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CHARGE DENSITY PROFILES AT CATALYST SURFACES

by

James Martin. MacLaren

Thesis submitted for the degree of Doctor of Philosophy of the University

of London and the Diploma of Imperial College

Solid State Theory Group Department of Physics

The Blackett Laboratory Imperial College of Science and Technology

London SW7 2BZ

August 1986

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ABSTRACT

Recent experimental and theoretical work has shown correlations between surface electronic structure and catalytic activity for certain catalysed reactions. Coadsorbates such as sulphur and potassium, a catalyst poison and promoter respectively, can dramatically influence catalytic properties. In an attempt to understand these phenomena, the surface electronic structure has been calculated for clean, poisoned and promoted transition metal substrates. In particular, the calculations have been related to the chemisorption, dissociation and methanation of carbon monoxide on nickel surfaces, for which there is a wealth of experimental evidence. Calculations presented in this thesis follow three schemes.(1) A cluster local density of states approach which samples directly changes in the surface electronic structure at the active site due to adsorbed impurity atoms. The flexibility of this method allows for the study of the lateral range of a catalyst poison. From these results, numerical simulations of catalyst activity as a function of coverage of poisons have been made and compared to experimental data.(2) A self-consistent cluster approach developed by Slater and Johnson, the Multiple Scattering-Xa method, has been applied to catalyst promoters, poisons and their influence on the bonding of carbon monoxide to nickel. From the results of these calculations, a detailed picture of the local mechanism of the poisoning and promotion of carbon monoxide adsorption has been found.(3) A self-consistent layer KKR program has been developed, which allows for the extended nature of surfaces to be considered. In contrast to previous schemes a full surface, rather than a collection of layers, can be created, by coupling to a reflection matrix for a semi-infinite solid, created by the layer doubling algorithm. The flexibility allowed by the stacking of layers into a solid, allows the study of surfaces, overlayers and interfaces. Whilst this method is not

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fully developed, some preliminary results will be presented.

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Title Page 1Abstract 2Contents 4Figure Captions 7Table Captions 12Acknowledgements 13Preface 14

CHAPTER 1. INTRODUCTION

1.1 Background 151.2 Motivation 191.3 The Blyholder model of molecular chemi­

sorption and the role of electronic structure 21

1.4 Theoretical approaches to catalysis andchemisorption 24

1.4.1 The Anderson model 241.4.2 Orbital symmetry considerations 251.4.3 Charge mobility 251.4.4 Geometric factors 251.4.5 Interactions between adsorbates 261.4.6 Semi empirical methods 27

n1.4.6.1 The Extended Huckel model 271.4.6.2 The LCAO model 281.4.7 Jellium models and the Effective Medium

Theory 281.4.8 Electronic structure calculations 301.4.8.1 Slab and layer calculations 301.4.8.2 Cluster calculations 301.4.8.3 Summary of electronic structure schemes 32

CHAPTER 2. THE CLUSTER MODELS

2.1 Introduction 332.2 The LDOS cluster method 332.2.1 Derivation of G+ 35

C O N T E N T SP a g e

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2.2.2 Calculation of the cluster reflectionmatrix 36

2.2.3 The model 382.2.4 Inputs to the calculation 402.3 The Multiple Scattering-Xa approach 43

CHAPTER 3. THE LAYER KKR METHOD

3.1 Introduction 473.2 Discussion of the layer KKR technique 483.3 Spherical wave expansion of G 523.4 Plane wave expansion of G+ 563.5 Core levels 583.6 Energy and integrations 583.7 Calculation of the Fermi level 623.8 The self-consistency loop 633.9 The muffin tin zero 643.10 The surface barrier 64

CHAPTER 4. RANGE EFFECTS IN CATALYST POISONING

4.1 Introduction 654.2 Poisoning on the (100) surface 664.2.1 Review of experimental and theoretical

studies for nickel (100) 664.2.2 Results of p(r,E) for various systems;

the role of electronegativity and surf­ace geometry 69

4.2.3 Comparison with the experiments of vKiskinova and Goodman 75

4.3 Poisoning on the (111) surface 824.3.1 Results and discussion 824.3.2 Conclusions and implications for

catalysis 874.4 Self-consistent layer calculations for

clean and sulphided nickel (100)

C O N T E N T SP a g e

89

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CHAPTER 5. LOCAL INTERACTIONS BETWEEN CO ANDPOISONED NICKEL SURFACES

5.1 Introduction 925.2 Results and comparison between the two

cluster approaches 945.3 The effects of sulphur on carbon mon­

oxide chemisorption on nickel (100) 985.4 The influence of sulphur on surface

charge mobility 1085.5 Discussion 112

CHAPTER 6. LOCAL INTERACTIONS BETWEEN CO ANDPROMOTED NICKEL SURFACES

6.1 Introduction 1146.2 Results for alkali metal promoters 1186.3 Discussion 128

APPENDIX A DERIVATIONS OF FORMULAE IN CHAPTER 2

a.l Calculation of shell scatteringmatrices 133

a. 2 A note on the matrices 135

APPENDIX B DERIVATION OF FORMULAE IN CHAPTER 3

b. l Spherical wave expansion of G+ 137b.2 Plane wave expansion of G+ 147b.3 Weights and ordinates for the energy

integration 150b.3.1 The semi-circular contour integral 150b.3.2 The Fermi energy contour integrals 151

C O N T E N T SP a g e

REFERENCES 152

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Figure 1.1 Schematic representation of the 20Blyholder model of CO chemisorption.

