Víctor Luaña Cabal
Quantum Chemistry Group

Ion  Wyckoff  Position 
A  (1b)  (1/2,1/2,1/2) 
M  (1a)  (0,0,0) 
X  (3c)  (1/2,0,0) 
Description: M cations are octahedrally coordinated by the X anions. The A ions appear in the center of a cube of MX6 octahedra. In the regular perovskites M is the high valence and A the low valence cation (e.g. SrTiO3 where A=Sr(2+) and M=Ti(4+)). There exist an small number of inverse perovskites for which the role of both cations is inverted.
Examples: KMgF3 (a=3.989 A at room temperature).
Ion  Wyckoff  Position 
A  (4c)  (0,1/2,1/4) 
M  (4b)  (0,0,0) 
X1  (4a)  (0,0,1/4) 
X2  (8h)  (1/4+u,3/4+u,0) 
Description: It is equivalent to the ideal perovskite if c/a=sqrt(2) and u=0. It can be described as the result of a (a0,a0,c) rotation of the ideal perovskite (Glazer notation).
Examples: RbCaF3 (a=6.2744, c=8.9125, u=0.0252 at 150K).
Ion  Wyckoff  Position 
A  (6a)  (0,0,1/4) 
M  (6b)  (0,0,0) 
X  (18e)  (x,0,1/4) 
Description: Tends to the ideal perovskite when c/a=sqrt(6) and x=1/4.
Examples: CaCO3 (a=5.092, c=16.28, x=0.2578)
The Atoms In Molecules (AIM) theory of R.F.W. Bader and cols. provides an unambiguous and rigorous method for defining the atomic regions (basins) in a molecule or crystal, based on the topological properties of the system wavefunction. Either theoretical or experimental wavefunctions could be used. Formal details can be consulted online, or better find the next excellent book in a Library near you:
The representation of the ionic basins provides new insight on the geometrical and structural properties of crystals. Let us examine the rocksalt phase of the lithium iodide (LiI) crystal. The image on the right presents the basins of Li(+) (small) and I() (large). The left image presents an approximation to the basins made of convex polyhedra, superimposed to the representation of the unit cell. Each face of the polyhedra corresponds to a bond. The square faces are associated to the LiI bond, whereas the hexagonal faces appear as a consequence of the II bond.
The value of several important scalar fields can be represented on the basin surfaces to get more insight on the crystal bonding. The color scale used goes linearly from blue (low values of the field) to red (high values).
Represented in this image we see the electron density of LiI on the surface of the ionic basins. High density values appear in the middle of the polyhedra faces, each corresponding to the existence of a bond. The accumulation of charge is larger in the hexagonal faces associated to the II bond than in the square faces of the LiI bond, thus suggesting that the II bond may dominate the crystal properties.
According to the AIM theory, however, the strength of a bond is not determined by the electron density but by the laplacian ("curvature") of the electron density. We see in the image to the left that the laplacian is very large for the LiI bond, the II bond being significantly less strong. Accordingly, the structural properties of the crystal will be dominated by the cationanion bonds, as it is usually assumed in crystal chemistry.
Even though not very interesting from the physical point of view, the representation of the gradient module of the electron density provides a clear picture of the critical points and regions (those in which the gradient of the electron density is zero) nicely colored in blue.
The question of how to distribute N points "evenly" on the surface of an sphere is a frequent topic on Usenet, and several interesting documents are available online:
I became interested in the problem while developing, in collaboration with A. Martín Pendás, the code to plot the atomic basins presented above. A difficulty that must be taken into account is that the basin surfaces are, in general, nonconvex polyhedra. The best method we have devised selects the sampling directions by distributing N points "regularly" on the surface of a sphere centered on the nucleus. The points are connected to form a convex polyhedron and then projected onto the basin surface.
We have tried several methods for generating the sample points on the sphere and, simultaneously, form the convex hull: