J. Phys. Chem. 1995,99, 9536-9550
9536
Molecular Mechanics Potential for Silica and Zeolite Catalysts Based on ab Initio Calculations. 2. Aluminosilicates JiSrg-R. Hillt and Joachim Sauer* M~-Planck-Gesellscha~, Arbeitsgruppe Quantenchemie an der Humboldt-Universitat, Jagerstrasse 1O/I I , D-10117 Berlin, Germany Received: January 12, 1995@
A consistent force field for the simulation of protonated aluminosilicates is presented. It has been developed on the basis of ab initio calculations on molecular models following a method proposed in a previous paper (J. Phys. Chem. 1994, 98, 1238). The molecular models consist of Si04 and protonated A104 tetrahedra connected to chains, rings, and cages. The ab initio calculations used a “double-5 plus polarization” basis set on the Si, Al, and H atoms and a “triple-I; plus polarization” one on the 0 atoms. The calculated structures of the finite models yield a structural model of Bransted acidic sites which is consistent with observed data but more complete than models derived from experiments. Compared with these ab initio results the derived force field predicts reasonable structures for aluminosilicates, but the errors are larger than the errors that our all-silica potential yielded for all-silica polymorphs. A new method is proposed for calculating the atomic charges as a function of the structure, and the corresponding potential results in better structure predictions. The force field is applied to the calculation of the local structures of different bridging hydroxyl groups in faujasite, and it is shown that results of the same accuracy as with a shell model potential are obtained. The predictions for a H-faujasite (Si/Al = 2.43) are consistent with mean bond distances and angles deduced from neutron diffraction data.
1. Introduction For molecules in the gas phase, quantum chemical ab initio techniques yield structures, properties, and reactivities with an accuracy which is comparable to that of experiments.’ In spite of substantial progress in computer power and computational techniques, the scaling of the employed algorithms with the size of the system prevents direct applications of such techniques to large chemical systems such as molecules in condensed phases, polymers, and solids. Fields of specific interest are biochemistry and heterogeneous catalysis. Even when some periodicity exists or can be assumed, the exploitation of periodic boundary conditions is not much help because of the enormous size of the unit cells. Hence, the only practical approach is to confine the ab initio treatment to the atoms of the site of interest and their neighbors. Such finite models of, on the molecular scale, essentially infinite systems, have had remarkable success. Nevertheless, there are classes of problems which require theoretical predictions of the structure of the whole system. These needs are served by classical potential functions, also known as force fields, which provide an analytical approximation to the energies of molecular systems as functions of atomic coordinates. Force fields become increasingly employed in calculating the structure and simulating the dynamics of complex organic, biomolecular, and polymeric systems and also of inorganic solids. The use of force fields is limited, however, by the lack of reliable parameters for these functions. Traditionally force field parameters are derived from observed data. For many but the simplest systems there is incomplete information for deriving detailed and accurate potential functions. A way out of this dilemma is deriving the parameters of such potential functions from quantum chemical ab initio calculations.
* Author to whom correspondence should be sent. FAX: +49-30-20192302. E-mail:
[email protected]. ’ Present adhress’: Byosym Teih>ologies, Inc., 9685 Scranton Road, San Diego, CA 92121-2777. &Abstract published in Advance ACS Abstracts, May 15, 1995.
This approach has several advantages. The set of data that may be included in the fit is in principle unlimited and there is a one-to-one correspondence between the “calculated” (by the force field) and “observed” (by the quantum chemical calculation) data, as “observables” serve the energy and its first and second derivatives with respect to the displacement of all nuclei. These “observables” are generated by quantum chemical calculations not only for stationary point structures but also for an, in principle unlimited, number of distorted configurations. This approach was pioneered by Clementi,2 who developed a strategy for deriving analytical potential functions for use in Simulations of liquids and solutions. More recently, a similar type-of approach has been adopted for the derivation of class II force fields for alkane molecules and the alkyl functional g r o ~ pand ~ , ~polycarbonate^.^ In a previous paper, we showed that this approach also yields viable potential parameters for silica, including dense (e.g. quartz) and microporous polymorphs (all-silica zeolites).6 Since silica is partially covalent and partially ionic, its structure and dynamics are traditionally described by ion-pair type potential functions and it was not obvious that a functional form such as that of organic molecules can also be successful. An urgent need for reliable potential functions also exists for zeolite catalysts which find increasing use as solid acids. The origins of their Bransted acidity are so-called bridging hydroxyl groups. These groups are formally formed when a silicon nucleus in a silica framework is replaced by an aluminum one and the resulting negative charge on the framework is compensated by adding a proton to any of the four oxygen atoms surrounding the aluminum atom. A complete structural model of an acidic zeolite catalyst including details of the bridging hydroxyl group as its active site cannot be derived from observed data. This is due to the low concentration of these sites in active catalysts and the fact that they are not ordered into unit cells. This is also the reason that periodic ab initio calculations on
0022-3654/95/2099-9536$09.00/0 0 1995 American Chemical Society
Silica and Zeolite Catalysts
J. Phys. Chem., Vol. 99, No. 23, 1995 9537 h
....................................
finite modcl
Figure 1. Atom types required for periodic solids and finite models. Those in the boxes belong to the subset required for aluminum-free systems.
Figure 3. 2-Aluminatetrasilicic acid and 2-(trihydroxysilyl)-2-aluminatrisilicic acid.
0 0
Figure 2. Aluminadisilicic acid, 1-aluminatrisilicic acid, and 2-aluminatrisilicic acid. these materials are still exceptional and have been completed for given structures of zeolites ~ n i y . ~The - ~ only ~ structure predictions made so far by lattice energy minimization techniques employed classical potential functions with empirically derived parameters which are connected with much uncertainty. Therefore, it is our aim to derive the parameters from ab initio calculations. As functional form we select the same “consistent force field” as adopted in our previous study on silica6 and as used for organic molecules and polymers. Using the same type of force field as used for other classes of molecular has the distinct advantage that interactions between different classes of systems can be treated in a natural way and, possibly, with less work in parameter derivation as far as built-in combination rules will work. As mentioned above ab initio calculations cannot easily be completed for periodic zeolite structures and, hence, finite molecular type models are adopted for which molecular codes can be employed. In the first part of this paper we present ab
Figure 4. Aluminacyclotetrasilicic acid and 1,3-dialuminacyclotetrasilicic acid. initio self-consistent field (SCF) calculations on finite models of bridging hydroxyl groups of zeolite catalysts. We show that the structural model derived from the calculations is consistent with the data known from different experiments. This is an important step, as it is a test of the reliability and quality of the data base which is subsequently used to derive the parameters of the analytical representation of the potential energy function. Once the reliability of the molecular models and the quantum chemical approximations has been established, the data base to be used in the parameter fitting is generated by performing ab initio calculations on a subset of these models which is referred to as the “training set”. The data include energies and forces on the nuclei for a large number of different structures of each model which are obtained by systematic distortions of the equilibrium structures as well as force constants for the
9538 J. Phys. Chem., Vol. 99, No. 23, 1995
Hill and Sauer
Figure 7. 1,9-Dialuminadodecahydroxydodecasilsesquioxane.
Figure 5. Aluminacyclohexasilicic acid and 1,3,5-trialuminacyclohexasilicic acid.
used for generating the data base but which are still accessible to ab initio structure predictions. This set of molecular models is named the “transferability test set”. The final test of the parameters is their use for predicting periodic zeolite structures. We will show that, following this strategy, structure predictions are possible for acidic zeolite catalysts which are solely based on ab initio Hartree-Fock calculations for molecular models without any adjustment to observed solid state or molecular data. Previous attempts to develop potentials for the simulation of zeolite catalysts with bridging hydroxyl groups were made by Kramer et aLi3and Schroder et al.I4 Kramer et al. used a rigid ion potential and replaced the bridging hydroxyl group by an effective atom at the position of the oxygen nucleus. Hence, a description of the local geometry of the bridging hydroxyl group was not possible. Schroder et al. used a shell model potential which accounts for the polarizability of the oxygen anions. The bridging hydroxyl group was described by a Morse function, but nevertheless the OH bond length obtained in the hydroxyl group was poor. Both previous attempts to model bridging hydroxyl groups employed modifications of an ion pair potential. This type of potential is the natural choice for ionic systems, but it does not account for mainly covalently bonded atoms as in bridging hydroxyl groups. An ion pair potential can also not easily be extended to include interactions of organic molecules with bridging hydroxyl groups in zeolites, which is essential for modeling of the catalytic process. Furthermore, both previous potentials are either largely empirical (Schroder et al.) or adjusted to experimental data (Kramer et al.). Moreover, there is a need for embedding quantum mechanically treated clusters in periodic zeolite structures modeled by classical potential function^.'^ The use of more or less empirical potentials in such an hybrid approach would not be consistent.