Figure 1.2 Schematic illustration of the influence of poisons and promoters on the Blyholder model of CO chemisorption. 22

Figure 2.1 Illustration of typical clusters, showing the division of atoms into shells for (a) a cluster with and (b) a cluster without a central atom. 34

Figure 2.2 Illustration of the four shell scattering matrices. 37

Figure 2.3 Nickel 3d phaseshifts. 41

F I G U R E C A P T I O N SP a g e

Figure 2.4 P phaseshifts for C, P, S and Cl. 41

Figure 2.5 The MS-Xa cluster for the gas phase CO 44molecule.

Figure 3.1 The layers of atoms used in (a) a surface and (b) an interface calculation. 49

Figure 3.2 Integration contours for charge density calculation. 59

Figure 3.3 Integration contours for Fermi level calculation. 61

Figure 4.1 Plan view of cluster used to investigate the poison/Ni(100) system. 68

Figure 4.2 Calculation of the changes in the LDOS for(a) C, (b) S, (c) P, and (d) Cl adsorbed on Ni(100). 70

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Figure 4.3 Calculation of the changes in the LDOS for a p(3x3) S overlayer. 70

Figure 4.4 Calculation of the changes in the LDOS for (a) P and (b) C adsorbed at various heights on Ni(100). 72

Figure 4.5 Comparison of the variation of 0 ^ with adatom coverage compared to the calculated catalyst "activity". 76

Figure 4.6 Comparison of the rate of methanationwith adatom coverage compared to the calculatedcatalyst "activity". 76

Figure 4.7 Comparison of the rate of methanation with adatom coverage compared to the calculated catalyst "activity", for various ranges of the interaction. 78

Figure 4.8 Comparison of the rate of methanation with adatom coverage compared to the calculated catalyst "activity", for different models of poisoning. 78

Figure 4.9 Plan view of cluster used to investigate the Ni(lll) and Rh(lll) / S systems. 81

Figure 4.10 Calculation of the LDOS~:for the clean and sulphided surfaces of (a) Ni(lll), and (b) Rh(lll). 81

Figure 4.11 The partial charge densities for clean and sulphided Ni(111). 83

Figure 4.12 Band structure for Ni. 85

F I G U R E C A P T I O N SP a g e

Figure 4.13 Sulphur p phase shift. 85

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Figure 4.14 MTDOS for the Ni(100) surface, calculated within the layer KKR scheme. 88

Figure 4.15 MTDOS for Ni(100) with a p(lxl) S over­layer, calculated within the layer KKR scheme. 90

Figure 5.1 Top view of the cluster used in the MS-Xa calculation for CO coadsorbed with S on Ni(100) and the cross-sections used for displaying wavefunctions. 93

Figure 5.2 Top view of the cluster used in the LDOS calculation, showing the near adsorption sites. 93

Figure 5.3 Comparison of the LDOS and MS-Xa levels, for the entire cluster and those with weight on the central Ni atom, (a) Clean Ni. (b) Ni with four co­adsorbed S atoms. 95

Figure 5.4 MS-Xa orbital energy levels of A^(a) and E(ti) symmetry for the entire cluster and those with weight on the central Ni atom for CO adsorbed on clean and sulphided Ni(100). 97

Figure 5.5 Wavefunction contour plots for the Ni/CO 5a, and the Ni/S/CO 5a, in both cross-sections. 99

Figure 5.6 Wavefunction contour plots for the Ni/COItx, and the Ni/S/CO In, in both cross-sections. 102

Figure 5.7 Wavefunction contour plots for the Ni/CO2k , and the Ni/S/CO 2n , in both cross-sections. 104

Figure 5.8 Schematic illustration of the Ni-S-CO interactions. 106

F I G U R E C A P T I O N SP a g e

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FIGURE CAPTIONS

Figure 5.9 Wavefunction contour plots for orbitals on the clean Ni surface, for (a,b) representative orbitals near the Fermi level and (c,d) representative orbitals close in energy to the 5a. 107

Page

Figure 5.10 Wavefunction contour plots for orbitals on the sulphided Ni surface, for (a,b) representative orbitals near the Fermi level and (c,d) represen­tative orbitals close in energy to the 5a. 109

Figure 5.11 Wavefunction contour plots of the CO 5a orbital in (a) the gas phase and (b) the clean Ni surface and (c) the sulphided Ni surface. Ill

Figure 6.1 Top view of the cluster used in the MS-Xa calculations for carbon monoxide and coadsorbed promot­ers on nickel (100), and the cross-sections used for displaying the wavefunctions. 116

Figure 6.2 Wavefunction contour plots the molecular and chemisorbed carbon monoxide levels 5a (a,b), lrt (c,d) and 2n * (e,f). 117

Figure 6.3 Direct comparison of the MS-Xa orbital energies and the partial density of states in the carbon muffin tin calculated by Wimmer et al (1985) for Ni/CO and Ni/K/CO. 119

Figure 6.4 MS-Xa orbital energy levels of Al(a) and E(n) symmetry for the entire cluster and those with appreciable weight on the central nickel atom for carbon monoxide adsorbed on a clean nickel (100) sur­face, and with coadsorbed lithium in the geometry of Figure 6.1. 121

