J. Phys. Chem. 1984, 88, 119-128
119
A Theoretical Model for the Index of Refraction of Simple Ionic Crystals Erik W. Pearson,+Mark D. Jackson, and Roy G. Gordon* Department of Chemistry, Harvard University, Cambridge, Massachusetts 02138 (Received: March 1, 1983; In Final Form: June 6, 1983)
A theoretical model for the index of refraction of simple ionic crystals, incorporating both electrostatic and short-range effects, is presented. It is shown to yield accurate values for the indices of refraction of alkali halide, alkaline earth chalcogenide,
alkaline earth halide, and alkali chalcogenide crystals. Alkali hydride crystals are treated somewhat less satisfactorily. Tables of ionic crystal polarizabilities and of free ion polarizabilities are given.
I. Introduction The ionic model for crystal indices of refraction assumes that a crystal is composed of distinct polarizable ions which may be assigned individual polarizabilities. If such a crystal has cubic symmetry, the high-frequency index of refraction, n, is related to the polarizabilities of its constituent ions by the ClausiusMossotti relation
Here t = n2 is the dielectric constant, V, is the volume per formula unit, and a, is the sum of the polarizabilities of the ions which compose a formula unit. There have been many empirical applications of this equation to simple ionic crystals, largely to the alkali halides but also to some other cubic crystals. In the simplest of these each ion is assumed to have a “crystal polarizability” which, although it may differ substantially from the gas-phase polarizability of the same species, remains constant from crystal to crystal. These crystal polarizabilities are then determined by fitting to the index of refraction data for a given set of crystals. A more sophisticated approach, developed by Wilson and Curtis3 and extended and refined by C ~ k e rallows ,~~~ the ionic polarizabilities to vary from crystal to crystal in systematic fashion, although at the expense of introducing additional variable empirical parameters. Coker’s papers provide a review of the empirical studies. Either of the empirical approaches is capable of producing a set of crystal polarizabilities which reproduce the index of refraction values for, say, the alkali halides, to reasonable accuracy. For example, varying eight polarizabilities (the polarizability of one ion, usually Li+, must be fixed to avoid ambiguity) Pirenne and Kartheuser* were able to reproduce the index of refraction data to within about 5%. Using an additional five parameters, four of which are related to the variation of halide polarizabilities from crystal to crystal and the fifth of which is a factor distinguishing NaCl from CsCl structure crystals, Coker4 obtained results which reproduce most of the experimental indices of refraction to better than 1%. In view of the extent of the empirical work, there has been surprisingly little theoretical work on predicting ionic crystal indices of refraction. Perhaps this is due to the difficulty of obtaining accurate values for ionic polarizabilities; the problems here are compounded by the need to consider the effects of the crystalline environment on the ions. Some time ago Ruffa6 proposed a simple model for estimating the effects of crystal electrostatic site potentials on ionic polarizabilities. He applied his model, which was based on a sum rule analysis of ionic polarizabilities and a simple interpretation of the effects of Coulomb potentials on the relative positions of electronic energy levels, to the alkali halides where he obtained reasonable agreement with the then extant empirical work. More recently, MahanZScalculated crystal polarizabilities for several ions using density functional techniques. He was able Present address: Battelle Northwest, P.O. Box 999, Sigma 5 , Richland, WA 99352.
0022-3654/84/2088-0119$01.50/0
to obtain indices of refraction accurate to roughly 15% using the polarizabilities thus obtained. In this paper we present a theoretical model, free of empirical parameters, for the index of refraction of ionic crystals. We test the model by applying it to the sodium chloride, cesium chloride, and zinc-blende structure alkali halide, alkaline earth chalcogenide, and alkali hydride crystals and to the fluorite structure alkaline earth halide and alkali chalcogenide crystals. We obtain the es for the ions in the crystal environment by applying a modified version of formulas due to B~ckingham’,~ to ionic wave functions which are obtained by Hartree-Fock calculations which incorporate, in a simple fashion first suggested by W a t ~ o nthe ,~ crystalline electrostatic site potential. Damping of the polarizabilities due to the effects of short-range nearest-neighbor and second-nearest-neighborinteractions is incorporated as a correction to the polarizabilites obtained in the first step. Our model is presented in detail in section 11, where we also indicate its theoretical limitations. In section I11 we present the results of the index of refraction calculations and compare the variation in ionic polarizabilities from crystal to crystal with empirical results. We also present a table of free ion polarizabilities which we compare with empirical and theoretical values. In a final section we summarize our results and conclusions and discuss the possible application of our model to more complicated crystals. 11. Theory
Given the validity of the ionic model, and consequently of the Clausius-Mossotti relation (eq l), calculation of the index of refraction of a cubic ionic crystal requires only determination of the ionic polarizabilities in the crystal environment. Here we present a model which allows the calculation of these polarizabilities with relatively little effort. Both electrostatic effects and short-range forces are included. In this section we present the model and briefly discuss its approximations and limitations. The next section contains the results of a test of the model on various cubic ionic crystals. Ionic polarizabilities are determined in three steps. First, Hartree-Fock S C F calculations are performed on ions in the presence of a perturbing electric potential selected to model the electrostatic potential felt by an ion in a crystal. Second, the polarizabilities of the perturbed ions are obtained from the SCF wave functions through application of a modified version of formulas given by B~ckingham.’,~Finally, by modeling the polarizable ions as charged harmonic oscillators in potential fields, corrections to. the ionic polarizabilities due to the effects of the interionic repulsive short-range forces are calculated. Below we (1) Tessman, J.’ R.; Kahn, A. H.; Shockley, W. Phys. Rec. 1953, 92, 890. (2) Pirenne, J.; Kartheuser, E. Physica 1964, 30, 2005. (3) Wilson, J. N.; Curtis, R. M . J . Phys. Chem. 1970, 7 4 , 187. (4) Coker, H . J . Phys. Chem. 1976, 80, 2078. (5) Coker, H. J . Phys. Chem. 1976, 80, 2084. (6) Ruffa, A. R. Phys. Reo. 1963, 130, 1413. (7) Buckingham, R. A. Proc. R . SOC.,Ser. A 1937, 160, 94. (8) Buckingham, R. A. Proc. R . Soc. Ser. A 1937, 160, 113. (9) Watson, R. E. Phys. Reo. 1958, I l l , 1108.
0 1984 American Chemical Society
Pearson et al.
120 The Journal of Physical Chemistry, Vol. 88, No. 1, 1984 TABLE I: Scaling Factors for the Buckingham Polarizabilitiesa atom He Ne Ar Kr Xe a
ob
1.3849 2.667 11.091 16.78 27.1
All values in atomic units.
d
IC
s,
1.247 1.645 6.661 6.897 10.315
1.111 1.621 1.665 2.433 2.627
Q
Values of ref 13.
a 2
Equation 5 .
