Density Functional Theory and the Local Density Approximation

What is the Functional?
E[ n]
T [ n]
+ V [ n]
V [ n].
=
e e!
e
+
ext
where T e [n] is the kinetic energy and V e-e is the electron-electron
interaction.
V ext [n] is trivial:
Vext [ n]
= V ext
! "
n(
r)
dr
where
Vˆ
ext
N at
Z#
= "!
| r " R
# #
|
T and V e-e need to be approximated !
E[n] – The Kohn Sham Approach
Write the density in terms of a set of N non-interacting orbitals:
The non interacting kinetic energy and the classical Coulomb
interaction
N
1
2
T
s
n]
= # ! $
i
" $
i
2
1 n(
r1 ) n(
r2
)
[ V H
[ n]
=
dr1dr2
i
2
!
r1
" r2
This allows us to recast the energy functional as:
E[ n]
= T [ n]
+ V [ n]
+ V [ n]
E [ n]
E
N
n( r ) = ! "
i
( r)
i
s ext H
+
where we have introduced the exchange-correlation functional:
xc
=
s
e!
e
!
[ n]
( T [ n]
! T [ n])
+ ( V [ n]
V [ n])
2
xc
H