J . Phys. Chem. 1989, 93, 3237-3240 in practice, since they would often lead to decomposition of the semiconductor materiaLs7 The model proposed in this paper provides a new and simple approach to the estimation of energy gap shifts in small colloidal particles containing excess charge or dopants. The model can be considered only an approximate one for very small particles containing few (or fractional numbers of) electrons or dopants, since the analogy with a heavily doped bulk semiconductor is only a rough one. For very small particles the simple bulk parabolic band picture loses its validity, and Coulombic interactions of eand h’, an increasing importance of surface atoms and states, and the onset of discrete energy levels rather than bands will be significant. (37) Bard, A. J.; Wrighton, M. S. J. Electrochem. SOC.1977,124, 1706. (38) The recent paper by Kamat, P. V.; DimitrijeviE, N. M.; Fessenden, R. W. J. Phys. Chem. 1988, 92, 2324 on I& colloids mentions this effect for photobleaching but does not suggest a quantitative model.
3237
Conclusions We have proposed a model to describe charging effects on energy level shifts in semiconductor particles. High electron concentrations created, for example, by strong irradiation to the particles can cause the semiconductor to become degenerate and raise the Fermi level into the conduction band. As a result, the energy necessary to excite electrons optically from the valence to the conduction band will be larger than the width of the forbidden energy gap, and thus, such particles will show an absorption edge shift toward the blue compared to the bulk semiconductor material. This charging will also affect the energies of photogenerated conduction band electrons and hence affect electrontransfer processes at irradiated particles and the measured potential for collection of such charges with an electrode or solution species.
Acknowledgment. The support of this research by the National Science Foundation (CHE8304666) is gratefully acknowledged. We thank E. S. Smotkin for sample preparation.
Chemisorption on Metals: The Method of Moments Point of View Jerzy Cioslowskit and Miklos Kertesz*yt Department of Chemistry, Georgetown University, Washington, D.C. 20057 (Received: May 23, 1988; In Final Form: October 31, 1988)
We show that the interaction of two adatoms, V,,, mediated by a semimetallic graphite surface is oscillatory as a function of the number of sites between the two chemisorbed species. The discussion is based on a method of moments expansion of the density of states of the locally perturbed system. In model calculations properly sized clusters or periodic clusters can be chosen by requiring the separation of the adsorbates to be large enough to separate all adatoms in one unit cell from the other unit cells by at least 4-5 intact sites in all directions. The results show a close connection between site preferences for a second adspecies and substitutional preferences in delocalized a-electron systems.
Interaction between chemisorbed species on metals, semimetals, and other systems with delocalized electrons contains an indirect component which decays slowly with distance. These long-range interactions are the solid-state analogues of the well-known rules of substitutional preferences for alternant hydrocarbons.’ Friede12has shown that in a metal a local perturbation causes an oscillatory decreasing potential
V(r)
-
cos ( ~ ~ r ) / ( z k ~ r ) ~
where kF is the Fermi wave vector. In conjugated hydrocarbons quite similar oscillations of charge densities and bond orders occur in response to a local perturbation. For instance Gutmans has shown that, if two centers are connected by only one path (series of bonds), then a perturbation at the first center causes a charge polarization at the other one. The sign of these charges alternate with a rapidly decreasing amplitude as the separation between the two sites increases. Due to the oscillatory charge distribution resulting from perturbation of one site, there has to be a nonbonded indirect interaction between two such perturbations: one expects a long-range oscillatory component due to the interaction of the two Friedel oscillations aroused by the two perturbations, e.g., the chemisorption of two H atoms on graphite. Unfortunately, no analytical results exist for a general molecule or surface such as the ones mentioned above for one path chain^.^ Recently LaFemina and Lowe4 have studied hydrogen atomic chemisorption on graphite at low coverages using the energy band formalism, which is based on periodic boundary conditions. They ‘Present address: Los Alamos National Laboratory, T-12, MS-J569, Los Alamos, NM 87545. ‘Camille and Henry Dreyfus Teacher-Scholar, 1984-89.
