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Subsections
3.2 Periodic systems
Exploiting the results of the previous section, we can now consider the
motion of noninteracting particles in a static potential, which is described
by the timeindependent Schrödinger equation 3.20.
In the study of bulk crystals, the system is infinite but periodic, and so
it is necessary to be able to reduce this problem to the study of a finite
system. This approach turns out to have several advantages so that it is
often easiest to study even aperiodic systems by imposing some false
periodicity. The system is contained within a supercell which is then
replicated periodically throughout space (see figure 3.1).
The supercell must be large enough so that the systems contained within each
one, which in reality are isolated, do not interact significantly.
Figure 3.1:
Using the supercell approximation, an isolated molecule can be studied using the same techniques which
are usually applied to crystals.

3.2.1 Bloch's theorem
See [45] for a fuller discussion of the proof outlined here.
We consider noninteracting particles moving in a static potential ,
which may be the KohnSham effective potential
(3.18).
In a perfect crystal, the nuclei are arranged in a regular periodic array
described by a set of Bravais lattice vectors . The system,
being infinite, is invariant under translation by any of these lattice
vectors, and in particular the potential is also periodic i.e.

(3.31) 
for all Bravais lattice vectors .
The Schrödinger equation which describes the motion of a single particle
in this potential is

(3.32) 
and we define translation operators
for each lattice vector
which act in the
following manner on any function of position :

(3.33) 
Since the potential and hence the Hamiltonian are periodic i.e.
, these operators
commute with the translation operators:

(3.34) 
i.e.
, and the translation
operators commute with each other i.e.
.
There must, therefore, exist a good quantum number corresponding to each
lattice vector , and it must also be possible to choose the
eigenstates of the Hamiltonian to be simultaneous eigenstates of all the
translation operators;



(3.35) 



(3.36) 
From the commutation relations of the translation vectors it follows that
the eigenvalues must satisfy

(3.37) 
We can define the eigenvalues for the three primitive lattice vectors
in terms of three complex numbers by

(3.38) 
Since all lattice vectors can be expressed in the form
, where the are integers,
it follows from equation
3.37 that

(3.39) 
which is equivalent to
where the
are the reciprocal lattice vectors satisfying
, and the are
complex numbers in general.
Thus we have shown that

(3.42) 
which is one statement of Bloch's theorem. Consider the function
.

(3.43) 
i.e. the function also has the periodicity of the lattice, and
so the wavefunction can also be expressed as

(3.44) 
where is a strictly cellperiodic function i.e.
.
We thus label the eigenstates of the Hamiltonian and the translation operators
where is the good quantum number labelling
different eigenstates of the Hamiltonian with the same good quantum vector
, related to the translational symmetry.
At this point we note that a periodic function can always be expressed
as a Fourier series i.e.

(3.45) 
where is reciprocal lattice vector
and the are integers.
Thus the state
can be expressed as a linear
combination of planewaves:
Instead of having to solve for a wavefunction over all of (infinite)
space, the problem now becomes one of solving for wavefunctions only
within a single (super)cell, albeit with an infinite number of possible values
for
. In order to simplify the problem to manageable proportions, it is
necessary to impose some boundary conditions on the wavefunction, which
restrict the allowed values of .
3.2.2 Brillouin zone sampling
We choose to model the infinite periodic system by a large number of primitive
cells
stacked together, with cells
along the direction, and we apply periodic or generalised
Bornvon Karman boundary conditions to the wavefunctions, which can be
interpreted by saying that a particle which leaves one surface of the
crystal simultaneously enters the crystal at the opposite surface. In
fact it can be shown [46] that the choice of boundary
conditions does not affect the bulk properties of the system. This
condition is expressed mathematically as

(3.48) 
Applying Bloch's theorem (3.42) gives

(3.49) 
so that the values of are restricted such that

(3.50) 
using equation 3.41.
Therefore the values of the are required to be real and equal to

(3.51) 
where the are integers, so that the general allowed form for
the Bloch wavevectors is

(3.52) 
Taking the limit to the true infinite perfect crystal (
)
we see that there is still an infinite number of allowed vectors,
but that they are now members of a countably infinite set.
Furthermore, we see that vectors which differ only by a reciprocal
lattice vector are in fact equivalent. Consider two such wavevectors related by
, then the corresponding Bloch states are also
related by
Since the expression in square brackets on the first line is a cellperiodic
function the whole expression is a valid Bloch wavefunction with
wavevector . Thus we can restrict our attention to those
vectors which lie within the first Brillouin zone, that
volume of reciprocalspace enclosing the origin which is bounded by the planes which
perpendicularly bisect lines from the origin to surrounding lattice points.
The situation now is that for each allowed vector within the first
Brillouin zone we must calculate the occupied Hamiltonian eigenstates in
order to construct the density. However, the wavefunctions and other
properties such as Hamiltonian eigenvalues vary smoothly over the
Brillouin zone [47] so that in practice only a finite set
of points need to be chosen, and methods for making efficient choices have
been developed [48,49,50,51,52,53]. From the calculation of the wavefunctions
at a certain set of points,
perturbation theory [54,55] can be used
to approximate the wavefunctions at other nearby points.
In this work, we are interested in the behaviour of very large systems. The
volume of the Brillouin zone
is related to the
volume of the supercell
by

(3.54) 
so that for large systems, the Brillouin zone volume is very small and only
a few points need to be considered to describe the variation
across the Brillouin zone accurately. In this work we therefore only
calculate the wavefunctions at the centre of the Brillouin zone, ,
known as the point. This has the added advantage that at this
point the wavefunctions can be chosen to be real (recall that there
is always an arbitrary global phase factor) without loss of generality.
Next: 3.3 The pseudopotential approximation
Up: 3. Quantum Mechanics of
Previous: 3.1 Densityfunctional theory
Contents
Peter Haynes