The Coulombic Lattice Potential of Ionic Compounds: The Cubic Perovskites E. Francisco, V. Luaiia, J. M. Recio, and L. Pueyo Universidad de Oviedo, 33007 Oviedo, Spain Coulombic models play a significant role in chemistry and physics. The essence of these models is to represent the particles of the system under study by point charges interactine throueh Coulomb's Law. In ionic solids the coulombic approach g&es rise to the coulombic lattice potential, Vdx.v.z). ,- , . , defined as the ootential created at the point of coordinates x, y, z by all theions in the lattice. This potential can he expressed as an infinite series of coulombic terms, the structure of the series depending on the type of lattice. The sum of the series is difficult to compute accuratelv due to its slow convergence. Madelung obtaihed for the first time the value of Vlat(x,y,z)a t the ionic sites of a crystal lattice ( I ) . Later on, several methods have been proposed to deal with the convergence problem. They have been reviewed by Tosi (2).Two well-known examples, the methods of Evjen (3)and Frank (4), have been commented recently in this Journal (5). . . l'hecoulombiclattice potential isa key piece in the theory of the ionic bond in crvstals. first devdooed bv Born and collaborators (6). In this theory the interionic interactions are represented by two terms: the long-range attractive energy deduced from the values of Vlat(x,y,z) at the ionic sites, and the short-range repulsive interaction today known as the Born-Mayer term (2,6). This simple theory can successfully describe the geometry, stability, and several thermal and optical properties of ionic crystals. Although the modern theory of the ionic bond demands a quantum mechanical description of the attractive and repulsive terms and involves additional contributions such as those coming from the polarization of each ion by the charge of the others, the simpler two-term model displays clearly the separate attractive and repulsive interactions, whose balance determines the stability of the bond (7). Further interest in point-charge potentials emerges from the work on coulombic models in chemical bonding recently reported by Sacks in this Journal (8). Use of the pointcharge approximation in quantitative discussions based on coulomhic models results (8) in adequate interpretations of structure types of molecules and crystals as well as in very reasonable predictions of observable quantities such as dipole moments or barriers to internal rotations. These arguments show that a detailed knowledge of V~.,(x.v.z) ..... " . is a matter of interest. not onlv on its own riaht but also because the large numberot'important applications. 'The relevance of this votential IS underlined in manv textbooks of inorganic and-physical chemistry, but practicdly no attention is given to presenting and discussing the shape of the potential and the type of useful information deducible from it. This may be due to the difficulty of computing accurate values of the lattice potential a t any point of the lattice. However, the computer programs available to do this job can efficiently be used on microcomputers, making immediately accessible a large amount of quite interesting information. Sunnlemented with adeauate m a ~ h i c a lrepresentations, the &udy of the coulomhic lattrce potentiacean be a stimulatine wav of uuderstandina manv aspects of the electronic struciurebf the ionic crystals. To show that this is possible is the main purpose of this paper. Taking a simple cubic perovskite (RhMnFJ as an example, we present the 6
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results of accurate calculations of the coulombic potential alone several crvstal directions. Althoueb the behavior of the potential can be somewhat intricate, particularly in some low-svmmetrv directions. we show that the oeculiarities of this potentiai can be easily understood by kxamining the effects of the ions closest to the point or direction considered. T o illustrate some t w e of information deducible from the shape of the lattice potential, we briefly discuss its relation with the cluster model. The basic assumption of this modelis that the observable properties of the crystal can he deduced from the analysis of a small group of representative ions, the cluster& the example considered here, the relevant cluster would he the [MnF6I4- complex ion. The cluster model has been one of the current frameworks for the theoretical analyses of the electronic structure of solids. Crystal-field theorv is a familiar example. On the other hand. the calculation ofsome properties of the crystal within the cluster model reauires the consideration of lattice effects in order to reach the desired accuracy and to interpret the small differences that a given cluster presents when located in different crystals. The coulombic lattice potential can he the vehicle for incorporating to a first approximation such lattice effects. Thus, to complete this work, we discuss qualitatively some lattice effects in RbMnF3, that is, some consequences of the lattice potential on the cluster-model predictions of properties such as equilibrium metal-ligand distances, covalency, and d-d electronic transitions of the Mn2+ ion. Finally, the computer programs required for (1)calculation of V&,y,z) at any point of the lattice, (2) making twodimensional plots of any section of Vlat(x,y,z),and (3) finding simple and useful analytical representations of Vlat(x,y,z),have been collected in a FORTRAN 77 package (9) available from us upon request. Coulombic Lanlce Potential of the RbMnF, Many A'MnL3 compounds, where A' is an alkali cation, M" a dipositive cation, and L a halide ion, show a cubic perovskite structure (Fig. 1). RhMnFs is an example (10). The unit cell contains five ions in the positions MN(O,O,O), L((a/Z,O,O), (O,a/Z,O), (O,O,a/2)), and A1(a/2,a/2,a/2), a being the cell parameter. In this structure, the MI1 ion is octahedrally surrounded by six ligands, located a t R(MD-L) = a12 (Fig. 2). The A' cations form an eightfold (cubal) arrangement around the Mu ion, with R(M1'-A') = 31/2a/2.The cell parameter of the RbMnF3 is 424.2 pm (10). In the following, we will use the MI1-centered coordinate system depicted in Figure 1. Using this system we compute the coulombic potential external to the M1'Ls cluster, defined by V,,,(X,Y,Z)= V&,Y,Z) - V,,,,,(~,Y,Z)
(1)
V,l,,w,(x,y,z) is the coulombic potential a t the point of coordinates x, y, z determined by the seven point charges representing the seven ions of the MI1Lscluster. Here we will work with the electrostatic enerm .. of an electron in the VeXt(x,y,z)potential, that is, V(x,y,z) = -eVeXt(x,y,z) = -VeXt(x,y,z) in atomic units (e = 1 ax.). In this work, V~,t(x,y,z)has been computed with a version of the EWALD
Figure 3. External lattice potential of RbMnF. in four directions of the xypiane.
