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:

    “I favor the policy of economy, not because I wish to save money, but because I wish to save people. The men and women of this country who toil are the ones who bear the cost of the Government. Every dollar that we carelessly waste means that their life will be so much the more meager. Every dollar that we prudently save means that their life will be so much the more abundant. Economy is idealism in its most practical terms.”
    —Calvin Coolidge (1872–1933)

    “Only the family, society’s smallest unit, can change and yet maintain enough continuity to rear children who will not be “strangers in a strange land,” who will be rooted firmly enough to grow and adapt.”
    —Salvador Minuchin (20th century)

    “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 good almost kill a man as kill a good book; who kills a man kills a reasonable creature, God’s image; but he who destroys a good book, kills reason itself, kills the image of God, as it were in the eye.”
    —John Milton (1608–1674)

    “No good poetry is ever written in a manner twenty years old, for to write in such a manner shows conclusively that the writer thinks from books, convention and cliché, not from real life.”
    —Ezra Pound (1885–1972)