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have considered a unit cell of 18 C atoms and two hydrogens, absorbed at a site indicated by * on Figure 1, and a second one at either of the sites specifically indicated. Since the unit cell is repeated in two directions in this periodic model, the interaction of the chemisorbed species is complicated by the presence of other hydrogens in adjacent cells. For example, for y-chemisorption (as pointed out in ref 4), a starred atom in a neighboring cell is separated by two empty sites while the starred one inside the cell is separated by four empty sites. As a result, such calculations based on periodic boundary conditions should include many more sites to obtain a clear picture concerning the “bare” interaction between two adspecies. The purpose of this paper is to present a new approach, based on the method of moments, with the aim to understand the nature of nonbonded adspecies-adspecies interactions mediated through the delocalized electrons of a surface. We first review the moment’s expansion of the energy expression for an extended system as perturbed by one or more chemisorbed species. The new formula so derived (eq 8) expresses the energy change in terms of the change in the number of electrons and in terms of changes in the moments, the latter being local if the perturbation is local. A test example is given demonstrating the favorable convergence of the method with respect to the size of the cluster going into the evaluation of the moments. Subsequently, the problem of two hydrogen atoms chemisorbed on graphite is analyzed by using a simple Hiickel Hamiltonian. The ( I ) Dewar, M. J. S.; Dougherty, R. C. The PMO Theory of Orgunic Chemistry; Plenum: New York, 1975. (2) (a) Friedel, J. Philos. Mug. 1952, 93, 526. (b) Harrison, W. A. Solid Srute Theory; McGraw-Hill: New York, 1970. (3) Gutman, I. Theor. Chim. Acto (Berlin) 1979, 50, 287. (4) LaFemina, J. P.; Lowe, J. P. J . A m . Chem. SOC.1986, 208, 2527.
0 1989 American Chemical Society
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Cioslowski and Kertesz
The Journal of Physical Chemistry, Vol. 93, No. 8, 1989
* P - P - P - P - P - P
,..
a#O a=o a-0 a-0 a50 a-0 Figure 2. Polyene with an adatom at the first site of the chain.
TABLE I: Results of Cluster Calculations for Perturbed Polyene (Figure 2) change in the total energy size of cluster a = 0.1 a = 0.2 (Y = 0.5 a = 1.0 (atoms)
Figure 1. Hydrogen chemisorption on graphite. The first atom is chemisorbed at site *, the second at any one of the others indicated by a Greek letter (after ref 4). (a’and 6‘ were not studied in ref 4.)
resulting “bare” adspecies-adspecies interaction, V,,, shows an oscillating decay with attractive and repulsive regions. Change of Total Energy Due to a Finite Number of Adsorbates. For systems described by effective Hamiltonians, such as Huckel or extended Huckel, the total energy is given by
E,,, = S E -m F g ( E ) Ed E
(1)
where EF is the Fermi energy and g(E) is the density of states. The function g(E) is normalized to the total number of electrons:
N = SE -m F g ( E )d E
(2)
Upon perturbation, the density of states changes. This changes the total energy and the Fermi energy. Some of this change may be associated with a change in the number of electrons:
E,,,
+ 6E,,,
lm
EF+~EF
=
M E ) + M E ) I E dE =
Etot
N
+ 6N =
s
+ g(EF)EFBEF + S E F 6 g ( E ) EdE -m
Ef+&EF
-m
[g(E)
(3)
+ 6g(E)] d E = N + g(EF)6EF il E F 6 g ( E )d E (4) -_
Combining eq 3 and 4 we arrive at 6E,,, = EF6N
+ SEF6g(E)(E- EF) d E -m
(5)
The first term in eq 5 is trivial because the Fermi level is the chemical potential of the system. The second term may be computed in practice by the following approximation. We approximate the function
o
(6)
over some interval ( a , b ) by the Legendre p ~ l y n o m i a l . ~ A somewhat similar use of orthogonal polynomials in the context of the method of moments is given by Burdett and Lee.6 This results in L
f ( x ) = X ( x - 1x1) z yzx - C a L j x z j j=O
(7)
where L is the order of approximation and aLj are appropriate coefficients independent of the chemical system. Using this result we obtain the main formula of this paper L
6E = EF6N
(5)
+ f/z6pI - j=O xajsL6pj
Cioslowski, J. Chem. Phys. Letr. 1985, 122, 234.
(6) Burdett, J. K.; Lee, S. J . Am. Chem. SOC.1985, 107, 3050.
