(Presented to the Second Electronic Computational Chemistry Conference, 1995)

Ions in crystals: The topology of the electron density in the cubic halide perovskites.

Víctor Luaña, Aurora Costales, and A. Martín Pendás
Departamento de Química Física y Analítica
Universidad de Oviedo, 33006-Oviedo, Spain
email: victor@carbono.quimica.uniovi.es

  1. Introduction
  2. Computational scheme
  3. Results
  4. Final remarks and conclusions
  5. Acknowledgements
  6. References

1. Introduction

The Atoms In Molecules (AIM) Theory by R.F.W. Bader and collaborators [1] has achieved, within the last years, widespread acceptation in the Quantum Chemistry community, and it has been applied to a large number of gas phase small molecules and metal clusters. Contrarily, very few studies have been published concerning solid state systems [2,3,4, 5]. Particularly, the complete characterization of the topological properties of the electronic density in a solid has not been done before, partly because of the great difficulty in determining all critical points (CP's) and of integrating over the atomic basins.

Several important differences appear, from the point of view of AIM theory, between gas phase molecules and crystalline solids:

  1. atomic basins in solids have well defined boundaries and encompasse a finite volume;
  2. whereas ring and cage CPs are rare in most organic molecules, they are quite abundant in crystals;
  3. as a consequence of the above, the molecular graph of a solid can be very complicated and, what is more important, it gives only a very poor image of the topological properties of the electronic density.

In this report we discuss some of the most significant features of the topology of the electronic density in 120 alkali-halide perovskites AMX3 (A: Li, Na, K, Rb, Cs; M: Be, Mg, Ca, Sr, Ba, Zn; X: F, Cl, Br, I). This work is part of an ongoing effort to determine and analyze the structural and chemical stability of the AmMnXo compounds. Consequently, one of our motivations in applying AIM theory to the analysis of the perovskites was to determine whether a connection between stability and electronic density topology exists or not.

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2. Computational scheme

To begin with the analysis of the electronic density we need to know the equilibrium geometry of the compound. Experimental geometries are not available for most of the perovskites analyzed in this work. Therefore, we have proceeded by determining, as a first step, the theoretical equilibrium geometry of every crystal.

The ab initio Perturbed Ion (aiPI) [6,7] method has been used to determine the best geometry for each crystal. Best available Slater Type function (STO) basis sets have been used for all ions. The unrelaxed hard Coulomb hole formalism (uCHF) has been used to estimate the correlation energy [8], but the crystal wavefunction has been obtained at the Hartree-Fock level, not being modified by the correlation estimation procedure. All coulomb interactions have been integrated exactly. Interionic exchange and projection operators, on the other hand, have been accounted for neighbors up to 25 bohr away.

The perovskite structure [Cubic, Pm3m, A at (1/2,1/2,1/2), M at (0,0,0), and X at (1/2,0,0), See Fig. 1] fixes all geometrical parameters but one: the cell side length a. We have optimized a for all the 120 compounds using a simplex method.

The ionic wavefunctions obtained from a final aiPI calculation at the optimal geometry have been used in the critic program [9] to determine the crystal wavefunction for each crystal. Ions up to 3 cell side lengths contribute to the wavefunction at any point. The program implements a new algorithm to locate automatically all critical points (CP's) of the electronic density. The algorithm works by first determining the irreducible zone of the Wigner-Seitz cell (IWSZ). The IWSZ is the smallest polyhedron that reproduces the complete crystal by the exhaustive application of the space group symmetry. Vertices of the IWSZ are guaranteed to be CP's. Similarly, every edge and face of the IWSZ contains at least one CP, and further CP's could be found within the IWSZ. The search algorithm explores the corresponding 1-D, 2-D and 3-D regions. Every time a new CP is found in a region, the region is splitted into two or more new ones and a further search is performed on each. As for the searching criterion, we have found that minimizing the square of the density gradient is far more efficient and less error prone than other navigation techniques based on the gradient and hessian of the function.

The automatic routine has worked remarkably well in most crystals. Early runs that included an insuficient number of ions in the evaluation of the wavefunction were unable to locate some CP's in some crystals. Once this problem was solved, the only failures of the automatic procedure were: (a) false positives of points that showed rather large gradients and that were easily discarded by inspection; and (b) remarkably, Wyckoff's 3c position, which is a CP fixed by symmetry (see below), was considered doubtful in some crystals by the automatic discard criteria. Morse relationships and similarities between compounds have been analyzed to make sure that all CP's were found in each case.

