Numerical Methods in Quantum Mechanics - Dipartimento di Fisica

Chapter 10
Pseudopotentials
In general, the band structure of a solid will be composed both of more or less
extended (valence) states, coming from outer atomic orbitals, and of strongly
localized (core) states, coming from deep atomic levels. Extended states are the
interesting part, since they determine the (structural, transport, etc.) properties
of the solid. The idea arises naturally to get rid of core states by replacing
the true Coulomb potential and core electrons with a pseudopotential (or effective
core potential in Quantum Chemistry parlance): an effective potential that
“mimics” the effects of the nucleus and the core electrons on valence electrons.
A big advantage of the pseudopotential approach is to allow the usage of a
plane-wave basis set in realistic calculations.
10.1 Three-dimensional crystals
Let us consider now a more realistic (or slightly less unrealistic) model of a
crystal. The description of periodicity in three dimensions is a straightforward
generalization of the one-dimensional case, although the resulting geometries
may look awkward to an untrained eye. The lattice vectors, R n , can be written
as a sum with integer coefficients, n i :
R n = n 1 a 1 + n 2 a 2 + n 3 a 3 (10.1)
of three primitive vectors, a i . There are 14 different types of lattices, known
as Bravais lattices. The nuclei can be found at all sites d µ + R n , where d µ
runs on all atoms in the unit cell (that may contain from 1 to thousands of
atoms!). It can be demonstrated that the volume Ω of the unit cell is given by
Ω = a 1 · (a 2 × a 3 ), i.e. the volume contained in the parallelepiped formed by
the three primitive vectors. We remark that the primitive vectors are in general
linearly independent (i.e. they do not lye on a plane) but not orthogonal.
The crystal is assumed to be contained into a box containing a macroscopic
number N of unit cells, with PBC imposed as follows:
ψ(r + N 1 a 1 + N 2 a 2 + N 3 a 3 ) = ψ(r). (10.2)
Of course, N = N 1 · N 2 · N 3 and the volume of the crystal is V = NΩ.
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