Modeling of infrared spectra of acidic groups in zeolites
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(2) Modeling of Infrared Spectra of Acidic Groups in Zeolites. PROEFSCHRIFT. TER VERKRIJGING VAN DE GRAAD VAN DOCTOR AAN DE. T ECHNISCHE U NIVERSITEIT E INDHOVEN , R ECTOR M AGNIFICUS , PROF. DR . R.A. VAN S ANTEN ,. OP GEZAG VAN DE. VOOR EEN COMMISSIE AANGEWEZEN DOOR HET. C OLLEGE VOOR P ROMOTIES. IN HET OPENBAAR TE VERDEDIGEN OP MAANDAG. 17 MAART 2003 OM 16.00 UUR. DOOR. V ELITCHKA V ELIKOVA M IHALEVA GEBOREN TE. K AVARNA , B ULGARIJE.
(3) Dit proefschrift is goedgekeurd door de promotoren: prof.dr. R.A. van Santen en prof.dr.ir. A. van der Avoird Copromotor: dr. A.P.J. Jansen. CIP-DATA LIBRARY TECHNISCHE UNIVERSITEIT EINDHOVEN Mihaleva, Velitchka V. Modeling of Infrared Spectra of Acidic Groups in Zeolites/ by Velitchka V. Mihaleva - Eindhoven: Technische Universiteit Eindhoven, 2003. Proefschrift. - ISBN 90-386-2884-4 NUR 913 Trefwoorden: heterogene katalyse; zeolieten / adsorptie / kwantumchemie / dichtheidsfuctionaaltheorie; DFT / infraroodspectra; Fermi resonantie / moleculaire vibraties / acetonitril / methanol / water Subject headings: heterogeneous catalysis; zeolites / adsorption / quantum chemistry / density functional theory; DFT / infared spectra; Fermi resonance / molecular vibration / acetonitrile / methanol / water. Printed at Universiteitsdrukkerij, Eindhoven University of Technology. This research has been financially supported by the Council for Chemical Sciences of the Netherlands Organization for Scientific Research (CW-NWO). The work described in this thesis has been carried out at the Schuit Institute of Catalysis (part of NIOK, the Netherlands School for Catalysis Research), Eindhoven University of Technology, The Netherlands..
(4) In memory of my mother.
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(6) Contents 1 General Introduction 1.1 Chemical Reactions and Catalysts . . . . . . . . . 1.2 Zeolites as catalysts . . . . . . . . . . . . . . . . . 1.3 Methanol as a raw material . . . . . . . . . . . . . 1.4 Objectives . . . . . . . . . . . . . . . . . . . . . . 1.5 Quantum chemical calculations . . . . . . . . . . 1.5.1 Harmonic oscillator . . . . . . . . . . . . . 1.5.2 Harmonic vs. anharmonic approximation 1.5.3 Hydrogen bonded systems . . . . . . . . . 1.5.4 Fermi resonance . . . . . . . . . . . . . . 1.5.5 The calculations, step by step . . . . . . . 1.6 Contents of this Thesis . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . 2 The Theory Behind the Calculations 2.1 Electronic structure calculations . . . . . . . . . 2.1.1 Choice of the DFT functional . . . . . . . 2.1.2 Models for the zeolite active site . . . . . 2.2 Vibrational Hamiltonian in internal coordinates 2.2.1 The kinetic energy . . . . . . . . . . . . . 2.2.2 The potential energy and dipole surfaces 2.2.3 Second quantization formalism . . . . . . 2.3 The vibrational Hamiltonian matrix elements . . 2.3.1 Gauss-Hermite quadrature . . . . . . . . 2.3.2 DVR and the Hamilton matrix . . . . . . . 2.4 Infrared absorption intensities . . . . . . . . . . . 2.4.1 Interaction of radiation with matter . . . 2.4.2 Integrated absorption intensities . . . . . 2.5 Isotopic effect . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . v. . . . . . . . . . . . .. . . . . . . . . . . . . . . .. . . . . . . . . . . . .. . . . . . . . . . . . . . . .. . . . . . . . . . . . .. . . . . . . . . . . . . . . .. . . . . . . . . . . . .. . . . . . . . . . . . . . . .. . . . . . . . . . . . .. . . . . . . . . . . . . . . .. . . . . . . . . . . . .. . . . . . . . . . . . . . . .. . . . . . . . . . . . .. . . . . . . . . . . . . . . .. . . . . . . . . . . . .. . . . . . . . . . . . . . . .. . . . . . . . . . . . .. . . . . . . . . . . . . . . .. . . . . . . . . . . . .. . . . . . . . . . . . . . . .. . . . . . . . . . . . .. . . . . . . . . . . . . . . .. . . . . . . . . . . . .. 1 2 2 4 4 6 7 8 9 10 12 13 13. . . . . . . . . . . . . . . .. 15 16 18 18 19 19 20 23 24 27 28 32 32 34 36 36.
(7) vi. CONTENTS. 3 Methanol Adsorbed on Chabazite 3.1 Introduction . . . . . . . . . . . . . . . 3.2 Computational details . . . . . . . . . 3.3 Results . . . . . . . . . . . . . . . . . . 3.3.1 Zeolite rings without methanol 3.3.2 Zeolite rings with methanol . . 3.3.3 Frequencies . . . . . . . . . . . 3.4 Discussion . . . . . . . . . . . . . . . . 3.4.1 Brønsted sites . . . . . . . . . . 3.4.2 Adsorbed methanol . . . . . . . 3.5 Conclusions . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . .. . . . . . . . . . . .. . . . . . . . . . . .. . . . . . . . . . . .. . . . . . . . . . . .. . . . . . . . . . . .. . . . . . . . . . . .. . . . . . . . . . . .. . . . . . . . . . . .. . . . . . . . . . . .. . . . . . . . . . . .. . . . . . . . . . . .. 39 40 41 42 42 44 47 48 48 50 53 54. 4 The heterogeneity of the hydroxyl groups in chabazite 4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . 4.2 Computational details . . . . . . . . . . . . . . . . 4.3 Results and Discussion . . . . . . . . . . . . . . . . 4.3.1 Zeolite cluster without adsorbate . . . . . . 4.3.2 Zeolite rings with adsorbates . . . . . . . . 4.4 Conclusions . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . .. . . . . . . .. . . . . . . .. . . . . . . .. . . . . . . .. . . . . . . .. . . . . . . .. . . . . . . .. . . . . . . .. . . . . . . .. . . . . . . .. 57 58 59 60 60 61 67 68. 5 Infrared Spectra of Water Adsorbed on Chabazite 5.1 Introduction . . . . . . . . . . . . . . . . . . . 5.2 Computational details . . . . . . . . . . . . . 5.2.1 Electronic structure calculations . . . 5.2.2 Dynamics . . . . . . . . . . . . . . . . 5.3 Results and Discussion . . . . . . . . . . . . . 5.3.1 Zeolite cluster without water . . . . . 5.3.2 Zeolite cluster with water . . . . . . . 5.3.3 Isotope effect . . . . . . . . . . . . . . 5.4 Conclusions . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . .. . . . . . . . . . .. . . . . . . . . . .. . . . . . . . . . .. . . . . . . . . . .. . . . . . . . . . .. . . . . . . . . . .. . . . . . . . . . .. . . . . . . . . . .. . . . . . . . . . .. . . . . . . . . . .. 71 72 73 73 73 78 78 79 85 87 87. . . . . . . .. 91 92 93 93 97 102 104 105. . . . . . . . . . . .. . . . . . . . . . . .. . . . . . . . . . . .. . . . . . . . . . . .. . . . . . . . . . . .. . . . . . . . . . .. . . . . . . . . . . .. . . . . . . . . . .. . . . . . . . . . .. 6 Infrared Spectra of Acetonitrile and Methanol Adsorbed on Chabazite 6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.2 Results and Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . 6.2.1 Zeolite cluster with acetonitrile . . . . . . . . . . . . . . . . . 6.2.2 Zeolite clusters with methanol . . . . . . . . . . . . . . . . . 6.3 Isotope effect . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.4 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ..
