Last Modified 17 December 1999

The ``Static'' Tight-Binding Program: Example III

In this example we use the tight-binding program static to determine the equilibrium configuration of Cu₃Au in the A15 (Cr₃Si) phase, which is not observed experimentally.

For more information about the construction of the input to the static program see e.g., Example 1 and the other examples.

The tight-binding parameters for the Copper-Gold system are under test and so are not available from our Tight-Binding Parameter Home Page. We have therefore temporarily stored the tight-binding parameters here. Click here to download the parameters and save them in a new working directory under the name cuau_par. Note that the header of the parameters is somewhat different than for a monatomic parameter file such as pd_par from Example 1, reflecting the fact that there are three types of interactions in this file (Cu-Cu, Au-Au, and Cu-Au). In addition, there are now 330 parameters, instead of the 97 used for a monatomic system. Also note that the static executable must be compiled with ``mkind > 1'' in P1 to be able to run this example.

• go to the Crystal Lattice Structure (http://web.archive.org/web/20111231202250/http://cst-www.nrl.navy.mil/lattice/index.html) page,
• click on Index by Strukturbericht Designation, and then
• on A15 (Cr₃Si) Structure (http://web.archive.org/web/20111231202250/http://cst-www.nrl.navy.mil/lattice/struk/a15.html).
• In this case we will use the space group file in Cartesian coordinates, so look for the Cartesian coordinate link.
• Save this file in the working directory under the name spcgrp.a15.
• We have stored some pre-generated k-point meshes at http://web.archive.org/web/20111231202250/http://cst-www.nrl.navy.mil/bind/kpts/. Space group files for the lattices associated with each mesh are also available from this site.
• To find an appropriate "k-point" mesh for the A15 crystal, note that the space group file has the same 48 rotational operations as the simple cubic lattice. The translational part of the space group does not matter in the calculation of the k-points. Therefore, click on simple cubic lattices about half-way down the page. A wide variety of k-point meshes are available here. K-point meshes with higher orders are denser, and so more accurate, than those of lower order. However, the A15 lattice has a large unit cell, and so a small Brillouin zone. So for this example we'll use the mesh labeled "Order 6" under the "REGULAR" heading. This file has 84 k-points, and includes the origin, Gamma. Save this file in the same directory as the tight-binding parameters, using the name sc.r06.

Now get a copy of the SKIN (Slater-Koster INput) file, and save it in your working directory under the name SKIN. Let's examine this file.

The first few lines of the file are nearly the same as in Example 1 or Example 2:

Mode=4
0.002  0.500
cuau_par

In this example we are interested in the energy as a function of volume, so we might as well determine the pressure as well. To do this we set ``Mode=4''.

The A15 structure is simple cubic with a basis. See the A15 page for more information. The structure part of the input file looks like this:

Cu_3 Au -- A15
Cu_3 Au  8.40        (20 character label)
 0.0       (Electrons above nominal copper charge)
 4         (sc lattice, read a)
 8.40      (lattice constant)
 0         (No strains)
 8         (Atoms in the unit cell)
 3 3 3
F          (Logical variable -- no internal displacements) 
 2 0.00  0.00  0.00  0 0 0
 2 0.50  0.50  0.50  0 0 0
 1 0.25  0.50  0.00  0 0 0
 1 0.75  0.50  0.00  0 0 0
 1 0.00  0.25  0.50  0 0 0
 1 0.00  0.75  0.50  0 0 0
 1 0.50  0.00  0.25  0 0 0
 1 0.50  0.00  0.75  0 0 0

There are a few changes from other examples:

• We want to determine the energy of a simple cubic lattice as a function of lattice constant. This is easily accomplished using lattice type ``3'' from the lattice types table.
• We'll do the first calculation at a lattice constant of 8.4 atomic units, and set the 20 character and lattice constant to reflect that.
• The "3 3 3" line controls the distance we search to find neighbor atoms. Note that in this case the search distance is, at a minimum, 3X8.5 = 25.5 Bohr, well beyond the cutoff distance of 16.5 Bohr. Note that we can get away with a shorter search range here because the primitive lattice vectors are so large.
• The atoms are located according to the positions set for them in the A15 crystal. The first two sites are occupied by gold atoms. Since gold is the second element in the parameter file, these atoms are designated "2". The six gold atoms are then designated type 3.

Next we have to generate a k-point mesh. We could let the program do this, since we have the spacegroup file, but, as in Example 1, we'll use a pre-generated k-point mesh.

This part of the file looks like this:

NEWSYM=T
LATTIC=1
spcgrp.a15
ILAT=F
-1313
sc.r06

• The first line is a flag indicating that a new k-point mesh will be generated
• The next line tells the program that this is a general lattice, i.e., we don't need to do anything special.
• The next line is the path to the space group file
• Since this is in Cartesian coordinates, we set "ILAT=F"
• "-1313" indicates that the k-point information is in an external file.
• And the last line is the name of the k-point file we saved above.

Since we'll keep the same k-point mesh for further calculations, the remainder of the SKIN file is much simpler. For example, to find the energy at a lattice constant of 8.625 Bohr, we use

Cu_3 Au -- A15
Cu_3 Au  8.60        (20 character label)
 0.0       (Electrons above nominal copper charge)
 4         (sc lattice, read a)
 8.60      (lattice constant)
 0         (No strains)
 8         (Atoms in the unit cell)
 3 3 3
F          (Logical variable -- no internal displacements) 
 2 0.00  0.00  0.00  0 0 0
 2 0.50  0.50  0.50  0 0 0
 1 0.25  0.50  0.00  0 0 0
 1 0.75  0.50  0.00  0 0 0
 1 0.00  0.25  0.50  0 0 0
 1 0.00  0.75  0.50  0 0 0
 1 0.50  0.00  0.25  0 0 0
 1 0.50  0.00  0.75  0 0 0
NEWSYM=F

The "NEWSYM=F" line indicates that the previous k-point mesh will be used.

After calculating the energy of Cu₃Au, we can also use the same procedure to calculate the energy of CuAu₃, again in the A15 phase. Since this is the same lattice, we don't need to recreate the k-point mesh. The first structure for CuAu₃ therefore has the form

Cu Au_3 -- A15
Cu Au_3  8.60        (20 character label)
 0.0       (Electrons above nominal copper charge)
 4         (sc lattice, read a)
 8.60      (lattice constant)
 0         (No strains)
 8         (Atoms in the unit cell)
 3 3 3
F          (Logical variable -- no internal displacements) 
 1 0.00  0.00  0.00  0 0 0
 1 0.50  0.50  0.50  0 0 0
 2 0.25  0.50  0.00  0 0 0
 2 0.75  0.50  0.00  0 0 0
 2 0.00  0.25  0.50  0 0 0
 2 0.00  0.75  0.50  0 0 0
 2 0.50  0.00  0.25  0 0 0
 2 0.50  0.00  0.75  0 0 0
NEWSYM=F

note that the Cu and Au atoms have now switched positions.

Now go to the output discussion to see what will happen.

Look at other examples.

Get other parameters from the Tight-binding periodic table.

Return to the static Reference Manual.