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Discussion of fuzzy evaluated force fields for NaX zeolites
Journal of
MOLECULAR STRUCTURE Journal
of Molecular
Structure
349 (1995)
235-238
Discussion of fuzzy evaluated force fields for NaX zeolites M. Kudraa, E. Geidelh, and H. Bijhliga
aInstitute of Physical and Theoretical Chemistry, University of Leipzig, Linnestr.3, D-04 103 Leipzig bInstitute of Physical Chemistry, University of Hamburg, Bundesstr. 45, D-20146 Hamburg Dedicated to Prof. Dr. G. Geiseler on occasion of his 80th birthday
1. INTRODUCTION To the spectroscopical characterization of zeolites, usually, the regions of frameworkand OH- stretching vibrations are considered. The corresponding infrared and RAMAN spectra are characterized by only few but broad bands. The fundamental papers of Flanigan et al. [l] lead to an assignment of absorption bands and RAMAN lines to vibration modes based on intra-tetrahedral and inter-tetrahedral vibrations. If additionally the validity of Blackwell’s decoupling postulate [2] is assumed, the vibrational behaviour of zeolites can be discussed on the basis of cluster models, e.g. with tetrahedral or ditetrahedral structure. Normal coordinate analyses are used to interprete the spectral frequency regions expecting less or more decoupled vibrations of tetrahedra. The frequency regions are overlapped and have no sharp defined boundaries. In this sense the spectral information are uncertainty ones. The uncertainty can be handled by using fuzzy sets introduced by Zadeh [3].
2. FUZZY EVALUATED FORCE FIELDS The broad and strongly structered bands in the vibrational spectra of NaX zeolites [4] cover broad frequency intervals. In these frequency regions one has to expect the tetrahedral vibrations and additional vibrations caused by coupling with neighbouring tetrahedra. We assume that (SiO4- and Al04-) tetrahedral vibrations of the cluster model are located in an experimentally given frequency interval but their positions do not necessarily correspond to the band maxima. The uncertainty of the frequency intervals can be modeled by special trapezoid functions. These functions arise by evaluation of frequencies in the middle of the intervals with a higher degree (e.g. 1) as frequencies at the boundaries evaluated e.g. with 0 and connecting the points by a straight line. (cp. Figures 1 and 2). 0022-2860/95/$09.50 0 1995 Elsevier SSDI 0022-2860(95)08752-4
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In the sense of fuzzy theory [3], these functions are membership functions of so-called fuzzy sets. On this way, any frequency belongs with a special degree - the so-called membership degree (a value between 0 and 1) - to the fuzzy set. In opposition to classical (sharp) sets, for example intervals, we cannot decide if any frequency belongs to the fuzzy set or not. We will call the fuzzy sets fuzzy observations of the experimental frequencies. Now, the membership degree can be used to evaluate sharp calculated eigenfrequencies with the fuzzy observations of the experimental frequencies. The higher the membership degree the better the eigenfrequencies approximate the fuzzy observations. To the normal coordinate analysis Olson’s [5] structural parameters of NaX and the bond lengths scaled force field (BLSFF) of Blackwell [2] are used. Taking into account a systematic variation of the parameters a, b and h of BLSFF, various force fields for the primary building units (SiO4- and Al04- tetrahedra) of NaX zeolites are derived. Additionally we consider interaction force constants introduced by Maroni [6]. With the arising force fields we carry out normal coordinate analyses based on Wilson’s [7] GFmatrix method. We evaluate a given force field by the least membership degree of the calculated eigenfrequencies to the corresponding fuzzy observations. To the discussion of the experimental vibrational spectra we choose such force fields with maximum fuzzy evaluation, so- called fuzzy-optimal ones.
3. RESULTS
AND DISCUSSION
3.1. Tetrahedra Figure 1 shows the modeled fuzzy observations of the experimental frequencies (trapezoids and triangels on the frequency intervals) containing the experimental information from the infrared and RAMAN spectra. To ensure the reliabiltity between experimental data and model vibrations, we have chosen different shape and height of the membership functions. Additionally, the calculated eigenfrequencies of BLSFF for the tetrahedra are drawn as vertical lines. The membership degree of a calculated frequency to a fuzzy observation arises from the crosspoint between frequency position and membership function.