(8) CONTENTS. vii. Summary. 107. Samenvatting. 109. List of publications. 111. Acknowledgements. 113. Curriculum vitae. 115.
(9) viii. CONTENTS.
(10) Chapter 1. General Introduction The purpose of this chapter is to present the topic of this thesis, which is the infrared spectra of acidic zeolites. The common approach in quantum chemical calculations of infrared spectra is a normal mode analysis. In this method the energy is assumed to depend quadratically on the atomic displacement. It will be shown, that a normal mode analysis cannot be used to get a quantitative agreement between the theoretically calculated and experimentally observed frequencies of the zeolite hydroxyl group when basic molecules like acetonitrile, water and methanol are adsorbed. To do so, it is important to take into account the mechanical and electrical anharmonicities. In this way the so-called Fermi resonance can also be described. Also, at the end of the Chapter, a short overview of the content of the thesis is given..
(11) 2. 1.1. General Introduction. Chemical Reactions and Catalysts. The needs of the modern society are constantly increasing and at present time manufacturers are pressed to produce more and more, and as cheap as possible. This often involves a lot of chemistry. However, at normal conditions many reactions have a low rate. One way to speed them up is to heat the reactants. This means extra energy expenses. Another way is to use chemical compounds which can make the reaction possible at milder conditions without appearing in the products. Such chemical compounds are called catalysts. Ostwald defined a catalyst as “a substance one adds to a chemical reaction to speed up the reaction without the catalysts undergoing a chemical change itself”. Catalysts do undergo chemical changes during the course of the reaction. It is just that the changes are reversible, so that the catalysts are not consumed as the reaction proceeds. Catalysts may make tremendous difference in reaction rate. For example, a hydrogen/oxygen mixture may be stable for years at C. However, if one inserts a platinum wire into the mixture, the mixture explodes. Sometimes, chemical substances can react in different ways and yield a range of products. Products that have no application are called waste. To reduce the amount of these unwanted products we can increase only the rate of the reaction for the desired products. In this case the catalyst are also selective. There are two kinds of catalysts: homogeneous and heterogeneous. A homogeneous catalyst is in the same phase as the reactants and the products. Examples of homogeneous catalysts are the enzymes which catalyze chemical reactions taking place in the bodies of living organisms. The advantage of the homogeneous catalysts is that they are very selective. The disadvantage is that it is difficult to separate them from the products. A heterogeneous catalyst is not in the same phase as the reactants and the products. Usually, the catalysts are solids and the reactants and the products are gases or liquids. Heterogeneous catalysts are widely used in industrial processes because they can be easily separated from the products. Over of all bulk chemicals and petroleum products are made via heterogeneous catalytic processes.. 1.2. Zeolites as catalysts. Zeolites are crystalline porous materials which have been used in many industrial processes as catalysts. Their crystal structure is build up from corner-sharing SiO tetrahedra. The central atom of such tetrahedra is called a T atom. There are many different ways for arranging these tetrahedra with respect to each other so that various structures are formed. So far, 136 different structures have been reported [1], of which about 1/3 are natural and the rest are synthetic. The different framework topologies contain cages and channels which can accommodate atoms, ions, and molecules. The pores have uniform size which is typically of molecular dimen-.
(12) 1.2 Zeolites as catalysts. 3. sions. For example, in Fig. 1.1 the 8T ring in chabazite and the 10T ring in ZSM-5 ˚ respectively. have a diameter of and A, Zeolites have a number of applications. These include simple application as molecular sieves, only allowing molecules of certain shape and size to pass through the pores. Their ability to hold ions means that they are often used for ion exchange. Thus, one of the most common situation in which zeolites are used is as water softeners in washing powder. Zeolites can not only hold different ions, but they can also exchange a Si atom with, for example, an Al atom. The charge originating from replacing the four valent Si atom with a three valent Al atom can be compensated by counter cation. If protons are used, then hydroxyl groups are formed as a bridge between one Si tetrahedron and one Al tetrahedron. These bridging hydroxyl groups are also called Brønsted or acid sites.. (a). (b). Figure 1.1: The structure of chabazite (a) and ZSM-5 (b). In the chabazite structure the largest pore is an 8T ring. In ZSM-5, a three dimensional channel structure of 10T rings is formed.. The acid form of the zeolites catalyzes reaction in which are involved. For example, dimethylether can be obtained from methanol in the presence of sulfuric acid. For the same reaction H-ZSM-5 can be used. The advantage of the solid acids is that the concentration of the protons can be easily controlled as the number of the OH groups is determined by the amount of aluminum atoms in the crystal structure. Zeolites are also extensively used in oil refining [2–5]. The main constituents of crude oil are paraffins which are unfortunately very unreactive. Different reactions like cracking, dehydrogenation, hydroisomerization, and aromatization are used to obtain hydrocarbons and hight-octane gasoline..
(13) 4. General Introduction. 1.3. Methanol as a raw material. Pure methanol is an important material in chemical synthesis. Its derivatives are used in great quantities for building up a large number of compounds, including synthetic fuel, synthetic dyes, resins, drugs, and perfumes. Large quantities are converted to dimethylaniline for dye stuffs and to formaldehyde for synthetic resins. It is also used in automotive antifreezes, in rocket fuels, and as a general solvent. Methanol is also a high-octane, clean-burning fuel. Unfortunately, it is toxic (taken internally methanol causes blindness) and forms explosive mixtures with air. However, methanol can be converted over zeolite catalyst to high-octane synthetic gasoline. This process is known as the methanol-to-gasoline (MTG) process. The process can be schematically represented as:. . . .
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(25) +-,#)#"$ ' . Methanol is made from synthesis gas (a mixture of carbon monoxide and hydrogen) which in turn is formed from steam reforming natural gas or gasification of coal. The methanol is then converted to an equilibrium mixture of methanol, dimethyl ether and water, which can be processed catalytically to either gasoline or olefins, depending on the catalyst and/or the process operation conditions. The usage of acidic zeolites as catalysts is strongly structure dependent. Zeolites with a /. 100 medium size of the pores, such as ZSM-5, yield generally hydrocarbons. hydrocarbons. The Small-pore zeolites, such as SAPO, usually generate SAPO zeolite has the same topology as chabazite, which is shown in Fig. 1.1, but has a different chemical composition. Its crystal structure is build of Al and P tetrahedra and the acid sites are introduces by replacing an Al or a P atom by a Si atom. The gasoline produced in this way does not need further refining.. 1.4. Objectives. Changes in the catalyst can influence the product distribution. To modify a catalyst in a particular way a knowledge of the mechanism of the reaction is needed. Information about the elementary steps of a chemical reaction can be obtained from different experiments. What experimental techniques will be used to monitor a reaction depends on the reaction itself. Infrared spectroscopy is a powerful method for studying chemical reactions because the position of the frequencies depends strongly on the changes in the geometry. By comparing the infrared spectra of the reactants and the products we can get an information about the transformation that takes place. Once we have assigned the observed frequencies to vibrations associated with different groups, we can also propose a mechanism for the reaction. In this thesis the first step of the conversion of methanol to high-octane gaso-.
