Description of Program BZ

Unlike molecules, whose eigenvalue spectrum consists of infinitely sharp lines (Kronecker-d type), the eigenvalues of solids form bands. These bands are conventionally represented by Brillouin Zones (BZ). For a polymer, the BZ consists of a line, for a layer structure, a surface, and for a solid, a three-dimensional shape. The dimensions of the BZ are inverse distance: if two polymers have unit cells of 10 and 8 Å, then the ratio of their unit cell translation vectors is 1.25:1, and the ratio of their BZ's is 1:1.25.

Although MOPAC can generate all the information necessary to generate band-structures, for ease of use this job is given to a special program called BZ. MOPAC is designed to run in batch mode, but drawing band-structures is an interactive operation.

Before BZ can be run, a MOPAC calculation must first be run in order to generate the data needed by BZ. Most of this data is invisible to the user, but the user must supply to MOPAC one or two data which BZ will use. These data are:

MERS=(n₁,n₂,n₃)

 
(See Program MAKPOL and MERS) BZ needs to know how the Fock matrix is organized, and MERS gives the number of unit cells in each direction. MOPAC uses the keyword MERS as the prompt to write out a file for use by BZ. Within MOPAC, MERS has no other use. Data sets using MERS should only be constructed using MAKPOL, because the sequence of the atoms is defined by the MERS keyword.  An implication of this is that the  MERS keyword should not be changed in the MOPAC data set.  If the keyword needs to be changed, it should be changed in the data set used by  program MAKPOL, and MAKPOL re-run.
In order to generate all points in the Brillouin zone, it is both necessary and sufficient for the number of unit cells in each direction to be 2.  That is, for a polymer, all points in the Brillouin zone can be generated by BZ if MERS=(2) is used; for a layer system, MERS=(2,2) and for a solid, MERS=(2,2,2).

Having said that, more unit cells might be preferred, in order to ensure that the cluster is large enough to allow MOPAC to use the gamma point as being representative of the entire band structure.  To verify that the cluster is large enough, increase the cluster size, and re-run the job.  If the heat of formation per unit cell and the charges do not change significantly, then the smaller cluster was sufficient for MOPAC. Regardless of how many unit cells are used, if the heat of formation per unit cell and the charges do not change significantly on changing the number of unit cells, then the band structure would also not change on changing the number of unit cells.
 

 

BCC (optional)

 
Like MERS, BCC is not used by MOPAC. BCC (see Keywords used by MAKPOL) indicates that the system is Body-Centered-Cubic; in other words, that every odd unit cell is missing. If the keyword BCC is present, MOPAC informs BZ that the system is Body-Centered-Cubic.

MOPAC will then write an unformatted file <fimename>.brz containing the following data:

Record 1:
NORBS: Number of orbitals (integer, 4 bytes)
Number of mers in each direction (3 integers, 4 bytes per datum)
BCC (logical, 4 bytes.  True if BCC is present, false otherwise)

Record 2:
Fock matrix, lower half triangle, (real, (NORBS*(NORBS+1))/2 data, 8 bytes per datum)

Record 3:
Translational vector lengths (9 real numbers, 8 bytes per number) For a polymer, ignore the last six data, for a layer, ignore the last three data.
Number of dimension (integer, 4 bytes) If  value = 1, then a polymer; if 2, then a layer, and if 3 then a solid.
Number of atoms (integer, 4 bytes)
Cartesian coordinates (real, 3 times number of atoms, 8 bytes per datum)

Record 4:
Orbital counters (integer, 2 times number of atoms, 4 bytes per datum) These are the starting and ending orbital indices for each atom.
 

BZ is an interactive utility program. It can perform three types of calculation:

• Points in k-space are to be studied.
• Lines in k-space are to be drawn.
• Surfaces in k-space are to be drawn.