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 (Bader 1994). 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 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 5.3). 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. 5.4). 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 on 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 5.3 Topological ionic charges averaged on the perovskites.
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 1.8256Fig. 5.4 Average ionic radii obtained from the volume of the attraction basin. The error bars indicate the variability of ionic volume from crystal to crystal.