Page 144 - Strategies and Applications in Quantum Chemistry From Molecular Astrophysics to Molecular Engineer

QUASICRYSTALS AND MOMENTUM SPACE 129

Here V is the volume of the Born-von Kármán region, i.e. that part of position space which
is repeated as a result of the fundamental periodic boundary conditions. The integration in
(II. 1) is carried out over that region, which we denote by BK.
For n = 1 we have for example,

i.e. the average density of the system.
Two densities and are said to be indistinguishable if all their correlation functions
and are identical. As shown by Mermin and collaborators their Fourier transforms
then have some very interesting properties. We can always expand a density in plane
waves,

The wave vectors k can be expressed in terms of any basis vectors we choose. At the
moment there is neither a direct nor a reciprocal lattice. Using (II.3a) in (II. 1) we see that
the Fourier components of two indistinguishable densities can differ only by a phase
factor:

The gauge function is linear in its argument:

A related concept is that of phase function which relates the Fourier components of
a density and those of a transformed density obtained by letting a point group
operation g work on r:

If g is an element of the point group of the material meaning that and are
indistinguishable for all elements g in that group, corresponding Fourier components can
differ only by a phase factor:

A "generalised" space group is specified by a point group and the associated phase
functions The ordinary space groups constitute special cases of these generalised
space groups.
Since (gh)k = g(hk) we get with (II.7)

which implies