What Program BZ can do (Back)

Program BZ is designed to be used mainly for exploring the Brillouin zone of a crystalline solid.  It can be used for working with the Brillouin zone for one-dimensional polymers such as polyethylene or cellulose, and for two-dimensional systems such as graphene, but the main use is for mapping the Brillouin zone of organic, organometallic, and inorganic solids. For this reason all the following discussion will be restricted to three-dimensional solids.

Structures in k-space

The volume within the Brillouin zone is referred to as k-space. Several regions in k-space are of interest, and all of these can be examined using BZ:

Types of energy band structure


Electronic band structures are calculated using the Fock matrix from a MOPAC calculation.  This is the simplest type of band-structure. The number of energy bands is equal to the number of atomic orbitals in the primitive unit cell, i.e., the smallest block that can generate the solid using only translations.


Vibrational band structures can be calculated if a FORCE calculation that includes the MERS keyword is run. The mass-weighted Hessian matrix generated in MOPAC is then written to <file>.brz.

Symmetry operations

If symmetry operations are supplied, via a file <file>.ops, then the electronic energy or phonon matrix is symmetrized.  In principle, these matrices should already have the correct symmetry, but small deviations occur due to incomplete geometry optimization and to the use of finite numerical precision in constructing the matrices, in particular the phonon matrix.  Symmetrizing the energy matrix results in the elimination of small artifacts, such as bands that should cross having a small band-gap.

When points are calculated, the characters of the symmetry operations of the associated eigenvectors are generated.  In diamond, for example, at the Γ point, each eigenvector belongs to an irreducible representation of the little group that is isomorphous with point-group Oh. At the X point, all irreducible representations are two-fold degenerate.

In general, and with the notable exception of the Identity operation, characters of eigenvectors are complex, and the imaginary component represents the momentum of the k-point.

Graphical representation

BZ is an interactive program.  While it's running, the band-structure, either a walk or a surface, is generated  so that you can see what the results look like.  This band structure can be captured using "snipping tool" or by screen capture.  The band structure is also written to a file in a form that is suitable for use by Excel.

Run in non-interactive mode

As an alternative to typing data interactively, a file can be written that contains all the typed responses.  Thus if the first response is the number "1" followed by (return), then the file would have as the first line the number 1.  Generating an aesthetically pleasing band-structure takes some trial-and-error.  If the keystrokes that were used in making such a graphic are put into a file, , and when such a