(26) 1.4 Objectives. 5. line catalyzed by acidic zeolites has been studied. Here a formation of hydrocarbons will not be shown as we will concentrate on the very first step of the reaction, namely adsorption of methanol on zeolites at low coverage. In the reaction two types of hydroxyl groups are involved, one from the catalyst and the other from the methanol molecule. The zeolite hydroxyl group has acidic character and the methanol hydroxyl group basic character. The interaction will affect the geometry of these groups. Thus, the reaction can be studied by looking at the changes in the infrared spectra associated with these hydroxyl groups. Let us define the relevant vibrations. A stretch is a change of the length of a bond. Two stretch coordinates have been used. The oxygen and hydrogen atoms have been moved along the zeolite and the adsorbates OH bond, and the adsorbate molecules have been moved along the hydrogen bond while keeping the adsorbate geometry fixed. A bending is a change of the angle between two chemical bonds having an atom in common, like in H–O–H. An in-plane bending of the zeolite OH group is a change of the angle between the O–H and O–Al bonds while the oxygen and the hydrogen atom are moving in the Si–O–Al plane. An out-of-plane bending of the zeolite OH group is a change of the angle between the O–H bond and the normal to the Si–O–Al plane. The frequencies are expected to be shifted compared to the frequency of the pure reactants. Unfortunately, the picture is not so simple because instead of a frequency shift, an appearance of three new bands, 0 denoted as an ABC triplet, at , , and cm0 is observed [7–11] . The triad is accompanied by a band at around cm . The changes in the spectrum of adsorbed methanol are puzzling because of the similarities with the spectrum of adsorbed water (compare the two spectra plotted in Fig. 1.2). Initially [8, 9, 12], the triad was attributed to asymmetric (A) and symmetric (B) OH stretch and H– O–H bending (C) 0 of protonated methanol. However, with this model the band at cm can not be explained as all hydroxyl groups in the system of adaround sorbed methanol were used in the assignment of the ABC triplet. Later, [13] the ABC triplet has been interpreted as a Fermi resonance of the zeolite OH stretch with the overtones0 of the in-plane and out-of-plane bending. The remaining band around cm can originate from the stretch vibration of the methanol hydroxyl group. The aim of the theoretical studies presented in this thesis is to propose a model for adsorption of methanol on zeolites at low coverage and to verify this model by comparing the computed infrared spectrum with the experiment. Two additional reactions (adsorption of water and acetonitrile) have been studied in order to look for a trend in the change of the infrared spectra when molecules with different basicity are adsorbed..
(27) 6. General Introduction. 2463. 3704 2877. 1358 1628. 3552. water 1430. 1680 3555. 2360 2760. methanol 1500. 2000. Figure 1.2: The difference spectra of coverage.. 1.5. 2500. . [6] and. 3000. . 3500. cm−1. [7] adsorbed on ZSM-5 at low. Quantum chemical calculations. As we have seen, sometimes the changes in the infrared spectra can be interpret in several ways. Here is where quantum chemistry can help. Assuming a particular structure we can calculate the infrared spectrum and compare it with the experiment. To do a quantitative comparison between calculated and observed frequencies we need to incorporate in the theoretical model different phenomena that may affect the spectra. Atoms in molecules even at absolute zero temperature vibrate with respect to each other. Because in molecules atoms form a bound system, the vibrational energy of the molecule is quantized. Molecules can absorb electromagnetic radiation that has energy which corresponds to a difference between the molecular energy levels. In this way a particular transition is excited. The separation of the vibrational energy levels corresponds to the energy of infrared light. To be able to predict what energy will be adsorbed we need to know the separation between the energy.
(28) 1.5 Quantum chemical calculations. 7. levels in the system of interest. We can find the vibrational energy levels if we know how the energy changes when the atoms are moving, i.e if we know the so-called potential energy surface.. 1.5.1 Harmonic oscillator The simplest approximation of the potential energy surface is a harmonic approximation, i.e. the energy depends quadratically on the displacements of the atoms from their equilibrium positions. In this approximation the solution of the vibrational problem is straightforward. The energy levels are then given by the formulae. . (1.1).
(29) where is the frequency of vibration, is the quantum number and is the Planck’s constant divided by . A one-dimensional example is depicted on the left side in . Fig. 1.3. Each vibrational level is characterized by an energy and a wave function. The square of the wave function corresponds to the probability density of finding a particular value of the coordinate when a measurement is done.. Figure 1.3: Harmonic (left) and anharmonic (right) potential energy surface as a function of atomic displacement. Each vibrational level is characterized by an energy (the horizontal lines) and a wave function ( ). Above the energy lines is plotted.. The ground state is the vibrational level with . . . The levels with are called excited states. Once we know the vibrational levels we can calculate what will be the frequency if a transition between two levels takes place. The frequency is given by the Bohr rule (1.2). where. . is the energy of the upper level and the energy of the lower level. Among these, by far the most important in the absorption spectra are the transitions between the ground and the first excited levels. These transitions are known.
(30) 8. General Introduction. as fundamental transitions and the corresponding frequencies as fundamental frequencies. Transition from the ground state to the second or higher excited state takes the name overtone. In a multidimensional case, if during a transition two (or more) quantum numbers are changed, the corresponding transitions are called combination bands. As can be seen from Eq. (1.1), the energy levels in harmonic approximation are equidistant. Thus, the transitions from the ground to the first has exactly the same energy as the transition from the excited excited level . Transitions between two excited levels are to the level level called hot bands since the lower excited state level must be thermally populated for a sufficient number of molecules to make the transition. Excited levels can be involved in a transition from a state excited in one mode to a state excited in another mode. The frequency then is the difference in the frequencies of the transitions from the ground state to these states and therefore these transitions are called difference bands. The intensity of a spectral line is determined by the probability of the transition which gives rise to the line. To get an estimation of it we need to know how the dipole changes when the transition occurs. For the harmonic oscillator the following selection rule holds. The only vibrational transitions that can occur are those, in which only one quantum number changes and that number changes by one unit only. In other words, on the basis of the harmonic oscillator, only fundamental frequencies should appear in the spectrum. Experimentally, it is found that the fundamental transitions are the most intense, but other transitions such as overtones and combinations also appear, sometimes quite strongly. It is evident that some of the assumptions we have made must be invalid. . . 1.5.2 Harmonic vs. anharmonic approximation Let us go back to Fig. 1.3 and assume that the plotted potential energy surface corresponds to a change of the energy when a distance between two atoms is varied. In the harmonic case, it does not matter whether we push the atoms together or pull them apart as the energy change is the same. We know that if the distance between atoms is decreased, the energy changes stronger compared to the energy change when the bond is elongated. To be able to describe this we need to add the cubic, quartic, etc., terms in the expression for the potential energy. In this way we take into account the mechanical anharmonicity of the vibration. Compared to the energy levels computed in harmonic approximation, the energy levels obtained with anharmonic potential energy are shifted and are not anymore equally spaced. The harmonic frequencies of the fundamental transition of a stretch vibration are always higher than the anharmonic frequencies. How large the difference will be, depends on the strength of the bond. The weaker the bond the large the anharmonicity is. The energy levels of the overtones are closed up. Also the wave function is not anymore symmetric with respect to the equilibrium position. At high excited levels the probability for measuring a longer bond distance is greater.
