Page 151 - Strategies and Applications in Quantum Chemistry From Molecular Astrophysics to Molecular Engineer

136 J. L. CALAIS
Using the inverse of (III.7) we can write the density of an arbitrary extended system as

which means that the Fourier component of the density can be written

This should be compared to (III. 18) where the role of k in (IV.5) is played by a reciprocal
lattice vector K.
Mermin's conceptual starting point is a set of vectors k in reciprocal space which
correspond to sharp Bragg peaks in the experimental diffraction pattern. The non
vanishing Fourier components are then to be found for wave vectors which can be
characterized as the set of all integral linear combinations of a certain finite set of D basis
vectors In an ordinary crystal D = 3 and the point group must be one
of the 32 crystallographic point groups. If we have a non crystallographic point group the
rank D of the lattice can be larger than three. Such a system is called a quasicrystal. A
system with a crystallographic point group and a lattice with a rank D higher than three is
called an incommensurately modulated crystal.
An important and interesting question is obviously whether for quasicrystals and
incommensurately modulated crystals there is anything corresponding to the Bloch
functions for crystals. Momentum space may be a better hunting ground in that connection
than ordinary space, where we have no lattice. Not only is there no lattice, one cannot
even specify the location of each atom yet [8].

A Bloch function for a crystal has two characteristics. It is labeled by a wave vector k in
the first Brillouin zone, and it can be written as a product of a plane wave with that
particular wave vector and a function with the "little" period of the direct lattice. Its
counterpart in momentum space vanishes except when the argument p equals k plus a
reciprocal lattice vector. For quasicrystals and incommensurately modulated crystals the
reciprocal lattice is in a certain sense replaced by the D-dimensional lattice L spanned by
the vectors It is conceivable that what corresponds to Bloch functions in momentum
space will be non vanishing only when the momentum p equals k plus a vector of the
lattice L.

The problem is to "translate" the fact that certain terms are absent in the expansion (IV.3)
to symmetry properties of the density in the sense of transformation properties under
certain operations. We have a density with non vanishing Fourier components only for
such wave vectors k which belong to the lattice L:

Mermin [9, 18] has given a recipe for the construction of a set of Fourier components for
a density characterised by a certain space group. The space group is then specified by a
point group G, a lattice of wave vectors in the sense discussed above, and a set of phase
functions one for each element of the point group.