Page 145 - Strategies and Applications in Quantum Chemistry From Molecular Astrophysics to Molecular Engineer

130 J. L. CALAIS

From this group compatibility condition Mermin and his collaborators have derived both
all the "ordinary" crystallographic and the quasicrystallographic space groups.

If gk = k, (II.7) implies that either the Fourier component vanishes or the phase
function isan integer or zero. Another way of expressing that important result is to
say, that given a phase function those wave vectors k, for which that function is not
equal to an integer or zero, determine a set of vanishing Fourier components The
number of vanishing terms in the Fourier expansion (II.3a) of the density is a kind of
measure of the degree of symmetry in the system.

3. Momentum space characteristics of crystals

The traditional characterisation of an electron density in a crystal amounts to a statement that
the density is invariant under all operations of the space group of the crystal. The standard
notation for such an operation is where R stands for the point group part (rotations,
reflections, inversion and combinations of these) and the direct lattice vector m denotes the
translational part. When such an operation works on a vector r we get

The details of the operation Rrcan be further specified by the matrix which represents
the operation R in a suitably chosen coordinate system [2], in which also the vector r is
expressed. For the operation on a function of r we need the inverse of the space group
operation,
We thus have for an arbitrary function f(r),

A crystal characterised by a space group G has an electron density p(r) which is invariant
under all elements of G:

The electron density is the diagonal element of the number density matrix , i.e the
first order reduced density matrix after integration over the spin coordinates,:

A transformation of the number density matrix N under a space group operation means that
both variables are transformed:

The following relation and its inverse hold between the elements of the number density
matrices in momentum and position space [21]:

The momentum space counterpart of (III.6) can therefore be written,