J. Phys. Chem. 1993,97, 6679-6685
6679
Structural and Electronic Features of a Br~nstedAcid Site in H-ZSM-5 Stephen J. Cook, Arup K. Cbakraborty,’ Alexis T. Bell, and Doros N. Theodorou Centerfor Advanced Materials, Lawrence Berkeley Laboratory, and Department of Chemical Engineering, University of California, Berkeley, Berkeley, California 94720 Received: January I I , 1993; In Final Form: March 12, I993
We report the results of local density functional theory calculations on a Brransted acid site of the zeolite H-ZSM-5. We have investigated the structural and electronic properties of the site. Comparison is made between our results and existing experimental data. It is shown that structural relaxation around an acid site must be performed to obtain accurate energetics for substitution of aluminum into the zeolite framework. The effects of cluster termination are studied by comparing results obtained for both isolated clusters and clusters embedded in a Madelung field generated by the zeolite framework. The properties of the electron density distribution in the region around an acidic proton indicate that the acidic moiety may be characterized as a rather soft acid.
Introduction Zeolites are microporous aluminosilicate crystals which are used as molecular sieves and/or solid acid catalysts in a wide variety of industrialprocessing applications, such as the conversion of methanol to gasoline.’ The catalytic activity of zeolites is associated with Brensted acid sites, which consist of protons bonded to oxygen atoms which bridge silicon and aluminumatoms in the zeolite framework. The local geometry of the site, as well as the number of aluminum atoms adjacent to the silicon atom associated with the site, affects the electron density distribution around the proton and hence its acidityS2While characteristics of Brensted acid sites have been studied experimentally,the effects of zeolite framework geometry, aluminum siting, and local geometry are still poorly understood. A desire to understand the relationshipbetween structure and the Brensted acidityof zeolites has prompted many researchers to perform quantum-chemical calculations. A recent review of quantum-chemical calculations applied to zeolites has been given by S a ~ e r .Most ~ of these studies have been performed using Hartree-Fock theory and have not included correlation effects. Some recent works in this area have included correlation at the MP2 level, but these have generally been limited to small cluster representationsof the zeolite due to computational limitation^."^ Many of the Hartree-Fock studies have also employed minimal basis sets, a practice which has recently been questioned.8 Many of the Hartree-Fock studies have been performed on monomeric or dimeric species which do not necessarily reflect the environment around an acidic proton in a zeolite crystal, since cluster termination effects may not be negligible. Those studies which were performed on larger crystal fragments, notably the work of Derouane and Fripiat on pentameric clusters of H-ZSM-5: did not include the effects of structural relaxation around an aluminum-substitutedframework site. Recmt workof Brand et ai.,however, has includedrelaxation effects for larger clusters.’ Finally, we note that the long-range electrostatic potential imposed by the zeolite lattice has rarely been included in its entirety. Rather, the Madelung potential due to the infinite zeolite lattice has often been represented as a finite collection of point charges distributed about the cluster which is treated explicitly using quantum-chemical methods.IO Kassab, Seiti, and Allavena have recently performed calculations using the Madelung field due to the infinite zeolite lattice under the assumption that the lattice is an array of point charges, but they haveargued that the matrix elements involving the Madelung potentialshould becalculated by assuming that thechargedensity
of the cluster is also represented by a collectionof point charges.’ Allavena et ai. have used this assumption to study the energetics of the NH3-faujasite interaction and found that proton transfer to ammonia was favored only when the effects of the Madelung potential were included in their calculation.11J2 The study did not, however, include relaxation effects under the influence of the Madelung field or correlation effects. We recently began a program of study designed to answer some of the questions pertinent to understanding the nature of a Brensted acid site in H-ZSM-5. We present results of KohnSham local density functionaltheory calculationsthat incorporate the Madelung field in the Hamiltonian and allow for full structural relaxation. We show that extensive structural relaxation occurs when an aluminum atom, accompaniedby a charge-compensating proton, is substituted for a silicon atom in the lattice. The Madelung field is shown to stabilize configurations around the acid site in which the proton may reside at longer distances from the bridging oxygen than have previously been observed.
