CHAPTER 2
Perfect crystal
We now consider the Kohn-Sham density functional theory for crystals. We as-
sume that the system under consideration has a crystal structure in the equilibrium
state: The nuclei positions form a lattice, denoted as L, with unit cell Γ. Recall
that the dual lattice and its unit cell are denoted as L∗ and Γ∗ respectively.
The nuclei provide a background charge to the system. The contribution from
each nucleus is represented by a compactly supported smooth or a fast decaying
function ma(· Xi), where Xi is the position of that nucleus. Here we are taking
a pseudopotential approximation, treating the core electrons as part of the nuclei
resulting in an effective charge distribution ma, only the valence electrons are al-
lowed to vary freely [18]. Therefore, the total charge contribution from the nuclei
is
(2.1) me(x) =
Xi∈L
ma(x Xi),
where the subscript e signals the equilibrium state. We also assume that the func-
tion ma respects the inversion symmetry: ma(x) = ma(−x). In addition, we
assume
R3
ma = Z.
This also implies
Γ
me = Z. We assume that there are Z (valence) electrons per
unit cell.
In Kohn-Sham density functional theory, it is assumed that the electrons in the
system follow an effective one-body Hamiltonian depending on the electron density.
The effective Hamiltonian is given by
(2.2) He(ρ) = −Δ + Ve(ρ).
The potential Ve depends on the electron density: If ρ is Γ-periodic, then
(2.3) Ve(ρ)(x) = φe(x) + η(ρ(x)),
where φe is the Coulomb potential, given by
(2.4) −Δφe = 4π(ρ me),
with the periodic boundary condition on Γ and the constraint that
Γ
φe = 0. For
the equation (2.4) to be solvable, we need the normalization constraint that
(2.5)
Γ
ρ = Z.
The current work is carried out under the assumption that the Kohn-Sham
equation for the undeformed crystal lattice has a smooth, periodic solution. Fur-
thermore, this equilibrium solution ρe satisfies some conditions that we will specify
below.
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