1.247 2.452 17.583 31.800 6 1 SO9 Equation 6.
e
s2
aje
s 3
1.111 1.088 0.631 0.528 0.44 1
1.486 4.228 24.467 42.585 83.014
0.932 0.622 0.453 0.394 0.327
Equation 8.
describe each of these steps in detail. The electrostatic, or Madelung, potential felt by an- ion in a crystal is modeled by that due to a uniform charged spherical shell centered at the lattice site occupied by the ion. The charge Q on the shell is equal in magnitude and opposite in sign to the ionic charge and the radius of the shell is determined by requiring that the Madelung potential at the lattice site be correctly reproduced by the charged shell. Thus, if 4 is the site potential, then the shell radius R, is given by Ra = IQI/I4I
(2)
This modeling of the site potential was originally suggested by Watson9 and has been investigated by Paschalis and Weiss" in the study of several properties of ions in crystals. It has recently been shown to be useful in electron gas model crystal structure calculations by Muhlhausen and Gordon." The perturbing potential arising from this model is given by
The SCF calculations on ions in the presence of the potential were carried out by using the Slater-type orbital extended double (basis sets given by Clementi12for all ions except I- and Te2- where, due to program limitations, one 5s function was dropped from the Clementi basis. The calculations were carried out with partial exponent optimization by using the program of Laws and cow o r k e r ~ , *modified ~ by Muhlhausen' to include the crystal potential. To determine the polarizability of an ion perturbed by a shell, we made use of formulas derived by Buckingham7.* using a variational approach. As these formulas are not accurate for atoms or ions with more than a very few electrons, we scale them so as to reproduce the correct values for rare gas atoms. The polarizability of a given perturbed closed shell ion, then, is calculated by applying one of the Buckingham equations, scaled by a factor determined by the polarizability of the isoelectronic rare gas atom, to the shell perturbed Hartree-Fock wave functions. Buckingham derived three forms for the polarizability of atoms and ions using a simple variational technique with functional forms of varying order. The resulting equations involve expectation values of powers of r for the electronic orbitals and cross terms reflecting the effect of polarization of one orbital on others. We give the formulas in order of increasing sophistication. If we define
( R 2 ) ,= (plr21p) -
1 I l ( ~ l x l ~+) II (~~ Y I P ) I ~ + I ( ~ z I P ) I ~ (4) I UfP
where p and u are indices running over the one electron orbitals, the first two Buckingham formulas are given by
(10) Paschalis, E.; Weiss, A. Theor. Chim. Acta 1969, 13, 381. (11) Muhlhausen, C.; Gordon, R. G. Phys. Reu. B 1981, 23, 900. (12) Clementi, E.; Roetti, C. A t . Data Nucl. Data Tables 1974, 14, 177.
Buckingham also gave simplified versions of these formulas appropriate to closed shell atoms. In Table I we give the results of application of these formulas to the rare gases and the scaling factors obtained by requiring agreement with the rare gas polarizabilities. The polarizabilities are given in atomic units, which are used throughout. The scaling factors for use with eq 5 , 6, and 8 are denoted S1, S2, and S3, respectively. Although the unscaled values obtained from eq 5 , 6, and 8 for the polarizability of a given atom or ion are quite different, as may be seen from the large variation in scaling factors, we found that, as will be shown in the next section, there is relatively little variation among the scaled values for a given species. A less drastic scaling of the type we use here has been successfully used by Coker4 to correct Hartree-Fock free cation polarizability calculations for failure to include correlation contributions. Short-Range Effects. Damping of the ionic polarizabilities by short-range repulsive forces, which may be expected to resist polarizing displacements of electrons from equilibrium positions, is introduced by modeling an ion in a crystal as a charged harmonic oscillator in a potential field. If an ion contains N polarizing electrons, then the oscillator force constant is related to the polarizability by
k =P / a
(9)
We now consider the modifications to the force constant (and hence the polarizability) caused by the presence of the short-range forces. As it is the outermost electrons which are most polarizable, and as, in a crystal, ions are sufficiently separated so that only the outermost electrons are involved in overlap with neighboring charge distributions, we take the entire short-range interionic potential as properly representing the potential felt by a shell displacing toward one of its neighbors. Let us first consider the effect of nearest-neighbor interactions in cubic crystals. In the crystals considered here, an ion has 4, 6, or 8 nearest neighbors symmetrically located about its position. Suppose that the interionic potential between a pair of ions is given by V(r), where r is the interionic separation. Then, if the nearest-neighbor distance is Rnn,Taylor expansion of the short-range potential felt by a displacing shell about its lattice site yields
Here C is the coordination number, x is the magnitude of the displacement, and cubic structure has been assumed. ,Thus, an
The Journal of Physical Chemistry, Vol. 88, No. 1, 1984 121
Index of Refraction of Simple Ionic Crystals TABLE 11: Effective Number of Electrons in the Drude
Ionic Charqe Densities: NaCl
Oscillators’ 8.
He Ne Ar a From ref
1.4327 3.9396 5.5058
Kr XI2
6.2091 7.6340
c
6. 4.
15.
adjusted force constant, which incorporates nearest-neighbor interactions, may be determined: -2. -4.
and the polarizability in the presence of the short-range forces becomes
-6.
0.00
2 = Nz/K
I
I
0.25
0.50
/\
I
0.75
1.00
R/Rnn
For some crystals the anions are sufficiently close to one another that second neighbor interactions are of comparable magnitude to nearest-neighbor interactions. In that case, proceeding analogously
again the polarizability may be obtained via eq 12. C, is the number of second neighbors and V, and V,, are the anion-cation and anion-anion short-range potential, respectively. For the repulsive short-range forces the second derivatives are positive and dominate the first-derivative terms. Indeed, the first-derivative terms may often be neglected in eq 10, 11, and 13 with little effect on the predicted polarizabilities. Consequently the corrections generally serve to reduce the polarizability. The interionic potentials used in these calculations were obtained from the modified electron gas model;14 they have been shown to be reliable for many systems for internuclear separations on the order of the crystal nearest-neighbor distances. The only parameter in the entire model is N , the number of polarizing electrons. Gordon and Kim15determined a suitable set of Ns for the rare gas atoms by requiring that the oscillator model correctly reproduce rare gas C, coefficients. This set is given in Table 11. For these calculations we chose, for each ion, the N appropriate for the isoelectronic rare gas atom. We conclude this section with a brief review of the major approximations we have made. These are four in number. First, of course, is the form we have chosen for the crystal potential. Second, we have omitted short-range compressive effects from the S C F calculations. To the extent that they are important, this will cause us to overestimate the size, and consequently the polarizability, of the various ions. Our third major approximation is the use of the shell model to evaluate the effect of short-range forces. Among other things, by omitting consideration of the fact that the ions polarize simultaneously, we may be overestimating the importance of short-range damping. Finally, we have used a reasonable but previously untested method of determining the polarizability of the perturbed ions.