900
700 cm-l
500
300
Figurel: Fuzzy observations and eigenfrequencies for the SiO4- and Al04tetrahedra (BLSFF: a = 1.87 mdyn cm-l A-l, b = 0.94 A, h = 0.41 mdyn A)
237
We see in Figure 1 the difficulty of BLSFF- application in the case of simple tetrahedra: Especially the Al04- stretching and SiO4- and Al04- bending vibrations are reproduced only unsatisfactory. Fuzzy optimal force fields remove these drawbacks of BLSFF shown in Figure 2. The calculated eigenfrequencies of the SiO4- and Al04- fuzzy-optimal force field are illustrated in Figure 2 in connection with the experimental fuzzy observations. A SiO4fuzzy-optimal force field arises from modification of the parameters a, b (Badger rule [8]) and h (Saksena rule [9]). In the case of Al04- tetrahedron, additionally, we have to consider interaction force constans by means of intercorrelation coefficients Ik, Ig and Ih
WI. hi]p degree
I 0 cm-l Figure 2. Fuzzy observations and eigenfrequencies for the SiO4- and Al04tetrahedra on the basis of a SiO4- and Al04- fuzzy-optimal force field (Si04:a= 1.86 mdyn cm-l A- t, b=0.96 A, h=0.51 mdyn A Al04:a=1.86 mdyn cm-l A-I, b=0.96 A, h=OSl mdyn A, Ik=-O.l,Ig=0.25,Ih=O.O)
3.2. Ditetrahedron with 02- bridging atom [5] The transfer of the results to a ditetrahedron leads to the results given in Figures 3 and 4. The triangle functions reflect the available fuzzy information of the experimental frequencies for ditetrahedra. The fuzzy observations from the isolated tetrahedra cannot be used, because in the case of ditetrahedral structures there are not pure SiO4- or Al04tetrahedral vibrations and additional (bridge- and torsion) vibrations appear. The vertical lines illustrate the calculated eigenfrequencies of the BLSFF (Figure 3) or of the SiO4- and A104 fuzzy-optimal force field (Figure 4). We discuss the results as an example on the 02ditetrahedron. As an surprising result we have to mention that the origin BLSFF is an acceptable solution in the fuzzy sense, too. Comparing with the SiO4- and Al04- fuzzy-optimal force field, we cannot decide which force field is a better one in the case of the 02ditetrahedron. The overlapping of various frequency intervals on the one side and the impossibility of a division of the cluster normal vibrations in symmetric and asymmetric stretching and bending vibrations (eigenvectors and PED) on the other lead to an ambigious assignment of calcuiated eigenfrequencies to corresponding fuzzy observations. A possible way to circumvent this difficulty seems to be the construction of a suitable fuzzy operator reflecting the existing knowledge about ditetrahedral vibrations.
238
1100
900
700 500 300 cm-l Figure 3: Fuzzy observations and eigenfrequencies for the 02- ditetrahedron (BLSFF: a = 1.87 mdyn cm-l A- l, b = 0.94 A, h = 0.41 mdyn A)
1100
900
1 0.8 0.6 0.4 0.2 0
700 500 300 cm-l Figure 4. Fuzzy observations and eigenfrequencies for the 02- ditetrahedron (SiO4- and A104- fuzzy optimal force field) (Si04:a=1.86 mdyn cm-l A-l, b=0.96 A, h=OSl mdyn A A104:a=1.86 mdyn cm-l A- l, b=0.96 A, h=OSl mdyn A, Ik=-O.l,Ig=0.25,Ih=O.O) We would like to thank the Fond der Chemischen work.
Industrie for financial support of this
REFERENCES 1. 2. 3. 4. 5. 6. 7.
E.M. Flanigan, H. Khatami and H.A. Szymanski, Adv. Chem. Ser., 101 (1971) 201. C.S. Blackwell, J. Phys. Chem., 83 (1979) 3251, 3257 L.A. Zadeh, Information and Control, 8 (1965) 338. E. Geidel, H. Bohlig and P. Birner, Z. phys. Chemie, 171 (1991) 121. D.H. Olson, J. Phys. Chem., 74 (1970) 2758. V.A. Maroni, Spectrochim. Acta, 40A (1984) 379. E.B. Wilson jr., J.C. Decius and P.C. Cross, Molecular Vibrations, McGraw-Hill Book Company, Inc. New York 1955. 8. R.M. Badger, J. Chem. Phys., 2 (1934) 128; 3 (1935) 710. 9. B.D. Saksena, J. Chem. Sot. Faraday Trans., 57 (1961) 242.