(31) 1.5 Quantum chemical calculations. 10000. 8000. bar e ze olit e et on itr ile. 4000. methanol 6000 Energy, cm−1. 6000. ac. Energy, cm−1. 8000. ter wa nol tha me. 2000 0 −0.2. 9. water. 4000. acetonitrile. 2000 bare zeolite. 0 0.2 displacement, Å. 0.4. 0.6. 0. −0.4. −0.2. 0 sin (ϕ). 0.2. 0.4. Figure 1.4: Changes of the potential energy along the zeolite OH stretch (left) and the inplane bending (right) coordinates in the presence of basic molecules. Note also the asymmetry of the potential energy in presence of methanol. In order to include the first overtone ˚ of the zeolite OH stretch, the zeolite OH bond had to be elongated by A.. than that to measure a short distance. Very often the effect of the anharmonicity is associated with lowering of the frequency of the fundamental and the overtones. This is true for stretch vibrations where the anharmonicities can be up to several hundreds of wave numbers. However, we can also observe the opposite effect, i.e. spreading out of the energy levels. For a low frequency out-of-plane bending vibration the potential energy is shallow around the equilibrium position and steep at larger amplitudes. This leads to increasing the distance between the overtones. The effect of the anharmonicity is not so pronounced as in the case of stretch vibrations because the increasing of the distance between the levels is in the order of several tens of wave numbers.. 1.5.3 Hydrogen bonded systems The interaction between a zeolite hydroxyl group and small basic molecules is via a hydrogen bond. zeolite–O–H B Consequently, the zeolite OH bond is weakened. The potential energy becomes flatter and the corresponding frequency is lowered. It is also said that the frequency undergoes a red shift. As a measure for the acidity and the basicity the shift of the OH stretch frequency will be used. When different acidic groups interact with the same base, the one that undergoes larger shift is said to be the most acidic. The same holds for the basicity. What will be the strength of the hydrogen bond between the Brønsted sites and the adsorbates can be predicted from the proton affinity (PA) of the adsorbates. The proton affinity is defined as the difference of the energy of a protonated molecule minus the energy of the neutral molecule. We have calculated the PA of acetoni trile of kJ/mol, methanol of kJ/mol and water of kJ/mol. So, the largest.
(32) 10. General Introduction. shift of the zeolite OH stretch frequency should be expected to be caused by acetonitrile. However, the calculated energies of adsorption show that methanol is the strongest bonded to the zeolite, kJ/mol, than is water, kJ/mol, and the weakest interaction is with acetonitrile, kJ/mol. One possible explanation for the discrepancy between the proton affinity and the adsorption energy could be the secondary hydrogen bond that is formed between the hydroxyl group of methanol or water with the zeolite fragment. The formed complex is then more stable compared to the complex with the acetonitrile where only a single hydrogen bond is formed. As can be seen from Fig. 1.4, the largest change of the potential energy surface for the zeolite OH 0 stretch vibration is caused by methanol. The corresponding are adsorbed, the zeolite OH frequency is cm . When water and acetonitrile 0 stretch frequencies are and cm , respectively. Thus, the basicity of the studied adsorbates is as follows:. (,#'#" , . . ,& #'( $" . In the hydrogen bonded complexes the zeolite OH bending vibrations are in direction perpendicular to the hydrogen bond. The movement of the atoms along the bending modes is more restricted, thus the potential energy surface steepens. The corresponding frequency shifts to higher values, i.e undergoes a blue shift. The shifts of the frequencies of the stretch and the bending are in opposite direction. In Fig. 1.4 an example of the in-plane bending is shown. The main change is between the potential energy of an isolated OH bond and after the hydrogen bond is formed. Among the hydrogen bonded complexes the differences are small. Still the in-plane bending frequency with adsorbed methanol is higher than the corresponding one when acetonitrile is adsorbed.. 1.5.4 Fermi resonance When a hydrogen bond between the zeolite acid group and the adsorbates is formed, the frequency of the zeolite OH in-plane bending shifts upwards. This also holds for the overtones. The energy levels of the fundamental stretch and the first overtone of the in-plane bending come close to each other and they combine into new states. This phenomenon was described for first time by E. Fermi when analyzing the Raman spectra of0 . This molecule has three fundamental frequencies, and cm , which are the bending, the symmetric and antisymmet. , would ric stretch, respectively. The first overtone level of the bending, 0 then lie quite near to the fundamental at about cm . A resonance between these nearby levels is to be expected, pushing them further apart and causing each of the actual levels to be of mixed character, each being part fundamental and part overtone. This mixing shows in the selection rules. Both levels combine with the ground state to give Raman lines of about equal intensities, whereas without resonance only one strong line would be seen. In Fig. 1.5 a schematic presentation of a Fermi resonance between the zeolite OH fundamental stretch and a bending.
(33) 1.5 Quantum chemical calculations. 11. Figure 1.5: Energy diagram (a) of the resonance between the fundamental stretch and the bending overtone levels and the contour diagrams of the corresponding wave functions (b) associated with these energy levels. The stretch coordinate is along the abscissa and the bending coordinate is along the ordinate. The wave functions of the isolated coordinates are a product of the wave function of the first excited state of the stretch with the wave function of the second excited state of the bending.. overtone is given. The energy levels of the fundamental stretch are represented by several closely situated levels because of the coupling with the low frequency mode, which is moving the adsorbates as a rigid molecule along the hydrogen bond. The wave functions of the new levels show the mixed character of the levels. In order to observe the splitting the wave functions of the non-perturbed states should have a compatible symmetry. In the represented example all wave func-.
(34) 12. General Introduction. Figure 1.6: Schematic representation of the calculations. tions are symmetric with respect to a plane that contains the abscissa. If the fundamental of the stretch lies between the overtone of the in-plane bending and the overtone of the out-of-plane bending, then a Fermi resonance with both bendings can be expected. To be able to describe a Fermi resonance, we need to go beyond the harmonic approximation because in harmonic approximation there are no terms in the expression for the potential energy, that account for the coupling between the fundamental stretch and the bending overtones.. 1.5.5 The calculations, step by step Let us summarize what we need to do in order to calculate the infrared spectra including the anharmonicities. We start with electronic structure calculations to find an equilibrium structure for the system we want to study. Then we calculate the energy and the dipole of structures that resemble the equilibrium structure. As a next step, we find an expression for the potential energy and the dipole as a function of the change of the coordinates from the equilibrium structure. This is enough to setup the vibrational Schr¨odinger equation. The solutions of the Schr¨odinger.
(35) 1.6 Contents of this Thesis. 13. equation are the energy levels and the wave functions ( ). How we find these solutions is described in Chapter 2. The energy difference between the levels will give us the frequency spectrum. The wave functions will tell us, if a certain transition takes place, which vibrations are excited. To find out whether we will see these transitions, we need to calculate the probability for them to take place. We can do that by combining the information about the dipole surface with the energy levels and the wave functions. . 1.6. Contents of this Thesis. This Thesis is organized as follows. The objectives for the conducted research have been outlined in this introductory Chapter 1. There also the main trends of the changes in the infrared spectra of the zeolite OH group upon adsorption of molecules of different basicity are shown. To read this chapter a theoretical background is not needed. The theory and the mathematical apparatus of the computations are presented in Chapter 2. Here, the main accent is on the evaluation of the vibrational Hamiltonian matrix elements within the discrete variable representation (DVR) basis set. Adsorption of methanol on zeolites has been theoretically studied for more than a decade. In Chapter 3 a comparison of the chosen model for representing the zeolite active site with previous calculations is done. Also, the results obtained with two different density theory functionals, PW91 and B3LYP, are compared. The frequencies are calculated in harmonic approximation. Sometimes different hydroxyl groups may be presents in the zeolites. How the structure of the zeolite affects the interaction between the zeolite OH group and water or methanol is shown in Chapter 4. We used as a measure for the acidity of the different OH groups in chabazite the shift of the zeolite OH stretch anharmonic frequency, calculated using one dimensional potential energy surface. The calculations were further extended to four dimensions by including for the zeolite OH group the in-plane and the out-of-plane coordinates, and the adsorbates center of mass stretch coordinate. This is necessary to explain the splitting and the width of the zeolite OH stretch band. The spectrum in the presence of water is discussed in Chapter 5, and that of methanol and acetonitrile in Chapter 6. The main results are summarized at the end of the thesis in English and Dutch.. References [1] Baerlocher, C.; Meier, W. M.; Olson, D. H., Atlas of zeolite framework types, 5th edn., Elsevier, Amsterdam, The Netherlands, 2001. [2] Maxwell, I. E.; Stork, W. H. J., Stud. Surf. Sci. Catal. 1991, 58, 571..