111. Results and Discussion: Indices of Refraction In this section we begin by presenting our results for the indices of refraction. We also discuss the ionic crystal polarizabilities given by the model and, finally, turn our attention to the question of free ion polarizabilities. As the various classes of crystals we have studied with our model display somewhat different behavior, ~
(13) Starkschall, G.; Gordon, R. G. J Chem. Phys. 1971, 54, 663 (14) Waldman, M.; Gordon, R. G . J . Chem. Phys 1979, 71, 1325. (15) Kim, Y . S.; Gordon, R. G. J . Chem. Phys. 1974, 61, 1 .
Figure 1. Ionic charge densities in NaC1: (-) CI- charge density from perturbed HF wave function; (---) N a + charge density from perturbed HF wave function.
the index of refraction results are presented by crystal type. We begin with the alkali halides, which are the prototypical ionic crystals. Alkali Halides. Table 111 gives the results of our calculations for the alkali halides. Values for CsCl, CsBr, and CsI are for the CsCl structure; all others are for the NaCl structure. Structural information used in the calculations, which has been derived from the lattice parameters tabulated by Wyckoff,16is given in Table IV. The second derivatives of the short-range potential evaluated at the interionic separations, required for the inclusion of short-range effects, are also included in Table IV. We present index of refraction results for our model using a3,the polarizability obtained from eq 8, both with and without the inclusion of short-range effects. Results using a2 (eq 6) and including short-range effects are provided to show that choice of polarizability formula makes relatively little difference. Results using a1 (eq 5 ) are usually intermediate between those given by eq 6 and 8. Experimental values are taken from the compilation of Tessman, Kahn, and Shockley,’ the CRC handbook,” and the compilation in the Landolt-Bornstein series.18 Agreement with experiment is quite good, with a3 providing slightly better results. Inclusion of the short-range correction improves the agreement with experiment; for these crystals the effect is modest. Second-nearest-neighbor anion-anion interactions, included in the results given in column 9, are almost negligible for these crystals. The CsCl structure crystals (CsC1, CsBr, and CsI) are modeled with the same degree of accuracy as the NaCl structure crystals. The small magnitude of the short-range effects for these crystals is an indication that a view of these crystals as consisting of isolated ions is quite realistic. In Figure 1 we show the Madelung perturbed charge densities of the Na+ and C1- ions for NaC1; as may be seen there is very little overlap. In general, the short-range corrections are small for crystals with monovalent anions. For crystals with divalent anions (and for the hydrides which behave more like the chalcogenides than the halides), where the anion charge density is more diffuse, shortrange corrections prove to be more significant. Alkaline Earth Chalcogenides. Index of refraction results for the alkaline earth chalcogenides are presented in Table V; the structural information and potential derivatives are contained in Table VI. Again we present results using a3 both with and without short-range corrections and, for comparison purposes, results using a2 with short-range effects included. For these (16) Wyckoff, R. W. G. “Crystal Structures”, Wiley: New York, 1963; Val. I, 2nd ed. (17) “Handbook of Chemistry and Physics”; West, R. C., Ed.; CRC Press: Cleveland, 1973; 54th ed. (1 8) “Landolt-Bornstein Zahlenwere und Funktionen aus Physik, Chemie, Astronomie, Geophysik und Technik”; 1962; I1 band, 8 teil.
Pearson et 'a1
122 The Journal of Physical Chemistry, Vol. 88, No. 1, 1984 TABLE 111: Alkali Halide Indices of Refraction 1.436 1.360 1.365 1.418 1.518
3.6 3.1 0.4 2.2 3.3
1.427 1.349 1.360 1.412 1.511
2.9 2.2 0.0 1.7 2.8
1.420
2.5
CUI-
1.386 1.319 1.360 1.388 1.469
1.455 1.349 1.365 1.413 1.513
4.9 2.3 0.4 1.8 3.0
LlCl NaCl KC1 RbCl CsCl
NaCl NaCl NaCI NaCl CsCl
1.658 1.525 1.477 1.477 1.621
1.714 1.562 1.494 1.503 1.675
3.4 2.4 1.2 1.7 3.3
1.697 1.546 1.485 1.495 1.667
2.3 1.4 0.5 1.2 2.8
1.688
1.8
1.725 1.566 1.498 1.504 1.677
4.0 2.7 1.4 1.8 3.5
LiBr NdBr KBr RbBi CsBr
NaCl NaCl NaCl NaCl CsCl
1.778 1.611 1.533 1.528 1.668
1.760 1.619 1.544 1.548 1.712
-1.0 0.5 0.7 1.3 2.7
1.746 1.607 1.534 1.539 1.705
-1.8 -0.2 0.1 0.7 2.2
1737
1.778 1.628 1.548 1.550 1.719
0.0 1.0 I .0 1.4 3.1
LII
NaCl NaCl NaCl NaCl CsCl
1.949 1.706 1.624 1.606 1.738
1.906 1.720 1.619 1.599 i ,776
-2.2
1.893 1.705 1.610 1.591 1.770
-2.9 0.0 -0.9 -0.9 1.9
1.917 1.734 1.633 1.611 1.783
-1.6 1.6 0.5 0.3 2.6
K1 RbI CSI
NC
0.8 -0.3 -0.4 2.2
NU
-2.3
'
Ne
% error
NaCl NaCl NaCl NaCl NaCl
UdI
?A error
'% error
striict
Rb 1
N6
% error
crvst
L1r NaF KT
a Cxpcriniental values takcn from ref I , 1 7 , and 18. Model index of refraction computed by using 01 = S p , without inclusion of sliortrangc effects. Modcl i n d e i of refraction computed by using 01 = S p , inodified by nearest-neighbor short-range effects. Modcl index of refraction computed by using oi = S p , with inclusion of nearest- and second-nearest-neighbor effects. e Model index of refraction computed b y using 01 = S,oi, \hith inclusion of nearest-neighbor effects.
