Time-Dependent Electron Localization Function - Fachbereich ...

1.2 The Kohn--Sham formalism
n(r) =
N∑
|φ i (r)| 2 , (1.17)
i=1
where the N single-particle orbitals φ i in Eq. (1.17) satisfy the Schrödinger equation
)
(− 2
2m ∇2 + v KS (r) φ i (r) = ε i φ i (r), (1.18)
and have the N lowest eigenenergies ε i . The total energy of the ground-state of the
non-interacting system is therefore
E KS =
N∑
ε i . (1.19)
Note that the value of this energy does not correspond to the ground-state energy
of the interacting system. According to the Hohenberg--Kohn theorem, it exists a
unique energy functional
∫
E KS [n] = T KS [n] + V KS (r)n(r) d 3 r. (1.20)
We note that T KS [n] is the kinetic energy functional of the non-interacting system
and it is therefore different from the functional T [n] in Eq. (1.14). In order to solve
the interacting system, we need to find a form of V KS , so that the ground-state
densities of the non-interacting and the interacting system are the same. Since we
are really interested in the interacting system, we rewrite Eq. (1.13) as
∫
E[n] = T [n] + W [n] + n(r)V (r) d 3 r (1.21)
= T KS [n] + 1 e 2
4πε 0 2
∫ ∫ n(r)n(r ′ )
|r − r ′ |
i=1
∫
d 3 r d 3 r ′ +
n(r)V (r) d 3 r + E xc [n].
Here, the second term is the direct, or Hartree, term and the last term is the so-called
exchange-correlation (xc) energy functional, defined as
E xc [n] = F HK [n] − 1 e 2 ∫ ∫ n(r)n(r ′ )
4πε 0 2 |r − r ′ d 3 r d 3 r ′ − T KS [n]. (1.22)
|
With this formalism at hand, one only needs to develop reasonable approximations
for E xc , which contain the electron–electron interaction beyond the Hartree term
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