MOLECULAR STRUCTURE Journal
of Molecular
Structure
349 (1995)
235-238
Discussion of fuzzy evaluated force fields for NaX zeolites M. Kudraa, E. Geidelh, and H. Bijhliga
aInstitute of Physical and Theoretical Chemistry, University of Leipzig, Linnestr.3, D-04 103 Leipzig bInstitute of Physical Chemistry, University of Hamburg, Bundesstr. 45, D-20146 Hamburg Dedicated to Prof. Dr. G. Geiseler on occasion of his 80th birthday
1. INTRODUCTION To the spectroscopical characterization of zeolites, usually, the regions of frameworkand OH- stretching vibrations are considered. The corresponding infrared and RAMAN spectra are characterized by only few but broad bands. The fundamental papers of Flanigan et al. [l] lead to an assignment of absorption bands and RAMAN lines to vibration modes based on intra-tetrahedral and inter-tetrahedral vibrations. If additionally the validity of Blackwell’s decoupling postulate [2] is assumed, the vibrational behaviour of zeolites can be discussed on the basis of cluster models, e.g. with tetrahedral or ditetrahedral structure. Normal coordinate analyses are used to interprete the spectral frequency regions expecting less or more decoupled vibrations of tetrahedra. The frequency regions are overlapped and have no sharp defined boundaries. In this sense the spectral information are uncertainty ones. The uncertainty can be handled by using fuzzy sets introduced by Zadeh [3].
2. FUZZY EVALUATED FORCE FIELDS The broad and strongly structered bands in the vibrational spectra of NaX zeolites [4] cover broad frequency intervals. In these frequency regions one has to expect the tetrahedral vibrations and additional vibrations caused by coupling with neighbouring tetrahedra. We assume that (SiO4- and Al04-) tetrahedral vibrations of the cluster model are located in an experimentally given frequency interval but their positions do not necessarily correspond to the band maxima. The uncertainty of the frequency intervals can be modeled by special trapezoid functions. These functions arise by evaluation of frequencies in the middle of the intervals with a higher degree (e.g. 1) as frequencies at the boundaries evaluated e.g. with 0 and connecting the points by a straight line. (cp. Figures 1 and 2). 0022-2860/95/$09.50 0 1995 Elsevier SSDI 0022-2860(95)08752-4
Science
l3.V
All rights
reserved
236
In the sense of fuzzy theory [3], these functions are membership functions of so-called fuzzy sets. On this way, any frequency belongs with a special degree - the so-called membership degree (a value between 0 and 1) - to the fuzzy set. In opposition to classical (sharp) sets, for example intervals, we cannot decide if any frequency belongs to the fuzzy set or not. We will call the fuzzy sets fuzzy observations of the experimental frequencies. Now, the membership degree can be used to evaluate sharp calculated eigenfrequencies with the fuzzy observations of the experimental frequencies. The higher the membership degree the better the eigenfrequencies approximate the fuzzy observations. To the normal coordinate analysis Olson’s [5] structural parameters of NaX and the bond lengths scaled force field (BLSFF) of Blackwell [2] are used. Taking into account a systematic variation of the parameters a, b and h of BLSFF, various force fields for the primary building units (SiO4- and Al04- tetrahedra) of NaX zeolites are derived. Additionally we consider interaction force constants introduced by Maroni [6]. With the arising force fields we carry out normal coordinate analyses based on Wilson’s [7] GFmatrix method. We evaluate a given force field by the least membership degree of the calculated eigenfrequencies to the corresponding fuzzy observations. To the discussion of the experimental vibrational spectra we choose such force fields with maximum fuzzy evaluation, so- called fuzzy-optimal ones.
3. RESULTS
AND DISCUSSION
3.1. Tetrahedra Figure 1 shows the modeled fuzzy observations of the experimental frequencies (trapezoids and triangels on the frequency intervals) containing the experimental information from the infrared and RAMAN spectra. To ensure the reliabiltity between experimental data and model vibrations, we have chosen different shape and height of the membership functions. Additionally, the calculated eigenfrequencies of BLSFF for the tetrahedra are drawn as vertical lines. The membership degree of a calculated frequency to a fuzzy observation arises from the crosspoint between frequency position and membership function.
