Periodic Boundary Conditions - Practical Implementation: Continuity and The Minimum Image Convention

Practical Implementation: Continuity and The Minimum Image Convention

To implement periodic boundary conditions in practice, at least two steps are needed.

The first is to make an object which leaves the simulation cell on one side enter back on the other. This is of course a simple operation, and could in code be e.g. (for the x dimension, assuming an orthogonal unit cell centered on the origin):

if (periodicx) then if (x < -xsize*0.5) x=x+xsize if (x >= xsize*0.5) x=x-xsize endif

The second is to make sure that every distance between atoms, or other vector calculated from one atom to another, has a length and direction which corresponds to the minimum image criterion. This can be achieved as follows to calculate e.g. the x direction distance component from atom i to atom j:

if (periodicx) then dx = x(j) - x(i) if (abs(dx) > xsize*0.5) dx = dx - sign(xsize,dx) endif

Naturally both operations should be repeated in all 3 dimensions.

These operations can be written in much more compact form for orthorhombic cells if the origin is shifted to a corner of the box. Then we have, in one dimension, for positions and distances respectively:

! After x(i) update without regard to PBC: x(i)=x(i)-floor(x(i)/xsize)*xsize !For a box with the origin at the lower left vertex ! Works for xs lying in any image. dx=x(j)-x(i) dx=dx-nint(dx/(0.5*xsize))*xsize

For non-orthorhombic cells the situation can be considerably more complicated.

In simulations of ionic systems considerably more complicated operations may be needed to handle the long-range Coulomb interactions.

Read more about this topic:  Periodic Boundary Conditions

Famous quotes containing the words practical, continuity, minimum, image and/or convention:

    “It is always a practical difficulty with clubs to regulate the laws of election so as to exclude peremptorily every social nuisance. Nobody wishes bad manners. We must have loyalty and character.”
    —Ralph Waldo Emerson (1803–1882)

    “There is never a beginning, there is never an end, to the inexplicable continuity of this web of God, but always circular power returning into itself.”
    —Ralph Waldo Emerson (1803–1882)

    “After decades of unappreciated drudgery, American women just don’t do housework any more—that is, beyond the minimum that is required in order to clear a path from the bedroom to the front door so they can get off to work in the mourning.”
    —Barbara Ehrenreich (20th century)

    “The whole world is an omen and a sign. Why look so wistfully in a corner? Man is the Image of God. Why run after a ghost or a dream? The voice of divination resounds everywhere and runs to waste unheard, unregarded, as the mountains echo with the bleatings of cattle.”
    —Ralph Waldo Emerson (1803–1882)

    “No convention gets to be a convention at all except by grace of a lot of clever and powerful people first inventing it, and then imposing it on others. You can be pretty sure, if you are strictly conventional, that you are following genius—a long way off. And unless you are a genius yourself, that is a good thing to do.”
    —Katharine Fullerton Gerould (1879–1944)