J. Phys. Chem. 1995,99, 3251-3258
3251
Comparison of Cluster and Infinite Crystal Calculations on Zeolites with the Electronegativity Equalization Method (EEM) Geert 0. A. Janssens,* Bart G. Baekelandt, Helge Toufar, Wilfried J. Mortier, and Robert A. Schoonheydt Centrum voor Oppervlaktechemie en Katalyse, Katholieke Universiteit Leuven, Kardinaal Mercierlaan 92, 3001 Heverlee, Belgium Received: June 30, 1994; In Final Form: November 30, 1994@
In the present work the validity of the finite cluster approximation is investigated for two zeolite types, faujasite (FAU) and zeolite A (LTA), by comparison of crystal calculations, with cluster calculations, both performed at the same level of accuracy within the EEM frame. The quantities under consideration are the electronegativity, the chemical hardness, the electrostatic potential, and the electrostatic field. A criterion for the determination of the valid region of a cluster is proposed. Also, a method is offered to approximate the structure specific, long-range Madelung potential used for embedding of clusters.
Introduction The dependence of the properties of the active sites of zeolites on structure type and chemical composition is generally Theoretical understanding of these dependencies requires calculations on full structures. The application of quantum chemical models to solid-state problems is hampered by the infinite dimensions of the solids. This is particularly serious for ab initio techniques.* Substantial progress is being made, however, with the so-called crystal orbital technique, which fully exploits the symmetry relations of the systems under con~ideration.~ In contrast, the use of finite cluster models appears to be simple, although approximate. The formation of the cluster is connected with the breaking of cluster-crystal bonds. Artificial surface states are formed, which disturb the charge distribution in the model. One of the most widely used methods to remove the consequences of this artificial scission of bonds involves saturation of the dangling bonds with hydrogen atoms.1° Two questions arise: (1) what is the minimum size of the cluster for a correct calculation of the properties of zeolites? (2) which properties can be calculated correctly on these minimum-sized clusters and which cannot? To understand the limitations of these finite models, comparison with the whole system, at the same level of sophistication, is needed. Up to now, the properties of mainly small clusters have been examined at the ab initio level.11.12A calculation at the 3-21 G level has been performed on a sodalite cage, terminated with hydroxyl groups on each silicon atom (108 atoms), making use of the TURBOMOLE code.13 This code fully exploits the high symmetry of the sodalite cluster. Generally speaking, there is a trade-off between the cluster size and the level of sophistication of the calculation with quantum mechanical methods.14 In this perspective various possibilities for making the cluster more realistic without extending its size have been considered. A valuable effort to take into account the effect of the longrange Madelung potential, generated by the remainder of the zeolite crystal structure, is embedding of the cluster in a point charge m a t r i ~ . ' ~ ,Sauer '~ has outlined the difficulties that accompany this method.8 In particular, the choice of the correct embedding potential is p r ~ b l e m a t i c , ' ~and ~ ' ~ further work remains to be done. @
Abstract published in Advance ACS Abstracts, February 1, 1995.