900
700 cm-l
500
300
Figurel: Fuzzy observations and eigenfrequencies for the SiO4- and Al04tetrahedra (BLSFF: a = 1.87 mdyn cm-l A-l, b = 0.94 A, h = 0.41 mdyn A)
237
We see in Figure 1 the difficulty of BLSFF- application in the case of simple tetrahedra: Especially the Al04- stretching and SiO4- and Al04- bending vibrations are reproduced only unsatisfactory. Fuzzy optimal force fields remove these drawbacks of BLSFF shown in Figure 2. The calculated eigenfrequencies of the SiO4- and Al04- fuzzy-optimal force field are illustrated in Figure 2 in connection with the experimental fuzzy observations. A SiO4fuzzy-optimal force field arises from modification of the parameters a, b (Badger rule [8]) and h (Saksena rule [9]). In the case of Al04- tetrahedron, additionally, we have to consider interaction force constans by means of intercorrelation coefficients Ik, Ig and Ih
WI. hi]p degree
I 0 cm-l Figure 2. Fuzzy observations and eigenfrequencies for the SiO4- and Al04tetrahedra on the basis of a SiO4- and Al04- fuzzy-optimal force field (Si04:a= 1.86 mdyn cm-l A- t, b=0.96 A, h=0.51 mdyn A Al04:a=1.86 mdyn cm-l A-I, b=0.96 A, h=OSl mdyn A, Ik=-O.l,Ig=0.25,Ih=O.O)
3.2. Ditetrahedron with 02- bridging atom [5] The transfer of the results to a ditetrahedron leads to the results given in Figures 3 and 4. The triangle functions reflect the available fuzzy information of the experimental frequencies for ditetrahedra. The fuzzy observations from the isolated tetrahedra cannot be used, because in the case of ditetrahedral structures there are not pure SiO4- or Al04tetrahedral vibrations and additional (bridge- and torsion) vibrations appear. The vertical lines illustrate the calculated eigenfrequencies of the BLSFF (Figure 3) or of the SiO4- and A104 fuzzy-optimal force field (Figure 4). We discuss the results as an example on the 02ditetrahedron. As an surprising result we have to mention that the origin BLSFF is an acceptable solution in the fuzzy sense, too. Comparing with the SiO4- and Al04- fuzzy-optimal force field, we cannot decide which force field is a better one in the case of the 02ditetrahedron. The overlapping of various frequency intervals on the one side and the impossibility of a division of the cluster normal vibrations in symmetric and asymmetric stretching and bending vibrations (eigenvectors and PED) on the other lead to an ambigious assignment of calcuiated eigenfrequencies to corresponding fuzzy observations. A possible way to circumvent this difficulty seems to be the construction of a suitable fuzzy operator reflecting the existing knowledge about ditetrahedral vibrations.
238
1100
900
700 500 300 cm-l Figure 3: Fuzzy observations and eigenfrequencies for the 02- ditetrahedron (BLSFF: a = 1.87 mdyn cm-l A- l, b = 0.94 A, h = 0.41 mdyn A)
1100
900
1 0.8 0.6 0.4 0.2 0
700 500 300 cm-l Figure 4. Fuzzy observations and eigenfrequencies for the 02- ditetrahedron (SiO4- and A104- fuzzy optimal force field) (Si04:a=1.86 mdyn cm-l A-l, b=0.96 A, h=OSl mdyn A A104:a=1.86 mdyn cm-l A- l, b=0.96 A, h=OSl mdyn A, Ik=-O.l,Ig=0.25,Ih=O.O) We would like to thank the Fond der Chemischen work.
Industrie for financial support of this
REFERENCES 1. 2. 3. 4. 5. 6. 7.
E.M. Flanigan, H. Khatami and H.A. Szymanski, Adv. Chem. Ser., 101 (1971) 201. C.S. Blackwell, J. Phys. Chem., 83 (1979) 3251, 3257 L.A. Zadeh, Information and Control, 8 (1965) 338. E. Geidel, H. Bohlig and P. Birner, Z. phys. Chemie, 171 (1991) 121. D.H. Olson, J. Phys. Chem., 74 (1970) 2758. V.A. Maroni, Spectrochim. Acta, 40A (1984) 379. E.B. Wilson jr., J.C. Decius and P.C. Cross, Molecular Vibrations, McGraw-Hill Book Company, Inc. New York 1955. 8. R.M. Badger, J. Chem. Phys., 2 (1934) 128; 3 (1935) 710. 9. B.D. Saksena, J. Chem. Sot. Faraday Trans., 57 (1961) 242.