(36) 14. REFERENCES. [3] van Santen, R. A.; van Leeuwen, P. W. N. M.; Moulijn, J. A.; Averill, B. A., eds., Catalysis: An Integrated Approach, 2nd edn., Elsevier, Amsterdam, The Netherlands, 1999. [4] St¨ocker, M., Microporous Mesoporous Mater. 1999, 29, 3. [5] Keil, F. J., Microporous Mesoporous Mater. 1999, 29, 49. [6] Lee, B.; Kondo, J. N.; Domen, K.; Wakabayashi, F., J. Mol. Catal. A: Chem. 1999, 137 , 269. [7] Kotrla, J.; Nachtigallova, D.; Kubelkova, L.; Heerbout, L.; Doremieux-Morin, C.; Fraissard, J., J. Phys. Chem. B 1998, 102, 2454. [8] Mirth, G.; Lercher, A., J. Chem. Soc. Faraday Trans. 1990, 86, 3039. [9] Mirth, G.; Lercher, J. A., Stud. Surf. Sci. Catal. 1991, 61, 437. [10] Parker, L. M.; Bibby, D. M., Zeolites 1991, 11, 293. [11] Zecchina, A.; Geobaldo, F.; Spoto, G.; Bordiga, S.; Ricchiardi, G.; Buzzoni, R.; Petrini, G., J. Phys. Chem. 1996, 100, 16584. [12] Marchese, L.; Chen, J.; Wright, P. A.; Thomas, J. M., J. Phys. Chem. 1993, 97 , 8109. [13] Pelmenschikov, A. G.; van Santen, R. A.; J¨anchen, J.; Meijer, E., J. Phys. Chem. 1993, 97 , 11071..
(37) Chapter 2. The Theory Behind the Calculations In this Chapter the employed theoretical and computational techniques used throughout the work on the thesis are discussed. The total energy and the dipole are calculated using the B3LYP density theory functional. The time-independent Schr¨odinger equation for the vibrations including the anharmonic potential is solved for a limited number of internal coordinates. The Schro¨ dinger equation is solved with the variational approach where the multidimensional wave functions are expanded in a products of Hermite discrete variable representation (DVR) basis set. Within the DVR basis set the potential matrix is diagonal and the kinetic matrix is sparse. An expression for the integrated absorption intensities is given and the effect of isotopic substitution on the frequencies and the intensities is discussed..
(38) 16. The Theory Behind the Calculations. 2.1. Electronic structure calculations. In Chapter 1 an idea about the calculation of infrared spectra was given. Here we will go into more technical details concerning the calculations. Scheme 1.6 shows that the quality of the computed spectra depends on the method used for calculating the total energy of the system. To find the total energy we need to solve the Schr¨odinger equation for the atomic nuclei and the electrons, which the molecules are build of. This can be done exactly only for a few very small systems. For larger systems the Born-Oppenheimer approximation is adopted, in which it is supposed that the nuclei, being so much heavier than an electron, move relatively slowly, and may be treated as stationary while the electrons move relative to them. We can therefore think of the nuclei as being fixed at positions , and then solve the Schr¨odinger equation for the wave function of the electrons alone. One of the fundamental approximations in quantum chemistry is the HartreeFock (HF) approximation [1], which is a wave function based method. An alternative approach was proposed by Hohenberg and Kohn [2], in which the aim is not to obtain a good approximation of the ground state wave function, but rather to find the energy of the system as a functional of the electron density . This method is known as the density functional theory (DFT) [3–7]. The concepts of DFT can be outlined by comparing the derivation of the Hartree-Fock equations with that of the Kohn-Sham equations [8]. HF(1928,1930). DFT(1964,1965). .
(39) . "!
(40) # . 0 0 0 20 1 3 . / 6 87 6 . . %$& (') )*,+- 4 +5+ . 0 . 6 97 6 . . . . Hartree-Fock equations:. Kohn-Sham equations:. . <; + >= * ? 0 A@ 0 : 0. . 0.
(41) :. <;. + > = *,+ 0 A@ 0 ?. The starting point in the derivation of the HF theory is the total energy expressed as a function of the total wave function and the positions of the atoms B . The derivation in DFT begins with the total energy written as a functional of the total electron density , for given positions of the atomic nuclei. In contrast to HF theory, DFT uses a physical observable, the electron density, as fundamental quantity. In HF theory, the total energy is expressed as an expectation value of the exact non-relativistic Hamiltonian using a Slater determinant as an approximation for the total wave function. In DFT, the total energy is decomposed in a formally ex.
(42) 2.1 Electronic structure calculations. $&. 17. act way into three terms, a kinetic energy term , an electrostatic or Coulomb (' , and a many-body term (*,+ , which contains all exchange and energy term $& correlation effects. This decomposition is constructed in such a way that corresponds to the kinetic energy of a system of non-interacting electrons that yield the same density as the original electron system. In the Kohn–Sham approach the total density is decomposed 0 into single particle densities which originate from oneparticle wave functions . DFT requires that upon variation of the total electron density, the total energy assumes a minimum. This leads to conditions for the oneparticle wave function in the form of effective one-particle Schro¨ dinger equations or Kohn-Sham (KS) equations. In HF theory, a variational principle applied to the Slater determinant leads to one-particle eigenvalue equations, which are known of the HF as the Hartree-Fock equations. The only formal difference in the form = = * and the KS equations is the term . In HF theory, this term , describes = * + in only the electron exchange effects. In contrast, the corresponding term DFT contains both the electron exchange and correlation effects. Approximations in DFT are introduced in the exchange-correlations operator while, in principle, there are no conceptual approximations made in the wave functions or any other place. One could say that the Hartree-Fock based methods converge to the exact solution of the many-body Schr¨odinger equation through systematic improvements in the form of the many-body wave functions, for example by configuration interaction expansions. DFT theory converges to the exact solution by improv= * + . This is equivalent to improving the ing the exchange-correlation operator "* + , because of the relation description of the total exchange-correlation energy = *,+ 6 (* + 7 6 . "* + have been sugDuring the last two decades numerous approximations for gested. In practice, the exchange and the correlation forms are sought individually and different combinations among them are possible. The simplest approximation (* + is known as local density approximation (LDA). In this approximation , of the electronic properties are determined as functionals of the electron density by applying locally relations appropriate for a homogeneous electron gas. For metallic systems with fairly constant electron density, the LDA comes very close to the exact solution, whereas this approximation can be expected to be less accurate for system with strongly varying electron densities. The next logical step is to use not only the density at a particular point but also the gradient of the electron den sity : , in order to account for the non-homogeneity of the true electron density. These functionals are known as generalized gradient approximation (GGA). Another type of functionals are the hybrid functionals where the exchange is represented as a linear combination between the HF exchange and the DFT exchange. Electronic structure calculations can be used for predicting many properties of molecules and reactions, such as geometries, reaction energies and pathways, atomic charges, dipole moments, polarizabilities and hyperpolarizabilities, infrared and Raman spectra. The efficiency of the DFT approach allows us to treat large molecules ( atoms and more) which occur in practical chemical reactions..
