First of all, we will examine how many critical points do appear and in which positions within the unit cell.

It can be easily shown that site symmetry
forces some special positions of the cell to be
CP's of the electron density. This is the case for the Wyckoff's
1a, 1b, 3c, and 3d positions in the Pm3m
space group of the perovskites. Three of these positions are occupied
by the crystal ions: *A* occupies 1b, *M* 1a, and *X*
3d. The 3c positions lie at the center of the unit cell cubic
faces, octahedrically surrounded by 4 equatorial *X* ions and two
axial *A* ions, and they play a very
important role in determining the overall topology of the electronic
density.

After analyzing all 120 halide perovskites, we have found that the
electronic density can be classified into one of seven differents
*topological schemes* (i.e. seven different arrangements of critical
points). Seven may appear to be a large number for a group of apparently
quite similar compounds. On the other hand, we have performed some numerical
experiments by placing two-electron imaginary ions, represented as
single 1s Slater type orbitals, in the ionic sites. By independently
varying the three orbital exponents we were able to generate a rather
large number of topological schemes, including the seven actually
obtained for the halide perovskites. This numerical experiment proved
that the physical limitation of the ion sizes effectively reduces to a
large extent the possible topological schemes.

The seven topological schemes can be organized into three families
according to the character of Wyckoff's 3c position: (a) 3c is a
bond CP in the **B** family; (b) a ring point in the
**R** family; and (c) a cage point in the **C** family.
The special positions of the Pm3m group have been included in
Table 5.1, and the number and positions of the
CP's in the seven topological schemes have been summarized in
Table 5.2.

The translationally invariant crystals are topologically equivalent to the 3-torus and, consequently, all topological schemes must satisfy Morse invariant relationships (Morse and Cairns 1969):

and

where n represents the total number of nuclei in the unit cell,
b the number of bonds, r the number of rings, and c the number
of minima or cages. On the other hand, the topological
schemes can be easily organized in terms of the
*total number of symmetrically different* CP: **T**.
All topological schemes found for the perovskites can be uniquely
identified by giving the family and value of **T**. Furthermore,
every scheme has a different **T**, except **B10**
and **C10**.

A single scheme, **B10**, forms the **B** family. The
scheme presents three different bond CP's: b4,
b1 and b2, corresponding to *A*-*A*,
*M*-*X*, and *A*-*X*
bonds, respectively. The existence of the *A*-*A* bond is the most
distinctive aspect of the **B10** scheme, and it is the
consequence of two combined factors: a very large *A* to *X*
size ratio, and a large cell side length a due to the large
size of *M*. This is an unusual combination and, in consequence,
only two out of the 120 crystals belong to this scheme:
CsSrF3 and CsBaF3.

The **C** family comprises three topological schemes,
all of them having an even number of different CP's:
**C8**, **C10**, and **C12**.
The **R** family comprises three topological schemes,
all of them having an odd number of different CP's:
**R9**, **R11**, and **R13**.
Both families maintain a close relationship and exhibit identical
mechanisms in going from the simplest to the most complex
scheme. Accordingly, the **C10** and **R11**
schemes are obtained from the **C8** and **R9**
ones, respectively,
by adding both a bond and a ring point at
Wyckoff's 12i position.
The addition of both a ring and a cage point at the 8g
position originates the **C12** and **R13**
schemes.

We see here a simple mechanism for increasing the complexity of a topological scheme: add new CP's in pairs, each new point being of a type with different sign in the Morse sum. Either bonds and rings, or rings and cages would do the trick. If both types of CP's appear in the same Wyckoff's position, or in two positions with identical multiplicity, the invariance required by the Morse relation is automatically fulfilled.

The **C8** and **R9** schemes, the simplest in
their respective families, present just two types of bond
CP's. The bond at Wyckoff's 6e position lies at the edges of the
cubic unit cell and constitutes a bond between the divalent metal
and the halide: *M*-*X*. The bond point in 12j, on the other
hand, represents an *A*-*X* bond. The other four schemes of the
**C** and **R** families also show a *X*-*X* bond
occupying the 12i position.

**R** is the most frequent family. Both, **R** and
**C**, families
show a decreasing number of ions in passing from the simplest
to the most complex topological scheme. The actual number of
crystals belonging to each scheme is: 21 (**C8**),
27 (**R9**), 15 (**C10**), 25 (**R11**),
12 (**C12**), and 18 (**R13**).

Wyckoff | Symmetry | Symmetry | Position |

1a | Oh | m3m | (0,0,0) |

1b | Oh | m3m | (1/2,1/2,1/2) |

3c | D4h | 4/mm.m | (0,1/2,1/2) |

3d | D4h | 4/mm.m | (1/2,0,0) |

6e | C4v | 4m.m | (x,0,0) |

6f | C4v | 4m.m | (x,1/2,1/2) |

8g | C3v | .3m | (x,x,x) |

12h | C2v | mm2.. | (x,1/2,0) |

12i | C2v | m.m2 | (0,x,x) |

12j | C2v | m.m2 | (1/2,x,x) |

24k | Cs | m.. | (0,y,z) |

24l | Cs | m.. | (1/2,y,z) |

24m | Cs | ..m | (x,x,z) |

48n | C1 | 1 | (x,y,z) |

Wyckoff | CsSrF3 | KCaF3 | LiCaF3 | KMgF3 | LiZnCl3 | CsBeI3 | LiBeI3 |

1a | n:Sr | n:Ca | n:Ca | n:Mg | n:Zn | n:Be | n:Be |

1b | n:Cs | n:K | n:Li | n:K | n:Li | n:Cs | n:Li |

3c | b4 | c1 | r2 | c1 | r2 | c1 | r2 |

3d | n:F | n:F | n:F | n:F | n:Cl | n:I | n:I |

6e | b1 | b1 | b1 | b1 | b1 | b1 | b1 |

6f | --- | --- | c3 | --- | c3 | --- | c3 |

8g | c2 | c2 | c2 | c2 | c2 | c2 | c2 |

" | --- | --- | --- | --- | --- | r4 | r4 |

" | --- | --- | --- | --- | --- | c4 | c4 |

12h | r5 | --- | --- | --- | --- | --- | --- |

12i | c5 | --- | --- | r3 | r3 | r3 | r3 |

" | --- | --- | --- | b3 | b3 | b3 | b3 |

12j | b2 | b2 | b2 | b2 | b2 | b2 | b2 |

24k | --- | --- | --- | --- | --- | --- | --- |

24l | --- | --- | --- | --- | --- | --- | --- |

24m | r1 | r1 | r1 | r1 | r1 | r1 | r1 |

48n | --- | --- | --- | --- | --- | --- | --- |

Scheme | B10 |
C8 |
R9 |
C10 |
R11 |
C12 |
R13 |

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