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Choice of Functional Form for new term in the Jastrow Factor

In the case of optimising , the choice of parameters to optimise was obvious as 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 tex2html_wrap_inline7137 is given by


In an attempt to improve on the above function it was decided to add an extra term into the exponential in Eq.(gif) to take account not only of the electron-electron separation, tex2html_wrap_inline7139 , but also of the individual positions of the electrons tex2html_wrap_inline5837 and tex2html_wrap_inline7143 . It was suspected that the region where correlation effects are likely to deviate most strongly from the symmetric correlation described by the standard Jastrow function will be close to the ionic cores. Therefore, the any new function, tex2html_wrap_inline7145 should be short ranged and centred on each of the ions. For simplicity, the tex2html_wrap_inline7145 function was chosen to be a function of the distances of the 2 electrons from the ion, tex2html_wrap_inline7149 and tex2html_wrap_inline7151 and the electron-electron separation tex2html_wrap_inline7139 , with no angular dependence. It must also obey the following conditions:-

  1. It should not cause the total Jastrow term to violate the cusp condition (see section gif), i.e. tex2html_wrap_inline7155 for the extra term tex2html_wrap_inline7145 .  
  2. tex2html_wrap_inline7145 should be well behaved as one of the electrons moves through an ion, i.e. there should be no cusp in tex2html_wrap_inline7145 or in the 1st derivative as tex2html_wrap_inline7103 . 
  3. tex2html_wrap_inline7145 should take the most general form possible, subject to the above 2 restrictions.
To check that condition gif was satisfied, the new term, tex2html_wrap_inline7145 was expanded about tex2html_wrap_inline7169


Now condition gif specifies tex2html_wrap_inline7171 , therefore


Finally, expanding tex2html_wrap_inline7173 gives


We chose to keep electron j fixed (i.e. tex2html_wrap_inline7177 ) and move electron i through it, to test the behaviour as tex2html_wrap_inline7181 . As the angle between tex2html_wrap_inline5837 and tex2html_wrap_inline7143 varies between 0 and tex2html_wrap_inline7187 the value of tex2html_wrap_inline7189 will vary smoothly between -1 and +1, (see figure gif).

Figure: Dependence of tex2html_wrap_inline7189 on the angle between tex2html_wrap_inline5837 and tex2html_wrap_inline7143

Therefore, the only solution to Eq.(gif) for all geometries of electrons is tex2html_wrap_inline7221 .

The final form chosen for the new short range function tex2html_wrap_inline7145 was therefore


The prefactor in Eq.(gif) tex2html_wrap_inline7225 performs the following functions; the tex2html_wrap_inline7227 term removes any terms independent of tex2html_wrap_inline7139 or terms linear in tex2html_wrap_inline7139 as specified above. The tex2html_wrap_inline7233 term is required to satisfy condition gif, 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 tex2html_wrap_inline7145 as one electron moves through an ion. The tex2html_wrap_inline7237 term enforces the short range nature of tex2html_wrap_inline7145 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 tex2html_wrap_inline7145 is a general Chebyshev expansion in all three variables; tex2html_wrap_inline7245 , and tex2html_wrap_inline7139 .

It should be noted that there are in fact two separate tex2html_wrap_inline7145 functions required, one dealing with the case where the spins of electrons i and j are parallel and one where they are anti-parallel. This has no effect on the choice of functional form, but it does mean that there are twice as many parameters to be optimised and this reduces the maximum possible number of terms in the Chebyshev expansion.

It is also worth noting that the final form for tex2html_wrap_inline7145 is very similar to that proposed by Mitas [41], see Eq.(gif). The difference between the functions is that Mitas only includes even powers of tex2html_wrap_inline7139 whereas the tex2html_wrap_inline7145 function used here contains odd and even powers and should therefore be more general.

next up previous contents
Next: Implementation of the new Up: Optimising the u Function Previous: Optimising the u Function

Andrew Williamson
Tue Nov 19 17:11:34 GMT 1996