Figure 6.5 Wavefunction contour plots for the Ni/CO 5a, and the Ni/Li/CO 5a, in both cross-sections. 122

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FIGURE CAPTIONS

Figure 6.6 Wavefunction contour plots of the Ni/COIn, and the Ni/Li/CO lTt, in both cross-sections. 124

Page

Figure 6.7 Wavefunction contour plots for the Ni/CO * &2 k , and the Ni/Li/CO 2rt , in both cross-sections. 125

Figure 6.8 Schematic illustration of the effects of Figs. 6.6 and 6.7, showing the influence of lithium on the ti bonding. 127

Figure b.l Schematic view of plane waves leaving layer i. The beam amplitudes are shown in paren­theses. 139

Figure b.2 Schematic view of the final set of plane waves in the interstitial region surrounding layer i.The beam amplitudes are shown in parentheses. 139

Figure b.3 Schematic view of plane waves leaving layer i in the - sense (towards the surface), including all multiple scattering between the layer and the bulk. The beam amplitudes are shown in parentheses. 141

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TABLE CAPTIONSP a g e

Table 1.1 Surface science analytical techniques. 18

Table 2.1 a parameters and occupancies of the p- levels of C, Cl, P and S. 43

Table 4.1 Pauling electronegativities of C, Cl, N, P and S. 67

Table 4.2 Maximum relative change in the sulphur potential. 91

Table 5.1 Orbital energies, with respect to the Fermi energy, of the 4a, 5a and In levels of CO on clean and sulphided Ni(100). 100

Table 5.2 Orbital weight and angular momentum break­down for the 5a. 101

Table 5.3 Orbital weight and angular momentum break­down for the In. 105

Table 5.4 Orbital weight and angular momentum break-*down for the 2n . 108

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ACKNOWLEGEMENTS

I would like to thank my supervisor Professor J.B. Pendry for his interest, help and encouragement during the course of this work. I am also indebted to Dr. R.W. Joyner for his advice and discussions relating to catalysis, and to Dr. D.D. Vvedensky for many stimulating discussions (and his coffee machine!). I have also benefited from the help and ideas of my colleagues at Imperial College. Some of the coding of the layer KKR program was done by Carl Larsson and Richard Blake, and I gratefully acknowledge their help in the early stages of this project. I would also like to thank Dr. M.E. Eberhart and Professor K.H. Johnson for making an updated version of the MS-Xa program available to me. Lastly, I would like to thank my parents for their support over the years, and my brother for his unique sense of humour. Financial support was provided by British Petroleum PLC.

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PREFACE

The organisation of this thesis is as follows. Chapter 1 outlines some background discussion on catalysis with particular reference to the methanation of carbon monoxide by hydrogen. For this system, we review the experimental results which highlight possible reaction pathways and rate determining steps. We also review, in this chapter, several models which have been used to probe surface chemical bonding. In chapters 2 and 3 the theory of the cluster and layer models, used in this thesis, is developed, including a discussion of the approximations, advantages and drawbacks of each method. The derivations of the formulae are in Hartree atomic units, unless stated to the contrary.

Chapters 4, 5 and 6 contain the results of these calculations. In chapter 4 we consider how perturbations in the surface electronic structure, induced by poisons, vary asxa function of both position and coverage. The results are compared quantitatively to experimental studies of poisoned Ni(100) surfaces. Chapter 5 looks at local effects, including the bonding of carbon monoxide to Ni(100) with and without coadsorbed poisons, and the influence of sulphur on the surface nickel charge. The last chapter, chapter 6, considers the influence of catalytic promoters, such as the alkali metals.' We examine changes in carbon monoxide bonding on promoted surfaces, looking specifically at the role of the carbon monoxide In and 2n molecular .* orbitals in the chemisorptive bond, the effects of electrostatics, and direct interactions on the promoted surface.

Some of this work has either been submitted, or accepted for publication, and is referenced in the text.

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CHAPTER 1

Introduction

1.1 Background

Catalytic processes occur in nearly all chemical and biological syntheses. Typically, industrial catalysis are made from small particles of active material dispersed on an inactive support with a large surface area. Many materials are used in commercial catalysts ranging from metals to insulators. In this brief review, however, we restrict ourselves to a discussion of metal catalysed systems (Bond 1962), and in particular to transition metal surfaces. Transition metals form the basis of many of the catalysts used in the chemical industry, for example in the Fisher-Tropsch sy. nthesis (FTS) of hydrocarbons (Ponec 1982), nickel and rhodium are frequently used (Bell 1981), whilst in the ammonia synthesis iron catalysts are common (Boudart 1981). In initial theories of transition metal catalysts (for a review see Spencer and Somorjai 1983) it was proposed that those metals which bind reactants and products either too strongly or too weakly do not make the best catalysts. This is an observation which has been found to be predominantly true. Taking the FTS synthesis as an example, molybdenum has a low activity towards methane formation since the bonding of carbon and oxygen to the surface is too strong, whereas the origin of the inactivity of copper is that the "binding is too weak (Ponec 1982), in fact copper does not dissociate molecu- larly adsorbed carbon monoxide (Broden et al 1976).