Computatiod Method System energetics are computed using density functional theory as formulated by Hohenberg, Kohn, and Sham.13J4 Although density functional theory has been used extensively to study problems in condensed matter physics and materials science, it has only recently begun to be used as a framework in which to studyissues of chemicalimportan~e.~~J6 We provide a brief review of the theory before giving the details of our calculations. Although the origins of the density functional theory lie in the early works of Thomas” and Fermi,18 density functional theory was placed on a firm theoretical foundation with the publication of the theorems of Hohenberg and Kahn." The first theorem of Hohenberg and Kohn proved that the Hamiltonian, and hence all ground-state electronic properties, is a unique functional of the electron density distribution, p ( i ) , within that system. This is so because the density, p ( i ) , is uniquely determined (within a trivial additive constant) by the external potential, v(i). The second theorem further showed that the energyobeysa variational principle; that is, the correct ground-state electron density distribution is that distribution which minimizes the total energy. The variational principle is thus stated as
where p is a Lagrange multiplier corresponding to the chemical potential and N is the number of electrons in the system. The
0022-3654/93/2097-66~9~04.00/0 Q 1993 American Chemical Society
6680
The Journal of Physical Chemistry, Vol. 97, No. 25, 1993
Cook et al.
TABLE I: Exponents of the Gaussian Basis Sets Used in tbe Local Density Functional Theory Calculations atom
orbital ~~
Si
0 H A1 acidic H
s P
s P s s P s P
~~~
0.0824,0.2264, 1.803 1 0.0624,O. 1708,0.4478 0.2788.0.8879, 11.4660 0.3725, 1.6684, 8.0472 0.1689.0.6239, 3.4253 0.0590,0.16 17, 1.3292 0.0447.0.1 158.0.2300 0.0637,0.2044,0.7299, 3.2242, 24.0493 0.0429,0.3240
energy functional of Hohenberg and Kohn is simply the sum of terms involving the kinetic energy of the electrons, electronnuclear interactions, and electron4ectron repulsions
If the energy functional were readily obtainable, variationally calculating the electronic structure of even complicated systems would be relatively easy. However, & ~ [ p ( i ) ] is not known accurately for inhomogeneous electron density distributions due to uncertainties regarding the kinetic energy functional, T [ p ( i ) ] , and the electron-electron energy functional, V,,[p(i)1. To circumvent part of this problem, Kohn and Sham reintroduced orbitals and used the kinetic energy operator appropriate for a system of noninteracting e1ectr0ns.I~ Electron-lectron interactions were represented using a mean-field expression. Corrections to the kinetic energy functional,as well as terms describing electron exchange and correlation, were combined into a term called the exchangecorrelationenergy (or potential). Corrections for electron exchange are required in this procedure since the KohnSham wave function is not explicitly constructed to be antisymmetric. Likewise, corrections for electron correlation account for nonclassical contributions to the electron-electron interaction not included in the mean-field representation. This procedure yielded a set of N independent particle equations with the form of the one-electron Schrainger equation. The KohnSham Hamiltonian for a molecular system with no externally imposed fields in thus
Figure 1. Pentameric cluster used to represent the local environment of a Bronsted acid site within ZSM-5.Boundary hydrogen atoms used to saturate dangling bonds are not shown.
The calculations presented here are, therefore, valence-level calculations, which require only modest basis sets to accurately reproduce molecular energies. Energy minimization is performed using Hellman-Feynman forces and the optimally conditioned optimization algorithm of D a ~ i d o n . ~Although ~ HellmanFeynman forces are not accurate in all-electroncalculationsunless very large basis sets are employed, they have been shown to be accurate2swhen smaller basis sets are used together with ab initio pseudopotentialssuch as those of Bachelet, Hamann, and Schliiter. Despite the computational efficiency of density functional theory, a calculation including full energy minimization of the 288-atom unit cell of silicalite is currently computationally intractable. To overcome this difficulty, we model the region around an acid site using an embedded cluster model. The atoms which exert the largest effect on the acid site, i.e., those nearest the site, are treated explicitly via density functional theory. The PO') di' 1 KohnSham Hamiltonian for this cluster, however, also includes (3) Hi = - 2V: u,,, lii- 3.1 + vxc a term that correspondsto theclassical electrostatic field imposed by the atoms of the infinite zeolite crystal that are not explicitly where unucis the potential due to atomic nuclei and uxcis the incorporated into the cluster. The cluster is shown in Figure 1. exchange-correlation potential. The KohnSham equations are The pentameric cluster of T-atoms, which has the formula solved by expanding