Molecular Heterogeneous Catalysis, Wiley (2006), 352729662X

6Chapter 1

of the molecular aspects that control heterogeneous catalytic systems due to major advances in in-situ spectroscopy, theoretical methods and computational power. Molecular and atomic scale descriptions of the fundamental physicochemical steps involved in the overall catalytic cycle are thus becoming possible. This same level of description has also advanced the description of biochemical systems. As we will see later in the book, detailed comparisons between chemical and biochemical systems can be very useful for aiding our understanding of the common features governing both systems and also the primary di erences. Nature is not only elegant in the materials it creates but in its process of discovery. The comparisons between biochemical and heterogeneous catalytic systems may therefore provide a wealth of information not only in ideas for new biomimetic materials but also in the development of novel catalyst discovery processes.

A detailed analysis of the chemistry alone reveals that catalysis is not comprised of a single elementary event, but is a complex phenomenon in which the elementary reaction as well as the catalyst can both take part in feedback cycles. We examine this unique catalyst feedback cycle by probing the possible mechanisms involved in the conversion of a catalytic material from its initial state towards its catalytically active state.

Earlier we gave the classical Berzelius definition for a catalyst as is found in most textbooks, i.e. a material which enhances a particular reaction but is not used or consumed in the process. Decades of fundamental research have shown that the working catalyst is a dynamic entity that can continuously change. The reactive surface structure is metastable and, in some cases, even mobile. Active sites are formed and consumed dynamically either through surface reactions or through structural changes in catalyst surface topology. Some have described the catalyst as a living system whereby active sites and ensembles must be continuously reborn.

During the course of a catalytic reaction many sites, regions on the catalyst surface or actual catalytic particles can interact. Under some conditions, these sites or regions can communicate with one another and thus lead to self-organizing phenomena that occur in both space and time. This provides information on the complexity in catalytic reaction system, that once again can be related back to phenomena that are well known in biocatalytic systems.

We will discuss the molecular basis of chemocatalysis in comparison with related aspects of biocatalytic systems. This will enable us to elucidate the molecular foundation and mechanisms for catalytic reactions and also provide a broader perspective to these principles. In many instances, the refinement of the biological system is often lacking in the chemo-systems. Current challenges in chemocatalysis are related to a more complete understanding of these issues.

The complexity of the catalytic reaction is a common thread through most of the chapters that follow. We describe the issues associated with the di erent time and length scales that underpin the chemical events that constitute a catalytic system. For example, a typical time scale for the overall catalytic reaction is a second with characteristic length scales that are on the order of 0.1 micron. The time scales for the fundamental adsorption, desorption, di usion and surface reaction steps that comprise the overall catalytic cycle, however, are often 10−3 sec or shorter. The time scales associated with the movement of atoms, such as that which must occur for surface reconstruction events, may be on the order of a nanosecond. The vibrational frequencies for adsorbed surface intermediates occur at time scales on the order of a few picoseconds. The di erent processes that occur at these time scales obey di erent physical laws and, hence, require di erent methods in order to calculate their influence on reactivity. In this book we will show how the

Introduction 7

description of these processes should be integrated in order to provide predictions on the performance of catalytic systems.

A proper kinetic description of a catalytic reaction must not only follow the formation and conversion of individual intermediates, but should also include the fundamental steps that control the regeneration of the catalyst after each catalytic turnover. Both the catalyst sites and the surface intermediates are part of the catalytic cycle which must turn over in order for the reaction to remain catalytic. The competition between the kinetics for surface reaction and desorption steps leads to the Sabatier principle which indicates that the overall catalytic reaction rate is maximized for an optimal interaction between the substrate molecule and the catalyst surface. At an atomic level, this implies that bonds within the substrate molecule are broken whereas bonds between the substrate and the catalyst are made during the course of reaction. Similarly, as the bonds between the substrate and the surface are broken, bonds within the substrate are formed. The catalyst system regenerates itself through the desorption of products, and the self repair and reorganization of the active site and its environment after each catalytic cycle.