Perhaps one of the simplest reactions with the greatest commercial potential is the FTS of synthetic fuels. This reaction, involving the hydrogenation .of carbon monoxide to produce alkanes, has had a long histo­ry. Methane production using a nickel catalyst was first reported in 1902 (Sabatier and Senderens 1902), and in the following years longer chain hydrocarbon synthesis was established. Interest in the FTS has, over the years oscillated. When oil displaced coal as the input to

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chemical and fuel syntheses, interest in the FTS waned, however with subsequent "oil crises" in the seventies interest was revived, in an attempt to be in part indepen­dent from oil. Indeed the SASOL company in South Africa produces one million barrels of fuel this way today (Dry and Hoogendoorn 1981; Spencer and Somorjai 1983).

Studies of catalytic systems were traditionally based on two themes: (1) isotopic labelling of certain reactants (Emmet 1972) and (2) kinetic measurements using, for example, gas chromatography to measure concentrations of reactants and products as functions of time, temperature and pressure. Then, this data was analysed to provide information about reaction pathways, intermediates and possible rate determining steps. More detailed probes into catalytic activity have been hampered by the lack of knowledge of local atomic geometry of the active sites and reaction intermediates on the catalyst surface. Recent SEXAFS studies (for a review of these results see for example Sinfelt et al 1984) of Osmium/Copper andRuthenium/Copper catalysts have gone some way to finding the average coordination and local environment about each of the components. In the FTS, mentioned above, there existed a controversy about the reaction intermediates. The route via surface carbon, formed by the dissociation of carbon monoxide was established by Araki and Ponec(1976). In a series of experiments, the catalyst surface

13 12was precovered with C. After exposure to H- and CO13the initial methane yield consists of mainly CHv, only

13 ~ q 12when most of the surface C had been exhausted did CH^start to be produced.

These experiments have shown that carbon monoxide dissociation is an important step in FTS. However, the dissociation of carbon monoxide, in common with many other diatomic molecules, is known to be structure sensitive occuring to a greater extent on open surfaces than on close packed surfaces, whilst the overall methanation rate is structure insensitive (Goodman 1984a, b). This clearly shows that the dissociation step cannot be rate limiting. Somorjai (1979) has postulated that active sites such as terraces, steps and vacancies should facilitate, in a

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similar manner to open surfaces, molecular dissociation.Bell (1981) has proposed a reaction sequence with the ratedetermining step as a hydrogenation stage as followsCH +yH - > CH , . to explain the kinetic dependence of x x*t*ymethane formation on partial pressures of carbon monoxide and hydrogen, though he notes that the identification of rate determining steps cannot be unambiguously obtained from kinetic data. To elucidate the breaking and making of C-H bonds hydrogen-deuterium exchange techniques, to measure the corresponding changes in reaction rates and products, have been used (Burwell 1973).

The development and application, over the lastfifteen years, of many surface science analytical tech­niques ,see Table 1.1 for a summary of the most frequently used techniques, have provided many insights into chemisorptive and catalytic properties of the single crystal surface (Joyner 1982). Moreover, thereproducibility of such studies has allowed a much clearer picture of surface reactions to be drawn. For example, the surface bonding geometry of many atoms and molecules at surfaces is now known.

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Table 1.1 Surface science analytical techniquesSurface Analysis Technique_________Information obtainedLow Energy Electron Atomic positions atDiffraction (LEED) surfaceAuger Electron Spectroscopy (AES)High-Resolution Electron Energy Loss Spectroscopy (HREELS)X-Ray and Ultra-violet Photoelectron Spectroscopy (UPS), (XPS)Inverse Photoemission (BIS)

Extended X-ray Adsorption Fine Structure (EXAFS)

Temperature Programmed Desorption (TPD)

Chemical identification of surface adsorbatesSurface bonding of adsorbates

Electronic structure of surfaces and adsorbates (occupied states)Electronic structure of surfaces and adsorbates (unoccupied states)Radial distribution function and local bonding geometryBinding Energies of adsorbates

In its simplest form the FTS reduces to the methanation of carbon monoxide by hydrogen:

CO + 3H2 -> CH4 +H20.The suitability of this reaction for study both by

the comparatively recent surface science techniques and theoretical approaches arises for the following reasons:(1) whilst microscopic reaction pathways are still not fully understood in any detail, sufficient information has been obtained from kinetic and isotopic studies to provide a basis for a more detailed theoretical and surface science approach;(2) the simplicity of the reactants allows the elucida­tion of surface bonding and the study of reaction interme­diates ; and(3) calculations of surface properties such as surface electronic structure and the nature of the chemisorptive bond, based on experimentally determined geometry, can be made.

Catalysts, unfortunately, do not have an indefinite lifespan, eventually they become poisoned by the

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impurities present in the reactant gases. Catalytic regeneration/replacement is then necessary, which can be both expensive and time consuming. It is therefore of technological importance, a fact reflected by the large volume of literature in this field, to understand how the poisoning arises with the aim of developing catalysts with a greater resistance. The opposite effect, namely cata­lytic promotion, is of equal importance. This usually involves adding other impurities to the catalyst to improve efficiency, selectivity or increase operational lifetime. Catalysts used in the methanation reaction are known to be extremely sensitive to poisoning, in particu­lar to electronegative elements such as sulphur (Oudar 1980; Hegedus and McCabe 1981). For example, coadsorption of 0.25 monolayers of sulphur is sufficient to totally inhibit methane formation (Goodman and Kiskinova 1981).

