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):
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).