Numerical Methods in Quantum Mechanics - Dipartimento di Fisica

A reciprocal lattice of vectors G m such that G m · R n = 2πp, with p integer,
is introduced. It can be shown that
G m = m 1 b 1 + m 2 b 2 + m 3 b 3 (10.3)
with m i integers and the three vectors b j given by
b 1 = 2π Ω a 2 × a 3 , b 2 = 2π Ω a 3 × a 2 , b 3 = 2π Ω a 1 × a 2 (10.4)
(note that a i · b j = 2πδ ij ). The Bloch theorem generalizes to
ψ(r + R) = e ik·R ψ(r) (10.5)
where the Bloch vector k is any vector obeying the PBC. Bloch vectors are
usually taken into the three-dimensional Brillouin Zone (BZ), that is, the unit
cell of the reciprocal lattice.
It can be shown that there are N Bloch vectors; in the thermodynamical
limit N → ∞, the Bloch vector becomes a continuous variable as in the onedimensional
case. We remark that this means that at each k-point we have
to “accommodate” ν electrons, where ν is the number of electrons in the unit
cell. For a nonmagnetic, spin-unpolarized insulator, this means ν/2 filled states
(usually called “valence” states, while empty states are called “conduction”
states). We write the electronic states as ψ k,i where k is the Bloch vector and
i is the band index.
10.2 Plane waves, core states, pseudopotentials
For a given lattice, the plane wave basis set for Bloch states of vector k is
b n,k (r) = 1 √
V
e i(k+Gn)·r (10.6)
where G n are reciprocal lattice vector. A finite basis set can be obtained, as
seen in the previous section, by truncating the basis set up to some cutoff on
the kinetic energy:
¯h 2 (k + G n ) 2
≤ E cut . (10.7)
2m
In realistic crystals, however, E cut must be very large in order to get a good
description of the electronic states. The reason is the very localized character
of the core, atomic-like orbitals, and the extended character of plane waves.
Let us consider core states in a crystal: their orbitals will be very close to the
corresponding states in the atoms and will exhibit the same strong oscillations.
Moreover, these strong oscillations will be present in valence (i.e. outer) states
as well, because of orthogonality (for this reason these strong oscillations are
referred to as “orthogonality wiggles”). Reproducing highly localized functions
that vary strongly in a small region of space requires a large number of delocalized
functions such as plane waves.
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