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 endifThe 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) endifNaturally 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))*xsizeFor 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:
“Great statesmen seem to direct and rule by a sort of power to put themselves in the place of the nation over which they are set, and may thus be said to possess the souls of poets at the same time they display the coarser sense and the more vulgar sagacity of practical men of business.”
—Woodrow Wilson (1856–1924)
“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)
“For me, the child is a veritable image of becoming, of possibility, poised to reach towards what is not yet, towards a growing that cannot be predetermined or prescribed. I see her and I fill the space with others like her, risking, straining, wanting to find out, to ask their own questions, to experience a world that is shared.”
—Maxine Greene (20th century)
“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)