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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