20 50 100 200
0.103 95 0.10411 0.104 17 0.10420
0.21 5 70 0.21637 0.21660 0.21672
0.594 95 0.59864 0.59995 0.60062
1.344 82 1.35571 1.35950 1.361 43
extrapolated
0.10423
0.21684
0.601 29
1.363 36
TABLE I 1 Results of Perturbational Calculations Using the Method of Moment Approach for Perturbed Polyene (Figure 2) change in the total enernv no. of moments a = 0.1 a = 0.2 a = 0.5 (Y = 1.0 2 4 6 8 10 12
0.103 13 0.10425 0.10441 0.10425 0.104 18 0.10420
0.21250 0.21665 0.21756 0.21690 0.21660 0.21671
0.578 13 0.60444 0.60628 0.601 22 0.59942 0.60035
1.31250 1.39497 1.37961 1.35393 1.35466 1.361 97
where the changes of the moments due to the perturbation(s), bp,, are given by
I_ +m
=
bg(E)(E - EFY dE
(9)
The moments are computed easily from topological considerations. This is done by assigning a weighted self-loop to the perturbed atom and weighting all the bonds that originate from this atom. The moments are calculated by counting weighted self-returning walks. Two points have to be noted. First, since any change in g(E) that originates from point defects is infinitesimally small compared to g(E) itself, eq 5 is exact even for large point perturbations. In the case of a bulk system the change in g(E) due to a point perturbation is of the order of N-I. Therfore, our first-order m since the treatment of g(E) becomes exact at the limit N second-order correction is of the order of N-’ as compared to the first-order term. Second, since the Legendre polynomials form m. a complete set, eq 8 is exact in the limit of L A Test Case: End Site Perturbation in a Chain. Let us consider a perturbed equidistant polyacetylene chain in the Huckel approximation (Figure 2). The perturbation cannot be accounted for by crystal orbital calculations because the system lacks the appropriate translational symmetry.’ The change in energy due to perturbation can be calculated by considering finite clusters and extrapolating the result to the bulk limit. Convergence of this procedure is quite slow (Table I). Diagonalization of 200 X 200 matrix is necessary for obtaining about 1% accuracy. The same accuracy is achieved by taking into account only the 12 first moments which requires considering only a 12 X 12 matrix (Table 11). The dramatic improvement can be explained by the fact that in the perturbational calculations the boundary effects are effectively neglected and that the present approach focuses on a difference due to the perturbation rather than on accurately calculating the difference of two large quantities. Hydrogen Chemisorption on Graphite. We also used our method for modelling the hydrogen chemisorption on a layer of graphite. This was previously considered by LaFemina and Lowe4 by applying extended Huckel band calculations based on periodic boundary conditions. In our calculations we use the simple Hiickel method and added two more configurations (CY’ and 8, see Figure
-
-
(8) (7) Alternative approaches to point perturbations in chains include Green’s function and transfer matrix techniques. See, e.g.: (a) Koutecky, J. Phys. Reu. 1957, 108, 13. (b) Economou, E. N. Green’sFunctions; Springer: Berlin, 1983. (c) Seel, M.; Ladik, J. Phys. Reu. E 1985, 32, 5124.
The Journal of Physical Chemistry, Vol. 93, No. 8, 1989 3239
Chemisorption on Metals
TABLE 111: Results of Simulated Chemisorption Calculations: Two Hydrogen Atoms on Graphite (in @ units) change in the total energy for site“ no. of moments p, n = 0 b,n= 1 a,n=2 a’,n = 2 t,n=3 t‘,n=3 2 4 6 8 10 12 14
-3.125 -3.464 -3.899 -4.020 -4.185 -4.206 -4.199
v,b
0.756
y,n=4
n=m
-3.750 -4.132 -4.639 -4.800 -5.01 1 -5.050 -5.045
-3.750 -4.010 -4.438 -4.532 -4.698 -4.7 16 -4.706
-3.750 -4.010 -4.547 -4.7 12 -4.924 -4.958 -4.951
-3.750 -4.010 -4.583 -4.755 -4.971 -5.018 -5.010
-3.750 -4.010 -4.583 -4.7 18 -4.927 -4.964 -4.955
-3.750 -4.010 -4.583 -4.706 -4.927 -4.963 -4.951
-3.750 -4.010 -4.583 -4.706 -4.945 -4.971 -4.955
-0.090
0.249
0.004
-0.055
0.000
-0.002
0
“Site labeling according to ref 4; see Figure 1. a’ and c‘ correspond to configuration not studied in ref 4. n indicates the number of sites between the two adspecies along the shortest path. V,, = E,, - E,. (b) --e
0.01
-6
-6
---E
-
-6
-7
t
t
-0.21
0.5
- -1.47 O’I
0.11 -1.51
--P
--P
Figure 3. Relative stabilities of the seven arrangements of two (H) adatoms on a graphite layer from (a) LaFemina and Lowe4 (EHT periodic), (b) moments calculation (Hiickel, two adatoms to an infinite surface), (c) simulated periodic cluster (Hiickel) assuming additivity of adatom-adatom interactions,and (d) periodic Hiickel model (Hiickel 0 = -2 eV).
Figure 4. Interaction energy, V,,, of two hydrogen atoms adsorbed at a graphite layer. n is the number of carbon sites between the two adatoms. Greek letters refer to the geometry as given in Figure 1 (in 0 units).
