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:
“No delusion is greater than the notion that method and industry can make up for lack of mother-wit, either in science or in practical life.”
—Thomas Henry Huxley (1825–95)
“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)
“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)
“The exile is a singular, whereas refugees tend to be thought of in the mass. Armenian refugees, Jewish refugees, refugees from Franco Spain. But a political leader or artistic figure is an exile. Thomas Mann yesterday, Theodorakis today. Exile is the noble and dignified term, while a refugee is more hapless.... What is implied in these nuances of social standing is the respect we pay to choice. The exile appears to have made a decision, while the refugee is the very image of helplessness.”
—Mary McCarthy (1912–1989)
“No convention gets to be a convention at all except by grace of a lot of clever and powerful people first inventing it, and then imposing it on others. You can be pretty sure, if you are strictly conventional, that you are following genius—a long way off. And unless you are a genius yourself, that is a good thing to do.”
—Katharine Fullerton Gerould (1879–1944)