2. Molecular Models
Figure 6. 1,6-Dialuminaoctahydroxyoctasilsesquioxaneand 1,3,5,7tetraaluminaoctahydroxyoctasilsesquioxane.
stationary points of the models. The potential function parameters are then obtained by fitting these data to the consistent force field functional form. The structure of the “training set” molecules provides a fiist check of the parameters by comparing the results directly determined by ab initio calculations with those predicted by the consistent force field using the derived parameters. The next step is the check of the transferability of the parameters. The parameters are employed in molecular mechanics calculations on molecular models which were not
The design of adequate models is prerequisite to the success of the ab initio description of zeolite catalysts and of the derivation of ab initio potential energy function parameters. The principles of designing such models have been outlined in detail.I6 The models have to represent the structural features of the system simulated as closely as possible, but they have to be limited in size. The smallest model for a bridging hydroxyl group in zeolite catalysts used in this study is aluminadisilicic acid (Figure 2), which includes a Si04 and an A104 tetrahedron which share a comer. By adding another si04 tetrahedron, the 1-alumina- and 2-aluminatrisilicic acid models are obtained (Figure 2). Adequateness is not the only criterion when designing models for deriving potential energy function parameters. Another criterion is whether the potential function expression for the models selected includes all the parameters which are needed to define a potential for a bridging hydroxyl group as part of an extended Si02 network (vide infra, cf. Figure
Silica and Zeolite Catalysts
J. Phys. Chem., Vol. 99, No. 23, 1995 9539
TABLE 1: Molecular Models Used
model Al-disilicic acid 1-Al-trisilicic acid 2-Al-trisilicic acid 2-Al-tetrasilicic acid
2-(trihydroxysilyl)-2-Al-trisilicicacid Al-cyclotetrasilicic acid
1,3-di-Al-cyclotetrasilicicacid Al-cyclohexasilicic acid
1,3,5-tri-Al-cyclohexasilicic acid 1,6-di-Al-octahydroxyoctasilsesquioxane 1,3,5,7-tetra-Al-octahydroxyoctasilsesquioxane 1,9-di-Al-dodecahydroxydodecasilsesquioxane a
Schlafli symbol of sbu" represented
sum formula AlSi07H7 AlSi2010H9 A N 2 0 loH9 AlSi3013Hll AlSi3013Hll A N 3 0 j 2H9 A12Si20dh AlSisO 1*HI 3 A13Si3OI gH I 5 A12Si6020H10 A14Si4020H12 Al~Si10030H14
point group
number of basis functions
SCF energy (hartree)
214 304 304 394 394 365 370 545 555 614 624 916
-1059.481 315 -1574.434 61 1 - 1574.430 952 -2089.390 455 -2089.407 319 -2013.360 363 -1966.939 153 -3043.212 723 -2950.378 124 -3722.609 299 -3629.177 694 -5630.358 998
C, C S C S
CS
CX
c1
4 4 6 6 46 46 4662
C2h CS c 3h
C, c 2
C2h
sbu = secondary building unit.
TABLE 2: Bond Lengths for the Molecular Models (pm) H 0-
SiAI-disilicic acid 1.AI-trisilicic acid 2.Al-trisilicic acid 2-AI-telrasilicic acid (Tl=Si) 2-trihydroxysilyl-2-AI-trisilicic acid AI-cyclo-tetrasilicic acid (Tl=Si) AI-cyclo-hexasilicic acid (TI=T2=Si) 1,6-di-Al-octasilsesquioxane (TI=Si, T2=AI)
161.3
163.0 162.5
159.1
158.7 160.1
1.1 I-di- AI-dodecasilsesquioxane (TI=Si)
Average Range of Distances 1,3-di-Al-cyclo-tetrasilicic acid 1,3,5-tri-Al-cyclo-hexasilicic acid 1,3,5,7-tetra-Al-octasilsesquioxane (Tl=T2=AI)
Average Range of Distances a
I
0-
Si-
AI-0-
166.3 168.1 166.8 165.8 167.1 167.9 168.6 169.5 169.5
197.9 192.1 194.3 199.1 195.6 195.4 194.3 192.7 192.7
169.1
193.4
170.8 170.0 172.2 171.2 170.8 171.6 171.5 171.5 170.8 170.8
0-
Si-
157.9 159.5 159.0 156.6 158.6 159.1 158.6 158.6 158.7 158.7
0-
T+
0-
Si-
161.2
160.1
163.2 162.1 162.5 162.5 164.i 163.4 162.2
159.7 161.2 162.3 162.5 158.4 161.2 160.7
161.9 164.7 164.1 159.3 162.4 162.4
160.2 158.9 159.3 164.1 158.8 158.8
157.8
172.4
172.5
157.7
T2
161.6
159.2
159.1
171.6
158.4
172.6
167.9 194.8 171.2' 160.7" 165.8 192.1 170.0 157.9 -169.5 -199.1 -172.2" -164.7" 171.6 170.8 172.6 li2.6
190.4 189.7 187.9 187.9
172.6 174.3 172.7 172.6 172.7
156.5 157.5 158.7 158.4 158.7
171.9 189.0 172.8" 158.0170.8 187.9 172.4 156.5 -172.6 -190.4 -174.3" -158.7"
All A10 and S i 0 bonds, respectively.
1). This is the reason for supplementing our set of models by 2-aluminatetrasilicic acid and 2-(trihydroxysilyl)-2-aluminatrisilicic acid (Figure 3). Molecular models containing one or more rings have advantages over chainlike molecules, as they contain fewer terminating hydrogen atoms. Therefore, we include the following models: aluminacyclotetrasilicic acid, 1,3-dialuminacyclotetrasilicicacid, aluminacyclohexasilicic acid, 1,3,5-trialuminacyclohexasilicic acid, 1,6-dialuminaoctahydroxyoctasilsesquioxane, 1,3,5,7-tetraaluminaoctahydroxyoctasilsesquioxane, and 1,9-dialuminadodecahydroxydodecasilsesquioxane (Figures 4-7). Table 1 lists all molecular models used and specifies the point groups, the number of basis functions, and the SCF energies. For a number of models, in addition to the symmetry restrictions imposed, some torsion angles had to be kept to avoid the formation of intramolecular hydrogen bonds. The tendency to form hydrogen bonds was much stronger than in the case of the all-silica models, which can be explained by the larger flexibility of the A 1 0 4 tetrahedron (vide infra). The equilibrium structures of all the models were determined within the SCF approximation by a standard algorithm. These calculations employed the semidirect SCF code TURBOMOLE." For the 1,9-dialuminadodecahydroxydodeca-
silsesquioxane the parallel features of TLTRBOMOLE were used on a cluster of five workstations. A "double-t; plus polarization" (DZP) basis set was used on the silicon, aluminum, and hydrogen atoms, and a "triple-t; plus polarization" (TZP) basis set, on the oxygen atoms.I8 3. Ab Initio Results on Molecular Models 3.1. Structures Tables 2-6 show the optimized structures of the molecular models studied. These data are of value on their own, as they provide structural information on the active sites of zeolite catalysts within the finite model approximation.I6 Moreover, the convergence of the finite model approximation can be investigated, because the molecules form a systematic series of models of increasing size. The most notable finding is that the AIOt,fidgebond length varies over a broad range of more than 11 pm. This bond is also significantly longer, by 21.4 pm on average, than the A10 bonds in nonprotonated AlOSi linkages. The Siaridge bond shows similar effects, but to a lesser extent. Its length varies over a range of nearly 7 pm, and it is 8.6 pm on average longer than S i 0 bonds in nonprotonated SiOAl linkages. The longest SiObridgebonds and the shortest AIObndgebonds are found in
9540 J. Phys. Chem., Vol. 99, No. 23, 1995
Hill and Sauer
TABLE 3: Average Bond Lengths of Si04 and A104 Tetrahedra and AIHbddgeNonbonded Distances for the Molecular Models (pm) Al-disilicic acid 1-Al-trisilicicacid 2-Al-trisilicic acid 2-Al-tetrasilicic acid
1,6-di-Al-octasilsesquioxane 1,9-di-Al-dodecasilsesquioxane
162.8 162.4, 162.3 162.6, 162.7 162.7, 162.3, 162.1 162.7 ( 2 x ) 162.4, 161.9, 162.1 162.9, 162.0 (2x), 161.8, 162.2 162.3, 162.0 ( 2 x ) 162.0, 161.6, 161.8
177.9 176.6 176.9 177.7 177.4 177.2 176.8 176.5 176.5
average range
162.3 161.6- 162.9
177.1 176.5- 177.9
241.3 235.7-249.4
1,3-di-Al-cyclotetrasilicic acid 1,3,5-tri-Al-cyclohexasilicicacid 1,3,5,7-tetra-Al-octasilsesquioxane
162.9 163.2 162.9 ( 2 x )
176.3 176.5 175.8
238.0 234.2 239.4, 239.1
average range
163.0 162.9- 163.2
176.2 175.8- 176.5
237.7 234.2-239.4
2-(trihydroxysilyl)-2-Al-trisilicicacid Al-cyclotetrasilicicacid Al-cyclohexasilicic acid
249.4 235.7 246.6 240.8 (236.3)” 238.8 236.1 242.0 240.6
For each non-symmetry-related tetrahedron. Distance is influenced by an intramolecular hydrogen bond and not included in the calculation of the average.
