Basic concepts for k-points and symmetry

For a small unit cell periodic system calculation, it is necessary to include k-points besides the Gamma point (sometime the Gamma point is not even included in the calculation). This is because, what you should really calculate is a much bigger cell, which contains many unit cells. One Gamma point in the bigger cell corresponds to (folded back to) many k-points in the Brilliouin zone of the (smaller) unit cell. This strategy is used to generate the k-points in the Kpt_gen subdirectory under the PEtot package. It is called the equivalent k-point method [S. Froyen, Phys. Rev. B 39, 3168(1989)]. In this method, the equivalent bigger cell (which contains many united cells, thus the bigger cell and unit cell must be commensurate) is given in the "kpgen.input" file. For example, if a m1xm2xm3 Monkhorst-Pack k-point set is needed, then the bigger cell dimension should be m1, m2, m3 times the unit cell 1,2,3, dimenstions.

The symmetry of the system can also be used by the PEtot program. If the system (atomic structure contained in atom.config file) has a certain symmetry, then the charge densities and potentials of the system should also have the same symmetry. If two k-points are related by a symmetry operation, they are equivalent, the eigen energies and eigen wavefunctions on those two k-points will be the same. Thus it is not necessary to repeat the calculation for these two k-points. As a result, we only need to calculate the non-equivalent k-points, and each non-equivalent k-point will have a different weight, corresponds to the number of equivalent k-points it is representing. This is called the reduced k-point set. The symmetry and reduced k-point set are checked and generated by the "kpgen" program in subdirectory Kpt_gen. Currently, Kpt_gen and PEtot only check and use the point group symmetry around the origin.

Note that, for small molecules, it is customary to analyze and classify the wavefunctions according to the class of the symmetry group of the system. Doing that can further reduce the computation by taking the advantage of the symmetry of the wavefunctions (they are related but not the same as the symmetry of the atomic structure). But this is too complicated to be used in a general planewave code.

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