the single-particle wave functions in an AlOHO3(SiO3H3)4, is terminated by hydrogen atoms which are appropriate basis set placed at a distance of 1 A from the oxygen atoms on the exterior of the cluster. The hydrogen atoms are positioned along the (4) bond axes connecting the boundary oxygen atoms to the surrounding silicon atoms in the unit cell which are not an explicit and by invoking the variational principle to obtain the basis part of the cluster. Although the lengths of the oxygen-boundary coefficients, Cij. Further details may be found in the excellent hydrogen bonds were not systematically relaxed to minimize the book of Parr and Yang.19 energy, the hydrogen atoms should be very close to the minimum In our calculations, the wave function is expanded in a linear energy configuration. Since the perturbation introduced by not combination of Gaussian-type orbitals (LCGTO) using the relaxing the hydrogen atoms would have to propagate across three minimal basis sets of Huzinaga et aL20 The basis functions used bonds to affect the properties of the acid site, we conclude that here, which are completely uncontracted to increase variational the effects of not relaxing the hydrogen atoms are negligible. flexibility, are given in Table I. Exchange and correlation are The electrostatic potential due to the framework atoms not calculated using the local density approximation (LDA) as explicitly included in the LDA calculation is obtained under the parameterized by Perdew and Zunger.21 The computational burden is reduced by fitting the Hartree and exchange-correlation assumption that those atoms constitute a lattice of point charges. potentials to auxiliary Gaussian basis sets. The Hartree potential The potential at a given point in space is computed by first using is fit in the manner of Dunlap, Connolly, and Sabin,22while the Ewald summation to calculate the potential generated by a exchange-correlation potential is fit on a set of icosahedral meshes perfect infinite crystal of ZSM-5. The framework used for the centered about each atom in the system using a least-squares Ewald summation is assumed to be completely siliceous. The contribution to the electrostatic potential due to the framework procedure. The computational requirements of the LCGTO atoms which are explicitly included in the quantum-mechanical method used here thus scale as the cube of the number of basis functions. Atoms heavier than hydrogen are treated using the cluster is then calculated using Coulomb's law and subtracted ab initio pseudopotentialsof Bachelet, Hamann, and S ~ h l i i t e r . ~ ~ from the value obtained from the Ewald summation.
+
+
s-
Features of a Bronsted Acid Site in H-ZSM-5 Although the hydrogen atoms on the boundary of the cluster are necessary to saturate dangling bonds on the frontier shell of oxygen atoms, they induce an artificial electrostatic potential at the center of the cluster. We compensate for this effect by following a procedure suggested by Kassab et The procedure works by assuming the charges of the first shell of silicon atoms within the unit cell not explicitly included in the cluster to be adjustableparameters. Chargesare assigned to thesesilicon atoms by minimizing the difference between the potential due to these -frontier" silicon atoms when they have the full charge of other silicon atoms in the zeolite framework, qs', and the potential which is created by the silicon atoms with adjustable charges plus that due to the hydrogen atoms which saturate the cluster. If this difference is minimized in some region surrounding the acid site, then the effect of the boundary hydrogen atoms should be mitigated. The minimization is carried out using a leastsquares procedure with the objective function
The set of charges q, represent the result of the minimization procedure. The partial charges of the boundary hydrogen atoms, qH, are taken from a Mulliken population analysis performed on the result of an LDA calculation on the siliceous cluster in the absence of the Madelung field. The charge of the framework silicon is derived self-consistentlyin the following way. The initial value of the framework silicon charge is set equal to the charge of the silicon atom at the central T-site in the isolated (no Madelung field) siliceous cluster. This charge is determined by Mulliken population analysis. A least-squares analysis is performed, the Madelung field is generated, and an LDA calculation is performed to obtain a new net charge on the central T-site. The self-consistentcycle was continued until the Mulliken population of the silicon atom at the central T12 site changed by less than 0.0lld between iterations. This process, which required three iterations to achieve convergence, yielded a net silicon charge of +2.1814 and a net oxygen charge of -1.09lel. Using the selfconsistent framework charge of silicon, the electrostatic potential was fitted in a 4-A cube centered on the acid site and was reproducedin this region with an average error of 1 X 1c5hurtrees. The maximum error at any point in the fitting region is 1 X l e 2 hartrees.