(43) 18. The Theory Behind the Calculations. 2.1.1 Choice of the DFT functional There are many ways to combine different exchange and correlation functionals. We are interested in describing the interaction between base molecules and the acid sites in zeolites. This interaction is via a hydrogen bond. The accuracy for describing hydrogen bonded systems of different gradient corrected and hybrid functionals has been studied and it has been shown [9, 10] that hybrid functionals, especially B3LYP [11], are the best for modeling such systems. Most periodic code are based on GGA functionals, mainly the PW91 functional [12]. We have compared this functional against the hybrid B3LYP functional and we have found that PW91 overestimates the zeolite OH bond distance. The corresponding stretch frequency with adsorbed methanol already in harmonic approximation is in the 0 cm , which are very low frequencies for an OH stretch viregion of bration. Therefore, the frequency calculations were based on electronic structure calculation using the B3LYP hybrid functional and a relatively large cluster consisting of 7T or 8T atoms.. 2.1.2 Models for the zeolite active site Several approaches have been used for theoretical studies of zeolites. The bonding within the zeolite framework is mainly covalent. Therefore, small fragments can be used to describe the active site. This is known as a cluster approach. The dangling bonds at the edges of the cluster are generally saturated by hydrogen atoms. Allowing the cluster to be completely unconstrained overestimates the freedom of relaxation. The structural constrains of the zeolite framework can be imitated by fixing the boundary atoms. The disadvantage of using clusters is that the longrange electrostatic effects are not accounted for. A way to include these long-range electrostatic effects is to embed the cluster in a set of point charges chosen to reproduce closely the electrostatic potential of the real zeolite structure. Another way is to use quantum mechanical - molecular mechanical methods (QM/MM),where the reactants and the zeolite active site are treated at quantum mechanical level, and the rest of the zeolite framework is described by molecular force field. The quality of this method depends on the parameterization of the force field and the way the QM and the MM parts are linked. Recently, adsorption of molecules on zeolites has been be studied in periodic approach, i.e the whole zeolite framework is treated at QM level..
(44) 2.2 Vibrational Hamiltonian in internal coordinates. 2.2. 19. Vibrational Hamiltonian in internal coordinates. 2.2.1 The kinetic energy. . Let us consider a molecule built of atoms. The internal coordinates are a set (for linear molecules ) coordinates which express the changes in of the interatomic distances or in the angles between chemical bonds, or both. These coordinates are unaffected by translation or rotation of the molecule as a whole. The internal coordinates are particularly important because they provide the most physically significant set for describing the potential energy of the molecule. The kinetic energy on the other hand is more easily set up in terms of cartesian coordi nates of the atoms. The Lagrangian of the system is defined as the difference between the kinetic $ ; energy and the potential energy . In cartesian coordinates is . A$. ; . . . .
(45) . 0. . ; . . . (2.1). The momenta conjugate to the cartesian coordinate ferentiation of the Lagrangian.. . . . . . are found by the dif-. (2.2). The classical Hamiltonian in cartesian coordinates is. A$ . ; . . .
(46) . 0 . <;. (2.3). We can express the internal coordinates as a function of the cartesian coordinates . and the reverse . Then . . (2.4). . (2.5). . . 0. . . . . . (2.6). The kinetic energy in internal coordinates becomes. $ .
(47) 0 0 0. . . . . . . . . . . (2.7).
(48) 20. The Theory Behind the Calculations. Let us define matrix. which has elements. The momenta . . . .
(49) 0 . . . . . . (2.8) . conjugate to the internal coordinates , . . . 0. are. . (2.9). The classical Hamiltonian in internal coordinates is then . . 0 0 0 . . . ; . . (2.10). The quantum mechanical Hamiltonian is obtained from the classical Hamiltonian 7 7 by substitution of the momenta with the operators . In cartesian coordinates or coordinates that are a linear combination of cartesian coordinates the substitution is straightforward. In internal coordinates the kinetic energy opera tor has a much more complicated form because the matrix elements are not constants but a function of the internal coordinates. The correct kinetic energy op; erator is described by Podolsky [13]. The potential energy is not a function of the momenta and will therefore be identical with its classical form.. 2.2.2 The potential energy and dipole surfaces We want to include the anharmonicities in the potential energy and the dipole. This can be done only for limited number of degrees of freedom. We fit the poten tial energy with a polynomial of coordinates that can be arbitrary functions of the cartesian or internal coordinates. For each coordinate we use five fit points. This number may seem quite low, but we have to keep in mind that in a multidi mensional case the number of the points will be , where is the dimensionality. In this thesis the highest dimension was four and a regular grid would consists of points. The total number of the grid points can be decreased by leaving out the points where more than one coordinate takes an extreme value. These points will have a too high energy to be important for the wave functions of lower energy we are interested in. The energy polynomial is given by. 0 .
(50)
(51)
(52) . . .
(53)
(54)
(55) . . . 0 . . . . . . . . . 0 . 3. (2.11). with being the number of the coordinates,
(56)
(57)
(58) are the coefficients for the 3 products of power of the coordinates, and is the order of the polynomial. The.
(59) 2.2 Vibrational Hamiltonian in internal coordinates. 21. Figure 2.1: Schematic representation of the coordinates used to compute the four dimensional spectra of the zeolite OH vibrations. Further explanation is given in the text.. highest oder of the polynomial employed in this thesis was four, also in the multidimensional case. The grid should be constructed in such way that it covers the area where the wave functions of interest have non-negligible amplitudes. It is difficult to estimate what should be the value of the coordinates to fulfill this condition. In one-dimensional case more points, for example 15, can be used to perform the fit, and from these, five points are chosen so that the ground state and several excited state energies are reproduced. These five points are further used for constructing the multidimensional grid. The four dimensional spectra of the zeolite OH vibrations were calculated using the following set of coordinates. 8. . 8 ) '. . ). . . (2.12). where is the deviation of the zeolite OH bond distance from the equilibrium structure, is the change of the in-plane angle , is the change of the out of-plane angle , and is the change of the distance between the zeolite oxygen atom and the center of mass of the adsorbate (a water molecule in Fig. 2.1) while keeping the geometry of the adsorbate fixed..
(60) 22. The Theory Behind the Calculations. . . In the least square approximation, the coefficients
(61)
(62)
(63) are sought such that the residual
(64) (2.13) 0. . . is minimal. Here is the energy value computed for grid point using Eq. (2.11). We are interested in obtaining good description of the ground state and a relatively small number of excited states. Therefore, we have given greater weights to grid points with lower energy. The weights are calculated as. .
(65) 0 . (2.14). such that where is the number of grid points and is an energy scale factor 0 the root mean square (r.m.s) error of the fit is smaller than ( is atomic kJ/mol). The value of used throughout unit of energy called0 Hartree, . For the chosen set of coordinates we can write a set of linear this thesis is equations that in matrix form is. ! , / ! $,# coefficients and grid points, then . or. (2.15) (2.16). Suppose there are is an matrix containing the products of the powers of coordinates and is a vector containing the energies of the grid points, multiplied with the square root of the weight of the grid point and is a vector containing the coefficients
(66)
(67)
(68) . If the number of equations is equal to the number of coefficients and there are no linear dependencies between the equations, then the Eq. (2.16) has a unique solu . In our case and we do not know in advance whether the tion with equations are linearly dependent. We can use singular value decomposition (SVD). method [14] for solving Eq. (2.16). With this method the matrix is de. composed into an column orthogonal matrix , an diagonal matrix , and the transpose of an orthogonal matrix . . "!. . (2.17). If the matrix were square and all elements of were non-zero, the inverse of can be easily computed with SVD 0 , 7$# ! ? ! (2.18) 0 # Eq. (2.18) shows that will be impossible to compute if some of the elements are zero or very small. The condition number of a matrix is defined as the ratio of.
