Use of Symmetry in Reciprocal Space

The relative cost of ab initio calculations depends on many variables, such as the Hamiltonian, basis set, accuracy requirement, size, and density of the system (see Appendix 2). The Fock or KS matrix diagonalization step during the solution of Eq. [25] can become the calculation bottleneck with a large basis set, when, for example, more than 1000 basis functions are used. Such a number of functions may correspond to about 100 atoms per cell, when a local basis set is used, but this is the usual size of plane wave calculations, even with a small unit cell. As many crystalline systems are highly symmetric, taking advantage of symmetry is, therefore, important for reducing computational time. 

Symmetry properties can be used both in the direct and in the reciprocal space, for example, to form matrices in direct space, such as F and or to diagonalize F(k) more efficiently. The application of symmetry to direct space 

The application of a point symmetry operator of the space group to a given point k in reciprocal space has two possible consequences  

Each block in the matrix on the right corresponds to a different irreducible representation (IR) of the so-called little co-group, and the number 

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