We report here the results of calculations which examine the effect of aluminum substitution at a particular T-site within the zeolite and the effect of explicitly considering the environment of the zeolite crystal within which the quantum-mechanicalcluster is embedded. In order to examine these questions, identical calculations were performed on an unsubstituted siliceouscluster and a cluster in which the silicon atom at the center of the cluster has been replaced by an aluminum atom. We have chosen substitution at the T12 site for our initial studies, since previous work has shown this site to be energeticallyfavorablefor aluminum substitution>%26 Chargeneutrality is maintainedin thealuminumsubstituted cluster by pacing a proton near 024. This oxygen, which bridges two T12 sites, has been shown to be an energetically favorable site for placement of the proton.26 Full structural relaxation of the inner AlH(OSi)4 subunit is performed on the cluster containing aluminum. The outermost oxygen atoms, as well as the boundary hydrogen atoms, were held fixed in positions determined from the X-ray measurements of Olson et uL2' GeometricShcture of the Acid Site. Hellman-Feynman forces were used to relax the innermost AlH(OSi)4 subunit of the aluminum-substitutedcluster both withand without theinfluence of the imposed Madelung field. The results of this structural relaxation are given in Table 11. The acidic proton is found to
The Journal of Physicul Chemistry, Vol. 97, No.25, 1993 6681
TABLE II: Result8 of Structural Relaxatiod quantity Si cluster isolated AI cluster embedded Al cluster r(Tl2-024) r(T12-H) LT12-024T12 LT12-024-H
1.59 143.4
-
1.02 1.78 2.39 131.9 114.1
1.02 1.79 2.39 131.7 114.0
* Distances are in A. Angles are in degrees. 20
-80
-40
0
40
80
Angle out of AI - 0 - Si Plane (Degrees) Figure 2. Energetics of an orbiting motion of the proton around the
bridging oxygen. Motion was constrained to the equilibriumbond length (1.02A) and to the plane bisectingthe AI-O-Sibond angle. Calculations were performed without including the effect of the Madelung field. reside at a distance of 1.02 A from the bridging oxygen. The bond angle between the substituted aluminum, bridging oxygen, and proton is found to be 114O. There is virtually no difference between the values calculated with and without the presence of the Madelung field. The T12-024 bond length increases from a value of 1.59 A in the siliceous crystal to a value of 1.78 A in the aluminum-substituted case. The aluminum-proton distance is calculated to be 2.39 A. The geometric data calculated here are in general agreement with previous computational and experimental work concerning the acid site. Sauer has previously calculated the oxygen-proton distance as 0.956 A,2gwhile Brand et al., who have recently performed Hartree-Fock calculationson clusters of various sizes, calculated this value as 0.97 A for the largest cluster studied? Derouane and F r i ~ i a t ?who ~ explored selecteddegrees of freedom of a dimeric cluster model of aluminum-substituted ZSM-5, calculated an aluminum-xygen bond length of 1.84 A. Brand et ul. calculate an aluminum-xygen bond length of between 1.93and 1.87 A.7 Our calculation of the aluminum-proton bond length, 2.39 A, is in reasonable agreement with the experimental value of 2.48 A obtained using solid-state NMR.30 Our calculations also show that the proton does not reside in the aluminum-bridging oxygen-silicon plane, as is commonly assumed in quantum-mechanical calculations of zeolite systems. In order to characterize the energetics associated with movement of the acidic proton out of the Al-oSi plane, we have performed LDA calculations for clusters in which the framework is held fixed while the proton orbits the bridging oxygen at 1.02 A. Movement of the proton is restricted to the plane which bisects the Al-OSi angle in order to maximize overlap of the proton with the unshared pairs of the bridging oxygen atom. The results of these calculations are shown in Figure 2. Although the minimum energy configurationis obtained when the proton forms an angle of loo with the Al-OSi plane, the energy difference between this configuration and other out-of-plane angles is very
Cook et al.
6682 The Journal of Physical Chemistry, Vol. 97, No. 25, 1993
TABLE IIk Net Proton Charge As M d by the Mulliken Procedure and Density Integrrrtion' configuration Mulliken charge integrated charge
l.oo~
r((FH ) = 1.02 A
!
8
Isolated Cluster 0.75
-Embedded Cluster
U 0
0
L 0.5
1 .o
1.5
Distance from Bridging Oxygen
2.0
(A)
Figure 3. Electron density along the bridging oxygen-proton bond as a functionof distance from the bridging oxygen. Charge is strongly polarized toward the bridging oxygen. The effectsof the Madelung field are shown to be small.
