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:

    “Despair, feeding, as it always does, on phantasmagoria, is imperturbably leading literature to the rejection, en masse, of all divine and social laws, towards practical and theoretical evil.”
    —Isidore Ducasse, Comte de Lautréamont (1846–1870)

    “Every generation rewrites the past. In easy times history is more or less of an ornamental art, but in times of danger we are driven to the written record by a pressing need to find answers to the riddles of today.... In times of change and danger when there is a quicksand of fear under men’s reasoning, a sense of continuity with generations gone before can stretch like a lifeline across the scary present and get us past that idiot delusion of the exceptional Now that blocks good thinking.”
    —John Dos Passos (1896–1970)

    “There are ... two minimum conditions necessary and sufficient for the existence of a legal system. On the one hand those rules of behavior which are valid according to the system’s ultimate criteria of validity must be generally obeyed, and on the other hand, its rules of recognition specifying the criteria of legal validity and its rules of change and adjudication must be effectively accepted as common public standards of official behavior by its officials.”
    —H.L.A. (Herbert Lionel Adolphus)

    “Maybe the bride-bed brings despair,
    For each an imagined image brings
    And finds a real image there....”
    —William Butler Yeats (1865–1939)

    “Mankind owes to the child the best it has to give.”
    —United Nations Convention on the Rights of the Child, 1989.