Action
Consider
Take the partial with respect
to ci
Interchange the derivative
order – usually O.K.
Assume that changes in ci
do not effect the boundary conditions, that is assume that
On the boundary 
and then integrate by
parts to find
If A is a minimum the partial
in is zero or for a complete set of c’s
This is
This
shows that if this potential is written as a function of a vector c, minimizing with respect to the constants in c is equivalent to solving
Action èPoisson’s equation
The
action that gives rise to Poisson’s equation contains an extra term
Again set the derivative with
respect to ci equal to zero
Integrate the first term in by parts assuming that 
to find
This is zero for arbitrary
values of 
if and only if
Poisson’s equation can be
solved with a sufficiently flexible F(x,c) –
satisfying the boundary conditions by simply finding the c for which is a minimum.
Action for Electron Density in a 1d system
Charge neutrality requires
Or
The correct F minimizes an action
defined by
The forms for F are restricted to those which satisfy the boundary conditions. The derivative of the last term with respect
to ci is
Or
So that with the usual
integration by parts
Energy
The ground state energy is always less than
The
derivative with respect to c contains a term from the denominator
With
the usual integration by parts
And
in we recognize the Schroedinger equation in the parenthesis that need to
be zero. The integral in is multi-3dimensional but with an intense three d character.
..\WaveFunction\Hartree.htm .doc
Variance
The
Schroedinger equation can also be solved by minimizing the error coefficient in
the Monte-Carlo estimate of <H>.
This
differs from minimizing the action by the presence of w(x)?? and by the fact
that the minimum is now zero rather than <H>. In multi-dimensional situations, using
In
curve fitting the variance is defined as