small. Furthermore, the energy difference between the minimum energy angle and the in-plane position is only 0.7 kcal/mol. Root et al. have recently performed NMR experiments which support our finding that the proton resides out of the AI-O-Si plane.31 Although Root et al. suggest out-of-plane angles of either 35.3" or 54.7", there is general agreement that the proton most likely does not reside in the AI-O-Si plane. The vibrational frequency for motion of the proton normal to the Al-O-Si plane is found to be 812 cm-l. This may be compared to the 0-H stretching frequency, which we calculate to be 3587 cm-1 in the absence of the crystal field. The latter value is in good agreement with the experimental value of 3610 cm-le3* Electronic Structure of the Acid Site. Atomic Charges. The electronic structure of the T12 acid site has been characterized by examining the polarization of the bridging oxygen-proton bond and the partial charge of the proton. Polarization of the electron density along the oxygen-proton bond is shown graphically in Figure 3 . This figure shows that the proton issignificantly depleted of charge and that the imposition of the Madelung field does not change the distribution significantly. We have also calculated the Mulliken populations of the proton in the two configurations shown in Figure 3 . The proton is found to have a Mulliken partial charge of +0.331el in the absence of the Madelung field, while the proton is found to have a Mulliken partial charge of +0.51(el at an oxygen-proton distance of 1.02 A under the influence of the Madelung field. The Mulliken charge of the proton is found to increase by 0.181eI when the effects of the Madelung field are self-consistently included in the LDA calculation. Considering the rather small change in the oxygen-proton electron density distribution shown in Figure 3 upon inclusion of Madelung field effects, the large change in the Mulliken charge appears to be an artifact of the Mulliken population analysis procedure. Deficiencies in the Mulliken procedure have been previously noted, especially when applied to systems containing significant ionic character.33 In order to characterize the amount of electronic charge associated with the acidic proton more accurately, we have integrated the electronic density around the proton. The minimum in the electron density between the bridging oxygen and proton is used to define the integration region. Defining the integration region in this way underestimates the amount of electronic charge associated with the proton, since the "tail" of the diffuse proton distribution extends farther into the oxygen region than does the "tail" of the less-diffuse oxygen distribution into the proton region. The net
0.62
0.33
no Madelung field 0.76 40-H) = 1.02 A 0.55 Madelung field a The chargeslisted are net positive chargesfor the proton: net charge = 1.&fractional electroncharge. Chargca are listed in unitsof the electron charge. charge of the proton, defined as the difference between unity and the electronic charge of the proton, will, therefore, be overestimated by density integration. Nevertheless, integration of the charge yields electronic populations more accurately than the Mulliken procedure. The results of the integration, as well as the corresponding Mulliken partial charges, are listed in Table 111. The values obtained suggest that the proton is moderately charged and that the presence of the crystal field increases the charge by 0.141el. Although the change in proton charge upon inclusion of the Madelung field is captured accurately by the Mulliken procedure, the Mulliken procedure underestimates the actual values of the charges. IonizationPotentialand ElectronAffinity. Previous quantumchemical studiesof the Bronsted acidityof zeolites have frequently associated this acidity with the deprotonation energy of the acid site, which is the energy required to completely separatethe acidic species, H+,from the zeolite framework. This is equivalent to characterizing the acid only by its dissociation constant. The desire to rationalize the vast amount of other experimental data which may be used to characterize acidity led Pearson and coworkers to develop the hard and soft acid-base principle (HSAB).3'37 The HSAB principlecharacterizes acids and bases as being either hard or soft by using properties such as charge magnitude, charge delocalization, and polarizability. Species which are weakly charged, and which arecharacterized by diffuse, polarizable charge distributions, are termed "soft". "Hard" species possess the opposite properties. The basic principle of HSAB is that like species interact more strongly with each other than do unlike species. The empirical generalizations of Pearson and co-workers have been placed on a more quantitative basis by Klopman within the framework of perturbation molecular orbital the~ry.~*-~~ Pearson and Parr have used the framework of density functional theory to show that a species may be characterized by its electronegativity, x, and its absolute hardness, q.u2 Within the framework of density functional theory, these quantities are defined as
I+ A 2 (7) where E is the energy, N is the number of electrons, u is the external potential, lis the ionization potential,and A is the electron affinity. The lower the value of the absolute hardness, the softer the species. Since the value of q reflects the HOMO-LUMO energy gap in a species, decreasing values of this energy gap imply increasing values of softness. Since the value of this gap may also be associated with quantities such as the polarizability of the charge distribution, Parr and Pearson's work serves to place the HSAB concept on a more fundamental basis. In order to make a connection with the HSAB concept, as well as perturbation molecular orbital theory, the ionization potential and electron affinity of the cluster have been calculated both with and without the effect of the Madelung field. These
Features of a Bronsted Acid Site in H-ZSM-5
The Journal of Physical Chemistry, Vol. 97, No. 25, 1993 6683
TABLE Iv: Values of the Electron Affinity, Ionization Potential, and Absolute Hardness for Clusters Representing the Br~rnstedAcid Site of ZSM-9 configuration r(O-H) = 1.02 A no Madelung field r(O-H) = 1.02 A Madelung field
electron affinity
ionization potential
0.0100
absolute hardness
-1.25
1.38
4.32
DNCb
4.11
DNCb
Values are shown for an isolated cluster in which the Madelung field was not applied and for a cluster in which the Madelung field was included in the Kohn-Sham Hamiltonian. All values are in units of eV. SCF procedure did not converge (DNC) for the zeolitic anion in the presence of the Madelung field.