This reorganization may be governed or, at least, aided, by self organization phenomena. In heterogeneous catalytic systems, molecular events that occur at di erent positions on the catalyst surface can interact through di usion or surface strain. These interactions can lead to complex self organization phenomena when the catalytic cycle proceeds through the elementary reaction steps that constitute this cycle. The time and length scales for these self organization phenomena are typically on the order of seconds and microns, respectively. Self organizing spatial patterns can be formed due to the coupling of autocatalytic reaction steps and di usion concentration gradients. Wave fronts in the form of spirals or pulses can also result. Ertl and co-workers were the first to demonstrate this experimentally for reactions catalyzed by single crystal transition-metal surfaces[10] . Self organizing systems are well known in biology and are related to systems that self replicate. Computational approaches suitable to the simulation of these processes are based on the concept of cellular automata, a technique which is just beginning to be used to simulate the kinetics for chemical catalytic systems.

As discussed in Chapter 9, perhaps the most elegant biocatalytic system which demonstrates the next higher level of organization is that for the immunoresponse system. This is a biological system that operates through a combinatorial chemical process. A particular catalytic antibody is selected and amplified in response to a specific reacting molecule, the antigen. If the shape of the antigen molecule is similar to the shape of the transition state for of the reaction intermediates involved in a chosen reaction, it can induce the production of catalytically active antibodies with high selectivity for a chosen particular reaction.

Interestingly, the inorganic chemistry and kinetics for template-directed zeolite synthesis is rather analogous to that of the biocatalytic system. Aluminosilicate oligomers are consumed during zeolite synthesis through the formation of a template aluminosilicate complex. The complex is unique for the zeolite system and subsequently crystallizes. The template can be chosen to resemble the transition state for a specifically desired reaction, thus serving as a catalyst designed to enhance the rate of a pre-selected reaction. This is directly analogous to an antigen-induced antibody synthesis. The di erence between the biological and chemical systems is that that the chemical system does not replicate.

8Chapter 1

1.3 Outline of the Book

While the primary focus of the book is on the molecular basis of heterogeneous catalysis, a final chapter (Chapter 9) is partially devoted to the description of self organizing catalytic systems that lead to the formation of (proto) cellular systems. The design of self organizing and replicating catalytic systems can be considered one of the ultimate goals of molecular heterogeneous catalysis. This chapter is concluded with a description of the physical chemistry of biomineralization and analogous processes to produce microporous systems with widely varying organization of their micropores.

Between the initial chapter and the final chapter, we present a more conventional treatment of molecular heterogeneous catalysis, with a focus on surface catalytic elementary reaction steps and their connection to overall catalytic behavior for a series of di erent substrates including metals, zeolites, metal oxides and metal sulfides.

As an introduction to the principles in molecular heterogeneous catalysis, we focus in Chapter 2 on basic elementary concepts in heterogeneous catalysis including catalytic reaction cycles, mechanisms for the regeneration of the catalytic active site and the Sabatier principle, which relates catalytic activity with free energy associated with reactant-catalyst interactions. These general concepts provide the constructs necessary to begin to establish molecular level theories and models. The lock-and-key model along with specific modifications to this model are described and used to compare with theories for steric control in homogeneous and heterogeneous catalysis. Steric control in homogeneous catalysis is established by ligand choice, an art that has became increasingly refined over the past few years due to the drive to design homogeneous catalysts with high enantiomeric selectivities. Steric control in heterogeneous surface catalysis is assisted by the coadsorption of shaped adsorbates. In an organizational sense, these shaped adsorbates can be considered as a primitive version of ligands attached to single-site organometallic catalysts.