TABLE 4: Bond Lengths for the TOH Groups (T = Al, Si) (pm) model Al-disilicic acid 1-AI-trisilicicacid 2-Al-trisilicic acid 2-Al-tetrasilicic acid 2-(trihydroxysilyl)-2-Al-trisilicicacid Al-cyclotetrasilicicacid AI-cyclohexasilicicacid
r(SiOt)” 161.2-162.5 161.3- 163.6 160.5-164.8 161.5- 163.9 (159.9-163.7)b 161.O- 163.4 161.4- 164.1 161.4- 162.2 161.3- 162.8
r(A1Ot)U 170.9-171.7 17 1.O- 172.4 171.3 170.9 169.2 170.6- 171.6 171.1 170.2 170.8
r(OH)U 94.1-94.4 94.0-94.7 94.1-94.4 94.2-94.6 94.1-94.6 94.2-94.6 94.1 -94.8 94.2-94.7 94.1-94.6
range
160.5- 164.8
169.2-172.4
94.0-94.8
95.27-95.57
1,3-di-Al-cyclotetrasilicic acid 1,3,5-tri-Al-cyclohexasilicic acid
1,3,5,7-tetra-Al-octasilsesquioxane
161.8 162.3 162.7-162.8
171.0 171.0 170.2-170.4
94.1-94.6 94.2-94.9 94.1 -94.6
95.38 95.64 95.34
range
161.8-162.8
170.2-171.0
94.1-94.9
95.34-95.64
1,6-di-Al-octasilsesquioxane 1,9-di-Al-dodecasilsesquioxane
r(OHbridae)
95.32 95.54 95.27 95.50 (96.40)” 95.52 95.53 95.50 95.57
a The smallest and the largest values are given. Bonds are influenced by an intramolecular hydrogen bond and not included in the calculation of the average.
ring type models with a SUA1 ratio of 1. In rings, the tendency to form very long AIOb,idgebonds is counteracted by ring strain. The distortion caused by substituting a silicon atom in an all-silica framework by an aluminum atom is very local. This becomes obvious when looking at the predicted structures for 1-aluminatrisilicic acid and the 2-aluminatetrasilicic acid models (Figures 2 and 3). In the Si-0-Si linkage of the A1-0-Si0-Si chain in 1-aluminatrisilicic acid the fist S i 0 bond is 159.1 pm long and the second 161.3 pm. In the Al-0-Si-0-Si chain of 2-aluminatetrasilicic acid the bond lengths are 170.0, 159.5, 161.2, and 160.1 pm (cf. Table 2). Only in the Si04 tetrahedron directly connected with the A 1 0 4 tetrahedron do the bond lengths change. Tables 5 and 6 show the bond angles. The range of values for the SiOSi angle, though large, is smaller than in the allsilica case. The SiOAl angle is even more flexible than the SiOSi angle; it varies over an even larger range than the SiOSi angle in the all-silica models (f21.4”, ref 6). The angles at the bridging hydroxyl group are less flexible. The SiOH(A1) angle is more rigid than the SiO(H)Al angle. The results obtained from the ab initio calculations for the equilibrium structures of the molecular models considered can be summarized as follows: (1) The S O 4 tetrahedron is comparatively rigid. (2) The A104 tetrahedron is less rigid. (3) The SiOSi and SiOAl angles are extremely flexible. (4) The
S a n d g e and the bonds show large variations depending on the model. ( 5 ) The S i 0 bond lengths in SiOSi linkages between Si04 tetrahedra alternate by about 1.5 pm. (6) The average bond length in a TO4 tetrahedron is constant, even if it contains a protonated bond. (7) Structure distortions caused by aluminum substitution are very local. Among the molecular models considered, two systematic series of increasing size can be identified. The first comprises models of the composition
HAl[OSi(OH),],(OH),-,
(x = 1, 2, 3)
i.e. aluminadisilicic acid, 2-aluminatrisilicic acid, and 2-(trihydroxysilyl)-2-aluminatrisilicicacid. The second consists of models of increasing chain length
(H0),Si0(H)A1(0H),[Si(0H),],0H(x = 0, 1, 2) i.e. aluminadisilicic acid and 2-aluminatri- and -tetrasilicic acids. Further, the influences of ring formation and ring size can be studied by comparing a four-membered chain (2-aluminatetrasilicic acid) with a four-membered ring model (aluminacyclotetrasilicic acid) and a four-membered ring with a sixmembered ring, respectively.
J. Phys. Chem., Vol. 99, No. 23, 1995 9541
Silica and Zeolite Catalysts TABLE 5. Bond Angles for the Molecular Models (deg) model
L(Si0Si)
L(SiOA1)
171.6
168.3 167.5 174.3 (145.4)" 171.2
133.5 137.3 132.8 141.2 144.4 135.4
114.8 117.7 115.0 114.7 112.9 118.3
111.7 105.0 112.2 104.0 102.7 105.1
156.9
146.3
109.8
103.9
156.3 154.2
134.4
115.8
109.6
166.4
136.1
116.0
107.9
154.2 -174.3
132.8 -146.3
109.8 -118.3
102.7 -112.2
169.3 128.1 158.3 157.1
140.3 145.O 134.0 133.9
111.6 109.5 115.4 114.9
108.1 105.5 110.6 111.1
128.1 -169.3
133.9 -145.0
109.5 -115.4
105.5 -111.1
AI-disilicic acid 1-Al-trisilicic acid 2-AI-trisilicic acid 2-Al-tetrasilicic acid
2-(trihydroxysilyl)-2-Al-trisilicicacid Al-cyclotetrasilicic acid Al-cyclohexasilicic acid
1,6-di-Al-octasilsesquioxane
1,9-di-Al-dodecasilsesquioxane
range
153.2 160.0 145.2 154.4 167.9 137.2 140.7 157.8 157.6 167.5 153.0 146.3 137.2 -171.6
1,3-di-Al-cyclotetrasilicicacid 1,3,5-tri-Al-cyclohexasilicic acid 1,3,5,7-tetra-Al-octasilsesquioxane range a
L(SiO(H)Al)
L(SiOH(A1))
L(AlOH(Si))
Bond angle is influenced by an intramolecular hydrogen bond and not included in the determination of the range.
TABLE 6: Bond Angles for the Molecular Models (deg) model
L(Si0H)"
L(A1OH)"
L(OSi0)"
L(OA10)"
1,6-di-Al-octasilsesquioxane 1,9-di-Al-dodecasilsesquioxane
124.2-126.0 119.4-122.6 119.3-128.3 115.5- 126.2 116.3- 125.4 117.8-127.2 117.7- 120.2 118.2-121.4 118.6-122.2
124.8-128.2 131.5- 132.2 128.2 128.7 130.2 123.9- 125.7 129.5 126.7 130.5
100.5-1 14.2 105.4-1 14.4 101.7-114.5 103.5-113.6 103.9-1 17.0 104.8-1 15.0 104.2-1 14.2 100.9-1 14.8 100.9- 114.3
93.9-120.4 91.3- 118.7 98.3-1 17.8 94.7-118.0 97.7-118.3 91.2-118.8 101.0-117.6 96.5-119.5 92.9-1 18.7
range
115.5-128.3
123.9- 132.2
100.5-1 17.0
91.2- 120.4
1,3-di-Al-cyclotetrasilicicacid 1,3,5-tri-Al-cyclohexasilicic acid 1,3,5,7-tetra-Al-octasilsesquioxane
119.3 118.7 117.9
129.6 128.8 130.3- 130.7
105.3-116.8 101.3-114.3 101.4-115.0
102.6- 119.4 105.3-112.6 96.2- 119.7
range
117.9-1 19.3
128.8- 130.7
101.3-116.8
96.2-1 19.7
AI-disilicic acid 1-Al-trisilicicacid 2-Al-trisilicic acid 2-Al-tetrasilicic acid
2-(trihydroxysilyl)-2-Al-trisilicicacid AI-cyclotetrasilicicacid Al-cyclohexasilicic acid
The largest and the smallest value are given.
It is important to note that no systematic trends emerge for the structure parameters in these series of models. It seems that the variations of the structure parameters within the ranges reported above are due to subtle differences in the conformations of the models rather than to electronic effects connected with the model size. There are, however, pronounced differences between models of isolated bridging hydroxyl groups and models with next nearest neighbor pairs of A104 tetrahedra. To the former belong all models that contain only a single bridging hydroxyl group or two bridging hydroxyl groups with A104 tetrahedra separated by at least two si04 tetrahedra, while the latter group contains 1,3-dialuminacyclotetrasilicicacid, 1,3,5-trialuminacyclohexasilicic acid, and 1,3,5,7-tetraaluminaoctahydroxyoctasilsesquioxane. In agreement with results obtained in a previous studyI9the next nearest neighbor models show shorter M o b r i d g e and longer Sia"dgebonds than the models for isolated bridging hydroxyl groups (Figure 8). Figure 8 summarizes the structural model of isolated bridging hydroxyl groups in zeolite catalysts emerging from the ab initio
calculations on finite models. The long N o b r i d g e distance and the larger flexibility of the SiObridgeAl bond angle compared with the SiOb"dgeH angle are consistent with the view that a bridging hydroxyl group is an internal silanol group located close to a AlO3 Lewis site. The zeolite framework can easily accommodate the perturbation connected with the creation of a bridging hydroxyl site because of the following reasons. First, whereas the d o b r i d g e bond is longer than the average S i 0 distance by 30-37 pm, the average distance of all four A10 bonds of the A104 tetrahedron is longer by only about 15 pm. Second, due to the large flexibility of the nonprotonated SiOSi and SiOAl linkages the framework can easily adjust its structure. Direct observed structural data for bridging hydroxyl groups are scanty. Figure 8 shows some of the few data derived from experiments. For the AIHb"dgenonbonded d i ~ t a n c e , * values ~-~~ between 236 and 252 pm have been inferred from different types of NMR experiments, in good agreement with the 236-249 pm range found in our calculations. This is an indirect confirmation of the long Alaridge distance predicted. Experimental structure refinements, even when based on neutron
Hill and Sauer
9542 J. Phys. Chem., Vol. 99, No. 23, 1995 10507/
H
100
substituted frameworks. The aluminum-substituted cyclotetrasilicic and cyclohexasilicic acids can formally be formed by the following reaction:
133O- 146O < 162.3 >
1971, Stevenson
~,~~
s104
isolated AI only
< 163.0>
and for the aluminum-substituted silsesquioxanes, the following reaction holds:
1992, Czjzek et al.
