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From: Thomas E. Cheatham, III <cheatham.chpc.utah.edu>

Date: Wed, 20 Apr 2005 08:53:38 -0600 (Mountain Daylight Time)

*> 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 net-charged, 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 net-neutral)

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 tin-foil

(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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Received on Wed Apr 20 2005 - 16:53:00 PDT

Date: Wed, 20 Apr 2005 08:53:38 -0600 (Mountain Daylight Time)

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 net-charged, 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 net-neutral)

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 tin-foil

(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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Received on Wed Apr 20 2005 - 16:53:00 PDT

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