AMBER Archive (2005)Subject: Re: AMBER: NonBonded Cutoff vs PME
From: Thomas E. Cheatham, III (cheatham_at_chpc.utah.edu)
Date: Wed Apr 20 2005  09:53:38 CDT
> on the subject...May I ask one more related question. When in PBCs we say
> "infinitely repeated", what does it mean, how calculations are done on
> infinite scale? I am not able to capture this concept...or its that only a
Infinity is indeed a difficult concept to grasp, execpts perhaps
mathematically.
for the Coulomb interaction, we can convert the sum over pairs to a double
sum over all unit cells and all pairs. This is an infinite sum. If the
system were netcharged, this total energy would diverge (however the
forces interestingly do not diverge). The sum is conditionally convergent
which means that the value in the limit (assuming it is netneutral)
obtained will be different depending on how you sum the series. In the
literature describing this, two means of summing are generally presented,
that of adding more and more 2D slabs or that of adding more and more
layers until the sum converges. Ewald converted this conditionally
convergent sum into a sum to two finite converging series, the screened
(erfc) direct space sum and the reciprocal sum. A missing factor is the
surface at the limit; almost all of the Ewald codes assume tinfoil
(conducting) boundary conditions at the surface. This avoids the shape
dependence. There are a few other details, but basically the sum
converges in the limit; our goal is to make sure that we calculate it to
sufficient accuracy. Implementations of PME, SPME, PPME, etc all can
reproduce the exact Ewald energetics/forces to whatever the desired
accuracy. With sander, we tend to look at force errors in the 10**6
range. For more detail, please continue to dig through the literature.

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