I ) . Chemisorption of hydrogen is simulated by setting to zero all the resonance ( p ) integrals pertaining to the bonded carbon atom. The results of calculation are given in Table I11 and Figure 3. Convergence within I % is reached by using 14 moments. The stability is the highest for the @-arrangementof atoms (see Figure 3, last three columns) and decreases in the following order: @ > a > (a’ > t’ >) y > t > 6. The main prediction that is different from the results from ref 4 relates to the arrangement a, which is associated with cleavage (or weakening) of six C-C bonds and is predicted in our calculations to be less stable than arrangement p which corresponds to breaking five C-C bonds, in accordance with chemical intuition. (We will return to the analysis of the last three columns of Figure 3 at the end of the paper.) We have also depicted the total energy relative to two infinitely separated adatoms of the system as a function of the number of sites separating the adsorbate, n (Figure 4). On this scale a,a’ corresponds to n = 2, @ to n = 0, y t o n = 4, 6 to n = 1, and t , ~ ’ to n = 3. While our model is clearly too crude for quantitative purposes, the oscillatory behavior of the “bare” interaction between two adsorbates occurs just as expected on the basis of the Friedel oscillations or alternacy rules. The rate of decay of the interaction is fast but cannot be quantitatively characterized because of the large differences of interactions with the same n (e.g., a and a’) which are due to their different topologies. Can our results be used to interpret the difference of the chemisorption preferences in our model (@ is most stable) and the
periodic adsorbate coverage calculations of LaFemina and Lowe? Since adsorbate-adsorbate interactions are not exactly additive and also there are differences between the two Hamiltonians, only a qualitative answer is expected to this question. A major difference is due to the effect of neighboring cells as pointed out above. Thus for the a-perturbation in addition to the intracell interactions there are also two intercell interactions with n = 4 per unit cell, not counting interactions beyond n = 4. For pperturbations there are two additional intercell interactions with n = 4; for y there is an additional one with n = 2; for 6 there is one with n = 3 and finally for e there are two with n = 3. In order to make a further connection with the results of ref 4, one might assume additivity of the V,,pairs interactions. For example, the energy of an a-perturbation would correspond to V,(a) 2V4. Assuming such additivty, this would result in the relative energies of the sites using the 18-carbon unit cell of ref 4 which are given in column c of Figure 3. The last column in Figure 3 shows the relative energies of the five periodic models used in ref 4 using a periodic (1 8 carbon site unit cell) calculation in conjunction with the Hiickel Hamiltonian, allowing a more quantitative comparison of the periodic boundary calculation and the method of moment’s calculations. (The energy of the y-configuration has been chosen as the zero point on the energy scale and the energy values refer to the 18-site unit cell.) Most remarkable is the restored similarity with the extended Hiickel periodic calculations of ref 4,with the noted exception of the reverse in the orders of the a- and @-configurations. The
-0.11 xs
X€
+
J . Phys. Chem. 1989, 93, 3240-3243
3240
changes of ordering between columns b and d, and the large reduction of the extra stability of configuration 6 in the periodic calculation (d) as opposed to the individual absorption calculation as given by the moments method (b) calls for caution regarding the use of periodic models in the theory of absorption. Interactions mediated by the surface are not a d d i t i ~ e . ~ (8) It seems that various forms of approaches based on periodic boundary conditions have been invented independently by several groups. These differ in the number of k points used (if more than one k point is used, this is essentially a "large unit cell" crystal orbital method), in the way the lattice summations are carried to convergency, and in the manner the Hamiltonian is approximated. For selected references, see: (a) Zunger, A. J. Chem. Phys. 1975,62, 1981. Zunger, A. J. Chem. Phys. 1975.63, 1713. (b) Kertesz, M.; Koller, J.; Azmann, A. Chem. Phys. Lett. 1978, 53, 446. Evarestov, R. A.; Lovchikov, V. A,; Tupitsin, I. I. Phys. Status Solidi B 1983, 117, 417. (c) Messmer, R. P. Phys. Rev. B 1977, 15, 1811. Skala, L. Phys. Status Solidi B 1982, 109, 733. Salem, L. J . Phys. Chem. 1985, 89, 5576. (d) Burdett, J. K. In Structure and Bonding in Crystals; OKeefe, M., Navrotsky, A., Eds.; Academic: New York, 1981. (e) Baetzold, R. C.; Mason, M. G.; Hamilton, J. F. J . Chem. Phys. 1980, 72, 366, 6820. S e d , M.; Bagus, P. S.; Ladik, J. J . Chem. Phys. 1982, 77, 3123. Deilley, B.; Ellis, D. E.; Freeman, A. J.; Baerends, E. F.; Post, D. Phys. Reu. B 1983, 27, 2132. Mason, M. G. J . Chem. Phys. 1983,27,748. (f) Saillard, J.-Y.;Hoffmann, R. J . Am. Chem. SOC.1984, 106, 2006. (9) Deik, P., Snyder, L. C. Phys. Rev.B 1987, 36, 9619.