0022-365419512099-3251$09.0010
Another possibility for estimating long-range effects is the electronegativity equalization method (EEM). 19-24 The EEM is a semiempirical density functional method, which allows for calculations on infinite crystals, whatever the structure type and chemical composition, when, at least, the structure and the calibration coefficients of all atoms in the structure are known. As a consequence, this frame is ideally suited to test the cluster calculations, as the results can be compared with total crystal calculations on the same structures and at the same level of sophistication. The present article compares, where possible, the electronic properties of the clusters of zeolite A (LTA) and faujasite (FAU) of different sizes, with those of the full lattice, thus providing insight in the validity of the cluster approximation. A method is offered to approximate the structure specific, long-range Madelung potential for cluster calculations. Methodology In order to assess the influence of the structure type only, the zeolite framework parameters (positional parameters and unit cell) were converted to a hypothetical Si02 composition using the DLS p r ~ g r a m , ~with ~ %prescribed *~ Si-0 distances r = 1.62 A and Si-0-Si angles of 145". In this idealized case the two zeolite structure types, LTA and FAU, only differ in the way that the sodalite cages are connected. In FAU the sodalite cages are interconnected via double six-rings, whereas in LTA the connection is accomplished via double f ~ u r - r i n g s . ~ ~ The following clusters were generated: Si(OH)4, four-ring, six-ring, eight-ring, twelve-ring, double four-ring, double sixring, sodalite cage @-cage), a-cage, supercage, and a series of clusters built up from four, five (SSod), ten, and fourteen (14Sod) sodalite cages, interconnected in the characteristic way of the zeolite types. The dangling bonds of the clusters are saturated with hydrogen atoms. The 0-H bond lies in the Si-0-Si plane, bisecting the Si-0-Si angle, as proposed by Jirak et a1.28The 0 - H bond length is set at 0.96 A. The protons are placed at all crystallographically different oxygen sites. These positions are different in both structures, and therefore otherwise identical clusters are slightly different. 0 1995 American Chemical Society
Janssens et al.
3252 J. Phys. Chem., Vol. 99, No. I O , 1995
Results
6.5
The Electronegativity. The density functional expression for the total molecular electronic energyz9 is given by
1
6.0
ne]+
where F[g] = V,,[Q]. e is the electron density, and T and V,, are respectively the kinetic energy and the electronelectron potential energy. The integral represents the electrostatic interaction energy of the electron density e with the electrostatic field v@) generated by all surrounding nuclear charges. In EEM two approximations are made: (1) spherical atoms carrying an effective charge qa = 2 - Nu and (2) F[Q] is written as a second-order Taylor expansion of the effective charge. The EEM energy is then
o
LTA
o
FAU
I
-
A
..-> CI
5.5 P)
c
e
-w
4
0 P)
5.0
141-5
Derivation with respect to the number of electrons, or the effective charge, gives the EEM electronegativity of an atom in a molecule: =
=
(e)v
where qa is the charge of atom a and Va expresses the potential generated by the surrounding atoms, a term that results from the electrostatic interaction between the electrons and nucleus of atom a with the electrons and nuclei of all the other atoms. The constant k is a conversion factor (1 esu = 14.4 eV) when the internuclear distance RM is expressed in angstroms. The expansion coefficients (E*,, q*J have been calibrated for several atom types so as to reproduce ab initio (STO-3G) data.30 For the solid-state application of EEM, the electrostatic potential is generated by a Madelung-type summation, using the Ewald technique.31 This method divides the summation of the electrostatic interactions into a summation in the direct lattice and a summation in the reciprocal lattice. A detailed description of this procedure is given by B e r t a ~ tby , ~Jackson ~ and cat lo^,^^ and by In earlier work35 it was reported that the electronegativity of the zeolite frameworks correlates with the framework density. These findings explicitly relate a structure type with the electronic (and therefore also the physicochemical) properties. For LTA and FAU the calculated crystal electronegativities are given in Table 1. Note that the difference with the earlier reported values of Van Genechten et al.35 results from a difference in the expansion parameters, due to the more extensive calibration set used. This recalibration does not however affect the general conclusions drawn in the paper by Van Genechten et al. The electronegativities of the clusters, isolated from LTA and FAU, were calculated using the SA program.36 The results are also graphically represented in Figure 1. The data show that the electronegativities of the clusters are larger than those of the infinite crystals, even for very large clusters containing up to 1224 atoms. Secondly, the electronegativity tends to decrease with increasing cluster size, although
