5. Results

5.2 Analysis of the molecular graph


Visualizing the arrangements of critical points and the connections between them is not an easy task given the huge number of points that lie within the unit cell (from 58 CP in the C8 scheme to 104 CP in the R13). The most significant and comprehensive form that we have found is by depicting the attraction basins for each nuclei.

The attraction basin of a nucleus is made out of all the points such that a trajectory that starts at the point and moves upward in the gradient field of the electronic density ends in the nucleus. The interior of an attraction basin is then a 3-dimensional region of the crystal with no other CP inside but the nucleus itself. All non-nuclear CP's lie, therefore, on the 2-dimensional zero-flux boundaries of the different attraction basins of the crystal. The latter are called atomic surfaces. The set of CP's situated on the atomic surface of a given nucleus is said to be related to it. Despite the general shape an atomic surface may display, its set of related CP's induce a homeomorphism between the surface and a polyhedron. This relation is established as follows: each cage point on the surface is defined as a vertex of the associated polyhedron. Every bond point is mapped into a face, made up of all the gradient lines ending up at that point. Last, but not least, we associate an edge to each ring point, defined by the two only gradient lines that die at it. If we draw onto the atomic surface all the vertices and edges of the associated polyhedron, we may compare the actual atomic surfaces with the ideal (planar faces and linear edges) mapped polyhedron. Though symmetry, when present, is a very powerful means of assuring planarity and linearity of the actual faces and edges, it is not uncommon that large curvatures do appear in some cases.

It is interesting to note that physicists and chemists tend to think of atoms in crystals as to slightly deformed spheres. Spheres, however, do not fill the three dimensional space without gaps and are thus inappropriate approximations to the attraction basins shape in crystals. The ideal polyhedra just defined do represent a much better approach, but we have to keep in mind that in passing from curved faces and edges to linear representations some fictitious facets may appear, that is, facets that are not associated to a bond point.

The attraction basins of Li, Ca and F in LiCaF3, a crystal with R9 topology, are depicted in Fig. 5.1. We can see that the basin of Li is a rounded rhombododecahedron. Each rhomb is associated to a Li-F bond. The F basin only has six real faces: four concave rhombi related to Li-F bonds, and two large and curved squares corresponding to Ca-F bonds. In a polyhedral approximation, every curved square would appear as a smaller square with four trapezoids connected to it by an edge each. Were they real, the trapezoids of the F basin would represent F-F bonds, that do not exist in R9. To avoid face to face contact between two neighbor F basins, this implying F-F bond, the Ca basins develop nearly bidimensional wings that perfectly envelop the F basins. Apart from those wings, the Ca basin could be described as a slightly engrossed cube. A similar behavior is observed in C8 and B10 crystals.

Fig. 5.1 Attraction basins of the ions in LiCaF3, a crystal having R9 topological scheme. Different colors are used for each ion: gold (Ca), green (F) and fuchsia (Li). The figure was created with critic and rendered with geomview.

The C10, R11, C12, and R13 schemes, on the other hand, do not suffer from this circumstance, and the face that separates two neighbor X ions is a true bond. Accordingly, the M basin is topologically equivalent to a cube: the basins show more or less acute spikes on the cube vertices for C10 and R11, and a round, nearly spherical, shape for C12 and R13 (see Fig. 5.2).

It is an important observation that the topological schemes having the smallest numbers of symmetrically different CP's are the most complicated ones in the geometrical sense. Contrarily, the schemes with T=12 and T=13 are the simplest ones and present shapes easily related to the classical concept of spherical cations. In all cases the anion basin stretches to fill in the gap between cations.

Fig. 5.2 Illustration of the ionic basins in different representative perovskites. The basins fill completely the space leaving no holes. Click on each image to get a larger version. The basins have been calculated using critic and rendered with POVRay 2.2.
KCaF3 C8 KMgF3 C10 CsBeI3 C12
LiCaF3 R9 LiZnCl3 R11 LiBeI3 R13

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Page dates: May 20, 1996 (creation), June 7, 1996 (last update).
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Copyright © 1996 Víctor Luaña, Aurora Costales and Angel Martín Pendás.