c 176.2>
NNN -AI only
Figure 8. Structure models (pm and deg) of bridging hydroxyl groups in zeolite catalysts emerging from ab initio calculations on molecular models (left) and inferred from experiments (right)?4,40,5'The ab initio calculations predict slightly different structures for isolated hydroxyl groups (aluminum atoms separated by at least two silicon atoms) and hydroxyl groups belonging to next nearest neighbor pairs of atoms. TABLE 7: Stabilities of Aluminum-Substituted Cyclosilicic Acids and Silsesquioxanes per Mole of TO Bonds (T = AI, Si) Formed (m and n Are Defined in the Reaction Equations) cyclosilicic acids m
n
3 2 5 3
1 2 1 3
~~
energy (kJ/mol) ~
-16.2 -24.2 -23.0 -10.8
silsesquioxanes
m 6
n 2
4 10
4 2
energy (kJ/mol) -24.6 -32.3 -26.3
diffraction, cannot distinguish between silicon and aluminum atoms and yield only average TO distances and TOT angles. The average TOT angle reported by Czjzek et al.24 falls into our predicted range. In order to compare with the average TO distance of 165.0 pm calculated from the TO values reported by Czjzek et al.*, for their H-Y(III) sample of protonated zeolite faujasite, we took the weighted average of our (SiO)s,04 and (A~O),A.Q mean values according to the reported SUA1 ratio of 2.43. Our predicted result, 166.6 pm, is 1.6 pm larger than the observed value. This points to a systematic overestimation of the calculated TO bond length. The average (SiO)s104bond length for various all-silica models is 161.5 pm.6 This is about 0.5-2.5 pm longer than the average S i 0 bond length in dense and microporous Si02 polymorphs, respectively (cf. Table 12). It is known that OH bond distances calculated with the same methodhasis set as used in the present study (SCF/triple-l; (O)/ double-l; (Si, Al, H) basis set plus polarization) are systematically too short by 1.2 f 0.2 pm.25 Hence, the value recommended on the basis of our calculations is 96.6 f 0.3 pm. This is a more realistic value than the OH bond distances of 83 f 2, 98 f 4,and 102 f 5 pm reported for zeolite H-Y(III)24 on the basis of neutron powder diffraction. In conclusion, the structural model derived from the calculations is consistent with known observed data. It is more complete and, considering known systematic deviations on the order of magnitude of 1 pm, more accurate than models derived from experiments. It is also worth mentioning that no major revision is necessary on the structural model suggested years ago16*26 on the basis of a careful discussion of results obtained for smaller models and/or with smaller basis sets. 3.2. Energies. Some of the molecular models studied can be used to obtain information about the stability of aluminum-
In these reactions HAl(OH)4 is a complex of water and Al(OH), which, among others, was used in previous studies as a primitive model of a bridging hydroxyl Table 7 contains the stabilities calculated according to these equations in the SCF approximation. These reactions belong to the class of isodesmic reactions for which the SCF approximation was shown to yield reliable results (Chapter 6.5.6 of ref 1). The results show that increasing the number of aluminum atoms increases the stability of the molecules containing four-membered rings. The opposite is true for the hexasilicic acids. Here the three aluminum atoms cause a significant distortion of the ring (cf. Figure 5 ) , because three 189.7 pm long AIOb"dge bonds have to fit into the ring. This causes ring strain and results in a lower stability of this molecule compared to the single-substituted one. Obviously, the stability decreases with increasing size of the ring if the ring contains the maximum number of aluminum atoms. This means that small rings containing the maximum possible number of aluminum atoms should be favored. The models containing only one aluminum atom or two aluminum atoms which are far from each other have nearly the same stability. This result can be further confirmed by looking at the formal substitution reaction of the form H (HO)~S~--OO-AI(OH)~
(H0)aSi-0--Si(OH),
I
0
I
I o
I
(HO),Si-O-Si(OH)2
+
I
0
I
I
0
I
(HO)zAI-O--Si(OH)~ H
-2
(HO)2Si-O-Si(OH)a 0
I
I
0
/
(HO),AI-O-Si(OH)z H
The reaction energy for this reaction is 27.2 kJ/mol; Le., the formation of the singly-substituted cyclotetrasilicic acid is not favored. This conclusion is not affected by assuming some symmetry-restricted optimization for the all-silica and the double-substituted four-membered rings. Relaxing the constraints will only decrease their energy and, hence, increase the reaction energy. This finding is contrary to the so-called Dempsey's rule known in zeolite chemistry, which assumes that aluminum atoms in a zeolite framework should be as far apart from each other as possible, but it confirms the results obtained by Schroder and Sauer concerning the stability of Al-0-Si-0-AI linkages in zeolite^.'^
4. Force Field Definition and Parameter Derivation 4.1. Consistent Force Field. The molecular mechanics potential adopted in this paper relies on the definition of atom types. A large number of parameters is required for acidic
J. Phys. Chem., Vol. 99, No. 23, 1995 9543
Silica and Zeolite Catalysts zeolites. This is due to the different atom types found in the periodic structure and the presence of terminal hydroxyl groups in the finite models (cf. Figure 1). The force field for silica polymorphs6 was based on the definition of the following atom types: silicon atom (Si), oxygen atom between two Si04 tetrahedra (OSS),oxygen atom in a terminal hydroxyl group connected to silicon (OSH), and hydrogen atom in a terminal hydroxyl group connected to silicon (Hos). To specify a force field for zeolites, we have to define additional atom types (Figure 1): aluminum atom (A), oxygen atom between an A104 and a Si04 tetrahedron not involved in a bridging hydroxyl group (OAS),oxygen atom in a bridging hydroxyl group (Ob), oxygen atom in a terminal hydroxyl group connected to aluminum (OAH),hydrogen atom in a terminal hydroxyl group connected to aluminum (HoA), and hydrogen atom in a bridging hydroxyl group (Hb). The consistent force field (CFF)3.27is defined by the following potential energy expressions:
+
+ +
+
E(pot) = ,?(bonds) &angles) E(torsions) E(out-of-plane) E(bond-bond) E(ang1e-angle) E(bond-angle) E(ang1e-angle-torsion) E(nonbond)
+
+
+
+
This second-generation force field3327is implemented (together with some additional cross terms not used in this paper) in the DISCOVER code of Biosym Technologies.28 For fitting the potential parameters, the PROBE code of Biosym Technologies was employed.29 To avoid parameter correlations and to get transferable parameters, linear force constants were used in the fit which were eliminated afterward. Since this procedure is not described in the generally available literat~re,~, we will explain it in section 4.2. 4.2. Avoiding Parameter Correlations. Molecular mechanics force fields use a redundant coordinate set. For example in a tetrahedral coordinated center such as Si04 all six bond angles Aal ... A m are used, but only five of them are independent. This leads to correlations between the force constants and the reference values of the coordinates, and therefore there is an infinite number of mathematical solutions for the parameters. Parameters which are derived from redundant coordinate sets are generally nontransferable and without well-defined physical meaning. The correlation problem between force constants and reference values can be avoided by adding a linear term to each diagonal term of the force field, e.g. for bond angles
(1) E(bonds) =
C [K2(b -
+ K4(b -
+ K3(b -
bonds
E(ang1es) =
(2)
C [ ~ ~- (eo)* e + ~ ~- ( + e angles
~ E(torsions) =
C
[v,[I - cos 41
~- (
0 (3)
+ v2[1 - cos(24>1+
The linear force constant H I is optimized instead of the reference value et. The optimum reference value 80 can then be computed from the trial reference value Or, and the optimized force constants, by solving the following equation:
Ele=e, ae = H , + 2 ~ , ( e ,- e,) + w3(e0- erl2+
torsions
V3[1 - cos(34)Il (4)
(5)
b
E(ang1e-angle) =
+
E = ~ h " ' A s ; ...
b
CeCvF e w (6 - e,)(el - 6;)
E(bond-angle) = x x F b & b
To eliminate the correlations between the linear force constants, an explicit relationship between them can be used. The energy is expressed once as a Taylor series in local symmetry coordinates, Asi:
- e,)
I
(7)
(8)
e
which are linear combinations of the redundant set of internal coordinates, Ar,:
Asi = c c g A r j j
E(ang1e-angle-torsion)
=
The higher-order terms have been avoided here, since they have no effect on the derivation. Substitution of eq 15 into eq 14 yields
+
E = C h " ' ~ c g A r j ... i
x
b, 8, 4, and are the bond lengths, bond angles, torsion angles, and out-of-plane angles in the molecule, rb are the distances between two atoms, and qi are charges. The dielectric constant E was set to 1.0. All other symbols denote parameters which are defined in Table 8. The charges qi are related to the bond increments d: of Table 8 by bonds on i
6;.