Fig. 1 Crystallographic (balls and sticks) and Wigner-Seitz (transparent box) unit cells of the cubic perovskites. Dark grey balls represents M ions, green balls X ions, and blue balls are A ions. Small red balls are placed at the Wyckoff's 3c position and will play a very significant role for the topological properties. The figure was designed with pairpot3 and rendered with POV-Ray. An high-quality jpg figure (57 kbyte) is available.

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3. Results

Topological structure of the electronic density

It can be easily shown that site symmetry determines that some special positions of the cell must 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 1, and the number and positions of the CP's in the seven topological schemes have been summarized in Table 2.

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

n - b + r - c = 1


n>=1, b>=3, r>=3, c>=1,

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: 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 in 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).

Table I Wyckoff's special points of the Pm3m space group.
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)

Table II The seven topological schemes of the perovskites. The different bonds correspond to: M-X (b1), A-X (b2), X-X (b3), and A-A (b4).
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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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 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, to each ring point we associate an edge, 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 no 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 LiCaF3, a crystal with R9 topology, are depicted in Fig. 2. 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 (see Fig. 3), and a round, nearly spherical, shape for C12 and R13 (see Fig. 4).

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. 2 Attraction basins of the ions in LiCaF3, a crystal having R9 topological scheme. Different colors are used for each ion: golden (Ca), green (F) and fuchsia (Li). The figure was created with critic [9] and rendered with geomview [13].

Fig. 3 Attraction basins of the ions in LiZnCl3, a crystal having R11 topological scheme. Different colors are used for each ion: golden (Zn), green (Cl) and fuchsia (Li). The figure was created with critic [9] and rendered with geomview [13].

Fig. 4 Attraction basins of the ions in LiBeI3, a crystal having R13 topological scheme. Different colors are used for each ion: golden (Be), green (F) and fuchsia (Li). The figure was created with critic [9] and rendered with geomview [13].

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Origin of the topological schemes

Once the different topological schemes have been described to a large detail, the question is what crystal exhibits which scheme and why. In doing this analysis, we will concentrate into C and R schemes, the B10 one being rare as it was described before.

We can define topological ionic radii as the distance from the nucleus to the surface of its attraction basin. It is evident from Figs. 2 to 4 that such topological radii do depend strongly on the direction being analyzed. Not all directions are equally important from the chemical point of view: the directions that contain bonds are the most significant ones for the crystal stability. In this way, the b1 CP, that represents a M-X bond, provides estimated ionic radii for M and X, that we will designate as R(M) and R1(X), respectively. Similarly, the b2 CP represents an A-X bond, and provides the R(A) and R2(X) radii. The b3 CP, found in many crystals, corresponds to a X-X bond and gives R3(X).

We have found, after some initial surprise, that the ratio of ionic radii along the M-X and A-X bond lines completely determines the topological scheme exhibited by the crystal. To show this we have prepared Fig. 5, in which we have plotted the ratio R(M)/R1(X) versus the ratio R(A)/R2(X), every crystal being represented by a symbol different for each type of topological scheme.

It is observed that the C and R topological families are decided according to the value of R(A)/R2(X). If the A ion is larger than about 0.88 times the X ion in the A-X bond line, the crystal belongs to the C family, and it belongs to the R family if it is smaller.

The R(M)/R(X) ratio, on the other hand, is in control of the topological complexity within each family: the larger M is in comparison to X, the simpler the topological scheme appears.

It is interesting to notice that althought the topological radii behave similarly to Shannon's empirical radii, only the topological values provide an exact criterion to predict the topological family.

Fig. 5 The topological schemes are determined by the ratio of topological radii R(A)/R2(X) and R(M)/R1(X).

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Ionic properties

The topological analyses offer detailed information regarding individual ions within the crystal. We will analyze here some of the properties obtained by integrating appropriate functionals on the attraction basins. More concretely, we will examine the ionic charge (Q) and the average ionic radius (R) defined as the radius of an sphere that would contain the same volume as the attraction basin of the ion.