A very useful procedure to analyze transition states in heterogeneous catalysis relates changes in the activation energy for a particular elementary reaction step with changes in the overall reaction enthalpy for that step over a family of similar catalytic materials. This is the Brønsted–Evans–Polanyi relationship. It is also analogous to the Hammett relationship pioneered in physical organic chemistry that linearly relates the activation barriers for a family of substituted aromatics with a substituent parameter. The Brønsted– Evans–Polanyi relationship as well as other linear free energy relationships hold when the molecules or catalytic materials fall within the same family. The entropic changes across a reaction family are either considered negligible in comparison with the enthalpic contributions or are linearly related to the changes in enthalpy. The entropic changes can be estimated by an understanding of the type of transition for the reaction in question and the di erence between early and late transition states.

The final section in Chapter 2 deals with the molecular aspects of transition-metal catalysis. It serves as an introduction to Chapter 3. A characteristic feature of the transitionmetal surfaces under catalytic conditions is their potential to restructure. Adsorbate overlayer adsorption can induce the surface to reconstruct with rapid di usion of the metal as well as the overlayer atoms. The state of the surface may start to resemble that of a solid state compound. The state of the surface is not only strongly influenced by the composition of the reactant gas, but can also be strongly a ected by the addition of promoters or other modifiers, that can result in alloy formation or new complex surface phases.

In order to describe the active sites and the associated kinetics, two predominant theories ascribed to Langmuir and Taylor have prevailed in heterogeneous catalysis. In the

Introduction 9

Langmuirian view, the active catalytic surface is comprised of a uniform distribution of static sites that do not interact with one another. This is sharply contrasted by the Taylor view, which proposes vacancies and topologically unique surface atom configurations as the centers of reactivity. The Langmuirian idea of a catalytically reactive surface leads to the ensemble e ect that ascribes the changes in the selectivity for an alloy surface to the dilution of multi-atom surface ensembles in the alloy induced by mixing inert components into the active surface. In this view, the selectivity of a particular reaction depends predominantly on the number of reactive surface atoms that participate in elementary reaction events.

Recent surface science discoveries, however, demonstrate that step edges and defect sites display markedly lower activation barriers than terrace sites, and thus promote the Taylorian view of catalysis. The selectivity can be strongly influenced by the specific poisoning of these step edge sites. For a number of hydrocarbon conversion processes, these steps will be the most active and lead to potential C–H and C–C bond breaking steps which can ultimately result in deactivation via the formation of surface graphene overlayers.

In Chapter 3, we extend the general concepts developed in Chapter 2 on chemisorption and surface reactivity to establish a fundamental set of theoretical descriptions that describe bonding and reactivity on idealized metal substrates in Chapter 3. There is an extensive treatment of the adsorbate transition-metal surface bond, its electronic structure, bond strength and its influence on its chemical activity. Attention is given to periodic trends in the interaction energy as a function of transition metal and also on the dependence in transition-metal structure.

To illustrate the use of this type of information, an extensive analysis of C–H and C–C bond activation and formation reactions is given. The chapter is concluded with a section that focuses on experiments and theories that explicitly consider lateral e ects between adatoms and molecules.

We transfer some of the general concepts developed for the chemical bonding on metals in Chapter 3 to describe the bonding and reactivity that occur in zeolites in Chapter 4. Zeolites are mesoporous systems that have well-defined atomic structures, in contrast with the ill-defined structures of supported metals. This well-defined structure allows them to benefit greatly from the close ties between theory and spectroscopy. This combination of theory and experiment helps to provide for a more detailed understanding of the intrinsic and extrinsic factors that control catalytic reactivity. In the analysis of reactions carried out on zeolites, it becomes clear that the micropores of the zeolite play an important role in dictating their catalytic performance. In order to understand the mechanistic factors that control the sorption and reactivity in zeolites, we focus on two general features in our analysis: the nature of solid acid acidity and the influence of the micropore size and shape on catalysis. Ab initio quantum mechanical calculations now allow for a detailed analysis of reaction intermediates and transition states for reactions of practical interest along with more realistic models of the active sites that capture the full pore cavity. Some of the key concepts developed in this chapter include the importance of pre-transition state stabilization, the screening of the charge separation when charged protons activate bonds, and physical e ects that relate to adsorption and di usion.

