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 do not remember anything which Confucius has said directly respecting man’s “origin, purpose, and destiny.” He was more practical than that. He is full of wisdom applied to human relations,—to the private life,—the family,—government, etc. It is remarkable that, according to his own account, the sum and substance of his teaching is, as you know, to do as you would be done by.”
    —Henry David Thoreau (1817–1862)

    “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)

    “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)

    “My image is a statement of the symbols of the harsh, impersonal products and brash materialistic objects on which America is built today. It is a projection of everything that can be bought and sold, the practical but impermanent symbols that sustain us.”
    —Andy Warhol (1928–1987)

    “The metaphor of the king as the shepherd of his people goes back to ancient Egypt. Perhaps the use of this particular convention is due to the fact that, being stupid, affectionate, gregarious, and easily stampeded, the societies formed by sheep are most like human ones.”
    —Northrop Frye (b. 1912)