We would like to point out that the effects of Friedel oscillationsZ on adatoms-adatom interactions are following from the present method of moments approach to chemisorption but should be also obtainable from very large cluster calculations (periodic or properly terminated finite) as well. In choosing properly sized clusters or repeat units, such that V, has decayed sufficiently, one may be able to use any of the more convenient approaches4,* based on periodic boundary conditions as advocated by many groups to extract useful adatom-adatom interaction information. It is advisable that the clusters or repeat units be chosen large enough to allow a sufficient decay of V, for intercell interactions. We recommend that at least 4-5 sites should separate the adatoms within the unit cell from the ones in neighboring cells in every direction.
Acknowledgment. We thank Prof. A. Graovac for useful discussions in an early stage of this project. This work has been supported by the Camille and Henry Dreyfus Foundation and by N S F Grant DMR8702148. Registry No. H,, 1333-74-0; graphite, 7782-42-5. (9) Burdett, J. K. (private communication), also finds that adatom-adatom interactions are not additive.
Improved Method for Dealumination of Faujasite-Type Zeolites with Sllicon TetrachlorMe B. Sulikowski: Gabriela BorbBly, Hermann K. Beyer, Central Research Institute of Chemistry of the Hungarian Academy of Sciences, Pusztaszeri ut 59-67, 1025 Budapest, Hungary
Hellmut G . Karge,* Fritz- Haber-Institut der Max-Planck-Gesellschaft, Faradayweg 4-6, 1000 Berlin 33, Federal Republic of Germany
and I. W. Mishin Zelinsky Institute of Organic Chemistry of the Academy of Sciences of the USSR, Leninski Prospekt 47 Moscow 117913, USSR (Received: June 3, 1988)
The dealumination of Na-Y zeolite with silicon tetrachloride vapor is a product-inhibited reaction because of the formation of sodium tetrachloroaluminate, which is nonvolatile at the reaction temperature. However, starting from Li,Na-Y prepared by ion exchange in aqueous solution, a nearly complete substitution of framework aluminum by silicon proceeds as evidenced by magic-angle spinning NMR and IR spectrometry. Also after contact-induced ion exchange performed by grinding a mechanical mixture of hydrated Na-Y and crystalline LiCI, the framework dealumination is not prematurely stopped by product inhibition. In contrast to Na[AICI,], the corresponding Li compound dissociates at the reaction temperature, the decomposition product (AQ) volatilizes, and only minor amounts of solid products not affecting the progression of the dealumination reaction are deposited in the zeolite pores.
Introduction Zeolites with high Si/Al ratios and especially crystalline silica isostructural with zeolites are known to be adsorbents with predominant hydrophobic properties. The S i 0 2 variety of faujasite is of special interest because this structure shows the largest pore space and, consequently, the highest potential adsorption capacity among all known zeolites. The removal of lattice aluminum from the framework of Y zeolite by treatment with H4EDTA1 (EDTA = ethylenediaminetetraacetic acid) gives products maintaining much of the crystallinity. High-crystalline Y-zeolite with Si/Al ratios over 100 could be obtained by repeated alternating hy'On leave from Institute of Organic Chemistry and Technology, Krakdw Technical University, 31-155 Krakbw, Poland.
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drothermal treatment and acid leachingz These procedures, however, create lattice vacancies or even a secondary mesopore system inside the zeolite crystallites, first reported by Ciembroniewicz et al.,3 and result in the formation of hydrophilic sites. In 1980 a new method for dealuminating zeolites by reaction with silicon tetrachloride was reported4 that involves, in contrast to the methods mentioned above, the direct substitution of framework aluminum by silicon atoms. However, the dealumi( I ) Kerr, G. T. J . Phys. Chem. 1968, 72,2594. (2) Scherzer, J. J. J. Catal. 1978, 54, 285. (3) Ciembroniewicz, A,; Zolcinska-Jezierska, J.; Sulikowski, B. Pol. J . Chem. 1979, 53, 1325. (4) Beyer, H. K.; Belenykaja, I. Stud. Surf. Sci. Cutal. 1980, 5 , 203.
0 1989 American Chemical Society
