# Is a 2-D periodic structure isomorphic with the surface of a torus, a sphere, neither or both?

When I was reading through the ADF-BAND tutorials, one of the toy systems presented was a 1-D periodic structure involving 3 collinear hydrogen atoms. The tutorial pointed out that, topologically speaking, this is cylindrically symmetrical (more specifically, it's ring symmetrical).

In the instance of a 2-D structure, can the calculation be considered a model for the surface of a torus (this seems logical), a sphere (I doubt this because if you fit a rectilinear grid to a sphere you end up with two poles and dissimilar meridians and parallels), or other?

Bonus questions: Has anyone used periodic calculations to model electronic structure/chemistry on the surface of a sphere or torus? Can you introduce a curvature term to account for these structures having a finite size?

• Yes, it's the same thing, at least according to physics.stackexchange.com/questions/21882/… May 7 '12 at 6:33
• @Manishearth - the same as which? May 7 '12 at 6:43
• I was saying that a repeating 2D space is the same as a torus (topologically) May 7 '12 at 6:58

Yes, a 2D-periodic space can be mapped to a torus, but that's more a question for the math.SE

Regarding your bonus question, why would there be? What would you do with it? Molecular structures are intrinsically 3D, so I don't see what you would do in a 2D (periodic or not) space? Even when we talk about planar or pseudo-2D structures (buckyball, nanotube, etc.) they are 3D objects with 3D electronic densities and wavefunctions.

Edit: 3D structures that are periodic in two dimensions and finite in the other one can be studied by many computational chemistry codes. They are often referred to as slab calculations or surface calculations. The most common issue is that of Coulombic interaction (or Poisson equation solver), which typically requires special treatment in the 2D case.

• By 2-D I mean structures which are periodic in 2 dimensions but finite in a third. A potential motivation is to model tubular structures that are too large to feasibly solve aperiodically. May 7 '12 at 7:05
• @RichardTerrett OK, I edited my answer accordingly… but I don't understand what you mean by “curvature term”, then.
– F'x
May 7 '12 at 9:44
• By this I mean an element of the calculation that corrects for the distortion(s) of the plane arising from being mapped to a torus with non-zero local curvature. May 7 '12 at 14:36
• @RichardTerrett then there is no need too… while the “mathematical” view of a 2D periodic space is akin to a 3D torus, I don't believe any technique out there would really physical cast the 2D structure onto a 3D torus to perform any simulation.
– F'x
May 7 '12 at 15:15

2D periodic systems can be mapped to toruses, but not spheres. This is easy to see because in a sphere, parallel lines always intersect. In the periodic system parallel lines never intersect.

Regarding your bonus question: I don't know of anyone who has tried to use a periodic model to study a sphere or torus. But people have kind of gone the other way around, and replaced a 3D periodic model with the surface of a 4D sphere. This allows you to avoid complications associated with long-range Coulomb interactions.