q; = j= 1
(16)
j
The energy can, equivalently, be expanded in a Taylor series that explicitly depends on the redundant coordinates Ar,, giving
E =c@)Arj
+ ...
j
By equating terms in eqs 16 and 17, the relationship among the linear force constants can be derived
9544 J. Phys. Chem., Vol. 99, No. 23, 1995
Hill and Sauer
TABLE 8: Potential Parameters bond Si-Ob Si-OAs Si-Oss Si-OsH Al-Ob
K2 K3 K4 bo (A) (kcaV(mol.A2)) (kcaY(m01.A~)) (kcaY(m~l*.&~)) bond 1.6581 370.5744 -569.4988 3102.2502 Al-OAs 1.6157 494.1042 -36.7016 2150.6836 A~-OAH 459.0786 1.6104 -672.4445 Ob-Hb 443.3651 1.6238 43 1.7825 -799.2202 OSH-HOS 4446.0952 1.9698 86.8129 26.3933 OAH-HOA 758.4299
bond angle
80
Ob-Si-OsH OSH-Si-OSH Ob-Al-OAs Ob-Al-OAH OAS-AI-OAH OAH-AI-OAH Si-(&- A1 Si-Ob-Hb Al-Ob-Hb Si-OsH-Hos Al- 0 ~ s -Si
(deg)
103.9653 11 1.3627 81.4988 97.9051 118.3370 107.0368 136.6645 119.2823 107.1200 118.9384 162.4000
torsion angle
H4 (kcaY (mohad4))
103.5467 58.2199 43.2829 27.6920 48.9048 76.4087 195.2950 43.0215 43.9725 20.0134 10.3454
-19.2943 96.8353 - 113.7790 -41.0318 180.6775 -274.1490 48.8575 -15.2099 -53.4364 -33.2675 12.6366
331.9679 527.5761 202.1476 35.4654 1557.5081 293.2546 185.3481 13.1396 103.3838 42.4295 9.0038
Vz (kcaYmol)
V3 (kcaymol)
0.6228 4.6176 1.0464 1.7740 -4.1073 -3.6990 -7.3022 -6.3428 2.1752 0.9847 -0.9767 - 1.8010 -1.1075 3.0650
1.0917 -1.1913 -1.5833 -0.0772 3.5859 -3.0937 3.1717 -3.6734 -0.9201 -0.7285 0.7745 0.6647 0.8045 0.3404
-0.1696 -0.1 141 -0.4804 0.0326 0.2635 0.8520 0.3184 -0.3907 1.1534 -0.5777 -0.9612 0.5364 0.3328 0.0146
Fbu (kcall
(mol.A2))
Ob-Si-OSH 131.7627 OSH-Si-OSH 28.5745 Ob-Al-OAs 84.3372 Ob-Al-OAH 37.6888 0 ~ s - A l - 0 ~ ~ 49.9526 OAH-A~FOAH 19.3587 Si-Ob-A1 130.8388 Si-Ob-Hb 18.0332 Al-Ob-Hb -18.4463 Si-OsH-Hos 9.6155 Al-OAs-Si 83.1658 angle-angle cross termo
H3 (kcaV (mol-rad3))
V1 (kcaYmol)
OsH- Si-ob- A1 OSH- Si-Ob-Hb Ob- Si-OsH-Hos OsH- Si-OsH-Hos 0b.s- Al-Ob- Si 0AS -A1-0b -Hb 0.A.HAl-Ob-Si OAH- AI- Ob- Hb Ob- Al-OAs- Si OAH-A1-OAS-Si Ob- AI- 0.w- HoA OAS-AI-OAH-HOA OAH-A~-OAH-HOA Al- 0 ~ s Si-OsH bond-bond cross term
H2 (kcal/ (mo1-rad2))
bond-angle cross term
FbB (kcaV F ~ (kcaU B (molehad)) (mol*A-rad))
Ob-Si-OSH -0.9347 OSH-Si-OSH 43.7331 Ob-Al-OAs 12.9574 Ob-Al-OAH 25.6722 0 ~ s - A l - 0 ~ ~ 87.4126 OAH-AI-OAH -6.2892 Si-Ob-Al 23.0450 Si-Ob-Hb 36.5384 Al-Ob-Hb -12.7238 Si-OsH-Hos 46.1485 Al-OAs-Si 9.1662
96.7623 20.0579 20.7253 15.6056 3.6805 53.8615 33.9457 25.1227 13.7478
F ~ s(kcaY . angle-angle- torsion (molrad2)) cross term
Kew# (kcaY
52.6194 60.1802 38.4444 52.3408 86.3954 -6.3030 -13.6331 17.4074 0.0000 -20.0594 -5 1.8916 36.0652 58.8895 44.4559 61.4849 75.8868 50.5839 13.6760 28.2428 -34.0834 -62.0442
213.6799 -2 1.4028 -9.3392 -2.9461 32.3177 8.8676 -5.9667 -22.3131 16.7797 7.8776 -6.3680 -52.1956 6.5018 -14.1105 -4.6863 8.4655 - 10.4191 2.3216 6.4184 -4.5150 11.8037
(mol-rad2))
K2 4 K4 (kcaY(mol*A2)) (kcaU(mo1-A3)) (kcaY(m01.A~)) 1.7193 328.6850 -341.0143 2 189.0349 1.7096 31 1.6053 -734.8390 6022.3970 0.9540 656.8465 -1627.4874 3684.1929 0.9458 707.2556 - 1531.5564 1603.1003 0.9408 730.0168 - 1566.4841 1703.5916 bo (A)
bond angle
60 (deg)
A~-OAH-HOA 131.3984 OAs-Si-OsH 118.5582 o ~ S - s i - 0 ~ ~ 112.4279 Oss-Si-osH 109.9424 Oss-Si-Oss 112.0200 Oss-Si-a 107.6066 Si-Oss-Si 173.7651 0 ~ s - A l - 0 ~ ~ 113.4000 OAs-Si-Ob 102.9438 OAS-Si-OAS 110.6120 torsion angle OAs-Si-OsH-Hos Oss-Si-OAs-A1 OAS-Si-Oss-Si OSH-Si-Oss-Si Oss-Si-OsH-Hos Oss-Si-Oss-Si Ob-Si-Oss-Si Oss-Si-Ob-Al Oss-Si-Ob-Hb OAs-Al-OAs-Si Ob-Si-OAs-Al Oas-Si-Ob-Al OAS-Si-Ob-Hb OAs-Si-OAs-A1 bond-bond cross term A~-OAH-HOA OAS-Si-OSH OAs-Si-Oss Oss-Si-osH Oss-Si-Ob Si-Oss-Si OAs-Al-OAs OAS-Si-Ob OAS-Si-OAS angle-angle cross term
HZ (kcaU (mo1-rad2))
H3 (kcaY H4 (kcaY ( m o l ~ a d ~ ) ) (mo1-rad4))
7.81 19 87.1068 88.0709 91.8675 81.9691 99.6715 20.7015 300.0520 103.8270 154.1860
-1.6786 167.1975 -57.0430 17.2882 -36.5814 -33.1512 27.5506 -32.5659 10.8990 -68.6595
53.6329 278.6751 92.4803 252.8231 116.9558 10.7204 10.9930 67.2591 0.6674 23.6292
VI (kcaYmo1)
VZ (kcaYmol)
V3 (kcaUmo1)
3.0272 2.1989 0.0397 0.1277 3.2254 0.0306 0.5350 0.1641 3.7091 6.0838 5.7178 -1.8325 5.7768 1.8597
-0.1573 0.7528 0.0133 0.0442 -0.2132 -0.0105 -0.4717 1.7914 -0.6560 -0.5702 0.1922 2.2280 -2.1457 -0.023 1
-0.2321 -0.4936 -0.2453 -0.0009 -0.4 145 0.0804 -0.0036 0.1970 0.0856 -1,4818 -0.3374 0.23 10 0.1001 -0.4345
Fbu (kcaY
bond-angle Fbo (kcaY cross term (mol&ad)) AI-OAH-HOA 38.1658 OAS-Si-OsH -54.0838 OAs-Si-Oss 20.1222 Oss-Si-osH 41.1242 Oss-Si-Oss 78.1239 Oss-Si-Ob 52.7776 Si-Oss-Si 9.2390 0 ~ s - A l - 0 ~ ~ 110.1523 OAS-Si-Ob 13.3600 OAS-Si-OAS 234.0716
Fue (kcal/ (mol-A-rad)) 31.21 14 63.3330 77.9618 75.6000
For (kcaV angle-angle- torsion (mol-rad2)) cross term
Kowo (kcaY (mol-rad2))
-19.2127 13.9083 46.1582 5 1.3039 40.7833 52.5469 -6.1120 39.4951 0.0000 98.5159 25.6772 -35.6378 31.8901 19.2372 0.0000 79.7785 0.0000 0.0000 -23.1336 0.0000 0.0000
-7 1.4139 15.1160 - 18.0184 17.2526 91.5202 17.5674 - 10.9279
(mol*Az)) -2.61 12 72.4616 45.7520 70.6866 63.3273 151.8742 -56.5050 63.3273 151.8742
12.2769 -8.3197
Silica and Zeolite Catalysts
J. Phys. Chem., Vol. 99, No. 23, 1995 9545
TABLE 8 (Continued) out-of-plane Si-(&-&Van der Waals parameters s1 o b
OSH AI OAS
i-j-kll
Kx (kcal/(mol-rad2))
x (den)
0.8407
-0.2235
AI
bond A, (((k~al/mol).A~)’/~) increments 432.3320 278.9970 150.7170 143.1830 282.2060
d i = -d; ( e )
Si-OsH HOS-OSH AI-OAH HOA-OAH
si-ob
is the coupling between the bond angles i-j-k
0.1303 0.0641 0.3013 0.0575 0.1392
Van der Waals parameters
OAH Hb Hos HOA oss
bond A, ( ( ( k ~ a l / m o l ) . ~ ~ ) ~ ~increment *) 168.1940 7.7660 20.9470 8.4760 239.6090
Al-Ob Hb-Ob Si-Oss Si-OAs Al-0~s
d i = -6; (e) 0.0284 0.0839 0.1309 0.1265 0.1694
and k-j-1.