1.2 Motivation

The final goal of any theoretical study of catalytic systems, and the modification by coadsorbates, is to be able to design better and more efficient catalysts. The development of theoretical approaches that can account for established experimental data, while having the flexibili­ty to explore possible reaction paths is therefore an important step in being able to design better catalysts.

The first step towards reaching this goal is an understanding of the salient features of catalytic sys­tems. We choose to study the methanation reaction, discussed above, with specific reference to carbon monox­ide chemisorption and dissociation. There is extensive experimental evidence for this system, on both static and dynamic properties of carbon monoxide adsorbed on clean, poisoned and promoted surfaces. Some of this evidence has already been discussed in the previous section and the rest will be reviewed in subsequent chapters of this work. By performing calculations, our aims are to be able to present a coherent picture firstly of carbon monoxide

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Gas Phase Adsorbed

Figure 1.1 Schematic representation of the Blyholder model of CO chemisorption, showing how the mefcal substrate facilitates transitions from the 5a to the 2ti . The open circles and small arrows indicate unoccupied and occupied electronic spin-orbit^ls respectively. The hopping rates 5a -> Ef and Ef -> 2rt are proportional to the density of metal states at the Fermi level, according to Fermi's Golden Rule.

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chemisorption, both on clean surfaces and those in the presence of coadsorbates. Then, secondly, to find fea­tures of the interactions which are not specific to this system, which can be used to predict the influence of coadsorbates on reactions involving dissociative steps.

The philosophy of our approach is to identify, using a variety of model calculations, properties of a surface which can be related to catalytic activity. We choose the surface electronic structure and show how it is influenced by coadsorbates and that features present in it are important for the chemisorption and dissociation of simple molecules, illustrating this with calculations for clean, poisoned and promoted surfaces both with and without coadsorbed carbon monoxide.

1.3 The Blyholder model of molecular chemisorption and the role of electronic structure

Blyholder (1964) proposed a model for diatomicIt

molecular adsorption on metal surfaces. Based on Huckel molecular orbital calculations, and using carbon monoxide on nickel as an example system, chemisorption arises from the gradual filling of an initially unoccupied

ieanti-bonding orbital (the 2 n for carbon monoxide) com­bined with the partial emptying of one of the filled molecular orbitals which is directed towards the surface (the 5a for carbon monoxide). This charge rearrangement ensures, to some degree, charge neutrality within the molecule, provides the binding to the surface and weaken­ing of the intramolecular bond. Filling of theanti-bonding orbital is a generally accepted mechanism by which the dissociation step occurs (N0rskov et al 1984). This model is illustrated schematically in Fig. 1.1.

Chemisorption and dissociation, within this model, is*related to charge transfer from the 5a to the 2ti orbital.

The influence of the metal catalyst, typically with a large density of states at the Fermi level, is to provide an easier route via a two-step model (Fig. 1.1). In particular, in a single electron picture, the electronic

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Ficrure 1.2 Schematic illustration of the influence of poisons and promoters on the Blyholder model of CO chemi­sorption (cf Fig. 1.1). The open circles and small arrows indicate unoccupied and occupied electronic spin-orbitals respectively. According to the indicated direction of charge flow, the poison decreases and the promoter in­creases the density of substrate levels at the Fermi energy.

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structure of the metal appears, through Fermi's GoldenRule, as the product of the local density of states at theFermi level and the matrix elements for the transitions 5a

*-> Ef and Ef -> 2n . Increased hopping rates, therefore,decrease the lifetimes for these states, particularly for *the 2ti because of its close proximity to the Fermi level.*This results in a broad molecular 2ti resonance, account­ing for the partial occupancy of this state for chemisorbed carbon monoxide. Although the Blyholder model and modifications thereof provide a useful framework for interpreting many of these results, the microscopic mechanisms by which coadsorbates either poison or promote catalytic reactions are still not well-understood. More detailed model Hamiltonians such as the Newns-Anderson model, which will be discussed in the next section, . also involve the Fermi level density of states in the broaden­ing of the molecular levels, and hence in the occupancy of molecular resonances.

Previous theoretical studies (Joyner et al 1984; MacLaren et al 1985, 1986 a-e; Feibelman and Hamann 1984, 1985), which we will discuss in the next section, have indicated that there is a correlation between perturba­tions in the electronic structure around the' molecular adsorption site and changes in catalytic activity. Specific emphasis has been placed on the Fermi level density of states, which for carbon monoxide adsorption can be easily illustrated within the Blyholder model (the influence of poisons and promoters is shown schematically in Fig. 1.2).

The argument given above provides some justification for the role of electronic structure in catalysis. However this is clearly an oversimplification, for al­though chemisorption, vibrational modes and hence energy exchange with the surface can be understood in terms of perturbations in the local electronic structure, the lack of knowledge of the microscopic details of catalytic reaction pathways makes detailed comparison to changes observed in product-yield difficult to quantify.

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1•4 Theoretical approaches to catalysis and chemisorption

Fundamental to any of the theoretical approaches to the properties of adsorbates bonded to surfaces is a description of the che'misorptive bond between reactants, products, intermediates and the catalyst surface. The discussion of the models is by no means exhaustive, but provides a brief overview of some of the techniques which have been applied to both chemisorption and catalysis. Those theoretical approaches, which we will review in this chapter, can be broadly classified into electronic struc­ture approaches, semi-analytic schemes based on model Hamiltonians, and arguments based on ideas such as the importance of orbital symmetry, bond mobility, and geomet­ric factors. Most of the discussion will be based around those models which have been directly applicable to catalysis, though for completeness the important models of chemisorption will also be mentioned.