(69) 2.2 Vibrational Hamiltonian in internal coordinates. #. 23. #. the largest of the ’s to the smallest of the ’s. A matrix is singular if its condition number is infinite, and it is ill-defined if its condition number is too large. When # 7$# is computing , if is singular or ill-defined for the very small elements , set to zero. (2.19) ! , 7$# ? ! . . As it was mentioned above, the matrix is not square. Its inverse can be found by filling up the matrix with zero columns until it becomes square. The # same . Applying SVD will give one zero or negligible for each we do with the vector row of zeros that we have added plus additional ones from any degeneracies in the equations. The , and -components of the dipole can be fitted in the same way as the potential energy surface.. 2.2.3 Second quantization formalism The Hamiltonian of anharmonic oscillator can have a complicated form. Therefore we will use a harmonic oscillator to illustrate the general principles and formalism related to the study of quantized oscillations. The Hamiltonian for an onedimensional harmonic oscillator along coordinate with reduced mass , force constant , and conjugate momentum is given by . . . . . . . (2.20). The derivation of the formula in the next Sections can be facilitated by expressing the coordinates and the conjugate momenta as a function of creation and anhilation operators [15]. This is also known as a second quantization formalism. These operators are defined as (2.21) . . . . . . (2.22). . 7. . and is the Planck’s constant divided by . The where is defined as operators and are Hermitian conjugates of each other. The coordinate and the conjugate momentum in turn can be expressed in creation and anhilation operators . . . . . . . . . . . (2.23). (2.24).
(70) The Theory Behind the Calculations. 24. The commutation relation for the creation and anhilation operators is. . . . . .3. (2.25) . 3. In Dirac notation, a harmonic oscillator eigenfunction is written as , where is the excitation level or the number quanta in the state. The effect of the operators and is to remove or add a quantum. . .3 3 .3 .3 3 .3 . (2.26) . . (2.27). The harmonic oscillator eigenfunctions form a set of orthonormal vectors. . . . 3 . (2.28). . which is the basis of a so called representation. From Eqs. (2.26) and (2.27) the representation matrices of the operators and in this basis set can be easily derived (2.29) .. . and . . . . . . . . . . . . . . . ... . (2.30). . . The Hamiltonian can then be written as . with. 2.3. 3 . . . 3. . . (2.31). the so-called number operator.. The vibrational Hamiltonian matrix elements. The Schr¨odinger equation is usually solved using the variational principle. In this 3 method the wave function is expanded as a linear combination of -dimensional orthonormal basis set . (2.32) . .
(71) 2.3 The vibrational Hamiltonian matrix elements. with. 1. . . . . . 0. 25. (2.33). The coefficients and the energy are the eigenvector and the eigenvalue of the Hamilton matrix which elements are given by. . . . . (2.34). When computing the matrix elements of the vibrational Hamiltonian the hardest part of the calculation is the evaluation of the matrix elements. ; . ; . . ; !. . . (2.35). (For the kinetic terms see Section 2.3.2). The superscript VBR in Eq. (2.35) stands for variational basis set representation. This name emphasizes that the eigenvalues of the obtained in this way Hamilton matrix are variational upper bounds to the exact ones. For anharmonic oscillator the potential energy can be a “complicated” function of the internal coordinates. Sometimes the computation of the matrix ; elements by an accurate numerical integration may take considerably more computational time than the full diagonalization of the Hamilton ; matrix. ; The potential be considered as a real function of the position ; operator ; . Thecan operator , i.e matrix representation of the position operator is.
(72) . . . . . . (2.36). Thus, the potential energy matrix can be considered as a function of the position matrix ; (2.37). . . where the superscript FBR stands for finite basis set representation. would be equal to if the basis set were complete. Since we are using a truncated basis set, however, we are introducing an error. This can be easily seen if the potential ; is expanded in a Taylor’s series in . . . ; . . 0 . . . . . . . (2.38). . where is the remainder if the series is truncated after terms. Since the ma trix represents the multiplicative operator , one may take the matrix as a representation of the operator . In a complete basis this procedure is exact. To for we insert the resolution of identity compute the matrix elements. . . . . 1. . 0 . . . . . .. (2.39).
(73) 26. The Theory Behind the Calculations. into. . . . . . 1 . . . . 0 1 0 ; in a Taylor’s expansion in So, the matrix representation ; 0 . . . (2.40) (2.41) (2.42). . (2.43) is. To find the matrix representation for the potential energy, the matrix diagonalized. . . and. . . . . . . . (2.44) has to be (2.45) (2.46). and similar for the higher powers of . Recall that the position operator can be expressed as a function of creation and anhilation operators (2.23)), the matrix 3 . . 3 . For an(Eq. 3 -dimensional is a tridiagonal matrix, that has elements basis the matrix elements for computed with this procedure are exact for 3 . , and is a diagonal The eigenvector matrix is an unitary matrix, i.e. 6 matrix of the eigenvalues , i.e . We will use Latin letters for numbering the basis functions and Greek letters for the grid points . Eq. (2.44) can be written as (2.47) 0 . . . . . . . .
(74). . . . . . The elements of the potential matrix in the finite representation basis set are given 1 by. ; .
(75) . ; 0. (2.48). This equation shows that the potential energy matrix can be evaluated if the transformation matrix and the eigenvalues are known. The last we find by diagonalization of the position matrix, which is independent of the potential. This method was used for first time by Harris et al. [16] for computing the matrix elements of one-dimensional anharmonic Hamiltonian in a basis set of harmonic oscillator eigenfunctions (Hermite functions). Later, Dickinson and Certain [17] have shown that for any basis set that consists of orthogonal polynomials (such as the Hermite functions, for example) this method is equivalent to Gauss quadrature..
(76) 2.3 The vibrational Hamiltonian matrix elements. 27. 2.3.1 Gauss-Hermite quadrature Suppose we want to calculate the integral of a function . . . . ! . . (2.49). is a some given weight function. Instead of doing this analytically we where can approximate the integral of the function by a sum of its functional values at a set of points , multiplied by weighting coefficients . 1. 3. 0 . (2.50). where is the number of the integration points. A particular Gauss quadrature form is characterized by the choice of the integration points and the weight function. For each choice of integration interval and a weight function a set of orthogonal polynomials can be defined. For the Gauss-Hermite quadrature these are (2.51) 0 0 (2.52) where is a Hermite polynomial of order . Eq. (2.52) is a recursive relations with . The harmonic 0 and oscillator wave functions (Hermite func tions) have the form. . . . . . . #" '.
(77). , " $' . " " )(, ! ,())' '. (2.53). The width of the wave functions is controlled by the parameter , which in turn is determined by the reduced mass and the force constant by the relation . . . . . . . . . . . (2.54). The precise form of the Hermite functions is . with. . . . 0 . . . being a normalization constant. . . . . . . . . . . . (2.55). . . (2.56). The Hermite functions are orthogonal. . . . ! 6. (2.57).