The integration on the ionic basins is computationally quite demanding. The algorithm implemented in the critic program [9] involves sampling a number of rays defined by its (theta,phi) spherical coordinates. The position of the basin surface is determined along each ray, and a Gaussian quadrature is used to integrate with respect to the radial coordinate for each ray. Site symmetry is used to improve the performance of this method. A grid of 40x40x60 points on (theta,phi,r) has been used in this work. This grid produces an accuracy better than 0.01 Å3 in the integrated total volume of the cell, and better than 0.001 a.u. in the cell total charge.

All perovskites behave clearly as ionic compounds, as indicated by the topological charges (See Table IV). The charge of the alkaline ions is very close to the nominal +1 value, the halides have an average charge Q(X)=-0.95, and the alkaline earth ions vary from Q(Be)=1.95 to Q(Zn)=1.8. The variability of the charge in the alkaline ions, on the other hand, increases as we descend the Periodic Table, effect that is not observed neither on the halides nor on the alkaline earth ions.

Several interesting facts are observed in analyzing the average ionic radii (See Fig. 6). First of all, the ionic size increases, within each group of ions, with the atomic number, as it should be the case if we want to claim any physical meaning for the topological property. Secondly, the dispersion of ionic radii increases, within each group of ions, as we descend in the periodic table, following the trend of the empirical scale of polarizabilities. With regard to the differences between groups, the alkaline earth ions present a rather constant radii along different compounds, with the noticeable exception of Ba, and halide ions exhibit a greater variability. The most interesting fact is, however, the large variability of radii observed for the alkaline ions, even larger than that found for the halides, in sharp contradiction with the polarizabilities of both groups. This circumstance suggests that the variability of R(A) has a different origin that the variability of R(M) and R(X). It appears that the A ions are weakly binded and easily vibrate within the cage made up of M and X ions.

Table IV Topological ionic charge obtained averaging to all the crystals.
 Ion   charge    Ion   charge        Ion   charge

  be   1.9541     li   0.9970         f   -0.9630
  mg   1.8908     na   0.9914         cl  -0.9466
  zn   1.8113     k    0.9896         br  -0.9490
  ca   1.8634     rb   0.9852         i   -0.9499
  sr   1.8321     cs   0.9990
  ba   1.8256
Fig. 6 Average ionic radii as obtained after the volume of the attraction basin.

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Stability of the perovskites

One of our objectives when we started this study was to determine whether the stability of the crystal is correlated or not with the topological structure. The lattice energy, which measures the difficulty in breaking apart the crystal into its isolated ions, is dominated by the electrostatic interactions and follows the simple Q/a Madelung law. Grouping together the crystals that have the same M(+2) cation, we may also observe, as a secondary effect, that the lattice energy is proportional to the 1/3 power of the value of the electronic density at the M-X bond CP.

On a more fundamental basis, the stability of the crystal should be addressed with respect to: (a) solid state reactions producing compounds of different stoichiometry; and (b) the existence of crystals with the same AMX3 stoichiometry but a different crystalline structure. To our knowledge there is neither experimental nor theoretical data regarding the thermodynamic stability of perovskites but for a handful of compounds.

The relation between the topology of the energetic hypersurfaces connecting several crystalline structures has been previously investigated for the alkali halides [14]. Based on the definition of topological ionic radii we were able to establish a single criterion that exactly predicted the stability of the B1 and B2 structures, defined in terms of the properties of the Gibbs free energy. The relation between phase transitions and topological structure of the perovskites, albeit very interesting, is beyond the scope of this work.

On the other hand, perovskites are usually prepared by melting a mixture of MX2 and AX crystals in 1:1 stoichiometric proportions and slowly cooling whereas the AMX3 crystal is simultaneously grown and extracted according to the Czochralski procedure. Even though, to the better of our knowledge, there is no systematic study on the energetics of the above reaction, most of the 120 AMX3 compounds have never been referred to in the literature, which suggests that most of the perovskites are unstable with respect to MX2+AX.

