[gmx-users] maximum cut-off in Minimum image convention

[Justin A. Lemkul]

Francisco Garcia wrote:
> Dear users,
> 
> I wish to ask a question regarding the maximum cut-off in the minimum
> image convention (MIC).
> 
> According to my understanding, MIC ensures that the separation
> between two particles i and j along each coordinate axes, namely,
> xi- xj, yi-yj, and zi-zj, is in the interval (-0.5L, 0.5L), where L is the box
> length (assumed the box to be cubic). This implies that
> distance rij can vary between 0 and 0.5L*sqrt(3).
> In order words the maximum separation should be less than half the diagonal:
>  for example if i is located at (0,0,0) and j is located at the corner
> (0.49L, 0.49L, 0.49L).
> 
> However, I read a couple of text books and online notes that claim
> that if r > 0.5L, then MIC is violated.
> Specifically, this means that 0.5L < rij < 0.5L*sqrt(3)  violates MIC.
> I find this confusing. I have been grinding hard to understand this
> point but I just
> cannot figure it out.
> 
> In short my claim is that in MIC, rij can be at most half the diagonal
> length whereas
> the textbook says that rij can be at most half the box length.
> 
> If I am wrong, can someone explain the flaw in my argument?
> 

The box diagonal is the longest dimension across the unit cell, so indeed it 
would seem that you can have r > 0.5L.  Simplify your problem to 2 dimensions 
and the answer becomes very clear.  If you have a particle in the center of this 
2-D lattice, and you have a cutoff > 0.5L, then you are counting interactions 
with other particles within the central unit cell and duplicating at least some 
of these interactions in the periodic image.  See, for instance:

http://cmt.dur.ac.uk/sjc/thesis_dlc/node60.html

Extend the radius in that image ever so slightly, and particle 1 is going to 
interact with *both* particles 4 and 7, the latter being the periodic image of 
the same particle.  Several textbooks also discuss this phenomenon in depth.

-Justin

> Thanks!

-- 
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Justin A. Lemkul
Ph.D. Candidate
ICTAS Doctoral Scholar
MILES-IGERT Trainee
Department of Biochemistry
Virginia Tech
Blacksburg, VA
jalemkul[at]vt.edu | (540) 231-9080
http://www.bevanlab.biochem.vt.edu/Pages/Personal/justin

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