TABLE IV: Alkali Halide Structural and Interionic Potential Dataa
Li I NdI KI RbT Cil'
NaCl NaCl N aC I NaCl NaCl
7.61 8.73 10.10 10.66 11.35
3.80 4.37 5.05 5.33 5.68
5.38 6.11 7.14 7.54 8.03
0.46 0.40 0.35 0.33 0.3 1
-0.01 48 -0.0 168 -0.0116 -0.0102 -0.0087
0.0343 0.0373 0.0240 0.0208 0.0172
LlCl h'lC1
NaCl NaCl NaC1 NnC1 CsCl
9.69 10.64 11.89 12.44 7.79
4.85 5.32 5.95 6.22 6.75
6.85 7.52 8.41 8.79 7.79
0.36 0.33 0.29 0.28 0.26
-0.0074 -0.0103 -0.0073 -0.0064 -0.0042
0.0148 0.0197 0.0135 0.0122 0.0062
KBI RbBr C4Br
NilCl NaCl NaCl NaCl CsCl
10.40 11.29 12.47 12.95 8.10
5.20 5.64 6.24 6.48 7.01
7.35 7.98 8.82 9.16 8.10
0.34 0.3 1 0.28 0.27 0.25
-0.0057 -0.0083 -0.0060 -0.0059 -0.0030
0.01 13 0.0158 0.0112 0.01 10 0.0070
LII NJI KI RbI Csl
NaCl NaCl NaCI NnCI CSCl
11.34 12.23 13.35 13.87 8.63
5.67 6.1 2 6.68 6.94 7.47
8.02 8.65 9.44 9.81 8.63
0.3 1 0.29 0.26 0.25 0.24
-0.0048 -0.0067 -0.0049 -0.0045 -0.0022
0.0088 0.0122 0.0089 0.0086 0.0048
KCl RbCl CtCl LiBr h'lB1
0.0 100 0.0040 0.0030
a All values in atomic units. From ref 16. Nearest-neighbor dictance. Second-nearest-nei~hbordistnncc. e First derivative of' ;inion-cation potential cvaluatcd a t nearest-neighbor separation. Second derivative of anion-cation potential cvaluated a t ncarcstSecond dcrivativc of anion-anion potential evaluated a t nearcst-neighbor separation. neighbor separation.
crystals, particularly the oxides and sulfides, short-range effects are very important. Without their inclusion, agreement with experiment is not particularly good. The greater importance of short-range effects in these crystals, as compared to the alkali halides, is not surprising as the charge distributions of the doubly charged anions is quite diffuse. In Figure 2 we show the atomic charge densities calculated for MgO. In contrast with the NaCl case (Figure l), there is substantial overlap of the anion and cation charge distributions. Column 9 of Table VI indicates that, for some of these crystals, second-nearest-neighbor effects do contribute to the damping of the anion polarizability. When short-range effects are included, the NaCl structure oxide and sulfide crystal model indices are in excellent agreement with the experimental values. For the selenide and telluride crystals agreement is reasonable, though not as good. These last two sets are the only crystals for which our model systematically underestimates the index of refraction by significant amounts. There are several possible explanations for the behavior of the model
in these cases. We consider it most likely that our HF calculations for these large, diffuse anions, particularly the telluride anion, are considerably less accurate than those for the smaller anions and the generally more compact cations. It is quite possible that more accurate calculations would restore systematic behavior of the model. As a final note, results for the zinc-blende structure crystals, BeS, BeSe, and BeTe, where the coordination is tetrahedral rather than octahedral, appear reasonable; we were unable to find experimental indices of refraction for these crystals. Fluorite Structure Crystals. Most of the alkali chalcogenides and five of the alkaline earth halides crystallize in the fluorite structure. Our index of refraction results for these crystals are presented in Table VII; the structural parameters and potential derivatives are given in Table VIII. Results for the fluoride and chloride crystals are in excellent agreement with the experimental values. Short-range effects, though modest, do improve the agreement. With the exception of Li20, index of refraction values for the alkali chalcogenides are not available. Thus, we are unable
The Journal of Physical Chemistry, Vol. 88, No. I , 1984 123
Index of Refraction of Simple Ionic Crystals
Scaled Buckinqharn Pol. Rb +
Ionic Charge Denratmr: MgO
I
6.
6* 4.
t
I
I
,
I
I
1
b.
11.5
/
'4
11.0 .c A
a 10.5 c 4
l:1 :;
,
I
' y,
--_
8.0 0.00
0.25
0.50
0.75
-0.45
1.00
-0.40 -0.35 -0.30 -0.25 -0.20
R/Rnn
Figure 2. Ionic charge densities in MgO: (-) 02-charge density from perturbed HF wave function; (---) Mg2+charge density from perturbed
HF wave function. to compare most of these results to experiment. Our result for LizO is quite good. We expect that the other results are of the same accuracy as those for the alkaline earth chalcogenides. Thus, the oxide and sulfide values are probably reliable to within a few percent while the selenide and telluride results should be within 10% of the actual values and are probably low. Alkali Hydrides. We conclue our presentation of model indices of refraction with the alkali hydrides. As may readily be seen from an examination of Table IX, we have saved the worst for last. Without inclusion of short-range effects, the results are completely inadequate, with errors in the index of refraction almost as large as a factor of two. Inclusion of the short-range effects dramatically improves the results. However, errors on the order of 10% remain. It is not surprising that the hydrides prove the most difficult case for our model, as our assumptions are least accurate for these crystals. The failure to include short-range effects in the S C F calculation is probably the primary cause of the large errors observed here. Because the hydride ion is very diffuse and, at the same time, very polarizable, the effect of the short-range potential on the electronic density may be quite large. In consequence the representation of the hydride ion by the wave functions we have calculated is likely to be rather poor. Indeed, the ability of our short-range correction to compensate for a major portion of the effects of neglect of short-range forces in the S C F calculations in this extreme case is quite encouraging.
IV. Results and Discussion: Ionic Polarizabilities Here we present the component ion polarizabilities obtained in the course of the index of refraction calculations. Variation of the ionic polarizability with site potential, the magnitude of short-range effects, and, finally, free ion polarizabilities are discussed. We also investigate the validity of our modifications of the Buckingham polarizability formulas in the case of anions, where, according to arguments presented in the literature, we may not be accounting for large correlation contributions. We begin by illustrating the general effects of site potential on ionic polarizability. There are three distinct cases: cations, monovalent anions, and divalent anions. The hydride anion is qualitatively similar to the divalent anions rather than the monovalent halides. In Figures 3-5 we present the polarizability as a function of site potential without consideration of short-range effects for Rb+, F, and 02-.As may be seen from the figures, cations, monovalent anions, and divalent anions exhibit qualitatively different polarizability curves. Cations are relatively little affected by the model perturbing potential. The reason for this is simple: for site potentials in the range sampled by the crystals considered here the model potential is a constant over the region in which almost all of the electronic density resides. In the limit of the shell radius going to zero, the
Site Potential
Figure 3. Scaled Buckingham polarizabilities for Rb+: (-) S30r3;(---) S20r2; S l a l . Vertical bars delimit range of site potentials in ru(-.-e)
bidium halide crystals. 10.0
-
Scaled Buckinqharn Pol. FI
I
1
0.4
0.6
\
8.0
7.
y, 8.0
.4
4N, k
g
-I
7.0
6.0
5.0
0.0
0.2
Site Potential
Figure 4. Scaled Buckingham polarizabilitiesfor F: (-) S3a3;(---) S2az;(-.-.) S l a l . Horizontal bars show polarizability in the free ion limit. Vertical bars delimit range of site potentials in alkali fluoride
crystals. Scaled Buckinqham Pol. 0 2-
30. 7.