To address this problem we have determined the energy of the reaction:

MX2(s) + AX(s) ---> AMX3(s)


E = E(AMX3) - E(MX2) - E(AX)

where E(compound) is the aiPI total energy. The fluorite structure [Cubic, Fm3m, M(+2) at (0,0,0), X(-1) at (1/4,1/4,1/4)] has been assumed for the MX2 compounds, and the rock-salt structure has been assumed for the AX compounds. Even though perovskite, fluorite and rock-salt are not necessarily the most stable structures of the three types of compounds, the difference should not be very large nor it should modify the general trends for the whole set of crystals.

Fig. 7 represents the reaction energy together with the topological scheme for the 120 AMX3 compounds. The first thing to notice is that the crystal stability is dominated by the identity of the M(+2) cation: Ba, Sr, and most Ca perovskites are unstable, whereas most Zn and Mg, and all Be perovskites are stable. For a given M(+2), stability is favored by having large A(+1) ions. The role of X(-1) is more complex, but it is seen that fluorides are different to the other halides, either more or less stable, the chlorides, bromides and iodides behaving rather similarly.

It is very interesting to observe in Fig. 7 that all crystals having R9 and B10, and most crystals having C8 topological schemes, are unstable. Contrarily, all C12 and most R13 crystals are stable. It appears that the existence of bizarre features, like wings, in the attraction basins works against the stability of the crystal. Alternatively, we can argue that the formation of X-X bonds is important to stabilize the structure. This sheds new light to the question of what is simple and what is complex in relation to the topological structure of the electronic density. The simple enumeration of critical points would suggests that those topological schemes having smallest values for T are the simplest ones. The shape of the attraction basins is simpler, however, in the schemes of large T and, as the analysis of the above reaction indicates, those schemes are more favored in energetic terms.

The above results refer to the difference between quantum mechanical total energies. It corresponds, thermodynamically, to null temperature and pressure. We have investigated the role of thermal effects, up to about 900 K, on the perovskite stability and have found interesting but small effects that do not modify the general trends discussed above.

Fig. 7 Stability of the perovskite structure with respect to the decomposition into MX2+AX. Different symbols are used for each topological scheme. Negative reaction energies imply that the perovskite is stable. The crystal numbering corresponds to three nested do loops: for M in (Be Mg Zn Ca Sr Ba) for A in (Li .. Cs) for X in (F .. I) {crystal AMX3}.

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Final remarks and conclusions

Nuclei act as the attractors in the electronic density gradient fields. Since the topological structure and stability are determined as a result of the balance between the attractors of the system, we should anticipate a significant degree of similarity in the topological properties of the AMX3 perovskites. The rich variety of topological schemes found is, in this regard, an interesting surprise. Our study has shown that the organizing principle behind that variety is the ratio between topological ionic radii along the main bond paths or, in other words, the competition between atomic basins to achieve a kind of preferred size.

Ionic radii along the bond paths follow systematic trends in passing from one crystal to another, even though the ionic radius of an ion in a crystal do strongly depend on the direction being considered.

Ionic basins have, essentially, a polyhedral shape, even though faces and edges present very large curvatures in some cases. The type and number of CP's in the unit cell is limited by the Morse relations. The type and number of CP's in the surface of an ionic basin is limited by the Euler relation. Both, together with the space group symmetry, posse strict requirements on the shape of the ionic basins. This point of view helps to explain some of the bizarre features exhibited by the basins in some cases, particularly the nearly bidimensional wings of the M(+2) basins in the C8, R9 and B10 topological schemes.

We have found a suggestive connection between the topological properties of the electronic density and the chemical stability of the perovskites. In particular, we have found that the crystal tends to be more stable, with respect to the decomposition into MX2 and AX, when there are X-X bonds and the attraction basins of the ions are free of strange features like wings and spikes. This view is in agreement with previous work [14] relating the phase stability of the B1 and B2 alkali halides to the ratios of ionic radii obtained through topological arguments.

As a final remark, the Atoms in Molecules theory may be used to give a rigorous foundation to important historical concepts like ionicity, index of coordination, coordination polyhedron or volume of an atom or an ion in a solid. Several interesting mappings between atoms and polyhedra can be built, some of which have been examined and exploited in this work. We think that there is room in solid state thinking for the new tools and concepts emerged from AIM, and that a judicious use of them, will give new ways to correlate chemical behavior and chemical structure in solids.


Finantial support was provided by the Spanish DGICyT of the Ministerio de Educación y Ciencia under Grant No. PB93/0327.


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