Complexity in zeolite catalysis can take on various forms. We focus on two of the challenging issues herein. The first involves the complexity of treating multicomponent systems. There is a strong non-additive behavior for the adsorption isotherms for multicomponent mixtures found in zeolites. This can dramatically a ect the selectivity of

10 Chapter 1

the zeolite for specific reactions. The complexity also obscures a fundamental analysis based on isotherms of individual molecular fragments. The second challenge relates to the analysis of catalysis by cationic complexes in zeolites which show a strong dependence on the chemical state of the cationic complex, as well as self organizing features. Theory and simulation have helped to gain insight into both of these challenges by its ability intrinsically to simulate multicomponent systems and the chemistry of cationic complexes in zeolites, respectively.

The results from this chapter on zeolite catalysis provide a good reference point for the discussion presented later Chapter 8 where we compare heterogeneous catalysis and biocatalysis. The similarity between the Michaelis–Menten kinetic expression for enzyme catalysis and the Langmuir–Hinshelwood kinetic models for heterogeneous catalysis are noted. This ultimately derives from the conservation in the number of active reaction centers for both systems. However, the more refined synergy of the activation of molecular bonds by the enzyme will become apparent as a major di erence between the two.

This general understanding of the similarities, as well as the di erences, between biological and heterogeneous catalysts has been the basis for numerous attempts in the literature to synthesize novel chemical systems that mimic enzymes. In Chapter 7, we will review some of the major advances that have taken place, which will also help to highlight further routes for new research. Another important characteristic for enzyme systems is that they are often part of an electrochemical bio-systems and can therefore be considered as bio-electrodes. This implies that electron-transfer catalysis governs their performance. Hydrogenation, for example, tends to occur in biosystems via the combination of electron transfer and reaction with protons. We present a short discussion on the bio-catalytic reduction of nitrogen, and subsequently compare it with the traditional heterogeneous nitrogen reduction by the iron heterogeneous catalyst.

We extend our understanding of the concepts of chemical bonding and reactivity learned in Chapter 3 on metals and Chapter 4 on zeolites to catalysis over metal oxides and metal sulfides in Chapter 5. The features that lead to the generation of surface acidity and basicity are described via simple electrostatic bonding theory concepts that were initially introduced by Pauling. The acidity of the material and its application to heterogeneous catalysis are sensitive to the presence of water or other protic solvents. We explicitly examine the e ects of the reaction medium in which the reaction is carried out. In addition, we compare and contrast the di erences between liquid and solid acids. We subsequently describe the influence of covalent contributions to the bonding in oxides and transition to a discussion on the factors that control selective oxidation.

Selective oxidation requires an understanding of the active metal cations in complex solid state matrices and their changes in complex reaction environments. The active systems are controlled by the oxidation state and the coordination number of the metal cations, the interaction between the oxide and the support, the domain size of the active cluster, the electronic properties of the active domain and the influence of the oxide support, the presence of defect sites, the surface morphology and surface termination. The metal cations are analogous to the single metal cation centers used in homogeneous catalysts but are now influenced by the presence of the oxide media in which they exist.

Many selective oxidation reactions demonstrate a strong synergy of combining both acid/base and redox functionality. Nature does this in a transparent way by strategically placing these functions in unique positions so as to enable both functions to act synergistically. The key to many oxidation reactions will likely also require a delicate balance between the strength as well as the specific spatial arrangements of acid, base and oxi-

Introduction 11

dation sites. The reactivities of reducible oxides or sulfides have many similarities with one another and also with coordination complexes. The close collaboration between theory, fundamental surface science studies and industrial experiments has helped to reveal the nature of the active sits for hydrodesulfurization (HDS). The current model of the active sulfide surface under practical sulfiding conditions is that the Mo edge and the sulfur edge of supported MoS2 particles are coordinatively saturated with sulfur. However, metal atoms which are just inside of this edge are shown to be partially reduced with vacant sulfur sites on the surface. These sites form a brim at the surface whereby sulfur-containing molecules can adsorb and undergo desulfurization. The addition of Co to the supported MoS2 is well know to promote HDS activity. The promotional e ects have been speculated to be the result of the weaker Co–S bond over that of the Mo–S bond.