quantities are determined by self-consistentlysolving the KohnSham equations for systems which contain either one more or one fewer electron than the neutral cluster and thus include electronic relaxation effects. Geometric relaxation was not performed for these systems. The ionization potential and electron affinity of the cluster are shown in Table IV, along with the resulting value of the absolute hardness. The ionization potential is 7.38 eV in the absence of the Madelung field and decreases significantly to 4.11 eV when the cluster is embedded in the field. The inclusion of the Madelung field clearly affects the electronic properties of the cluster much more than its structural features (see Table 11). The electron affinity for the cluster is negative, which indicates that the anion is less stable than the neutral cluster. The values listed in Table IV reveal that the zeolite has a very small HOMO-LUMO gap, sochargeshould bevery polarizable within thezeolite. Thecluster is characterized by a hardnessvalue of 4.32 eV when the Madelung field is not included in the Hamiltonian. This value may be compared to the hardness of 4.6 eV which characterizes the soft chlorine molecule.43 This is a strong indication that the acid site is a soft species. Efforts to obtain a self-consistent solution to the N + 1electron system while under the influence of the Madelung field were unsuccessful. The HOMO-LUMO gap for this configuration, therefore, could not be calculated. We note that an ad-hoc lengtheningof theoxygen-proton bond from 1.02 to 1.5OA yielded a configuration which could be iterated to self-consistency. Attempts to prompt convergence of the configuration with r ( 0 H) = 1.02 A by systematically reconverging configurations wherein the oxygen-proton bond length was reduced in small jumps from r(O-H) = 1.50 A failed. We suggest that this result is indicative of the increased acidity caused by the crystal field. The presence of additional electron density, whether from a basic molecule or from formal creation of the zeolite anion, prompts a lengthening of the oxygen-proton bond; i.e., it promotes proton transfer to the base. Our inability to obtain an SCF solution for the short bond length may be indicative of the instability of this configuration. The Fukui function is closely related to the HOMO-LUMO gap and to the absolute hardness. This function, which was developed by Parr and co-~orkers,4~ measures the change in a system's chemical potential to a change in the external potential or molecular environment. The function is thus defined as
where fi is the chemical potential, u ( i ) is the potential felt by the electrons, p ( i ) is the electron density, and N is the number of electrons. Since electron number is constrained to integer values, the derivative of electron density with repsect to electron number defined in eq 8 is discontinuous. That is, there will be distinct values of the Fukui function from the left,f -,and from the right, f +,about some integral value of N. These functions can be
0.0075
t K-
O .c
0
5
0.0050
.-
c
Y 2
3 LL
0.0025
n 1
2
3
4
Distance from Bridging Oxygen (A) Figure 4. Fukui function,?, along the bridging oxygen-proton bond as a function of distance from the bridging oxygen. The acidic proton is located at 1.02 on the abscissa. Addition of extra charge is shown to occur away from the oxygen and toward the pore region, which is located at increasing values of the abscissa.
calculated using the following relations: (9)
where pi is the electron density associated with the system containing V" electrons. The Fukui function,JY, thus measures reactivity toward a nucleophilic reagent and indicates regions of increased acidity within the zeolite. The Fukui function has been shown previously to be a useful measure of zeolite acidity45 and can be of use here in the characterization of the Bransted acid site. Although the Fukui function,f(i), is a continuous function which can be visualized in three dimensions, it is often advantageous to characterize the reactivity of a region in space represented by the Fukui function as belonging to a particular atom or group of atoms within a molecule. Since the Fukui function is defined as the difference of two electron densities, this assignment may be accomplished using the techniques which are commonly employed to assign charge density to individual atoms within a species. Values resulting from this procedure are said to represent the ycondensed' Fukui function. Sincewe have shown that the Mulliken procedure is of limited value here, we used density integration to obtain values of the condensed function. The Fukui function has been calculated as a position-dependent quantity along the oxygen-proton bond in the absence of the Madelung field and is shown in Figure 4. The function, which is characteristic of the LUMO density, exhibits a positive region on the side of the oxygen-proton bond facing the zeolite pore system. Although this result is not unexpected, it displays the utility of the Fukui function in identifying regions of acidic character. The value of the condensed Fukui function obtained by integrating within the region attributed to the proton is found to be 0.07H. The Fukui function for the cluster under the influence of the Madelung field could not be calculated due to the convergencedifficultiespreviously described for the zeolite anion. It we were able to converge this configuration, we expect that the condensed Fukui function would show an increased tendency toward nucleophilic attack in the vicinity of the proton when compared to the results obtained in the absence of the Madelung field. We note that convergence of the SCF procedure could be obtained for the zeolitic anion under the influence of the Madelung