x*,,
0
100
200
300 400 500 80010001200
number of atoms Figure 1. Electronegativity (eV/e) versus the number of atoms in clusters of LTA and FAU. TABLE 1: Electronegativity versus the Number of Atoms in Clusters of LTA and FAU name no. of atoms x(LTA) x(FAU) Si(OH)4 4R 6R D4R 8R D6R 12R 4Sod 5Sod lOSod 14Sod
9 24 36 36 48 54 12 108 216 3 84 468 864 1224
crystal
m
P-cage a- and superc:age
5.81 5.68 5.13 5.61
5.13 5.62 5.44
5.18 5.43 5.54 5.46
5.41 5.60 5.30 5.13
5.35 3.14
5.31 5.31 3.94
the actual electronegativities oscillate around this decreasing tendency. This oscillation is due to differences in density of the different clusters, with a lower electronegativity corresponding to a denser framework, and to different Si:O:H ratios of the clusters. Small differences in electronegativities of identical clusters of different structure types are due to different positions of the terminating hydroxyls. A clear distinction between the two structures cannot, however, be made. When only considering the smaller clusters, containing less than 108 atoms, the data might erroneously give reason to suggest that the electronegativity of FAU is lower than that of LTA. This conclusion is manifestly contradicted by crystal calculations. An important feature remains the reason for the difference between the electronegativity of the crystals and the clusters. We will demonstrate below that for the larger clusters (those built up from several sodalite cages) this difference is entirely due to a constant difference in the electrostatic potential caused by the remainder of the zeolite structure. Notice that it is important to account for this discrepancy, especially when studying molecular interactions, as the electronegativity difference is the driving force for the electron flow. The Hardness. The hardness, which is defined as the second derivative of the energy with respect to the number of
Comparison of Crystal Calculations with the EEM 10 I 9
J. Phys. Chem., Vol. 99, No. 10, 1995 3253
I
a 7
1
6 t
=
5 n
5
4 4r
1
I
0
LTA
0
FAU
3 2 1
0 '
0
"
'
'
"
'
200 400 600 800 1000 1200 1400 number of atoms
Figure 2. Hardness (eV/e*)versus the number of atoms in clusters of LTA and FAU. TABLE 2 : Hardness versus the Number of Atoms in Clusters of LTA and FAU name of cluster no. of atoms e(LTA-cluster) e(FAU-cluster) 8.57 Si(OH)4 9 8.49 4R 6R D4R 8R D6R 12R
,&cage a- and supercage 4Sod 5SOd lOSod 14Sod
24 36 36 48 54 72 108 216 384 468 864 1224
4.91 3.99 4.04 3.36 2.42 1.70 1.32
4.92 3.91
o 0.00
3.31 2.60 2.42 1.68 1.19 0.97
0.80
~
,
l
0.05
, 0.10
,
,
0.15
l
,
0.bO
l/sqrt(number of atoms) Figure 3. Hardness (eV/e2)versus the reciprocal square root of the number of atoms in clusters of LTA (dotted line) and FAU (full line). A very satisfying correlation coefficient Rvd was found for the clusters of both structures when performing a power fit: LTA : q = 23.47n-0.49
R,, = 0.998
(8)
FAU : q = 21.78n-0.47
R,, = 0.997
(9)
electrons,37is given by
I
ld48\ 1
The exponent close to -0.5 gives reason to suggest that there might be an inverse relation between the global hardness and the square root of the number of atoms. A graphic representation of this relation is presented in Figure 3. The accompanying equations are LTA :
where f is the atomic equivalent of the Fukui function:
fa =
dqa -(=)"
The data, given in Table 2 and Figure 2, show that the hardness is a continuously decreasing function of cluster size with small differences due to structure type. This decrease demonstrates that larger and thus softer systems are more appropriate to redistribute an added charge among their composing atoms.38 It is easily demonstrated that this is to be expected from the definition of the global hardness. Due to the normalization condition for the electron density distribution function the Fukui function is normalized to 1 and can be approximated as
Inserting this in eq 4b immediately shows the inverse relation between the global hardness and the number of atoms:
where d, is the topological density.
