Page 152 - Strategies and Applications in Quantum Chemistry From Molecular Astrophysics to Molecular Engineer

QUASICRYSTALS AND MOMENTUM SPACE 137
The Fourier components of the density are then obtained from the expression

Here f is a function on the lattice satisfying and such that f(k) is the Fourier
transform of a function with no symmetries whatever. That last condition is imposed in
order to avoid that the density obtained from (IV.7) gets any symmetries which are not
associated with the point group G, and also to prevent from vanishing on a set of wave
vectors so large that the lattice is thinned out to a sublattice for which the space group
would have a different character. The components (IV.7) transform under the elements of
the point group according to the fundamental rule (II.7).
An effective one electron Schrödinger equation with a local potential V(r) in position
space, (atomic units),

corresponds in momentum space to the following equation [19],

Wave functions in position and momentum spacce are related as in (III. 16), and the
Fourier component of the potential is

In density functional theories the potential is determined by the density, and consequently
its Fourier components are related to those of the density. One can therefore connect the
symmetry properties of the momentum functions, in other words the transformation
properties of under the operations of the point group, with those of the Fourier
components of the density, (11.7).

What has been sketched here is obviously just the bare framework of a general
investigation of the symmetry properties of momentum space functions in quasicrystals.
With all the information available in the papers by Mermin and collaborators it should
however be a very tempting enterprise to go ahead along the lines sketched and learn
about the details of the symmetry properties of those wave functions - both in momentum
and in positition space - which will be needed in quasiperiodic extended systems.

References

1. D. Shechtman, I. Blech, D. Gratias and J.W. Cahn, Phys. Rev. Letters, 53, 1951
(1984).
2. J. F. Cornwell, Group Theory in Physics. Vol. 1, Academic Press, London (1989).
3. See e.g. Electrons in Disordered Metals and at Metallic Surfaces, P. Phariseau,
B.L Györffy and L. Scheire Eds., NATO Advanced Study Institute Series,
Series B: Physics, Volume 42, Plenum Press New York and London (1979).
4. M.E. Esclangon, C.R. Acad. Sci. (Paris) 135, 891 (1902).
5.a S. Tanisaki, J. Phys. Soc. Japan , 16, 579 (1961).
b Y. Yamada, S. Shibuya and S. Hoshino, J. Phys. Soc. Japan , 18, 1594 (1963).