(78) 28. The Theory Behind the Calculations. 1. The quadrature approximation of this scalar product is. Let us define. 0 6. (2.58). . 0 7 7 (2.59) The elements of the positional matrix in the basis set of Hermite functions are . . . . . 1. 1 . . Thus, the matrix written as. . 0. 1 but. (2.60). 0 . . 1 0 0 . is unitary,. . . (2.61). . (2.62). . 6. (2.63). 0 . In matrix notation, Eq. (2.60) can be. . . (2.64). The comparison of this equation with Eq. (2.45) shows that, indeed, the method used by Harris et al. [16] is equivalent to the Gauss-Hermite quadrature approxi . mation of the matrix elements, with The Gauss-Hermite quadrature points are actually the roots of the Hermite 1 . There are several ways to calculate the quadrature points and polynomial the corresponding weights. One of the easiest ways to find the roots of the Hermite polynomials is by diagonalizing the matrix . The weights can be computed from Eq. (2.59) /. 7 . (2.65). . There exists also a routine
(79) [14] that computes the quadrature points and . These quadrature the weights. Note that in this routine the scale factor is points and weights are also tabulated [18].. 2.3.2 DVR and the Hamilton matrix. . . So far we have shown that in the so called FBR basis set the potential matrix can be written as a unitary transformation of a diagonal matrix. Lill et al. [19] have .
(80) 2.3 The vibrational Hamiltonian matrix elements. (a). 29. (b). Figure 2.2: The Hermite functions (a) and the corresponding DVR functions. proposed to apply the unitary transformation to the basis set so that the potential matrix is diagonal. 1. 0 . . . . 3. (2.66). . The new basis is called discrete variable representation (DVR) basis set. As can be seen from Fig. 2.2 the Hermite functions are delocalized whereas the DVR functions are localized at the grid points. The two basis sets are related via the unitary transformation . Thus, we can write the function as. . . . 0. . . . 0. . (2.67).
(81) 30. The Theory Behind the Calculations. The coefficients . . and . . are related via. . . . . 0. . . . . 0. . (2.68). This shows that we can always switch from one representation to another. The values of the DVR functions at the quadrature points are. 1. 1 0 0 . . . . . . . . . 6. . (2.69). The potential matrix elements in the DVR basis set are given by. ; . .; .. 1. . 0. (2.70). ; . 1. . (2.71). 6 ; 6 0. (2.72). ; 6 . (2.73). . The potential matrix is diagonal in the DVR basis but the kinetic matrix is not. The kinetic matrix elements are easy to compute in the FBR basis set:. . $ . . $8. . . . . . . . . . . (2.74). . . . . .. (2.75). . . . . . . (2.76). . . The matrix can be easily calculated using the matrix representation of the creation and anhilation operators. The matrix is then obtained from by applying the unitary transformation. . . . . . (2.77). For one-dimensional problem the resulting Hamilton matrix is not sparse. As the number of degrees increases, the sparsity of the Hamilton matrix in a DVR basis set increases as well. This is because the only source of non-zero off-diagonal matrix elements are the kinetic energy terms. The kinetic energy operator in cartesian coordinates has the form $
(82) (2.78) 0 . . . . . .
(83) 2.3 The vibrational Hamiltonian matrix elements. . 31. . The multidimensional wave function is expanded in products of one-dimensional DVR basis .. . 1. 0 . . 1. . . . . 0 0 0 . (2.79). The kinetic energy matrix elements are then. . 0 . . . . $& . 0 . . . . . . . . . . 0. 0 . 0. . . . . . 0 . . . . . . . 6 . . . (2.80). (2.81). The resulting matrix is sparse because of the deltas. For example, for a twodimensional problem we 0 can construct using Eqs. (2.76) and (2.77) the one . If we use DVR points along each coordinate, dimensional matrices and the kinetic matrix for the two-dimensional problem is then constructed as a sum of a Kronecker tensor product of these one-dimensional matrices with an identity matrix .. . . . . . 0. . . . . . . . 0 0 0 0 0 0 0 0 0 0 0 0 0 . 0 0 0. . . . . 0 0 0 0 . . . . . (2.82). 0 0 0 . . . . 0 . . (2.83). . As the number of degrees of freedom increases, the number of zeros in these matrices increases exponentially. Evaluating the Hamilton matrix elements within the DVR basis set makes it possible to express the kinetic energy operator in cartesian coordinates where the cross-terms of the matrix are zero. When evaluating the potential matrix we can transform the coordinate system from cartesian into internal and then calculate the value of the potential energy at the DVR points. The cartesian coordinates are defined as follows. The unit vector is along the equilibrium zeolite OH bond, the unit vector is perpendicular to and is in the Si–O–Al plane, and the unit vector is perpendicular both to and the Si–O–Al plane. The displacements are then written as
(84) .
(85)
(86)
(87) , where .
(88) are the cartesian coordinates. If . and are DVR points for the zeolite OH stretch, in-plane and out-of-plane.
(89) The Theory Behind the Calculations. 32. bending, respectively, then the coordinate transformation is. . ) '. . . . . . . (2.84). . . . &-). . . . . (2.85). .
(90). (2.86). . with being the equilibrium OH bond distance. The water center of mass coordinate is not affected by this coordinate transformation and remains unchanged. The potential matrix for the vibration of the zeolite OH bond is then. * * * . ; . * * * . ; )
(91) $' &-) 6 6 6 . . (2.87). When harmonic oscillator eigenfunctions are used to obtain the DVR points, the points are situated symmetrically around the origin. We have seen in Chapter 1 that the potential energy surface for the zeolite OH stretch coordinate in presence of adsorbates is strongly asymmetric with respect the equilibrium distance. As a result, we will calculate the potential energy for large negative values of the coordinate, which are in a region we are not interested in. We can avoid this by starting with many ( ) points which will give practically converged one-dimensional eigenfunctions. The eigenfunctions are linear combination of the DVR functions 3 ) eigenfunctions with eigenand are not symmetric anymore. We take all ( values below a certain energy cut-off and use them to calculate the matrix (Eq. (2.60)). From the diagonalization of we obtain new set of DVR points which are not symmetric anymore. We use this set in the multidimensional problem. The basis set obtained in this way is known as potential optimized DVR [20]..
(92) . 2.4. Infrared absorption intensities. 2.4.1 Interaction of radiation with matter. . lower Consider a system that has two stationary states with upper energy (1 all at constant temperature $ and coupled to a radiation field and energy with radi$ . The radiation density is defined by Planck as ation density 0 $ (2.88) . . # is the energy of radiation in the frequency interval between and # where and is the speed of light in vacuum, and is the Boltzmann constant. are three possible processes that can change the state of the system from )1 There )1 ; absorption, spontaneous emission, and stimulated to and from to. . .
(93) 2.4 Infrared absorption intensities. 33. hν absorption. hν spontaneous emission 2h ν. hν. stimulated emission Figure 2.3: Schematic presentation of the system before (left) and after (right) interaction with a photon. The black circle shows in which state the system is.. emission. These processes are schematically represented in Fig. 2.3. The probability that the system will absorb a quantum of energy and undergo transition from the lower state to the upper state in a unit of time is. 1 1 . . (2.89). "1 is the Einstein coefficient of absorption. The probability that the syswhere tem in the upper state will undergo transition to the lower state with emission of radiant energy is a sum of two parts, one of which is independent of the radiation density and the other is proportional to it . . . 1 . . 1 . 1. . (2.90). 1 is the Einstein coefficient of spontaneous emission and 1 is the where Einstein coefficient of stimulated emission. Since the entire system is in thermal 81 systems are in state 3 and systems are in state , then the equilibrium, if 3 91 1 1 , is equal to the number of trannumber of transitions from to , 3 1 1 1 . The ratio between 81 and is sitions from to , given by the Boltzmann distribution . . . . . . . . . .
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