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

10 1 Density functional theory (DFT)
E0 ≤ E[ñ(r)] and δE[ñ(r)] = 0 (1.15)
Assuming the E[n] in eq (1.13) is differentiable, the ground-state density must also satisfy
the stationary principle in eq. (1.9) according to the variational principle (1.15). The
Euler-Lagrange equation (1.10) becomes
μ = vext(r)+ δEHK[n]
(1.16)
δn(r)
The great advantage of this theory is the reduction of the degrees of freedom from
3N, if the energy is represented as a functional of wave function E[Ψ], to only 3 degrees
of freedom since the electron density depends only on space vector r. Because of that, a
huge step was performed toward productive calculations in computational physics.
Hohenberg and Kohn didn’t treat the kinetic energy term in Thomas and Fermi model,
they only reformulated the total energy as functional in density which minimizes the energy
functional in an external potential and describes all electronic properties of the system.
The difficulty which was and remained is in calculating the kinetic energy of the interacting
electrons. Many approximations were done, but they didn’t give enough accuracy which
encourages further hard work on that. The solution came one year after Hohenberg and
Kohn published there article about DFT. In 1965 Kohn and Sham were able to solve of
one part the problem by assuming a non-interacting electron system, with kinetic energy
TKS[n] which can be accurately computed and covers a large part of the exact T [n]. This
is explained in the next section.
1.3 The Kohn-Sham equations
Kohn and Sham proposed a kinetic energy, that can be calculated accurately, by introducing
wavefunctions of non-interacting electrons into the problem[61]. It is assumed that
these wavefunctions lead to the same ground state density as the many body wavefunction
Ψ. The correction to this kinetic energy can be handled separately through an exchange
correlation energy term. Equation (1.13) becomes
�
E[n] =TKS[n]+ n(r)vext(r)dr + EH[n]+EXC[n] (1.17)
where
TKS[n] =
N�
i
〈ψi|− 1
2 ∇2 |ψi〉 (1.18)
is the kinetic energy introduced in terms of single particle orbitals ψi. Since TKS is density
functional, satisfying Pauli principle, this leads that