In the case of optimising ¸, the choice of parameters to optimise was obvious because of the way the function is expressed as a Fourier expansion. In the case of the Jastrow function, this choice is not so clear. The current Jastrow factor has the form
where 
is given by
In an attempt to improve on the above function it was decided to add an
extra term into the exponential in Eqn.(
) to take account
not only of the electron-electron separation, 
,
but also of the individual positions of the electrons 
and
.
It was suspected that the individual positions of the electrons,
will only have an effect close to the ions. Therefore, the
new function, should be short ranged and centered on each of the ions.
For simplicity, the function was chosen to be spherically symmetric,
i.e. it is a function of the distances of the 2 electrons from the
ion, 
and 
as well as the electron-electron
separation 
.
It must also obey the following conditions:-
),
i.e. 
for the extra term .
was satisfied, the new term,
was expanded about 
.
Now condition . specifies 
therefore
Finally expanding 
gives
We chose to keep electron j fixed (i.e.
)
and move electron i through
it, to test the behaviour as 
.
As the angle between 
and 
varies
between 0 and 
the value of 
will vary smoothly between -1 and
+1, see Figure .
Figure: Dependence of 
on
the angle between 
and 
Therefore, the only solution to () for all geometries of
electrons is 
, i.e. the new term cannot contain
any overall constant or power of 
.
The final form chosen for the new short range function was therefore
The prefactor in Eqn.()
performs the following functions; the 
term removes any
overall constant and power of r as specified above. The
term is required to satisfy condition
, namely that the function be well behaved as one of
the electrons moves through the ion. The need for this term in the
prefactor was established by performing small simulations using
different forms of as one electron moves through an ion.
The 
term enforces the short
range nature of by forcing it to decay to zero with zero gradient when
one of the electrons is a distance L from the ion.
The remaining part of is a general Chebyshev expansion in all 3
variables; 
.
It should be noted that there are in fact two separate functions required, one dealing with the case where the spins on electrons i and j are parallel and one where they are anti-parallel. This has no significant effect on the choice of function, but it does mean that there are twice as many parameters to be optimised and this reduces the maximum possible power in the Chebyshev expansion.
It is also worth noting that the final form for is very similar to
that proposed by Mitas [5] ,see Eqn.().
The difference between the functions
is that Mitas only includes even powers of 
whereas the
function used in this report uses all powers and should therefore be
more general.
