 LDA: Local Density Approximation 
Definition:  A method to
approximate both the
exchange and
correlation energy
in density
functional methods 
Explanation:  One of the principle
ideas of the density
functional theory
(→DFT)
methods is to
convert the
many body problem
into N
singleparticle
equations in an
effective field. As
a consequence,
electronic exchange
and correlation
interactions are not
included and have to
be considered
separately. Exact
functional for these
electron–electron
interactions is
unknown except for
the hypothetical
free electron gas.
For this theoretical
model, the exchange
energy can be
calculated
analytically, while
the correlation
energy can be
derived from
perturbation theory.
Therefore,
the exchange and
correlation energy
can be
calculated as a
function of gas
density. LDA uses
tabulated
data of this
function to
approximate the
electronic
correlation and
exchange energy of
small volume
elements of the
crystal as the
correlation and
exchange energy of
the free electron
gas with the
corresponding
density. The total
exchange and
correlation energy
of the crystal is
then estimated as
the sum over all
volume elements.
Despite such a crude
simplification,
the LDA method
yields surprisingly
good results. The
LDA method can be
improved as the
generalized gradient
method (→GGA)
which
additionally
takes into account
the density
gradient.
The LDA is an
approximation of the
exchangecorrelation
functional where the
latter is
represented as a
local functional of
the electron
density. Within the
LDA, the electron
gas
is considered to be
homogeneous, and the
exchangecorrelation
energy, E_{xc}[n],
is
an integral over all
space with the
exchangecorrelation
energy density,
ε_{xc}^{hom},
at each
point assumed to be
the same as in a
homogeneous electron
gas with that
density, which can
be expressed as in
the equation below.
Therefore, the only
necessary
information is the
exchangecorrelation
energy of the
homogeneous gas as a
function of density.
The exchange energy
of the homogeneous
gas can be given
analytically; the
correlation energy
can be calculated
with a
great accuracy with
MonteCarlo (→MC
) methods [1].
The equation below
is obviously valid
when the electronic
densities vary
slowly over space,
but real atoms in
solids violate this
condition; thus, the
LDA usually
underestimates
lattice constants
and overerstimates
bulk
moduli of solids.
As mentioned above,
the more enhanced
approximation is the
GGA.

Picture / Figure / Diagram:  
SFBLink:  LDA methods are not used in the SFB since they are not reliable for a magnetic iron. 
References:  R. Dronskowski, Computational Chemistry of Solid State Materials, WileyVCH, Weinheim, 2005. [1] R. M. Martin, Electronic Structure: Basic Theory and Practical Methods, Cambridge University Press, 2004 