Electrocatalysis by chemical systems is extensively discussed in Chapter 6. The chapter provides a direct comparison between reactions occurring at the gas-solid interface with those occurring at the liquid-solid interface. Systems are specifically chosen that have been studied at both the gas-solid interface as well as in the liquid phase only. The latter case involves catalytic reactions carried out with organometallic coordination complexes. This provides an opportunity to compare catalysis at the surface-liquid interface with homogeneous catalysis with coordination complexes. More specifically, we describe the oxidative acetoxylation of ethylene to form vinyl acetate. Results from both homogeneous catalytic reactions carried out in solution as well as over ideal metal surfaces exist to provide guidance and help to interpret computational results. We subsequently transition into electrocatalysis over a metal substrates. In order to illustrate the comparison between reactions that are run in the gas phase with those run electrocatalytically, we specifically examine ammonia oxidation since there is an extensive data base for both gas phase as well as electrocatalytic reactions. In the presence of protic or aqueous medium, the solvent can enhance catalytic reactions by stabilizing polar transition states as reported in most physical organic chemistry text books. The solvent molecules can also specifically participate in the reactions themselves. An important example is the proton-transfer reaction. Consequences of the latter e ect are extensively discussed.

The concluding chapters in the book attempt to draw analogies between the concepts regarding the self-assembly and the catalytic propagation of the elementary structures in the origin of life and the analogous active self-sustaining features required for heterogeneous catalysis. As mentioned earlier, an important reason to include this topic in the book is that it o ers insights into novel strategies for the development of new catalytic systems. More specifically, we describe biomineralization strategies that have been used to synthesize silica materials with pore sizes architecture over di erent length scales.

The book is concluded with a final chapter that summarizes the key concepts presented in the book in a concise way. To some extent, Chapter 10 can be used as a glossary highlighting the important concepts presented throughout the book that govern catalysis.

1.4 Theoretical and Simulation Methods

One of the ultimate goals in modeling heterogeneous catalytic reaction systems would be the development of a multiscale approach that could simulate the myriad of atomic scale transformations that occur on the catalyst surface as they unfold as a function of time, processing conditions and catalyst structure and composition. The simulation would establish all of the elementary physicochemical paths available at a specific instant

12 Chapter 1

in time, determine the most likely reaction paths by which to proceed and then accurately calculate the elementary kinetics for each process along with the influence of the local reaction environment internal on the simulation. In addition, the simulation would predict how changes in the particle size, shape, morphology, chemical composition and atomic configurations would influence the catalytic performance, including activity, selectivity and lifetime. Modeling the spatial surface and gas phase composition along with the temperature would enable us to follow self organization phenomena also.

This is obviously well beyond what we can currently simulate. Even subsystems of this would be quite di cult to carry out with any meaningful accuracy. This does not mean, however, that theory is of little use and should be abandoned. On the contrary, one of the primary goals of this book is to highlight the impact that theory has made in establishing governing catalytic principles important for the science of catalysis. Many of these ideas could not have been conceptually or quantitatively obtained without the help of state of art computational methods.

The detailed prediction on the state of adsorbed species can be validated by comparison with experimental studies on well-defined model surfaces and model catalytic systems under controlled reaction conditions for which adequate theoretical modeling techniques are available. This includes the prediction of adsorbate surface structure, their properties and their reactivity. This can be determined by comparing the surface structure of adsorbed intermediates under idealized conditions measured through scanning tunneling microscopy (STM), and low energy electron di raction (LEED), vibrational frequencies from high resolution electron energy loss spectroscopy (HREELS) or reflection adsorption infrared spectroscopy (RAIRS), and their adsorption and reactivity measured from temperature programmed desorption (TPD) and temperature programmed reaction (TPR) spectroscopy, or microcalorimetry. This provides quantitative information on the elementary adsorption and reaction steps that occur on these model surfaces.