=I 25.
BN, m
n.a 20. 15.
10. 0.6
0.8
1.0
1.2
1.4
S i t e Potential
Figure 5. Scaled Buckingham polarizabilities for 02-:(-) S30r3;(---) Sp2; S,a,.Vertical bars delimit range of site potentials in alkaline earth oxides. (-e-.)
Pearson et al.
124 The Journal of Physical Chemistry, Vol. 88, No. 1, 1984 TABLE V: Alkaline Earth Chalcogenides Index of Refraction Results cryst
1vexp tu
Nb
% error
N C
5% error
Ne
% error
MgO
1.736 1.838 1.810 1.980
2.178 2.070 2.079 2.210
25.5 12.6 14.9 11.6
1.887 1.879 1.95 1 2.054
8.7 2.2 7.8 3.7
1.845 1.836 1.955 2.060
6.3 -0.1 8.0 4.0
1.972 1.950 2.009
13.6 6.1 11.0
SrS BaS
2.271 2.131 2.107 2.155
2.684 2.649 2.406 2.312 2.285
16.6 12.6 9.7 6 .O
2.324 2.313 2.187 2.128 2.146
1.8 2.3 1 .o -0.4
2.292 2.275 2.201 2.135 2.148
0.2 3 .O 1.3 -0.3
2.36 1 2.365 2.208 2.175 2.113
4.1 3.3 3.2 -1.9
BeSe MgSe Case SrSe Base
2.440 2.274 2.220 2.268
2.136 2.680 2.404 2.309 2.276
9.8 5.7 4 .O 0.3
2.4 12 2.398 2.226 2.15.0 2.159
-1.7 -2.1 -3.2 -4.8
2.379 2.43 1 2.236 2.152 2.160
-0.4 -1.7 -3.0 -4.8
2.398 2.380 2.217 2.14 1 2.078
-2.5 -2.5 -3.6 -8.4
BeTe CaTe SrTe BaTe
2.510 2.408 2.440
2.824 2.479 2.500 2.332
-1.2 3.8 -4.4
2.56 I 2.342 2.338 2.239
-6.7 -2.9 -8.2
2.481 2.5 13
-1.1 4.4
CaO
SrO BaO BeS MgS
cas
’
% error
N‘
2.581
Model index of refraction values computed by using a: = S,a, unmodified by shorta Index of refraction values from ref 1, 1 7 , and 18. Model index of refraction values computed by using @ = S3a3corrected for nearest-neighbor interactions. range effects. Model index of refraction values computed b y using cy = S,a3 corrected for nearest- and second-nearest-neiehbor interactions. e Model index of refraction values computed by using a = S p , corrected for nearest-neighbor interactions.
‘
TABLE VI: Structural and Potential Parametersa for Alkaline Earth Chalcogenides cryst
struct
lattice6 param
CaO SrO BaO
NaCl NaCl NKI NaCl
7.96 9.09 9.75 10.44
126.0 187.8 231.8 284.2
3.98 4.55 4.88 5.22
SrS Bas
ZnS NaCl NaCl N aC1 NaCl
9.17 9.83 1p.75 11.38 12.07
192.5 237.7 310.9 368.0 439.7
BeSe MgSe CaSe SrSc BaSc
ZnS NaCl NaCl NaCl NaCl
9.58 10.30 11.17 11.77 12.47
BeTc CaTe SrTc BaTc
ZnS NaCl NaCl NaCl
10.47 11.99 12.23 13.20
BeS MgS
cas
site potent]
v’(R,,)~
v”(R,,)~
5.63 6.43 6.89 7.38
0.878 0.769 0.7 17 0.670
-0.072 -0.053 -0.044 -- 0.03 4
0.139 0.087 0.056 0.053
0.014 0.005 0.000 -0.001
4.58 4.92 5.38 5.69 6.04
6.48 6.95 7.60 8.04 8.53
0.763 0.71 1 0.650 0.614 0.579
-0.070 -0.038 -0.034 -0.029 -0.022
0.135 0.068 0.052 0.046 0.034
0.010 0.005 -0.002 -0.001 0.000
219.9 273.3 348.3 408.0 485.0
4.79 5.15 5.58 5.89 6.24
6.77 7.28 7.90 8.32 8.82
0.730 0.679 0.626 0.594 0.560
-0.066 -0.033 -0.029 -0.027 -0.021
0.122 0.058 0.047 0.043 0.03 1
0.010 -0.004 -0.001 0.000 0.000
286.9 43 1.0 456.9 575.2
5.23 6.00 6.1 1 6.60
7.40 8.48 8.64 9.33
0.668 0.583 0.572 0.529
-0.055 -0.024 -0.028 -0.0 18
0.099 0.034 0.043 0.026
-0.005
niolar
VOI
R,,C
~~~d
‘
v”(R~~)~
a All values in atomic units, Reference 16. Nearest-neighbor distance. Second-nearest-nei~libordistance. e First derivative of anioncation potential cvaluatcd at nearest-neighbor separation. f Second derivativc of anion-cation potential evaluated a t nearest-neighbor scparation. g Second derivative of anion-anion potential evaluated a t second-neighbor separation.