1.4.1 The Anderson model

A model Hamiltonian, proposed by Anderson (1961) to discuss the formation of local magnetic moments, has been the most popular of those used to study chemisorption. The localised orbital in this model, which belongs to the adsorbate, is coupled to a continuum of metal states resulting in a broadened and down-shifted resonance and in some cases partial occupancy. Newns (1969) applied the model to hydrogen chemisorption on transition metal surfaces. Generalisations to carbon monoxide chemisorption by including two localised orbitals was done by Doyen and Ertl (1974). Matrix elements needed for a quantitative analysis of the model have been estimated .by Doyen and Ertl for the case of carbon monoxide on nickel. The model was then used to predict the binding energy of carbon monoxide on various sites and various crystal faces. This model is particularly useful for examining trends in chemisorption, though intrinsic difficulties in evaluating matrix elements limit its quantitative use.

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1.4.2 Orbital symmetry considerations

Woodward and Hoffman (for a review see Woodward and Hoffman 1968) demonstrated that orbital symmetry factors, as well as energy differences, can have a dramatic influ­ence on reaction rates of organic and inorganic systems. Banholze et al (1983) have applied a similar analysis to the decomposition of nitrous oxide on platinum surfaces. Symmetry restrictions will be less stringent on a surface, however, since there are typically many orbitals of the appropriate symmetries to couple with the molecular levels. The conclusions of these studies, in agreement with the experimental data, are that the dissociation of nitrous oxide occurs on (100) and to a limited extent on the (110) surfaces, but the (111) surface is inactive for nitrous oxide bond breaking. An extension to active sites such as stepped surfaces has been done. The authors conclude that those surfaces which have (100) terraces should be most active.

1.4.3 Charge mobility

An alternative philosophy has been proposed by Haydock (1981). In his model of catalytic molecular dissociation, the relevant quantity is the ease of ’’trans­fer of bonds” in the coupled system. He argues that surfaces whose bonds are more mobile should, as a conse­quence, be catalytically active. The "transfer of bonds”in, for example carbon monoxide chemisorption, arises from*charge transfer from the 5 a to the 2ti via the metal. We shall explore this viewpoint in more detail in chapter 5 , where we consider that one of the possible mechanisms by which coadsorbed sulphur poisons catalytic reactions is .to decrease surface charge mobility.

1.4.4 Geometric factors

Marks and Heine (1985) have advocated that local

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geometry can be important in determining catalytic activi­ty. Those catalytically active surfaces are noted to be surfaces which reconstruct easily and so the reconstruc­tion itself may be of importance. The reconstructions of noble and transition metal surfaces are determined between competing sp and d electron forces (Heine and Marks 1986). The mobile sp electron gas is forced away from its equi­librium density by the pairwise d electron interactions which are repulsive (attractive) for filled (partially filled) d bands. At surfaces, rehybridisation between s, p and d electrons alleviates these stresses and as a con­sequence induces electronically driven reconstructions. Poisons and promoters, within this framework, deplete and fill the d-band. respectively and thus induce reconstruc­tions. Experimental observations of Edmonds and McCarroll (1977) provide evidence of poison and promoter induced local reconstructions of nickel surfaces. We see evidence of hybridisation effects at surfaces, in our calculations, states which are antibonding tend to escape into the vacuum, whilst those which are bonding are kept predomi­nantly within the crystal. Changes in the hybridisation, induced by coadsorbed sulphur help to illustrate the dramatic perturbation induced, and the decreased surface reactivity which accompanies it.

1.4.5 Interactions between adsorbates

Indirect, as well as direct, electronic interactions mediated by the substrate conduction electrons are impor­tant in deciding the geometry and bonding of adsorbates on surfaces> ranging from the properties of promoted and poisoned systems to the formation of ordered overlayers. Grimely (1967a, b), Einstein and Schrieffer (1973) exam­ined the interactions between adsorbates on a surface using a tight binding model. The results of their calcu­lations showed an oscillatory variation of the interaction energy with separation, the analogue of bulk Friedel oscillations, which can be used to predict the structure of stable ordered overlayers on surfaces. The interaction

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is frequently repulsive/attractive on nearest/next nearest neighbour sites respectively thus explaining the absence of the p(lxl) structure for many adsorbates. A good example of this is sulphur adsorption on the (100) surface of nickel. Both the p(2x2) and c(2x2) (the saturation coverage) overlayers are observed in LEED experiments. However no p(lxl) structure has ever been identified. Although this scheme has been applied to the interactions between simple identical adsorbates, the analysis has not been extended to more realistic substrate densities of states or to more complex coadsorption systems such as chemisorbed carbon monoxide on a sulphided nickel surface. Muscat and Newns (1980), based on a cluster multiple scattering calculation, were able to compute the interac­tions, as a function of separation, between hydrogen atoms on a nickel (111) surface. The model, which included both sp and d electron interactions, showed that the energy of interaction of hydrogen atoms on a p(lxl) lattice would be repulsive, in agreement with the overall conclusions of the work of Einstein and Schrieffer.