The understanding of catalysis, however, will require modeling the di erences in surface structure between the ideal single crystal surfaces studied under ultrahigh vacuum (UHV) conditions and those likely present for the supported particles used industrially. This is typically called the materials gap in surface science. In addition, the understanding of catalysis will also require a move from the ideal conditions of the UHV and those modeled quantum mechanically to industrially relevant conditions. The di erence in pressure for experiments carried out under UHV and those under catalytic conditions can be as high as ten orders of magnitude. This can significantly alter the surface coverages and composition and thus lead to significant changes in the rate. This is known as the “ressure gap”.

The disparate time and length scales that control heterogeneous catalytic processes make it essentially impossible to arrive at a single method to treat the complex structural behavior, reactivity and dynamics. Instead, a hierarchy of methods have been developed which can can be used to model di erent time and length scales. Molecular modeling of catalysis covers a broad spectrum of di erent methods but can be roughly categorized into either quantum-mechanical methods which track the electronic structure or molecular simulations which track the atomic structure (see the Appendix).

The ability to calculate the intrinsic catalytic reactivity of bond-breaking and bondmaking events requires a full quantum-mechanical description of these events. The simulation of catalyst structure and morphology or reaction kinetics, on the other hand, would be more easily simulated via atomistic scale simulations, provided the appropriate interatomic potentials or intrinsic kinetic data exist. Over the past decade, it has become possible to derive such data from ab initio calculations, thus allowing for a hierarchical

Introduction 13

approach to modeling.

There are a number of excellent reviews and discussions about the advances that have taken place in quantum-mechanical method development and their ability to calculate a host of di erent material properties[11] . We therefore, do not go into this in detail in this book. Instead, we present a short overview of covering salient features of the di erent theoretical and computational methods and their application to catalysis. This is presented in the Appendix along with references to more detailed reviews on the di erent methods.

The path taken over the past decade for most reseachers modeling catalytic systems has been to move to first-principle quantum mechanical methods to describe bond-breaking and -making steps since they provide a reliable degree of accuracy. While there is still work on the development of semiempirical methods to describe transition metals more accurately, many of those modeling catalysis have abandon semi-empirical approaches, at least for the near future. High-level coupled cluster ab initio methods which attempt to simulate the wavefunction accurately exist and, in principle, provide the highest level of intrinsic accuracy as they can predict the heats of formation that are on the order of 4 kJ/mol or less in terms of accuracy. They can only treat, however, systems with less than about 10 heavy atoms. This is the level of accuracy that is required in order to predict rates of reaction with a significant degree of confidence. The computational requirements necessary even to come close to this level of accuracy for reliable catalytic models are currently well outside the reach of even today’s fastest multiprocessor computers.

Two other approaches have been taken in order to model the active site and its environment. The first has been to use somewhat less accurate quantum-chemical methods to obtain a more qualitative understanding of the key surface states, reaction pathways and mechanism. The key parameters can then be refined by the use of high level theory and/or experiments on model systems which are much smaller. The main benefit of theory then has been the design of a physically justifiable microscopic description of the catalytic system, with a qualitatively correct conceptual understanding.

The second approach has been to develop a model of the interactions that occur between the reactant intermediates and the catalyst surface using a force field that has been empirically or theoretically obtained using a well-defined model system. Molecular mechanics and molecular dynamics studies can then be used to simulate properties of the system which can be compared with experiment. This is the more conventional approach in enzyme catalysis[12].

Most of the calculations on heterogeneous catalytic systems today use ab initio density functional theoretical methods. DFT (density functional theory) is fairly robust and allows a first-principle-based treatment of complex metal and metal oxide systems whereby electron correlation is included at significantly reduced CPU cost. DFT can be used to calculate structural properties and typically reports accuracies to within 0.05 ˚A and 1- 2◦, overall adsorption and reaction energies that are typically within 20-35 kJ/mol and spectroscopic shifts that are within a few percent of experimental data.