cation polarizability must reach that of the isoelectronic rare gas atom. In the case of Rb’, illustrated in the figure, that would require an increase from a = 10 to a x 17. Almost all of this increase, however, takes place for site potentials higher than those present in the crystals. The region of site potential present in the rubidium halides has been indicated in the figure. Because the change in cation polarizability (and charge distribution) due to the site potential is quite small, it is not surprising that the crystallographic evidence reviewed by Coker4 indicates that cations are, if anything, on balance compressed rather than expanded by the crystalline environment. Short-range compression may easily dominate the small expansive Madelung potential effects on cations. In contrast, both monovalent and divalent anions are quite strongly affected by the Madelung potentials present in the cubic ionic crystals. Figure 4 shows the polarizability of F as a function of site potential. The model polarizability of the monovalent anions has a well-defined asymptotic free ion value, indicated on the figure by the horizontal bars, which is maintained for small site potentials. When the site potential reaches a critical region, the polarizability suddenly begins to decrease rapidly, presumably reaching the
appropriate rare gas value when the shell radius reaches zero. Again, the range of site potentials present in the alkali fluorides has been indicated on the figure by vertical bars. This range lies within the region of rapidly varying polarizability. The divalent chalcogenide anions behave rather differently. Figure 5 shows our polarizability curves for 02-.For 02-, and the other chalcogenides, the variation of polarizability with site potential is more rapid, both absolutely and relatively than for the halide ions. Indeed, the model polarizabilities may vary by more than a factor of two within this class of crystals for a given anion. The site potentials for MgO and BaO, which delimit the range of oxide site potentials found in the alkaline earth chalcogenides, have been indicated on the figure. In contrast to the halides, within the region of the curves we studied there is no evidence of an approach to a free ion polarizability. This is in accord with the fact that 02-is not a stable species. As a conclusion to this overview of the site potential dependence of the polarizability for the various classes of ions, we comment on the difference between a,,a*,and a j . As may be expected, the spread between the three polarizabilities increases with a decrease in the magnitude of the site potential. For the site
The Journal of Physical Chemistry, Vol. 88. No. 1, 1984 125
Index of Refraction of Simple Ionic Crystals
TABLE VI]: Fluorite and Antifluorite Crystals Index of Refraction Results
_-
cry\t
-
NexPta
Nb
1.64
1.790 1.629 1.693 1.806
L1,O N:;,O K,O Rb,O
Nc
% error
1.662 1.532 1.624 1.731
8.9
Nd
% error
1.1
1.767 1.602 1.672 1.764
Li,S K,S Rb,S
1.905 1.728 1.779
1.808 1.672 1.722
1.868 1.727 1.773
Li,Se Na,Se K,Se
2.041 1.797 1.729
1.942 1.716 1.685
2.006 1.769 1.732
Li,Te Na,Tc K ,T?
2.079 1.823 1.739
2.005 1.766 1.706
2.083 1.815 1.758
70 error 7.5
CLIl:, Srl', ELI I ,'
,
1.43 1.44 1.47
1.467 1.469 1.529
2.3 1.9 3.7
1.453 1.455 1.518
1.4 0.9 2.9
1.469 1.464 1.505
2.4 1.5 2.1
SKI, BaCI,
1.65 1.73
1.715 1.702
4.0 -1.6
1.697 1.686
2.8 -2.5
1.711 1.686
3.7 -2.5
'
Model index of a 1:roni rcf I , 1 7 , and 18. Modcl index of refraction computed by using 01 = S,a, unmodified by short-range forces. refraction coniputcd by using 01 = S p , corrected for nearest-neighbor interactions. d Model index of refraction computed by using a: = S p , cor rcct cd f o r ncarcat-ncighbo r cffec t s. TABLE VIII: Fluorite and Antifluorite Crystals Structural and Potential Parameters'
___
site potential
lattice' parain
RnnC
Raad
anion
cation
C"(Rnn)e
V"(R,,Jf
V"(R,a)g
3.78 4.54 5.27 5.52 4.67 6.05 6.26
6.17 7.42 8.60 9.01 7.63 9.88 10.22
0.87 0.72 0.62 0.59 0.70 0.54 0.52
-0.47 -0.39 -0.33 -0.32 -0.38 -0.29 -0.28
-0.0367 -0.0347 -0.0205 -0.0187
0.0723 0.0583 0.03 16 0.0280
0.0065
Rb,S
8.73 10.49 12.16 12.74 10.79 13.97 14.46
-0.0216 -0.0142 -0.0141
0.0380 0.021 2 0.0209
Li,Se Na,Se K,Sc
11.35 12.87 14.51
4.91 5.57 6.28
8.02 9.10 10.26
0.67 0.59 0.5 2
-0.36 -0.32 -0.28
-0.0 180 -0.0196 -0.0121
0:03 11 0.0318 0.0179
Li,Te
12.29 13.82 15.41
5.32 5.98 6.67
8.69 9.77 10.89
0.62 0.55 0.49
-0.33 -0.29 -0.26
-0.0136 -0.0158 -0.0099
0.0236 0.0253 0.0150
0.0005 0.0005
Srl. FLIl'>
10.32 10.96 11.72
4.47 4.75 5.07
5.16 5.48 5.86
0.39 0.37 0.35
-0.73 -0.69 -0.65
-0.0285 -0.0256 -0.0215
0.0603 0.0530 0.0414
0.0159 0.009 1 0.0046
SrCI, BKI,
13.18 13.87
5.71 6.01
6.59 6.94
0.3 1 0.29
-0.57 -0.55
-0.0146 -0.01 19
0.0268 0.0216
0.006 1
cryst Li ,O Na,O
K,O Rb,O LizS K2S
N;i ,Tc
K,TC
c;II,,
'
a All v;11ucs in atoinic units. Reference 16. Nearest-neighbor distance. Second-nearest-neighbor distance. e First derivative of anion-cation potential evaluated a t nearest-neighbor separation. Second derivative of anion-cation potential evaluated at nearest-neighbor Second dcrivativc of anion-anion potential evaluated at nearest-neighbor separation. scparntion.
