Numerical Methods in Quantum Mechanics - Dipartimento di Fisica

potential, we can hope to obtain a set of electronic levels that can describe the
main features of the crystal. A rigorous basis for such approach can be provided
by Hartree-Fock or Density-Functional theory. In the end, the basic step is to
solve to the problem of calculating the energy levels in a periodic potential.
We haven’t yet said anything about the composition and the periodicity of
our system. Let us simplify further the problem and assume a one-dimensional
array of atoms of the same kind, regularly spaced by a distance a. The atomic
position of atom n will thus be given as a n = na, with n running on all integer
numbers, positive and negative. In the jargon of solid-state physics, a is the
lattice parameter, while the a n are the vectors of the crystal lattice. The system
has a discrete translational invariance, that is: it is equal to itself if translated
by a or multiples of a. Called V (x) the crystal potential, formed by the
superposition of atomic-like potentials: V (x) = ∑ n V n(x − a n ), the following
symmetry holds: V (x + a) = V (x). Such symmetry plays a very important role
in determining the properties of crystalline solids. Our one-dimensional space
(the infinite line) can be decomposed into finite regions (segments) of space, of
length a, periodically repeated. A region having such property is called unit cell,
and the smallest possible unit cell is called primitive cell. Its definition contains
some degree of arbitrarity: for instance, both intervals [0, a[ and ] − a/2, +a/2]
define a valid primitive cell in our case.
9.1.1 Periodic Boundary Conditions
Before starting to look for a solution, we must ask ourselves how sensible it is
to apply such idealized modelling to a real crystal. The latter is formed by a
macroscopically large (in the order of the Avogadro number or fractions of it)
but finite number of atoms. We might consider instead a finite system containing
N atoms with N → ∞, but this is not a convenient way: translational
symmetry is lost, due to the presence of surfaces (in our specific 1D case, the
two ends). A much more convenient and formally correct approach is to introduce
periodic boundary conditions (PBC). Let us consider the system in a box
with dimensions L = Na and let us consider solutions obeying to the condition
ψ(x) = ψ(x + L), i.e. periodic solutions with period L >> a. We can imagine
our wave function that arrives at one end “re-enters” from the other side. In the
one-dimensional case there is a simple representation of the system: our atoms
are distributed not on a straight line but on a ring, with atom N between atom
N − 1 and atom 1.
The advantage of PBC is that we can treat the system as finite (a segment
of length L in the one-dimensional case) but macroscopically large (having N
atoms, with N macroscopically large if a is a typical interatomic distance and
L the typical size of a piece of crystal), still retaining the discrete translational
invariance. Case N → ∞ describes the so-called thermodynamical limit. It is to
be noticed that a crystal with PBC has no surface. As a consequence there is no
“inside” and “outside” the crystal: the latter is not contemplated. This is the
price to pay for the big advantage of being able to use translational symmetry.
In spite of PBC and of translational symmetry, the solution of the Schrödinger
equation for a periodic potential does not yet look like a simple problem.
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