1
= 23.53-
+ 0.07
Rval= 0.999
(IO)
-I- 0.13
R,, = 0.996
(11)
L l
FAU : 11 = 22.91-
1
h
The universal character of this relation and its dependence on structural parameters need further investigation. In the Appendix a theoretical explanation is suggested for the relation between the global hardness and the number of atoms in three limiting cases: a one-dimensional line of atoms, a twodimensional array, and a three-dimensional ordering. Potential and Field of FAU The charges on the atoms generate an electrostatic potential and an electric field which can be calculated for any point in space with the aid of eq 3. Both quantities have been calculated along the nxn-axis for the different clusters of FAU and the full crystal. The results of three clusters, a sodalite cage and two clusters of respectively five and fourteen sodalite cages, are graphically shown in Figures 4-6, respectively. Whatever the size of the clusters, the shape of the electrical potential curve along the xn-axis is the same as that of the full crystal. However, its absolute value is shifted upward with respect to the electrical potential of the full crystal by a constant factor. The shape of the potential in the clusters reproduces exactly
Janssens et al.
3254 J. Phys. Chem., Vol. 99, No. IO, 1995
2 1.5
1
0.5
a
'g 8
8
0 -0.5
-1 -1.5
-2 I
xxx
T
0.8 0.6
3s
0.4
Fidd ( C w l )
PI
F W (sodrllk
caw)
a :'O -0.2
d
D m m Plot -0.4 -0.6 -0.8 -1
xxx
Figure 4. Comparison of electrostatic potential (V) and field (V/A) along a 3-fold =-axis in the sodalite cage cluster and crystal of FAU.
the shape of the potential in the crystal near the inner part of the clusters. At the border of the clusters deviations occur, mainly due to the terminating H atoms. The constant potential difference between cluster and full crystal suggests that the potential of the crystal can be thought of as composed of two parts, :v and vz, representing respectively the cluster potential and the potential due to the remainder of the crystal. :v has a local character with a characteristic variation along the xxx-axis. v z is a constant. It is this constant, ,:v which gives rise to the difference in electronegativity between cluster and crystal as obtained before. Equation 3 can now be written as
potential and the crystal potential. In Figures 5 and 6 this region is marked by vertical dotted lines. For a study of molecular interactions with the surface, the molecule must be sited in that valid region of the cluster to get meaningful theoretical results from cluster calculations. The electrostatic field is obtained numerically as
Em -=-*vm ~
x
(15) x
x
It is shown in the lower parts of Figure 5 . In the valid region of the cluster the electric fields of the clusters coincide with that of the crystal. This is expected, as we can write
with
xc' = x;
+ 271;qa + v:
(13)
and xcr
- xcl =
:v
where A v ~ A = R 0,~ as vcr is a constant.
Discussion (14)
This derivation is valid for clusters with at least 468 atoms. When vf - v z is plotted, a curve is obtained which coincides exactly with the full lattice potential in the central region of the cluster. This is the valid region of the cluster. It can clearly be obtained by making a difference plot between this corrected
One of the pumoses . . of this work was to define a so-called minimum cluster size or the minimal cluster needed for calculations to be representative for the whole crystal. It seems now that a universally valid minimum cluster size cannot be defined. The electronegativity of a cluster, by itself, can never be equal to the electronegativity of the crystal, because of the
J. Phys. Chem., Vol. 99, No. 10, 1995 3255
Comparison of Crystal Calculations with the EEM
A 4.
Figure 5. Comparison of electrostatic potential (V)and field (V/A) alc
a 3-fold xu-axis in the 5Sod cluster and crystal of FAU.
contribution of an isotropic constant potential due to the atoms surrounding the cluster. The electrostatic potential and the associated field are generally expected to be accurately described only by an infinite or at least large number of atoms.I0 The present work demonstrates that indeed crystal Madelung potential calculations are necessary to determine this extra constant potential. The electrostatic field, however, is found to be represented accurately by the cluster itself. The character of this field in the zeolite pores has a major influence on the properties of sorbed molecules and thus on the nature of the reactions that occur. Its importance has been stressed by Rabo and co-workers a long time ago.39-41 Every cluster has border effects, which can be avoided if the cluster is taken large enough. In that case a representative region in the cluster can be defined as the region in which the border effects are negligible and the constant isotropic potential can simply be added to obtain the crystal values of electronegativity and potential. In the case of faujasite the minimum size of the cluster for which this reasoning is valid is 468 cluster atoms. Therefore, the crystal electronegativity can be imposed upon the cluster by attributing to the cluster atoms this additional, constant, extracluster potential.