Figure 1. Reference coordinate system for a cubic pwovskite.
Figure 4. External lattice potential vs. the 8 coordinate for the 4 = 0' plane. Figure 2. The octahedral M"b cluster in a cubic perovskite.
program (11). In Figure 3 we show V(xy,z) (solid lines) along several directions in the xy plane (0 = 90°), from r = Irl = 0 to 5 hohrs (1 hohr = 52.91771 pm). In Figures 4 and 5 we present the values of V(x,y,z) versus the polar coordinate (0' 5 0 5 180') for r$ = O0 and 45', respectively. The symmetry of this system ensures that the value of V(x,y,z) a t any point of the M1'Ls cluster can he deduced from known values within the volume determined by the 100, 110, 111 directions. These directions are shown in Figure 1. Let us see now how the behavior of V(x,y,z), as depicted in Figures 3-5, can be understood in terms of the action of the
nearest neighbors. In Figure 3 we can see that V(x,y,z) is very flat for r 5 3 hohrs. Within this range of r , the deviations of V(x,y,z) from V(0,0,0) are negligible in any direction, not only in the xy plane. For r > 3 hohrs, V(xy.z) shows anisotropy: it decreases along the 100 direction (4 = 0", Fig. 3), due to the action of a M" cation a t (a,O,O) (Fig. 2), and increases along the 110 direction (4 = 45", Fig. 3) mainly because the presence of two anions at (a,a/2,0) and (a/2,a,0). Along the 100 direction, V goes uniformly to -m as r goes to the (a,O,O) point. Along the 110 direction, V shows a maximum, not displayed in Figure 3, and then it goes to -m as r goes to the (a,a,O) point. The behavior of V in the xz plane (@ = OD)is depicted in Volume 65 Number 1 January 1988
7
Figure 5. External lanice potential vs. the 0 coordinate for me
d = 45'
plane.
Figure 4. Maxima a t 8 = 45O and 135' correspond to points located a t axes equivalent to the 110 one, and the minimum at 8 = 90° coincides with a point on the 100 axis. In agreement with Figure 3, the anisotropy increases with r. When 4 = 45O we find the potential shown in Figure 5. Again, the maximum at 8 = 90' corresponds to points on the 110 axis. Minima a t 8 = tan-I 2'12 = 54.74' and 8 = tan-' (-2I") = 125.26O reveatthe presence of the A' cations at (a1 2,a/2,a/2) and (a/2,a/2,-ah), respectively. Thus, we observe that the behavior of V along the highly symmetric 100,110, and 111 directions can be readily understood by looking a t the first ion encountered outside the cluster. A little bit of analysis is required to interpret the variations of V along lower symmetry directions. The smaller maxima a t 8 25' and 8 155' in Figure 5 are examples of this. They correspond to the 113 (8 = tan-' (2lI2/3) = 25.24O and 113 (0 = tan-' (-2'"/3) = 154.76") directions. Along the 113 direction the first ion outside the cluster is the A1(a/2,a/ 2,3a/2). This cation should produce a minimum in V. However, one has to consider the actions of the M" cation at (O,O,a) and the three L-' anions at (a/2,0,a), (O,a/2,a), and (0,0,3a/2), as shown in Figure 6. For instance, at the point P of coordinates (a/3,a/3,a) the contribution of these three = 6.83/a, hut the conanions is ((12.5'12)/5 (~i.l7~/~)/17)/a tribution of the A' and Mn cations is -(3.2IlZ (6.111/2)/11)/a = -6.05/a, as can he readily deduced from Coulomb's Law. The combined action of these five ions becomes a positive interaction, and it gives rise to the observed small maxima.