TABLE IX: Alkali Hydride Index of Refraction Results cry51
Nexpta
Nb
X error
Nc
7~error
LiH N;iH KH RbH C sH
1.615 1.470 1.453
2.751 2.256 1.905 2.036 2.088
70.4 53.5 31.1
1.844 1.632 1.656 1.736 1.853
14.2 11.0 14.0
a V;ilucs from rcf 17. Model index of refraction computed by using 01 = Sp, M ithout correction for short-range effects. Model indc\ of refraction coniputcd by using 01 = Sp, corrected for ncarc~t-nci~libor cffccts. f~
potentials of interest there is, save for the TeZ-ion, no more than 5% variation between the various curves. In the free ion limit there is as much as 10% variation for the largest ions (I-, Cs'); more typical is the 5% variation between a1and az for C1-. There is no clear a priori theoretical basis for selecting one or the other of the scaled polarizability formulas as the most reliable; the index of refraction results suggest that, for our model, a3is to be pre-
ferred. It is, of course, true that, in the absence of scaling, G~ is calculated via the highest order formula and would therefore be preferred. Actual Crystal Polarizabilities. In Tables X-XI1 we give the actual ionic polarizabilities calculated for the ions in the various crystal environments. Table X covers the alkali halide crystals, Table XI deals with the alkaline earth chalcogenides, and Table XI1 contains the data for the fluorite and antifluorite structure crystals. Included in each table are values for the cation and anion polarizabilities both with and without nearest-neighbor short-range corrections. Where second-neighbor anion-anion interactions are important, as for most of the alkaline earth chalcogenides, values of anion polarizabilities adjusted for both first and second neighbor effects are also included. For comparison purposes we include empirical polarizabilities for F and C1- in the alkali halides and 02-in the alkaline earth chalcogenides. We have selected the results obtained by Coker4 using a simple form for the polarizability as a function of crystal structure: a = aO[CRn[3]-1
(14)
126 The Journal of Physical Chemistry, Vol. 88, No. 1, 1984 TABLE X: Alkali Halide Crystal Polarizabilitiesa 6 crpst %at acatC Llf 0.19 0.19 NaT 1.11 1.1 1 KF 5.84 5.80 10.00 RbT 10.08 Csl: 18.07 17.91 LlCl 0.19 0.19 NaC1 1.1 1 1.10 ‘5.75 KCl 5.77 RbCl 9.91 9.95 17.59 17.66 CSCl
d
Pearson et al. f
h
&an 6.68 7.65 7.92 8.14 8.41
“ane 6.53 7.44 7.77 7.99 8.28
“an 6.44
“ang 4.19 4.66 4.93 5.07 5.00
“an 5.99 6.91 1.7 1 7.96 8.23
21.13 22.16 23.48 23.96 24.76
20.79 21.66 23.08 23.58 24.49
20.58
23.24 24.45 25.26 25.60
19.44 20.97 22.54 23.09
26.82
33.30 34.72 36.00 36.41
LiBr NaBr KBr RbBr CsBr
0.19 1.10 5.76 9.87 17.71
0.19 1.10 5.75 9.83 17.59
27.40 29.03 30.78 31.33 31.99
27.05 28.48 30.33 30.87 31.56
LiI NaI KI RbI CTI
0.19 1.10 5.75 9.79 17.40
0.19 1.10 5.74 9.76 17.35
40.48 42.05 44.09 44.68 46.73
40.08 41.45 43.60 44.19 46.31
a All values in atomic units, Cation crystal polarizability (Sp,) uncorrected for short-range effects. Cation crystal polarizability corrected for nearest-neighbor interactions. Anion crystal polarizability (S,a,)uncorrected for short-range effects. e Anion crystal polarizability corrected for nearcst-neighbor interactions. Anion crystal polarizability corrected for nearest- and second-nearest neighbor interEmpirical anion crystal polarizabilities (ref 4). actions. Theoretical results of ref 25.
His various empirical forms yield quite similar results for the polarizabilities in the crystal environments, although they can give somewhat different free ion polarizabilities. For the alkali halides and alkaline earth chalcogenides, the theoretical anion polarizabilities of Mahan25are also given. Several generalizations may be drawn from the tables. First, in accord with both empirical results and other evidence reviewed by C ~ k e rthere , ~ is very little variation of a given cation’s polarizability from crystal to crystal. The largest variation is that displayed by Cs; between CsF and CsI it varies by only about 4%. In contrast, the polarizabilities of the monovalent anions vary typically by on the order of 15-20% over a series of crystals while for the divalent anions the polarizability may vary by as much as a factor of two within a series. Second, inclusion of short-range forces has minimal effect on cation polarizabilities. Third, for the alkali halides and the fluorite structure alkaline earth fluorides and chlorides, short-range effects are also rather small; second neighbor effects are negligible. However, for the alkaline earth and alkali chalcogenides, short-range corrections are quite important; anion-anion second neighbor interactions are significant for the small cation alkaline earth oxides and sulfides. Finally, the model polarizabilities, with the inclusion of short-range effects, are in good quantitative agreement with the empirical results. Typically, as is reflected in the index of refraction results, the model polarizabilities are a few percent larger than the empirical values. Finally, it appears that the current method yields somewhat better results than does the density functional method of Mahan. Both the values and the variation of anion polarizabilities between crystals calculated here are in better accord with the empirical results than are Mahan’s. Free Ion Polarizabilities. In Table XI11 we present the free ion polarizabilities for the alkali and alkaline earth cations and the halide anions calculated with the modified Buckingham equations. The first three columns give the results for the various formulas (eq 5,6, and 8). The fourth and fifth columns are the empirical results obtained by Coker in studies on ionic crystals4 and solvation effect^.^ Column six gives various Hartree-Fock level results, column seven contains a scaled version of the Hartree-Fock result^,^ and column eight contains the results of various recent configuration interaction calculations.1g~20 A quick glance at the table reveals that the cation polarizabilities calculated with the modified Buckingham equations are in good agreement (19) Diercksen, G. H. F.; Sadlej, A. J. Chem. Phys. Lett. 1981,84, 390. (20) Diercksen, G. H. F.; Sadlej, A. J. Mol. Phys 1982, 47, 33.
with Coker’s empirical results and with Hartree-Fock calculations. Indeed, this agreement serves as much to support the correctness of our scaling procedure as to modify already established values. However, for the halides, the configuration-interaction (CI) results are substantially larger than our results, and all the other empirical and theoretical values. In analyzing the large discrepancy between CI results and those obtained by other approaches, there are two questions to be answered. First is the theoretical question as to which approach more accurately yields the polarizability of a truly free ion. Second is the practical question as to which set of values is more useful in various applications. The distinction between the practical and theoretical questions arises because, in many physical situations, an ion is perturbed by electrostatic potentials due to other ions. Thus, if the theoretical polarizabilities are not valid in such a context (as appears to be the case in ionic crystals if the CI calculations are correct), they should not be used. In the following discussion we concentrate for the most part on the fluoride ion, as it has been the most thoroughly investigated from the CI point of view. Our remarks, however, should be applicable to the other halides as well. In a series of papers,20~22~23 Sadlej and co-workers have studied the polarizability of the fluoride anion. The most recent calculation*Oyields a value for the total polarizability of 15-16 au; the correlation contribution is positive and is about 45% of the Hartree-Fock value. In these fluoride ion studies, as well as one on the chloride anion,lg various explanations have been offered for the dramatic difference between the CI values on one hand and H F and empirical results on the other. These arguments may be briefly summarized as follows. Hartree-Fock results neglect correlation and therefore will yield low values for the ionic polarizability. Scaling of the H F polarizabilities by rare gas results will not give correct results because the correlation contribution to the polarizabilities of the rare gases is proportionately much smaller than is the correlation contribution to the polarizability of anions. This claim, which would also apply to our results, is based both on the CI work on F and C1- and on the fact that the correlation contribution to the polarizability of H- is pro(21) Lahir, J.; Muhkerji, A. Phys. Reu. 1967, 153, 380. (22) Sadlej, A. J. J . Phys. Chem. 1979, 83, 1653. (23) Wilson, S.; Sadlej, A. J. Theor. Chim. Acta 1981, 60, 19. (24) E. Laws and the Walpole Computer Programmers, Atomic SCF Program, 1971. ( 2 5 ) Mahan, G. D. Solid State Ionics 1980, I , 29.
