Ab initio investigations of magnetic properties of ultrathin transition ...

8 1 Density functional theory (DFT)
ˆH =
N�
(− 1
2 ∇2 N� � Zα 1 � 1
i ) −
+
|ri − rα| 2 |ri − rj|
i=1 α
j�=i
i=1
(1.3)
Atomic units are employed, the length unit is the Bohr radius (a0=0.5292 ˚A), the charge
unit is the charge of the electron, e (e =1.602 · 10 −19 C), and the mass unit is the mass of
the electron, me. The first term in equation (1.3) is the sum of the kinetic energy operators
for all electrons in the system. The second term is the sum of the electron-nucleus Coulomb
attractions. The third term is the sum of the electron-electron Coulomb repulsions.
For solids N is larger than 10 23 . This makes the solution of such equation with 3N special
and N spin variables unobtainable obtainable without using approximations. Many numerical
methods and approaches were developed to solve this equation. The most powerful
theory was introduced by Hohenberg and Kohn in 1964[48], where they proposed that the
ground-state properties of the many-particle system can be determined by the ground-state
particle charge density n(r). This means that the degrees of freedom are reduced to be
only three instead of 3N. This theory is called ”Density Functional Theory (DFT)”, it
avoids the complicated many-body wavefunction and uses the electronic density for bandstructure
calculations. DFT is designed to calculate the total energies for small systems
(up to few hundreds of nonequivalent atoms) at zero Kelvin.
1.2 Origin of DFT
In the 1920s, Thomas and Fermi could approximate the electronic distribution of an atom
by assuming a uniform distribution of the electrons in a field of an effective potential[59],[60].
They could represent the electronic total energy ETF as a functional of the uniform electrons’
density n(r).
E[n(r)] = TTF[n(r)] + EH[n(r)] + Uext[n(r)] (1.4)
where n(r) is the density of electrons in the solid.
�
TTF[n(r)] = CF n(r) 5/3 dr (1.5)
is the Thomas-Fermi kinetic energy of electrons of density n(r). CF is a constant,
EH[n(r)] = 1
� � ′ n(r)n(r )
2 |r − r ′ | drdr′
(1.6)
is the Hartree energy, or the Coulomb interaction of the electron density with itself, and
Uext[n(r)] = − �
�
Zα
n(r)dr (1.7)
α
rα