Within the local density approximation to density functional theory, the total energy of an electronic system is
with
where
and
Minimizing the above total energy is equivalent to solving the Schrödinger equation
with
where
In terms of the electron energies 
above, an alternative form for
the total energy is
This first problem of this homework develops the software to compute
the exchange-correlation terms in the above equations. The literature
always gives 
in the form 
.
In terms of 
, the quantities we shall need are
and the exchange-correlation correction energy,
where