The implication of this finding is that such minimum-sized clusters are necessary for meaningful calculations on active sitemolecule interactions, where the active site and the molecule must be in the valid region. In this way one can be sure that no artificial potential gradients on the molecule are exerted, which are due to cluster ending effects. This does not imply that quantum mechanical calculations on small clusters (sodalite cage or smaller) are meaningless. They are indeed invaluable tools to obtain insight into the bond properties and minimum energy configurations of these entities. The hardness was found to be a continuously decreasing function of the number of atoms. A minimum-sized cluster to reflect this quantity therefore cannot be defined, especially since we are not able to determine the hardness of the crystals as a reference. This is due to the fact that the normalization condition for the Fukui function, within one unit cell, cannot easily be brought in accordance with the normalization condition for the full crystal. This is at variance with the crystal electronegativity, which can correctly be calculated by the very fact that the total charge of the unit cell as well as that of the crystal sums up to zero. The proposed fitting equations might provide a method to estimate the hardness of crystals, Le.
Janssens et al.
3256 J. Phys. Chem., Vol. 99, No. IO, 1995
-Potentlrl (Crystal) - _ - _Potential(14Sod)
_ _ Potential(14Sod) correctedfor
Ok&O~aUvity
dlffennce Difference Plot
-Field (Crystal) _ _ - _Field (14Sod) Dlffersnce Plot
Figure 6. Comparison of electrostatic potential (V) and field (VlA) along a 3-fold xrx-axis in the 14Sod cluster and crystal of FAU.
-
systems with an infinite number of atoms. For n 00 the hardness assumes a very small value. We are not yet in a clear position to make firm statements about the significance of the predicted crystal values, neither in an absolute or relative way. Acknowledgment. G.O.A.J. is grateful to the Institute for Scientific Research in Industry and Agriculture (I.W.O.N.L.) and the Flemish Government (I.W.T.). B.G.B. thanks the Research Council of the Catholic University of Leuven (Onderzoeksraad) for a Postdoctoral Mandate (PDM (93/57). H.T.is thankful to the European Community (Human Capital and Mobility Program) for a Postdoctoral Mandate. Many fruitful discussions with Prof. K. Sen are gratefully acknowledged.
+
A
Linear
Square
0
Cube
9
0 ' 0
44
88
132
176
220
number of atoms
Appendix The dimensionality dependence of the hardness can be seen from the D-l dependence of eq 4b. This dependence has been assessed for a linear, a planar, and a three-dimepional hypothetical all-Si structure with Si-Si distances of 2 A. The
Figure 7. Geometrical dependence of the hardness (eV/e2)versus number of atoms relation.
results of the application of eq 4b are shown in Figure 7, and the following power fits were obtained:
Comparison of Crystal Calculations with the EEM
q = 22.35n-0,75
1. linear case 2. square case
R,, = 0.999
q = 15.57n-0.46
J. Phys. Chem., Vol. 99, No. 10, 1995 3257 (Al)
R,, = 0.999
(A2)
R,, = 0.999
(A3)
as only the positive sign is physically meaningful. One obtains from eq A4
q a n-"* 3. cubic case
11 = 14.51n-0.37
with exponents close to -1/2/2, -112, and -1/3 for the linear, two-dimensional, and three-dimensional cases, respectively. The chemical global hardness 11 measures the response of a system to the addition or subtraction of charge. This response depends on the atom type through the parameters v*a and on geometry through In the present case, with all atoms being equal, mainly the latter is of importance, and one can write eq 4b as
u-'.
