Page 149 - Strategies and Applications in Quantum Chemistry From Molecular Astrophysics to Molecular Engineer

134 J.L. CALAIS
It is important to distinguish between symmetry properties of wave functions on one hand
and those of density matrices and densities on the other. The symmetry properties of wave
functions are derived from those of the Hamiltonian. The "normal" situation is that the
Hamiltonian commutes with a set of symmetry operations which form a group. The
eigenfunctions of that Hamiltonian must then transform according to the irreducible
representations of the group. Approximate wave functions with the same symmetry
properties can be constructed, and they make it possible to simplify the calculations.
In the case of a perfect crystal the Hamiltonian commutes with the elements of a certain
space group and the wave functions therefore transform under the space group operations
according to the irreducible representations of the space group. Primarily this means that
the wave functions are Bloch functions labeled by a wave vector k in the first Brillouin
zone. Under pure translations they transform as follows

This implies that a density built up from such Bloch functions [cf (III.5) and (III.14)] is
invariant under all such translations [the "little" period]:

Corresponding relations for arbitrary space group elements ' ~ ' show that if the
orbitals which make up the density transform asthe irreducible representations of
the space group, the density is invariant under all the operations of that group.

It is also of interest to study the "inverse" problem. If something is known about the
symmetry properties of the density or the (first order) density matrix, what can be said
about the symmetry properties of the corresponding wave functions? In a one electron
problem the effective Hamiltonian is constructed either from the density [in density
functional theories] or from the full first order density matrix [in Hartree-Fock type
theories]. If the density or density matrix is invariant under all the operations of a space
group, the effective one electron Hamiltonian commutes with all those elements.
Consequently the eigenfunctions of the Hamiltonian transform under these operations
according to the irreducible representations of the space group. We have a scheme which
is selfconsistent with respect to symmetry.
The symmetry properties of the density show up experimentally as properties of its
Fourier components If those components vanish except when the wave vector k
equals one of the lattice vectors K of a certain reciprocal lattice, the general plane wave
expansion of the density,

reduces to (III.17a). Since times an integer, we then have

If (III.24) holds we get the corresponding result for arbitrary space group elements