A comparison between experimental adsorption energies for di erent adsorbates on di erent metal surfaces estimated from UHV temperature-programmed desorption studies and those calculated using density functional theory is shown in Fig. 1.1a. Although this is a very useful first step, the di erences are certainly not within the 5 kJ/mol engineering accuracy that one would like. Figure 1.1b shows a comparison between HREELS and DFT calculated vibrational frequencies for maleic anhydride adsorbed on Pd(111).

The success in modeling catalytic systems depends not only on the accuracy of the

14 Chapter 1

Figure 1.1. (a) Comparison of calculated and experimental chemisorption energies for di erent adsorbates on di erent metal surfaces and (b) vibrational frequencies for surface adsorbates such as maleic anhydride bound to Pd(111)[13].

methods employed, but also on the reality of the model chosen to mimic the actual reaction system studied. A single metal atom, for example, would be a poor choice for modeling a transition-metal surface regardless of the accuracy of the method used. The model can not capture the metal band structure. This leads to errors that are at least as large as those from the accuracy of the method.

There are three di erent techniques that are currently used to model the structure at the active site, known as cluster, embedded cluster, and periodic methods. Each method has its own set of advantages and disadvantages. Characteristic models for each of these systems are presented in Fig. 1.2.

In the cluster approach, a discrete number of atoms is used to represent only the very local region about the active site. The basic premise is that chemisorption and reactivity are local phenomena, primarily a ected only by the nearby surface structure.

In the embedded cluster approach, a rigorous QM method is used to model the local

Introduction 15

region about the active site. This primary cluster is then embedded into a much larger model which simulates the external structural and electronic environment. The outer model employs a much simpler quantum-mechanical treatment or an empirical force field to simulate the external environment but still tends to treat the atomic structure explicitly. This minimizes cluster-size artifacts. The outer model can subsequently be embedded in yet a third model, which is made of point charges in order to treat longer range electrostatic interactions and the Madelung potential.

Figure 1.2. Three approaches and examples for modeling chemisorption and reactivity on surfaces. (Left) cluster approach, maleic anhydride on Pd; (center) embedding scheme: ammonia adsorption in a zeolite cage; (right) periodic slab model: maleic anhydride adsorption on Pd(111).

The last approach is the periodic slab method. In this approach one defines a unit cell which comprises a large enough surface ensemble. Periodic boundary conditions are then used to expand the cell in the x, y, and/or z directions, thus providing the electronic structure for linear, slab (surface), and bulk materials, respectively.

In later chapters we will meet applications of all of these approaches in the solution of a number of the example systems described.

On transition metals it has been suggested that a well-chosen cluster of 20-30 atoms enables one to simulate the interaction of an isolated molecule with a transition-metal surface provided that there are no atoms in the surface which form bonds to the adsorbate that are left coordinatively unsaturated.[14]

In ionic systems it is essential to chose clusters that are electrostatically neutral, otherwise electrostatic boundary e ects tend to dominate computed results. In the application of embedded methods care has to be taken to correct properly for boundary e ects between cluster and medium. Madelung electrostatic field simulations have to be done with proper choice of charges and dielectric constants. For these reasons periodic calculations, when feasible, are typically preferred for ionic systems.

Ab initio quantum-chemical methods can be used to calculate a range of relevant properties for homogeneous and heterogeneous catalysts. The size of the system that can be examined, however, is still quite small in comparison with the features that make up the actual system.

Structural Monte Carlo simulations can explore significantly larger system sizes due to the fact that they only treat interatomic interactions with no focus on the electronic structure[15] . The interatomic potentials that are necessary for structural simulations can be derived either from experiment or from rigorous QM methods. Potentials of choice for ionic systems are typically additive potentials such as the rigid ion potentials, that depend on charges, bond distances, and bond angle or shell model potentials that also contain terms that describe polarization. Potential parameters can be deduced by comparison with theory[16] or experiment.