Since local symmetry coordinates normally used obey &ci, = 0, it follows from eq 18
This relationship has to be used explicitly to avoid correlation between the linear force constants. 4.3. Data Base Generation and Parameter Derivation. The parameters of the nonbond energy expression (eq 10) are determined independently of the parameters of the internal coordinates. The Ag parameters of the repulsive van der Waals term are obtained by test particle calculations (cf. refs 31 and 32) as in the all-silica case. In these calculations, the repulsive nonbond interactions of the atoms in 2-aluminatrisilicic acid were probed with a nitrogen atom in its valence state as test particle. This particle was chosen because it proved most suitable when comparing different atoms (cf. refs 31 and 32). The Bv parameters of the dispersive van der Waals term were set to zero, since the SCF approximation does not account for dispersion. The charge parameters of the force field are directly taken from a population analysis based on occupation number^^^-^^ in the ab initio calculations. The use of charges in this type of force field is somewhat artificial, because it has its origin in a Taylor series expansion which accounts for all interactions. Point charge interactions are only introduced to describe interactions between atoms or atom groups which are not directly connected by bonds. In our previous paper6 we found that the best structure predictions are achieved when only half of the values obtained from the ab initio calculations were used in the force field. This factor of is often found when charges from population analyses are compared with charges obtained by fitting crystal data.37 The remaining majority of the parameters were fitted to a collection of ab initio data for the molecules of the training set which were generated in the following way. Starting from the equilibrium structures, all internal coordinates of the first six molecular models in Table 1 were systematically distorted, and for the resulting structures the SCF energies, SCF gradients, charges, and dipole moments were computed. The magnitude of typical distortions was k 1 2 pm for bond lengths in steps of 3 pm and A16O in steps of 4’ for bond angles. The number of distorted structures considered in the fit was 45 for aluminadisilicic acid, 20 for 1-aluminatrisilicic acid, 171 for 2-aluminatrisilicic acid, 35 for 2-aluminatetrasilicic acid, 13 for 2-(trihydroxysilyl)-2-aluminatrisilicic acid, 17 for aluminacyclotetrasilicic acid, 8 for 1,3-dialuminacyclotetrasilicicacid, and 11 for each of the aluminaoctahydroxysilsesquioxanes. In addition, the entire training set used for the derivation of the all-silica potential6 was included. The number of distorted structures available was much larger, especially for the alumi-
TABLE 9: Parameters for Geometry-Dependent Charges i Si Hos AI HOA Si AI Hb Si Si AI
j OSH
OSH OAH OAH ob o b o b
oss OAS OAS
a,, (e/& -0.8836 0.2644 0.0973 0.2258 0.2912 0.0036 0.2820 0.3391 0.2945 0.2666
6: (e) 0.1412 0.0679 0.3208 0.0571 0.1360 0.0286 0.0828 0.1311 0.1371 0.1725
nadisilicic acid, but difficulties were encountered by using them in the fit (vide infra). The number of parameters which had to be determined by a fit was 336. Since 47 468 quantum chemical “observables” were available, the dakdparameter ratio was 141.3. As start values for the parameters, force constants were used which were determined from analytical second-derivative calculations (SCF approximation) of the smallest models. It turned out that carefully chosen coupling constants were of great importance for the success of the fit. The coupling constants are more important here than in the all-silica case, because the large range of possible bond lengths, e.g. for the A10 bond in the bridging hydroxyl group, cannot be described by the diagonal terms in the potential alone. Furthermore, it was necessary for a successful fit to exclude all configurations from the data base with energies more than 84 kJ/mol above the corresponding minimum. The large number of parameters also required a large number of ab initio data, so that the training set of the present study is large compared to the transferability test set. The latter contains only the two hexasilicic acids and the substituted dodecasilsesquioxane as well as the transferability test set of our previous paper.6 The resultipg force field parameters were used to determine by molecular mechanics energy minimizations the structures of the molecular models used for fitting (the training set) and of the molecular models not used for fitting (the transferability test set). The DISCOVER code was used. The transferability of our new potential was tested by structure predictions for different bridging hydroxyl groups in faujasite and comparison with results previously obtained. These calculations used the DISCOVER code and, for the periodic structures, its Ewald summation technique for the Coulomb interaction. All computations were done on IBM RISC Systed6000 and Silicon Graphics Iris workstations.
5. Results of the Potential Parametrization Table 8 shows the final parameters, Table 10 the root mean square (RMS) and maximum deviations between the ab initio structures and the structures obtained with the parameters, and Table 11 the RMS of the positions of the non-hydrogen atoms
Hill and Sauer
9546 J. Phys. Chem., Vol. 99, No. 23, 1995
TABLE 10: Root Mean Square and Maximum Deviations between the Equilibrium Structures Obtained from Molecular Mechanics and ab Initio Calculation (pm and de& training set
RMS
transferability test set
RMS
maximum
maximum
a
b
a
b
a
b
a
b
0.18 0.87 0.52 0.89 0.1 1 0.06 0.27 0.03 0.29 0.65 1.01 1.55 0.75 2.03 1.42 0.77 1.70 0.20 0.83 0.29 1.50 0.68 1.76 0.33 2.75 0.68 3.91 6.1 1 13.08 5.24 9.00 7.71 5.17 1.14 1.49 0.53 7.61 1.88 13.74 2.21 4.61 5.88 2.24 1.55 0.70 2.66 2.77 10.51 4.81
0.17 0.89 0.46 0.80 0.11 0.06 0.26 0.02 0.27 0.64 0.90 0.89 0.67 1.85 0.97 0.89 1.24 0.20 0.56 0.26 1.46 0.63 1.23 0.33 2.67 0.66 3.45 5.14 6.41 4.99 8.32 4.57 4.38 0.99 1.59 0.49 3.26 1.64 12.98 1.76 1.52 4.43 1.74 1.68 0.76 2.21 2.45 9.49 4.75
3.79 -4.93 -2.98 7.50 - 1.02 - 1.67 3.04 -0.91 -5.56 5.95 3.62 -18.82 1 1.24 23.99 -14.51 4.49 13.71 6.17 10.79 5.59 - 11.79 -8.30 8.61 -9.11 26.28 -16.74 10.97 -38.02 11 1.42 -36.55 -41.44 81.59 36.44 - 10.46 -11.53 - 17.93 81.87 107.44 - 119.08 108.13 67.81 64.79 24.59 -19.00 9.83 -33.94 33.32 112.10 33.06
3.55 -5.08 -3.06 -6.70 - 1.03 1.42 3.02 -0.93 -5.49 6.02 3.44 -9.32 9.80 22.04 -6.61 6.46 -7.07 -7.84 8.18 -5.17 -11.98 -8.04 6.55 -7.99 26.13 -16.10 -10.00 -32.59 33.28 -34.24 -39.40 34.40 31.34 7.49 11.88 15.22 35.25 -88.80 - 120.26 86.89 19.13 35.61 19.55 -17.05 9.73 -28.26 29.04 113.57 -32.20
0.08 1.78 1.27 4.10 0.11 0.04 0.39 0.02 0.11 1.55 1.80 4.42 0.68 7.48 1.49 2.17 1.87 0.13 1.64 0.85 2.68 2.73 1.90 0.14 3.89 0.69 4.15 0.77 22.18 8.91 0.00 6.04 0.33 0.33 0.77 0.75 4.46 1.35 12.98 5.34 7.88 4.10 1.19 1.19 0.28 0.28 21.05 10.59 0.00
0.08 1.80 1.22 3.50 0.09 0.03 0.37 0.02 0.1 1 1.58 1.47 3.65 0.50 6.98 1.07 2.15 1.55 0.11 1.49 0.83 2.63 2.94 1.90 0.16 3.78 0.58 6.61 0.70 24.09 11.30 0.00 8.15 0.34 0.34 0.70 0.69 5.82 0.96 11.82 3.25 6.63 5.60 1.17 1.17 0.52 0.52 7.96 9.73 0.00
4.62 -5.02 -4.75 12.44 0.34 1.14 -1.51 -0.60 -5.55 -6.15 2.55 -12.81 27.32 -39.96 4.87 6.76 -5.90 -6.29 6.17 -5.82 11.84 -10.53 -6.86 2.32 14.46 20.17 10.38 -3.09 44.42 -29.74 0.00 24.24 -1.31 -1.31 3.09 20.88 -26.48 -78.44 53.87 -72.25 -29.53 15.70 4.84 4.84 - 1.02 - 1.02 -78.92 -40.05 0.00
5.07 -5.28 -4.69 9.57 0.26 1.12 -1.38 -0.60 -6.24 -6.77 2.08 - 10.90 26.58 -38.78 -4.00 6.43 -4.29 5.30 5.44 -7.57 11.62 -11.69 -7.23 3.82 13.86 19.96 -16.51 2.80 48.19 -37.17 0.00 32.58 -1.35 -1.35 2.80 -19.69 -34.62 -77.59 50.33 7 1.09 -27.88 26.56 4.90 4.90 -2.10 -2.10 -54.06 -37.27 0.00
a Fixed atomic charges. Atomic charges change with geometry. 0, denotes a "terminal" oxygen atom in a TOH group (T = Al, Si). These types of internal coordinates are not present in solids.
