. the perfect crystal)
with subsequent removal/addition of atoms in specific positions; the code allows
you to do so and also it checks on the run whether any position is translationally
equivalent to the already accepted ones. Mind, it is tricky to make a 3D step
or kink system, believe it or not! 

The main menu options are:

\begin{enumerate}
\item Generate geometry file from scratch: you can either key in all the lattice vectors
or choose them from a set of options with subsequent extention (if needed).
When you are back in the main menu, you can continue working on the geometry
/ \( \mathbf{k} \)-point files.
\item This is the same option as the previous one except that a self-explanatory file
\texttt{tetr.inp} with Cartesian coordinates of all atoms in the generated cell
is printed out and the code stops. Then, you can edit/remove/add atoms in this
file using a text editor, and then run \textbf{tetr} again using the next option
3.
\item The file \texttt{tetr.inp} is read in and checked. You can continue working
on the cell using other options of the main menu. 
\item An extremely powerful option! It allows you to modify your existing geometry
in a number of ways and build up your final periodic cell. You will be given
an extended menu with self-explanatory options. You should be able to change
your lattice vectors; extend your cell; rotate, shift the system; add, remove,
shift atoms; change their species; construct a slab for the surface calculation
using e.g. Millers indices, etc. What is more, after every step a \texttt{geom.xyz}
(xyz-format) file with the system geometry is written so that you can preview
your cell on the fly as you build it using some molecular viewer (e.g. \texttt{\textbf{XMOL}})!
In addition, you can preview it as an extended cell if you like. Every step
can also be undone. 
\item For existing geometry, an appropriate \textbf{k}-point sampling for the DOS,
ground state and band-structure calculations can be generated (the \textbf{k}-point
generator). In the case of the DOS options, the point group symmetry is explored
and symmetry nonequivalent \textbf{k}-points are produced; a file \texttt{brill.dat}
is written which contains an important data for the \textbf{lev00} routine which
calculates and previews the DOS/LDOS. At the same time, when you quit there,
a geometry file will be produced:

\begin{enumerate}
\item \texttt{\noun{fort.15\_}} for \textbf{CETEP} \texttt{}
\item \texttt{\noun{{[}Seed{]}.coord\_}} for \textbf{CASTEP} \texttt{}
\item \texttt{CONTCAR\_} and \texttt{KPOINTS\_} for \textbf{VASP}
\end{enumerate}
and the code stops.

\item Coulomb potential at any desired point in the cell is calculated in the framework
of the point-charge model; you will be asked to provide the charges on atoms
either separately for every atom in the cell or only for different species;
a file \texttt{madel.dat} is produced with all your results within the option.
\item For molecules only: the symmetry of vibrations is analysed and the distorted
geometry corresponding to a chosen vibrational mode and the amplitude is generated
in a new geometry file. Translational and rotational degrees of freedom (normally
six; there will be five degrees of freedom for a linear molecule) are eliminated
automatically by projecting them out from all the coordinates. The strength
of distortion from equilibrium is also to be given. Using the geometry file
corresponding to distorted geometry, you can run the PW code again saving energies
in a file to be used in the next option.
\item For molecules only (the follow-up option to the previous one): using the group-theoretical
information about molecular vibrations, the energies both in equilibrium and
distorted positions (see option 7) and atomic masses, vibrational frequency
is calculated for the given active mode.
\item A file \texttt{coord.dat} containing Cartesian coordinates of all atoms is generated. 
\item XYZ-type \texttt{geom.xyz} file with Cartesian coordinates is generated; more
than a single cell can be produced and then previewed with any molecular viewer
program: you will be prompted to breed your primitive cell before writing the
file.
\item Distances between atoms in the cell are studied taking into account images in
the adjacent cells; you can explore either the shortest distances for \textit{all}
atomic pairs or a set of distances for a \textit{given pair} of atoms; in the
latter case distances up to a desired order of neighbours are provided in the
ascending order; a file \texttt{distance.dat} is produced tracing all your actions
within the option.
\item This option allows you to check any points in space if they are equivalent to
any atoms in the system.
\item Only for \textbf{CASTEP}: an XYZ-type file \texttt{geom\_film.xyz} is produced
from the \texttt{SEED.coord} file.
\item An atom can be displaced from its positions, and all other atoms which are equivalent
to it due to the \emph{point-group} symmetry of the cell (detected automatically)
will be moved appropriately to keep the symmetry. This is useful if one wants
to calculate the potential energy surface (PES) with respect to full-symmetry
displacements of atoms in the system.
\item Quit the program
\end{enumerate}

\subsection{k-point generator}


\subsubsection{Symmetry operations}

If the \textbf{k}-point sampling is generated for the DOS case, the point-group
symmetry of the cell is automatically determined and only non-equivalent \textbf{k}-points
with appropriate weighting factors are finally written into the file containing
\( \mathbf{k} \)-points. Symmetry is also used in some other options of \textbf{tetr}
as stated above.

The complete list of the symmetry operations is shown in Fig.1 and Fig.2
\begin{figure}
{\par\centering \resizebox*{15cm}{!}{\includegraphics{Oh.xfig.ps}} \par}


\caption{Symmetry operations of the \protect\( O_{h}\protect \) group.}
\end{figure}


\begin{figure}
{\par\centering \resizebox*{15cm}{!}{\includegraphics{D6h.xfig.ps}} \par}


\caption{Additional symmetry operations due to the D\protect\( _{6h}\protect \) group.}
\end{figure}



\subsubsection{Additional information about the k-point generator (DOS)}

If the symmetry and/or special constraints imposed on atomic coordinates have
not been used at the stage of atomic relaxation while running the PW code, the
symmetry generator may fail to determine the correct symmetry group due to numerical
errors caused. Therefore, to make your symmetry group as rich as possible, you
may sometimes need to generate \textbf{k}-points for the symmetry you expect