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.

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.

KCaF3 C8 |
KMgF3 C10 |
CsBeI3 C12 |

LiCaF3 R9 |
LiZnCl3 R11 |
LiBeI3 R13 |

Admin.: