Density Functional Theory and the Local Density Approximation

The First Principles Approach
it is free of adjustable parameters
it treats the electrons explicitly
cost of the calculation limits system size and simulation time
electron
nucleus
Hˆ
({ R },
{ r }) = E ({ R },
{ r })
N , e!
N , e J i N , e!
N , e
J
i
where
H ˆ
N e
= Tˆ
N
+ Tˆ
e
+ Vˆ
N ! N
+ Vˆ
N ! e
+ Vˆ
,
e!
e
Born-Oppenheimer Approximation
Electron mass much smaller than nuclear mass:
Timescale associated with nuclear motion much slower than that
associated with electronic motion
Electrons follow instantaneously the motion of the nuclei,
remaining always in the same stationary state of the Hamiltonian
#
Hˆ
({ R },
{ r }) = " ({ R }) ({ r })
N , e J i N J
!
({ R },
{ r }) = E ({ R },
{ r })
N , e!
N , e J i N , e!
N , e
i
J
depends only parametrically on {R J }
i
Born-Oppenheimer approximation decouples the electronic problem
from the ionic problem. The electronic problem is:
({ r }) = E ({ r })
H ˆ ! !
i
i