1.4.6 Semi-empirical methods

» i1.4.6.1 The Extended Huckel model

The increased difficulty, imposed by the reduction of symmetry at the surface, has meant that initial chemisorption models tended to be based on semi-empirical

i tapproaches. One such scheme is the Extended Huckel (Hoffman 1963) method applied by Blyholder (1964) in his model of molecular chemisorption, and by Andreoni and Varma (1981) in a calculation of the binding energy .of carbon monoxide on many transition metal surfaces. The approach is non-self-consistent, and. is based on casting the Schrodinger equation into the form of a secular determinant whose basis functions are the exponential Slater orbitals. Matrix elements of the Hamiltonian are restricted to nearest neighbour interactions, and are

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parameterised in terms of atomic ionisation data. Diffi­culties in the reliable calculation of matrix elements, particularly for transition metals, and the lack of self-consistency are the major criticisms of the approach.

1.4.6.2 TheLCAO model

One of the first calculations to include both sub­strate, adsorbate and carbon monoxide was performed on a single layer of iron by Benziger and Madix (1980). This LCAO (Linear Combination of Atomic Orbitals) calculation showed that adlayers of sulphur, carbon and oxygen reducedcarbon monoxide binding to iron, by competing with the

*same d-orbitals that the carbon monoxide 2n level inter­acted with on the clean surface. Potassium, on the other hand, increased the binding energy of carbon monoxide by adirect interaction between the potassium 4s levels and the *2n . In chapters 5 and 6, of this thesis, we present calculations of carbon monoxide on clean, poisoned and promoted surfaces. Potassium and lithium do indeed show a direct orbital overlap effect. However, we find that electrostatic effects play a major contribution in in­creasing the bonding of carbon monoxide to the surface. Electronegative adsorbates do compete for the d-orbitals required by carbon monoxide to form chemisorptive bonds. In contrast to the calculation of Benziger and Madix, however, sulphur atoms interact almost exclusively with the 5a and the nickel orbitals in this energy range, whichmix to form the chemisorbed 5a level, and not to any great*extent with the 2n .

1.4.7 Jellium models and the Effective Medium Theory

The difficulties of reduced symmetry at surfaces can be circumvented by using the jellium model, in which the positive ion cores are smeared out into a uniform positive background. The potentials of the ion cores can then, if desired, be added through first order perturbation theory.

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Nprskov and Lang (1980) proposed that the influence on the chemisorption of simple molecules from electronic pertur­bations induced by electropositive and electronegative adsorbates on catalyst surfaces could be described in terms of an Effective Medium Theory. The theory, as first presented (Ndrskov and Lang, 1980) was applied to estimat­ing the binding energy of a single adsorbate on a jellium surface, and then extended to catalytic systems in a series of papers by N0rskov et al (1984) and Lang et al (1985).

In this model the Kohn-Sham (1965) density functional equations are solved self-consistently. Changes in the electrostatic potential are used to predict the influence of coadsorbates on molecular chemisorption and dissocia­tion. The argument for a molecule with nett-charge transfer away from the surface, for example carbon monox­ide, proceeds as follows. Electropositive coadsorbates (such as potassium) would be expected to lower the elec­trostatic potential, the work function and the energy barrier to dissociation and thus stabilise molecular chemisorption. Sulphur adsorption, on the other hand, would show the reverse, weakening chemisorption and inhibiting dissociation. Molecular chemisorption involv­ing nett-charge transfer towards the surface should be poisoned (promoted) by electropositive and (electronegative) adsorbates respectively. The model is clearly an oversimplification, since most metallic cata­lysts are transition rather than s-p metals, with d-electrons, which are poorly represented within the jellium approximation. Changes in electrostatic potential occuring with alkali metal coadsorption are much greater than the corresponding changes seen say in coadsorbing sulphur, hence the model is expected to be more realistic for the former systems. Indeed this is shown to be the case both in our calculations in chapters 5 and 6, and in the experimental studies of alkali metal covered surfaces.

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1.4.8 Electronic structure calculations

1.4.8.1 Slab and Layer calculations

Layer approaches are usually extensions of the bulk counterparts from band structure theory, and of the many methods utilised in bulk calculations, the Pseudopotential, Linearised Augmented Plane Wave (LAPW), Korringa (1947), Kohn and Rostocker (1954) (KKR) multiple scattering model and Linearised Muffin-Tin (LMTO) schemes have been applied to surface calculations (for a detailed review see Inglesfield 1982) . In chapter 3 we discuss a technique for solving electronic structure using a layer KKR approach, therefore we leave a discussion of many of these models to this chapter.

LAPW calculations of catalyst surfaces have been performed by several groups. Feibelman and Hamann (1984, 1985) extended the LAPW technique of bulk band structure to calculate surface electronic structure, and in particu­lar the Fermi Level Local Density of States (FLLDOS). Changes in catalyst activity, in their model, are related to perturbations in the FLLDOS.

In the previous section, the role of electronic structure, in particular the FLLDOS was discussed, hence we will not repeat it here. Results obtained for poisoned and promoted surfaces are consistent with experimental data for the systems they considered. A more detailed discussion of their work will be left until later chapters where a comparison to our calculations will be made.