6684 The Journal of Physical Chemistry, Vol. 97, No. 25, 1993
field when the oxygen-hydrogen bond length was increased to 1.5 A. Since the anion mimics the extra charge which would be present due to a basic molecule sorbed on the acid site, we suggest that our ability to converge a configuration with a longer oxygenproton bond length indicates incresed stability for such a configuration; i.e., the presence of the Madelung field promotes detachment of the proton from the zeolite framework. Since we have shown the charge density distribution between the bridging oxygen and the proton to be only slightly altered by the presence of the Madelung field, however, we expect that this effect may not be significant. Stabilization of Charged Species. During the course of geometry optimization for the aluminum cluster under the influence of the Madelung field, a low-energy state was found which is characterized by a bridging oxygen-proton bond length of 1.55 A and an aluminum-bridging oxygen bond length of 1.71 A. This state was found to be only 2 kcal/mol less favorable in energy than the state characterized by the shorter oxygen-proton bond length of 1.02 A. The aluminum-proton distance for this configuration is 2.85 A. The proton was found to reside out of the Al-OSi plane by 14.5". Since the discovery of this additional configuration was unexpected, we have examined our method of embedding the quantum-mechanical cluster todetermine if this method is capable of producing stabilization of a charged species as an artifact of the procedure. Our search focused on two questions: First, although the electrostatic field can be fit with small error within the region of the acid site, could local errors within the fitted field create regions of artificially low potential energy? Second, we alsoobserved that the electron density distributionalong thebond connecting the outermost oxygen atoms to the hydrogen atoms saturating the dangling bonds of the cluster was significantly distorted by the presence of the imposed Madelung field. This distortion is characterized by depolarization of the charge away from the oxygen and toward the boundary hydrogen and is primarily caused by the presence of positively charged framework silicon atoms outside the cluster located approximately 0.6 A from the boundary hydrogen atoms. Mulliken partial charges for the embedded boundary hydrogen atoms are, in fact, negative rather than positive. Wequestioned whether this artificial charge redistribution at the cluster boundary could be responsible for the energetic stabilization we observed. To address the first question, we have calculated the errors in thefittedfieldat thelocationofthenucleiforbothoftheembedded cluster states and find that the state characterized by the longer oxygen-proton bond length is artificially stabilized by approximately 1.7 kcal/mol. This value is obtained by calculating the potential energy of the cluster nuclei due to the fitted Madelung field, which includes the effects of the boundary hydrogen atoms, and the true Madelung field resulting from the hydrogen-free silicalite framework. This procedure yields an estimate of the error in the total energy for each configuration due to errors in the fitted Madelung field. The difference between the two values yields the stabilization energy due to errors in the field. The error term corresponding to the electron density distribution was not calculated but would likely reduce the stabilization energy from 1.7 kcal/mol, since the charge density and ionic nuclei are oppositely charged. The stabilization energy may be compared to the difference in energy between the two configurations in the absenceof the Madelung field, 5 1.9 kcal/mol. Since thedifference in energy between the two configurations in the presence of the Madelung field is approximately 2 kcal/mol, the 1.7 kcal/mol error is too small compared to the stabilization energy of approximately 50 kcal/mol to significantly alter our findings regarding the energetics of the configuration with an oxygenproton bond length of 1.55 A. In order to address the second question, we have investigated whether charge redistribution at the boundary of the cluster could
Cook et al.
c 1.oo
mf E c
8, .-a cn
0.75
c
Q)
0.50
P c
8
iij
0.25
0
0.5
1 .o
1.5
2.0
2.5
Distance from Bridging Oxygen (A) Figwe 5. Electron density along the bridging oxygen-proton bond at^ a function of distancefrom the bridging oxygen. Resultsare shown for two
embedded clusters. The charge density distribution of the cluster Characterized by an oxygen-proton bond length of 1.55 A is more diffuse than that of the cluster characterized by an oxygen-proton bond length of 1.02 A.