after superimposing the ab initio and the force field structure. The quality of the fit is a little bit worse than for the all-silica modek6 The AIObndgebond lengths of the training set show a maximum deviation of 7.5 pm. This is the result of the mathematical form of the potential. It is very hard to describe a bond which can have a length between 188 and 199 pm depending on the model using one and the same reference bond length. For the transferability test set the maximum deviation is even larger, more than 12 pm. Nevertheless, for all bonds, the Rh4S deviations are below 4.1 pm and, for the entire training set, even below 1 pm, which shows that only some of the predicted bond lengths deviate strongly from the ab initio results. It is surely necessary to apply a more appropriate mathematical form of the potential to reduce these errors. The bond angles also have large maximum deviations (26 and 40" for the training and the transferability test sets,
respectively). These large deviations occur for AIOtH and AlOSi angles, which again show a large range of possible values in the ab initio calculations, 124-132" and 128-174", respectively. The flexibility of the AlOSi angle is not sufficiently reproduced by the potential parameters. Very large maximum deviations also occur for the torsion angles. As far as the terminal hydroxyl groups are concerned, this does not affect the quality of solid simulations, because these types of torsions do not occur there. Large deviations found for the other torsion angles either can be explained by symmetry restrictions on them in the ab initio calculations, but not in the molecular mechanics calculations, or are a consequence of small changes of nearly linear bond angles involved. If one of these bond angles changes from 179 to 181", the corresponding torsion angle will change by 180". For example the maximum deviation for the OAlOSi torsion in Table 10 can
Silica and Zeolite Catalysts
J. Phys. Chem., Vol. 99,No. 23, 1995 9547
TABLE 11: Superposition of the Equilibrium Structures Obtained from Molecular Mechanics and ab Initio Calculationu charge model Al-disilicic acid I-Al-nisilicic acid 2-Al-trisilicic acid 2-Al-tetrasilicic acid Al-cyclotetrasilicic acid
fixed
9.1 7.2 10.1 7.7 7.8 1,3-di-Al-cyclotetrasilicicacid 6.6 Al-cyclohexasilicic acid 25.4 1,3,5-tri-Al-cyclohexasilicicacid 20.8 1,6-di-Al-octahydroxyoctasilsesquioxane 12.2 1,3,5,7-tetra-Al-octahydroxyoctasilsesquioxane 23.0 1,9-di-Al-dodecahydroxydodecasilsesquioxane 10.9
variable 7.5 6.1 3.2 4.9 7.6 3.8 3.6 19.3 11.6 20.2 8.3
"The root mean square deviations of the positions of the nonhydrogen atoms are listed (pm).
be understood if one considers that the AlOSi bond angle changes from 145.4' in the ab initio calculation to 190.8' in the force field calculation. The deviations of the out-of-plane angles are also a consequence of symmetry restrictions in the ab initio calculations. Nearly all of these calculations (the only exception was the aluminacyclotetrasilicic acid) assumed a planar bridging hydroxyl group. This type of restriction could not be imposed in the molecular mechanics calculations, so that there are differences possible. Nevertheless, the R M S deviations of the non-hydrogen atom positions are rather small. The more important transferability test set molecules which contain the essential structural elements of zeolites (rings and cages) are reproduced well, so that transferability of the potential can be assumed. A major drawback of all force fields currently used is that they employ fixed charges on the atoms. Quantum chemical calculations show, however, that-for a given method of calculating charges-the atomic charge is a function of the structure. Since the quantum chemical calculations made for deriving the potential parameters resulted in a large number of data, including charges, it was possible to look into the details of this dependence. A previous study showed that only the change of a bond length has a significant effect on the atomic charge.38 Therefore, the expression for the calculation of atomic charges from bond increments (eq 11) is replaced by the more realistic relation38
The 6~ and d: parameters have been determined by a fit of quantum chemically calculated charges. rU is the actual length of the bond i-j, and ri is the reference bond length of the potential (Table 8). The new parameters 60 and d: are summarized in Table 9. Since the DISCOVER program was unable to handle this new potential expression, an additional program was used to calculate the charges as a function of the structure and feed them back into DISCOVER. Geometry optimizations were possible in this way, but for technical reasons so far only for nonperiodic systems. The RMS and maximum deviations obtained with this method for all models are included in Table 10. Despite that there was no refit of all other parameters, the use of geometry-dependent charges improves the structure predictions significantly. The largest errors for the Al@,*dge bond length (12.44 pm in the transferability test set) and for the OAlOSi torsion angle (111.42' in the training set) are
reduced most (to 9.57 pm for the AIOb*dge bond length and 33.28' for the OAlOSi torsion angle). The use of geometrydependent charges seems to be essential for getting a good potential. for aluminosilicates. Further improvements may be expected when geometry-dependent charges are used in the fitting procedure.
6. Predictive Power of the Potential The present potential is also applicable to all-silica structures as a special case of aluminosilicates. Hence, the silica part of the potential can be tested by comparing the periodic structures obtained by lattice energy minimizations under constant pressure with experimental data of different dense and microporous silica polymorphs. Table 12 shows the results. The aluminosilicate potential gives all-silica zeolite structures with the same accuracy as for the special all-silica potential derived in part 1.6 For the unit cell lengths the average error is 3%; the largest deviation is found for a-quartz (5%), and the smallest, for mordenite (1%). Experimental structure refinements of aluminum-containing zeolites almost always average over all T sites (T = Al, Si) and do not give details about the aluminum distribution in the lattice. Only very few data for the local structure of bridging hydroxyl groups are available (vide supra). So we can only compare the structure obtained with our potential with results of previous theoretical predictions and make sure that our results are not contrary to the few experimental findings. Ab initio calculations on a large finite model were used to predict the structure of the O(1)H bridging hydroxyl site in f a ~ j a s i t e .We ~ ~ repeated this structure determination using our potential. Comparison of the results (Table 13) shows that the potential predicts too short Al0bnd.g and Si&ldgebond distances, which also results in a too short AIHbndgedistance, regardless of whether the charges were fixed or variable. The average bond lengths per tetrahedron are, however, close to the results of the ab initio calculation. This points to a problem with the prediction of the bond lengths of the TObndgebonds, which was already noted for the training and transferability test sets (vide supra). The OH bond length and the bond angles in the bridging hydroxyl group are predicted very well. We then performed lattice energy minimizations under constant pressure for a faujasite which had one bridging hydroxyl group per unit cell ( S U N ratio of 191), but in different locations. Table 13 shows the results. The average bond length per Si04 tetrahedron of about 163 pm found in the aluminumcontaining faujasites is the same as in all-silica faujasite (cf. Table 12). The TOT angles, however, are significantly smaller in the aluminum-containing faujasite, which shows that the structure responds to the longer A10 bonds by decreasing these very flexible angles. The sequence of the angle values, 0(1) < O(4) O(3) 0(2), is the same for the SiO(H)Al bonds (Table 13) as for the SiOSi bonds in all-silica faujasite (135.0', 144.8', 161.9', 164.6'). The size of the unit cell is virtually not changed if one silicon atom per unit cell is substituted by an aluminum atom. The lattice energy minimization of the aluminum-free faujasite predicts a unit cell length of 2484 pm (cf. Table 12), while the four different single-aluminumsubstituted faujasites show unit cell lengths of 2483-2485 pm. The cell angles always remain 90'. Incorporation of an isolated bridging hydroxyl group changes the structure close to the aluminum atom and influences the neighboring Si04 tetrahedra only. This result agrees with the findings in the molecular models (vide supra) and shows that the lattice is flexible enough to accommodate the distortions caused by an aluminum atom. We can also compare the predictions of the aluminosilicate potential with the results obtained with a shell model p~tential,'~
Hill and Sauer
9548 J. Phys. Chem., Vol. 99, No. 23, 1995
TABLE 12: Results of Lattice Energy Minimizations on All-Silica Zeolites Using the Potential Derived in This Paper (pm and deg) SilAl ratio a b c a=j3 Y (r(Si0)) (L(Si0Si)) obsd" calcd obsdb calcd obsd' calcd obsdd calcd obsd' calcd obsdf calcd obsdg calcd obsdh calcd
sodalite faujasite zeolite rho zsm-22 zsm-5 mordenite a-quartz /3-quartz
883 906 2426 2484 1488 1520 1742 1806 1990 2060 2052 207 1
no A1 no A1 7.0 no A1 299 4.4
1386 1438 2002 2066 1809 1834
no A1
49 1 516 500 517
no AI
504 529 1338 1384 752 769 540 567 546 566
90.0 90.0 90.0 90.0 90.0 90.0 90.0 90.0 90.0 90.0 90.0 90.0
159 162 161 163 163 163 160 162 159 162 162 162 161 163 159 163
120.0 120.0 120.0 120.1
90.0 90.0 90.0 90.0
160 162 144 151 146 153 151 163 155 161 153 159 144 158 153 158
Reference 43. Reference 44. Reference 45. Reference 46. e Reference 47. f Reference 48. g Reference 49. Calculated from data in reference 50.
