Wavevector point-symmetry group

The set of one-electron functions transforming according to the n j-dimensional irrep d( ) is called the shell. For molecules, these shells are connected with irreps of the point-symmetry group. For a crystal, f3 = ( fe,7) - full irreducible representation of space group G, defined by the star of wavevector k and irrep 7 of the point group of this vector. Taking into consideration the spin states a a) we have 2n/ one-electron states in the shell. The functions ( )Q ([Pg.110]

For crystaUine solids the translation symmetry of the Hamiltonian is taken into account in any electronic-structure calculations as it allows calculations to be made for the basis connected only with the primitive unit cell. In the translation-symmetry-adapted basis the matrix JI has a qnasidiagonal structure with identical blocks related to an irrep k of the translation symmetry group T. As the latter is Abelian its irreps are one-dimensional. The translation s3Tnmetry adapted functions are known as Bloch functions and numbered by wavevector k. Use of the point symmetry of a crystal allows the number of Bloch functions calculated to be decreased and further block-diagonaUzation of Hamiltonian of a crystal to be made. 

Thus these points in a small but well-defined region of k space include all possible irreducible representations of the translation group the vectors of the reciprocal lattice transform points in the Brillouin zone into equivalent points. The Brillouin zone therefore contains the whole symmetry of the lattice, each point corresponding to one irreducible representation, and no two points being related by a primitive translation. The smallest value of k ki, k2, kz) belonging to the rep is called the reduced wave-vector. The set oi reduced wavevectors is called the first Brillouin zone. 

This procedure gives the labels of the induced rep q, / ) in the k basis corresponding to those in the q basis, ie. the results of the reduction of the induced rep over irreps of the group Gfc. All the information obtained can be specified by listing the symmetry (the labels of irreps) of the Bloch states with wavevectors k corresponding only to a relatively small number of k points in the BriUouin zone forming a set K. The set K contains the inequivalent symmetry points of the BriUouin zone and one representative point from each inequivalent symmetry element (symmetry line or symmetry plane) if the latter does not contain the points of higher symmetry. 

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