- +
+
Effects 01 the Lattlce Potentlal on Cluster Properlles Let us see now some qualitative ideas deducible from the external lattice potential on the RhMnF3 (Figs. 3-5). We have observed that Vis nearly flat for 0 5 r _< 3 bohrs. If the M" cation is a 3d ion, like the Mn2+ ion in RbMnF3, we can conclude that the action of Vext(x,y,z)on the is, 2s, 2p, 35, and 3p core orbitals, with electron densities very small for r > 3 hohrs, will he negligible. Only the valence 3d, and the unoccupied 49, and 4p AO's will he sensitive to VeXt(x,y,z). The 4p electronic density concentrates along the 100 direction. Given the shape of Vert(x.y.z) along this axis (Fig. 3), we can expect that the lattice potential would reduce the 4p orhital energy. Under the octahedral symmetry of the M" 8
Journal of Chemical Education
Flgure 6. Penurbation of the A'(a/Z.a/2,30/2)and ~lI(0.0,a) cations and the L(0.0.3a/2),L(O.al2.a) and L(al2,O.a)anions on the point P(s/3,a/3.a)in RbM"F,.
site, the 3d AO's do split into 3deKand 3dtzg AO's. These orbitals are directed along the 100 and 110 directions, respectively. Since VeXt(x,y,z)decreases (increases) along the 100 (110) axes it would reduce (increase) the orbital energies of the 3de, (3dt2,) AO's. In consequence, electronic transitions involving changes in the 3deKandlor 3dt2, electron population will be affected by VeXt(xy,z).This turns out to he the case for the familiar transition known as A or 10 Dq, that can be defined as the difference between 3deg and 3dtz, orbital energies: 10 Dq = r(3deg)- 43dt2,). We can conclude that the theoretical cluster-in-vacuo (V,&,y,z) = 0) value of 10 Dq will he reduced by the action of V&,y,z) in the cluster-in-the-lattice calculation for RhMnF3. The auantum mechanical details of the metal-liaand bond can also he modified by the action of the lattice pitential. The MIL-liaand intcmction, as meaiured by the molecular orhital mixing coefficients of the cluster wavefunction, changes with the metal-ligand separation. The lattice potential affects this mixing in different amounts for different metal-ligand distances because V&,y,z) is a function of this coordinate. Thus, the metal-ligand covalency and its variation with the metal-ligand distance are cluster properties sensitive to the action of V&,y,z). In general, given the gradient of this potential near the ligand nuclei, we can conclude that those properties localized around the ligand sites, such as the s-p ligand hybridization, will be very sensitive to V&,y,z), in contrast with those localized near the MI1nucleus, such as the spin-orbit constants of the metallic orbitals. Finally, the lattice potential can also modify the metalligand equilibrium distance predicted by the cluster-in-vacuo calculation. These effects would be particularly obvious in a point-charge description of the cluster itself. Then, for instance, the 4 = O0 curve in Figure 3 tell us that the lattice potential of the RhMnF3 would increase the size of the octahedral [MnFs]4- cluster because the ligand fluorides, located
along this and the equivalent 010 and 001 directions, are more stabilized by Vert(x,y.z) at larger metal-ligand distances. In a quantum-mechanical description of the cluster the prediction is not so straightforward because the positive ligand cores (bare nuclei plus innermost electrons) suffer an inwards shift under V,&,y,z), but the valence electrons experience the opposite effect. The detailed answer must be obtained from the quantum-mechanical, cluster-in-the-lattice calculation, that, in turn, involves the use of VeXt(x,y,z). Acknowledgment
EF, VL, and JMR want to thank the Ministerio de Educa-
ci6n y Ciencia for postgraduate grants. Financial support from CAICyT (project No. 2880183) is gratefully acknowledged. 1. Madeluog,E.Phyaik.2. 1918,19,524. 2, Tosi,M,P,SDlidStotePhya,1964, 16,1, 3. E+L H.M.~ h y aR. ~ D1%2,.39,675. . 4. Frank, F.C. Phil. M a g 1950.41.1287. 5, Elert,M,:Kaubek,E. Chem .Edu c, uO, 6. B ~ " M . . ; H U ~K. ~ .~ ~ n o m i e~ho~ryaicrystollotticos; ol Clmendan:Oxford. 1954. 7. Berry,R.S.; Rba, S. A.; Ross,J. Physicol Chemistry; Wiley: New Yark, 1980. 8. Saelrs, L.J. J. Chom.Edue. 1986,63,286,373,487. 9. L U ~ v.; ~ ~, ~ E.; ~ e e~i oJ., M.:PU~YO, ~ L. ~ camput. ~ Phw m Commun., , submitted. 10. Wa1ker.M.B.;Stevenson, R.W. H . R m Phys. Soc. (London1 1966.87.35. 11. piken,A. G.;"an G ~ I W. , F O , ~~ e hRSP. . 1968. ~~68.10.
Volume 65
Number 1
January 1988
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