TABLE 13: Comparison of the Results of Lattice Energy Minimihations on Different Faujasite Structures Using the Potential Derived in This Paper with Ion Pair Potential and ab Initio Calculations (pm and deg) ref
r(AlOb)
direct ab initio39 force field, charges fixed force field, charges variable
O(1)H O(1)H O(1)H
192.9 187.7 186.8
force field (this work)
O(1)H O(2)H O(3)H O(4)H O(1)H O(2)H O(3)H O(4)H
185.8 185.7 189.7 184.0 191.0 190.3 193.0 190.6
O(1)H O(2)H O(3)H
187.3 186.5 187.8
shell model potentialI4
force field (this work)
r(si0b)
T(ObH) L(siOb.41) L(SiObH) Finite Model 170.9 95.9 134.2 115.7 167.4 95.3 136.0 117.8 167.6 95.3 135.0 118.1 Periodic Faujasite (SUA1 Ratio 191) 167.1 95.2 133.7 117.9 168.1 95.9 141.3 113.6 167.3 95.5 138.7 117.9 167.7 95.6 136.5 115.1 169.4 100.0 131.1 123.0 168.7 100.4 142.4 117.6 169.7 100.2 138.7 120.7 168.8 100.1 134.0 121.4 Periodic Faujasite (Si/A1 Ratio 2.43)b 167.9 95.3 135.8 118.1 168.2 95.6 140.6 115.7 167.7 95.3 137.0 117.8
r(AIHbn,jee)
(r(Al0))
(r(Si0))"
243.0 232.4 232.5
176.5 175.1 175.2
163.1 162.6 162.6
233.3 222.6 23 1.2 225.1 238.6 229.6 233.2 236.3
174.7 175.7 175.2 175.6 173.8 174.3 174.2 174.3
162.4 163.3 163.2 162.8 160.3 160.7 160.7 160.3
231.8 226.6 23 1.6
175.9 176.8 175.7
163.0 163.0 162.8
The average is taken over the Si04 tetrahedron which is part of the SiO(H)Al linkage. O(4)H is not occupied as in experiment; averaged over all corresponding sites.
which are also included in Table 13. There are two differences: The shell model potential yields unrealistically large OH bond distances, while our force field yields too short Al%ridge bond lengths, which means that also the nonbonded AIHb"dge distances are too short. All other structural features are very similar. For example the predicted SiO(H)Al bond angles are very similar and show the same sequence of values for the four different oxygen sites. Finally, we performed lattice energy minimizations under constant pressure for a protonated faujasite with an SUA1 ratio of 2.43. This is the SUM ratio of the faujasite samples for which neutron powder diffraction experiments were reported.24,* The A1 atoms were distributed in the lattice according to a model suggested by Klinowski et al.?' and the protons were distributed among the four different oxygen sites as O( 1)/0(2)/0(3)/0(4) = 32:8: 16:0, which is very close to the population deduced from the neutron diffraction data. The local structures of the bridging hydroxyl groups for the 0(1), 0(2), and O(3) sites (Table 13) show no major differences compared with the isolated groups in the silicon-rich faujasites (SUA1 = 191). The calculated bond lengths and angles cannot directly be compared with observed data. The refinements of the neutron diffraction data do not make a difference betwee! aluminum and silicon atoms and assume the space group Fd3m. Hence,
TABLE 14: Comparison of the Results of Lattice Energy Minimizations Using the Potential Derived in This Paper with Shell Model Potential Calculations and Experimental Findings for H-Faujasite (H5&156Si13603&1)and with All-Silica Faujasite (pm and deg) Hs6A156Si1360384 Si02 this shell this work model4* obse40 work obsd4 (r(TO))
0(1) 168.3 168.9 O(2) 165.7 163.0 O(3) 167.5 165.6 O(4) 165.0 162.3 (L(TOT)) 0(1) 139.1 136.3 O(2) 152.7 151.8 O(3) 147.2 145.6 O(4) 146.8 143.0 2531 2484 unit cell length no PI P1 space group
167.7(6) 163.2(7) 165.4(6) 163.6(7) 135.7(3) 144.6(5) 139.8(4) 143.9(5) 2477 Fd3m
163.3 162.3 162.8 162.6 135.0 164.6 161.9 144.8 2484 P1
160.7(2) 159.7(2) 160.4(2) 161.4(3) 138.4(2) 149.3(2) 145.8(2) 141.4(2) 2426 Fd3m
The composition of the experimentally studied sample was Na3H53A156Si1360384.
only average bond distances are obtained, which are shown in Table 14. For comparison, we took the arithmetic mean of all TO@) distances and TO(x)T angles of the calculated structure regardless of whether the particular 0 atom was protonated or not. Although this is not the same averaging as that involved in the refinement of the neutron diffraction data, a comparison
Silica and Zeolite Catalysts may still give useful information. Table 14 shows not only the observed data and the data predicted with the present force field but also a structure predicted with the shell model potential4* Both potentials yield very similar mean bond lengths and angles. When the results are ordered according to increasing values, the same sequence of the four oxygen sites is obtained. The differences between the different oxygen sites are primarily due to the different degree of protonation. This implies for the mean bond lengths O( 1) > O(3) > O(2) and for the bond angles O( 1) < O(3) .C O(2) as protonation lengthens the TO bond and narrows the TOT angle. The predicted and observed structures indeed show these sequences (Table 14). As none of the protons is located on the O(4) site, its mean bond angle should be larger and its mean bond length shorter than the corresponding values for the O(2) site. This is not the case for the bond angles (the value for the O(2) site is the largest) and also not for the observed bond length (the value for the O(2) site is the smallest). The reason is that the four TO bond lengths and TOT bond angles are different already for the all-silica faujasite structure. The TO(2) bond is shorter than the TO(4) bond, and the TO(2)T angle is wider than the TO(4)T angle. The effect of protonation is not large enough to reverse this ratio. Hence, the potential derived for aluminosilicates reproduces the observed pattern of mean bond lengths and angles for the four oxygen sites in protonated faujasites. On average, the TO bond lengths and TOT angles obtained with the present force field are larger by 1.6 pm and about 5" than the results derived from the diffraction data. These deviations are similar to those observed for the all-silica structures (cf. Table 12) and for the ab initio structures of finite models (section 3.1) and, hence, may be attributed to systematic errors of the ab initio approach used for parametrizing the potential rather than to deficits of the fit. The advantage of the theoretically predicted structures is that they distinguish between silicon and aluminum atoms. An unique feature of the present force field is that it provides a reliable structure for the acidic OH group itself. A weakness of our potential, which is connected with its force field functional form, is the relatively large errors of the predicted AIOb"dge bond distance.
7. Summary and Conclusions The method of deriving reliable potential function parameters for periodic structures from quantum chemical calculations on finite models developed previously for silica was applied to protonated aluminosilicates. Their structures proved much more flexible than those of the silica polymorphs. In particular, the length of the A10 bond in the SiO(H)Al bridge varies widely. This feature is not easily reproduced by a force field type potential as adopted here. Nevertheless, a transferable set of parameters was obtained, but the errors of the predicted aluminosilicate structures are slightly larger than that of the predicted all-silica structures. The structures of dense and microporous silica polymorphs are reproduced with the same accuracy as previously achieved with a special silica potential. The predicted structure of a H-faujasite is in accord with the average information from neutron diffraction refinements but provides more accurate data on the OH group itself and resolves differences between Si and A1 sites. The present force field potential predicts the structures of zeolites with bridging hydroxyl groups roughly with the same accuracy as the empirical shell model ~0tential.I~ Conceptually it has two advantages: It is completely derived from ab initio calculations, and it can be easily extended to include the
J. Phys. Chem., Vol. 99, No. 23, 1995 9549 interaction of organic molecules with zeolite frameworks. This is prerequisite to simulations of sorption and diffusion processes in catalysis. Acknowledgment. This work is part of a common project with the Theoretical Chemistry Group of Prof. R. Ahlrichs at the University of Karlsruhe supported by the "VolkswagenStiftung". We are particularly greatful to Prof. R. Ahlrichs and his group for providing the most recent versions of their TURBOMOLE code. We also thank the "Fonds der Chemischen Industrie" for financial support. We thank Biosym Technologies, Inc. for making available their DISCOVER and PROBE codes and Dr. Jon Maple for valuable discussions. The structure model of the H-faujasite with SUAl = 2.43 was kindly provided by Dr. K.-P. Schroder. References and Notes (1) Hehre, W. J.; Radom, L.; Schleyer, P. v. R.; Pople, J. A. Ab initio molecular orbital theory; Wiley: New York, 1986.
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