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:

    “Theorists may say what they like about a man’s children being a continuation of his own identity, but it will generally be found that those who talk in this way have no children of their own. Practical family men know better.”
    —Samuel Butler (1835–1902)

    “Continuous eloquence wearies.... Grandeur must be abandoned to be appreciated. Continuity in everything is unpleasant. Cold is agreeable, that we may get warm.”
    —Blaise Pascal (1623–1662)

    “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)

    “As every pool reflects the image of the sun, so every thought and thing restores us an image and creature of the supreme Good. The universe is perforated by a million channels for his activity. All things mount and mount.”
    —Ralph Waldo Emerson (1803–1882)

    “Every one knows about the young man who falls in love with the chorus-girl because she can kick his hat off, and his sister’s friends can’t or won’t. But the youth who marries her, expecting that all her departures from convention will be as agile or as delightful to him as that, is still the classic example of folly.”
    —Katharine Fullerton Gerould (1879–1944)