The Journal of Physical Chemistry, Vol. 88, No, I , I984
Index of Refraction of Simple Ionic Crystals
127
TABLE XI: Alkaline Earth Chalcogenide Crystal Polarizabilities" b
d
f
L?
cryst ME0 CaO SrO BaO
"cat 0.57 3.47 6.65 12.89
"catc 0.56 3.44 6.58 12.67
"an 16.15 19.99 22.46 25.43
"ane 13.30 17.16 20.24 22.47
Olan 12.82 16.88 20.30 22.6 1
an 8.71 11.08 11.96 12.90
BeS MgS CaS SrS BaS
0.05 0.56 3.42 6.50 12.12
0.05 0.56 3.40 6.38 12.06
30.95 37.33 42.25 45.54 49.3 1
27.25 33.09 28.07 41.10 45.42
26.87 32.53 38.37 41.23 45.50
40.13 43.17 45.12 48.23
BeSe MgSe Case SrSc BnSe
0.05 0.56 3.42 6.44 12.1 1
0.05 0.56 3.40 6.38 1 2.06
35.87 43.40 47.71 51.16 55.29
32.27 39.41 43.76 46.97 51.71
31.86 39.91 44.01 47.06 5 1.74
50.93 54.38 56.81 59.72
BeTe CaTe SrTe BuTc
0.05 3.42 6.40 12.10
0.05 3.40 6.37 12.06
44.28 61.64 63.05 69.95
44.57 58.38 58.78 66.61
43.87
~
ff
"an h 11.54 15.43 17.73 20.06
a All values in atomic units. Cation crystal polarizability (Sp,)uncorrected for short-range effects. Cotion crystal polarizability corrected for nearest-neighbor interactions. Anion crystal polarizability (Sp,)uncorrected for short-range effects. e Anion crystal polnriAnion crystal polarizability corrected for nearest- and second-neighbor interactions. zability corrected for nearest-neighbor interactions. g Theoretical values of ref' 25. h Empirical anion crystal polarizability (ref 4).
TABLE XII: Fluorite and Antifluorite Crystal Polarizabilitiesa b
d
cryst
Qcat
Li,O Na,O K2O Rb,O
0.19 1.1 1 5.81 10.06
acatc 0.19 1.10 5.79 10.00
"an 16.46 22.25 29.64 32.92
&ane 14.32 19.11 26.43 29.39
Li,S K,S Rb,S Li,Se Na,Se K,Se Li,Tc N;i,Tc K,Tc
0.19 5.77 9.81
0.19 5.75 9.76
34.64 53.29 56.05
31.85 49.46 51.86
0.19 1.10 5.76
0.19 1.10 5.75
44.41 52.02 61.15
41.39 47.77 57.72
0.19 1.10 5.75
0.19 1.10 5.74
57.84 66.70 76.44
55.14 62.86 73.35
c;1I.' SrI,', B;iF2
3.46 6.6 1 13.80
3.42 6.5 I 13.52
7.38 7.65 7.92
7.17 7.44 7.75
SrCI,BKI,
6.43 12.88
6.38 12.74
23.69 24.45
23.17 24.00
portionately much larger than is the correlation contribution to the polarizability of He. The empirical results yield unreliable free ion polarizabilities on extrapolation because the crystal environment is qualitatively different than the free space in which a free ion is, by definition, located. Crystal polarizabilities, parameterized by the nearest-neighbor distance, may be valid for the limited range of R,, present in crystals, but are not valid on extrapolation to R,, = m . In this view the agreement between our results and the scaled HF results is to be expected; the agreement between these sets of values and the empirical results is fortuitous. The range of ions and crystals successfully treated in our model and the good crystal by crystal agreement between our theoretical polarizabilities and the empirical values obtained in the R,, parameterization approach suggest that invocation of coincidence may be too facile a resolution of this matter. Although it is not impossible that there is a systematic rough cancellation between the correlation contribution to anion polarizabilities and various short-range effects omitted in our model, we feel that it is as likely that the CI studies need careful examination. The papers on the fluoride ion give the impression that the energy surface of the polarized ion is quite flat, thus indicating that great accuracy in the numerical work may be necessary to obtain correct results. In addition, the strong dependence of the C I polarizabilities on basis set22)23 and the dramatic effect on the results caused by changes in the order of the c a l ~ u l a t i o nsuggest ~ ~ ~ ~the ~ need for
"
All valucs in ;itomic units. Cation crystal polarizabilities (S,a,) uncorrected for short-ranpc cffccts. Cation crystal pol;irizii bil i t ies c o r r cc t cd f o r ncii rest -neigh bo r c f fcc t s. A n ion c r y s t ;il polnrizabilitics (S,a,) uncorrcctcd for short-ranpe effects. e Anion crystal polariz.abilitiev corrected for nearest-neighbor interactions.
TABLE XIII: Free Ion Polarizabilitiesa
-
ion Ll+ N:i+
K' Rb+
c S+
BCZ+ Mp2+ C;12+
Sr2+ BaZt 1:-
CI R r1-
0 20 1.15 5.82 9.96 17.90 0.05 0.59 3.47 6.63 12.5 1
9.24 27.72 35.5 1 50.64
" A l l values i n .itoniic units. I
QZC
.lb
Cnlpiri~alviilues of ref 5. Refcrencc 20.
'
"3
%*pe
0.20 1.11 5.66 9.49 16.17 0.05 0.56 3.29 6.1 I 11.39
0.19 1.10 5.76 9.75 17.33 0.05 0.56 3.42 6.4 1 11.91
11.68
9.65 28.77 31.75 55.15
9.9 1 28.46 37.05 53.81
9.3 1 26.59 35.23 52.70
aemp
5.14 9.52 16.33
9.99 27.74 36.64 52.84
QCHFg
"SCHFh
0.19 0.95 5.32
I .07 5.85
10.62i 25.37
10.66 27.87
&GI
15.li 37.5J
Sc;ilcd vi~lticfroni cq 5. Scalcd \~;ilue from e(] 6. Sc;ilcd value from eq 8 . e l