affect the properties of the acid site by performing a calculation on a cluster not subject to the Madelung field but which is terminated by fluorine rather than hydrogen. Fluorine terminationmimics thechargewithdrawal whichoccursat theboundary of the cluster and allows us to distinguish this effect from the host of effects which might be caused by a self-consistent calculation including the Madelung field. Although the presence of the fluorine caused a charge redistribution at the boundary of the cluster very similar to that found in embedded cluster calculations, charge redistribution along the bridging oxygen-proton bond was virtually zero. The pentameric cluster employed here is large enough to mitigate the effects of boundary termination. We therefore conclude that neither of the likely artifactual c a w of the stabilization of charge speciesis active and that the additional geometric structure we have observed is a physical result. The additional configuration characterized by an oxygenproton bond length of 1.55 A is characterized by an increased net charge on the acidic proton. The electron density distribution around the proton in this state, which is compared to that of the embedded cluster with an oxygen-proton distance of 1.02 A in Figure 5, has been integrated and subtracted from unity to yield the net positive charge on the proton. This value is found to be +0.8114, which may be compared to the value of +0.76H found for the embedded cluster with r(O-H) = 1.02 A shown in Table 111. The net charge of the proton is, therefore, slightlyincreased, although it is evident from Figure 5 that the charge density distribution has become more diffuse. Sankararaman et al.have recently reported experimental results which support the idea that the electrostatic field of the zeolite lattice canenergetically stabilizecharged ~ p e c i e s These .~ authors found a greatly increased lifetime for back charge transfer during the formation of charge-transfer complexes within the zeolite NaY and concluded that stabilization of the charge-transfer complex by the electrostatic field of the zeolite may play an important role. The rate of charge transfer within the chargetransfer complex is proportional to the chemical potential difference of the electroniccharge between the two species which comprise the complex. The presence of the zeolite Madelung field apparently decreases this difference by favoring the formation of the charged species. The energetic stabilization we observe is analogous in that the Madelung field decreases the energy difference between different cluster configurations characterized by increasing charge separation.
Features of a Bronsted Acid Site in H-ZSM-5 We have investigated the electronic properties of the configuration with r(O-H) = 1.55 A by selficonsistently calculating the ionization potential and electron affinity of this configuration. The ionization potential was found to be 1.85 eV, and the electron affinity was found to be 4 . 0 8 eV. The absolute hardness of this configuration is then calculated to be 2.97 eV, which may be compared to the value of 4.32 eV obtained for the cluster with r(O-H) = 1.02Aintheabsenceof theMadelungfield. Although it is not possible to separate the effects of the increased oxygenproton bond length from those induced by the presence of the Madelung field, it is apparent that the codiguration produced under the influence of the Madelung field with r(O-H) = 1.55 A is considerably softer than the configuration with r(O-H) = 1.02 A produced in the absence of the field. This is consistent with the more diffuse nature of the charge density distribution shown in Figure 5. This stabilization presumably occurs by a mechanism similar to the stabilization of charged speciesin polar solvents. Although the zeolite certainly should not be likened to a substance such as water, it seems clear that the zeolite possesses dielectricpropertiessignificantlydifferent than those of a vacuum. We note that the dielectric constant of quartz is approximately 4 and may be suggestive of the magnitude of the dielectric effect within the zeolite.
Conclusions We have examined the influences of geometry relaxation and inclusion of Madelung field effects on quantum-mechanical simulations of zeolite acidity within H-ZSM-5. We find that substantial geometrical rearrangements occur in the immediate vicinity of an aluminum atom substituted into the framework of ZSM-5 and that the relaxation has important consequences for the electronic properties of the acid site. The nature of Bransted acidity within ZSM-5 was studied using the hard and soft acidbase principle and perturbation molecular orbital theory. The proton is found to be a soft species characterized by a diffuse electron density distribution, a small HOMO-LUMO gap, and a moderately high partial charge near +0.714. The Madelung potential of the infinite zeolite crystal has been included in the KohnSham Hamiltonian. Inclusion of the Madelung field affects the equilibrium geometry and electron density distribution of the acid site within ZSM-5 only slightly. One of the effects of the Madelung field is the creation of at least one additional binding site for the charge-compensating proton. This binding site is characterized by a bridging oxygen-proton bond length of 1.55 A and a more diffuse electron density distribution than the configuration characterized by a short oxygen-proton bond length of 1.02 A. This additional binding siteis a consequenceoftheability of thezeolite tostabilizecharged species within the pore system by acting as a dielectric. Another, and perhaps more important, effect of the Madelung field is a significant decrease in the ionization potential. The changes in the electronic properties of the cluster suggest that the acidity of Bransted sites with the same local chemical constitution could be influenced by the framework of the zeolite of which they are a part, as a consequenceof the Madelung field!’ We are currently performing studies of the interaction of ammonia with the Bransted acid site in order to further characterize the nature of acidity within H-ZSM-5.
The Journal of Physical Chemistry, Vol. 97, No. 25, 1993 6685
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