For a single orbital containing f elections, the full density-functional-theory energy function, including the Hartree and exchange-correlation energies, is
where
is the solution of Poisson's equation, and
is the exchange-correlation energy function.
In our matrix language, this is
where 
,
with the `*' operator defined as point-wise multiplication of pvectors
as in HW#10, conj(v) representing complex conjugation of all
components the vector v, and representing a pvector
each of whose components is the application of the 
function applied to the corresponding component of the pvector n.
Finally, 
, and we can determine
from our previous algorithm for the solution of Poisson's
equation, 
.
In the same language, the gradient of E is
Here, 
where
where