3. Cubic System. For the cubic orientation the exponent of eq A3 is close to -0.33. By analogy with the planar system, a shell is now defined as a plane in a square-pyramid on which all atoms have xi yi zi = constant. The number of atoms per shell is equal to the square of the shell-number, s2. Again, the distance is approximated to be proportional to the shellnumber. The total number of atoms contained in s shells is, for the cubic ordering, equal to
+ +
n= with S(n) CR-'. With the following reasoning a justification for this type of fitting relation can be provided. When an electron is added to a molecular system, it will be divided among its n composing atoms. There will be repulsion between the atoms if charges of equal sign are added. This repulsion is dependent on the atom type and on the geometrical configuration, which is confined in the symmetry over the reciprocal interatomic distances, S(n). For all cases a theoretical evaluation of the S(n) term is presented, which confirms the trend observed above. We therefore investigate the relation between the number of atoms and the size of the systems under consideration. 1. Linear system. For each additional interatomic distance there is an additional atom. Thus, DC
S(n) =
C-R1 = -11 + -21 + -31 + ... = c,
E
J, -ln(n)
(A5)
and inserting into eq A4 leads to Mn) V n For this type of fitting relation an excellent correlation coefficient is obtained:
q = 16.56-
n
R,, = 0.999
2. Square System. The exponent, being close to -112 in eq A2, can be rationalized in the following way. The Si atoms are place at equal distances of 0.2 nm in one quadrant of the xy plane. All atoms i with an equal sum of coordinates, (xi yi), are connected by a straight line called a shell. Each shell contains as many atoms as the shell-number, s, indicates. The total number of atoms is
+
If it is assumed, as a rough approximation, that the average distance to the atoms of the same shell is proportional to the shell-number, one obtains
Combination of eqs A8 and A9 gives for large n,
(All)
s(s
+ 1)(2s + 1) 6
One obtains
G)
S(n)=i+4-
+9
-+...= 1+ 2 + 3
For large n one has n therefore
s313; s
+ ...=-s(s 2+ 1) ('413)
(3n)ll3; S(n) = (3n)U3;and
The coefficients - 112 and - 1/3 are close to those calculated for the two zeolite structures, the difference being due to the difference in geometry of an ideal Si lattice and the real Si02 lattices in the LTA and FAU frameworks. References and Notes (1) Mortier, W. J. J. Catal. 1978, 55, 138. (2) Barthomeuf, D. J. Phys. Chem. 1979, 83, 249. (3) Jacobs, P. A. Catal. Rev. Sei. Eng. 1982, 24, 415. (4) Haag, W. 0.;Lago, R. M.; Weisz, P. B. Nature 1984, 309, 589. (5) Mortier, W. J.; Sauer, J.; Lercher, J. A,; Noller, H. J. Phys. Chem. 1984, 88, 905. (6) Haag, W. 0.;Lago, R. M.; Mikovsky, R. I.;Olson, D. H.; Hellring, S. D.; Schmitt, K. D.; Kerr, G. T. Stud. Sui$ Sci. Catal. 1986, 28, 677. (7) Weisz, P. B. Ind. Eng. Chem. Fundam. 