Page 146 - Strategies and Applications in Quantum Chemistry From Molecular Astrophysics to Molecular Engineer

QUASICRYSTALS AND MOMENTUM SPACE 131

Thus the point group part of the operation works on the momentum coordinates and the
translation part gives rise to a phase factor. We notice that this phase factor reduces to 1 in
the diagonal elements, or in general when the difference between the the two arguments of
is a reciprocal lattice vector.
If the elements of the number density matrix in position space are invariant under all
operations of the space group, i.e. if

we get with (III.8), that their momentum space counterparts satisfy

The momentum distribution, i.e. the diagonal element of (III. 10) then satisfies

The reciprocal form factor [22] is the Fourier transform of the momentum distribution,

Using (III. 11) we see that the reciprocal form factor of a crystal which is invariant under a
space group, satisfies the relations,
for all point group elements R of the space group.

We notice that neither the momentum distribution nor the reciprocal form factor seems to
carry any information about the translational part of the space group. The non diagonal
elements of the number density matrix in momentum space, on the other hand, transform
under the elements of the space group in a way which brings in the translational parts
explicitly.
The number density matrix for a crystal with translation symmetry can be written in terms
of its natural orbitals [23, 24], as

This is the most general expression obtained from a set of natural spin orbitals written in
spinor form as

The orbitals are Bloch functions labeled by a wave vector k in the first Brillouin
zone (BZ), a band index µ, and a subscript i indicating the spinor component. The
combination of k and can be thought of as a label of an irreducible representation of the
space group of the crystal. The quantity is the occupation function which measures

the degree of occupation at wave vector k in band