1986, 25, 53. (8) Sauer, J. Chem. Rev. 1989, 89, 199. (9) Pisani, C.; Dovesi, R.; Roetti, C. Lecture Notes in Chemistry, Vol. 48, Hartee-Fork ab Initio Treatment of Crystalline Systems; Springer: Berlin, 1988. (10) Beran, S. In Theoretical Aspects of Heterogeneous Catalysis; Moffat, J. B., Ed.; Van Nostrand Reinhold Catalysis Series: New York, 1990; Chapter 5. (11) Fripiat, J. G.; Berger-Andre, F.; Andre, J. M.; Derouane, E. G. Zeolites 1983, 3, 306. (12) van Santen, R. A.; van Beest, B. W. H; de Man, A. J. M. Nato workshop: Physicochemical properties of zeolitic systems and their low dimensionality; April 1989. (13) Ahlrichs, R.; Bkr, M.; Haser, M.; Kolmel, C.; Sauer, J. Chem. Phys. Lett. 1989, 164, 199. (14) Catlow, C. R. A.; Price, G. D. Nature 1990, 347, 243. (15) Kassab, E.; Seiti, K.; Allavena, M. J. Phys. Chem. 1988,92,6705. (16) Allavena, M.; Seiti, K.; Kassab, E.; Ferenczy, G.; Angyan, J. G. Chem. Phys. Lett. 1990, 168, 461. (17) Vetrivel, R.; Catlow, C. R. A.; Colboum, E. A. J. Phys. Chem. 1989, 93, 4594. (18) Gale, J. D.; Catlow, C. R. A,; Cheetham, A. K. J. Chem. SOC., Chem. Commun. 1991, 179. (19) Mortier, W. J.; Van Genechten, K.; Gasteiger, J. J. Am. Chem. SOC. 1985, 107, 829. (20) Mortier, W. J.; Ghosh, S. K.; Shankar, S. J. Am. Chem. SOC.1986, 108, 4315. (21) Mortier, W. J. In Theoretical Aspects of Heterogeneous Catalysis; Moffat, J. B., Ed.; Van Nostrand Reinhold Catalysis Series: New York, 1990; Chapter 4. (22) Baekelandt, B. G . ;Mortier, W. J.; Lievens, J. L.; Schoonheydt, R. A. J. Am. Chem. SOC. 1991, 113, 6730. (23) Mortier, W. J. Struct. Bonding 1987, 66, 125.
3258 J. Phys. Chem., Vol. 99, No. 10, 1995 (24) Baekelandt, B. G.; Mortier, W. J.; Schoonheydt, R. A. Srruct. Bonding 1993,80, 187. (25) Meier, W. M.; Villiger, H. Z. Kristallogr. Kristallgeom. 1976,129, 411. (26) Baerlocher, C.; Hepp, A,; Meier, W. M. DLS-76 Manual 1977. (27) Deem, M. W.; Newsam, J. M. J. Am. Chem. SOC.1992,114,7189. (28) Jirak, Z.; Vratislav, S.; Zajicek, J.; Bosacekm V. J. Catal. 1977, 49, 112. (29) Parr, R. G., Yang, W. Density-Functional Theory of Atoms and Molecules;The International Series of Monographs on Chemistry 15; Oxford Univ. Press: New York, 1989. (30) Uytterhoeven, L.; Mortier, W. J.; Geerlings, P. J. Phys. Chem. Solids 1989,50,479. (31) Ewald, P. P. Ann. Phys. 1921,64, 253. (32) Bertaut, F. J. Phys. Radium. 1952,13, 499.
Janssens et al. (33) Jackson, R. A,; Catlow, C. R. A. Mol. Simul. 1988,1, 207. (34) Tosi, M. P. Solid State Phys. 1964,16, 1. (35) Van Genechten, K.; Mortier, W. J.; Geerlings, P. J. Chem. Phys. 1987,86, 5063. (36) Baekelandt, B. G. Ph.D. Thesis 224, Faculty of Agronomy, K. U. Leuven, 1992. (37) Parr, R. G.; Pearson, R. G. J. Am. Chem. SOC. 1983, 105, 7512. (38) Berkowitz, M.; Parr, R. G. J. Chem. Phys. 1988,88, 2554. (39) Rabo, J. A.; Poutsma, M. L. Adv. Chem. Ser. 1971,102, 284. (40) Rabo, J. A.; Kasai, P. H. Prog. Solid State Chem. 1975,9, 1. (41) Rabo, J. A. In Zeolite Chemistry and Catalysis; Rabo, J. A,, Ed.; American Chemical Society: Washington, DC, 1976. JP941640P