Wimmer et al (1985) applied the LAPW technique to include on the surface chemisorbed carbon monoxide. In chapter 6 we provide a detailed comparison between density of states and energy eigenvalues obtained from our cluster calculations.

1.4.8.2 Cluster calculations

The philosophy behind cluster calculations is that

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whilst the distribution of energy levels in a cluster only resembles the bulk density density of states for a very large number of atoms, the chemisorptive bond is spatially localised and as such a reasonable model of it can be made using small clusters to represent the local bonding geometry. The fundamental information obtained from an electronic structure calculation is the orbital energies and wavefunctions of those levels involved in the bonding.

Cluster formulations of chemisorption systems (for a review see Messmer 1979, 1985; Kunz 1980) include:(1) the semi-empirical approaches (discussed in the previous sections);(2) simple treatments, based on Local Density Functional Theory (LDFT), such as Green function Local Density of, States (LDOS) calculations (Joyner et al 1984; MacLaren et al 1986a-c), the multiple scattering models of Muscat and Newns (1980) and the Multiple Scattering-Xa (MS-Xa) model (Slater and Johnson 1972; Johnson and Smith 1972);(3) Hartree-Fock calculations; and(4) sophisticated Configuration Interaction (Cl) calcula­tions.

The LDOS and MS-Xa LDFT models are both used in this o-Pwork; details^the methods and the approximations used are

reviewed in chapter 2. The Hartree-Fock equations, obtained from a variational minimisation of the total energy with respect to an anti-symmetric product of single particle wavefunctions, result in a non-local exchange potential. It is the non-local nature of the exchange potential which leads to the incorrect isolated atom limit. Local exchange-correlation potentials, such as Slater statistical Xa form, ensure that this limit is reproduced. Modifications of the Hartree-Fock equations to take account of this have been suggested. Calculations by Rosen et al (1976) for carbon monoxide chemisorbed .on nickel, and by Paul and Rosen (1982) for carbon monoxide chemisorbed on copper use this Slater form of the exchange- correlation potential in the so called Hartree-Fock-Slater approximation. The Schrodinger equation is cast into a secular equation using a Slater orbital basis. Matrix elements of the full potential are calculated using the

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discrete variational technique of and Ellis and Painter (1970). The model of carbon monoxide chemisorption, which is consistent with the Blyholder model, based on these calculations shows the same overall features as our results for carbon monoxide on nickel.

The degree of sophistication gained by including electron correlation, however, would not be expected to change qualitatively the conclusions of our studies on clean, poisoned and promoted surfaces, since the effects we observe are large chemical perturbations. Two of the schemes which do include electron correlation are Cl type calculations, in which the ground state wavefunction is made from, for example, the Hartree-Fock ground and low lying excited state wavefunctions. Hermann et al (1984) have considered carbon monoxide chemisorption on a nine atom copper cluster using an multi-configurational SCF molecular orbital basis, the computational difficulties of this scheme restrict the size of cluster which can be used. The Generalised Valence Bond approach of Goddard has been applied to calculate the binding energy carbon monoxide on a nickel surface (Allison and Goddard 1982).

1.4.8.3 Summary of Electronic Structure Schemes

In all the electronic structure schemes described above, there is a balance between accuracy and complexity. The more sophisticated cluster (layer) calculations .are restricted to small clusters (surface unit cells) and lose flexibility as they gain accuracy. Since catalytic promotion and poisoning is accompanied by fairly large perturbations in electronic structure, it is therefore not necessary to use the very accurate type of cluster calculation if only the overall, rather than the detailed, features of the interaction are required.

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CHAPTER 2

The Cluster Models

2.1 Introduction

In this work we have used three models to calculate surface electronic structure. The two cluster models are discussed here, whereas the extended layer approach, which is still being developed, will be reviewed in detail in chapter 3. Each method is best suited to one particular aspect of the influence of surface impurities on catalytic activity. The local density of states (LDOS) cluster approach, which is non self-consistent, is suitable for studying the adsorption of electronegative elements such as sulphur on metal surfaces and examining the extent of the lateral electronic perturbations induced on the surface due to these adsorbates. The self-consistent multiple scattering-Xa (MS-Xa) approach (Slater and Johnson 1972; Johnson and Smith 1972; Johnson 1973) ,on the other hand, is more sophisticated and is able to look in detail at molecular adsorption and the interactions between coadsorbates, the substrate and chemisorbed molecules. It is difficult, however, with this model' to calculate the lateral extent of electronic perturbations since large clusters are awkward to deal with. A review of the theory, and the advantages and disadvantages of these cluster models will form the basis of this chapter; the applications of these models will be left for later chapters.

2.2 The LDOS cluster method

The electronic structure of the catalyst surface is characterised, within this model, by the local density of states (LDOS) p(r,E). The method used to calculate the density of states uses the relationship

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Figure 2.1 Illustration of a typical cluster showing the division of atoms into shells for (a) a cluster with a central atom and (b) a cluster with no central atom.

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+p(r,E) = -1/n Im G (r,r,E) { 2 . 1 }

where G is the Green function of the cluster, obtained by solving the Dyson equation

This is

+ + + _ + G = G + G T G o o o { 2 .2 }

Gq is the free space Green function, ie the Green function in the absence of any scatterers. T is a reflection coefficient, which describes the scattering of electrons placed in the solid at a point r with energy E. In solving the Dyson equation the potential is approximated by its muffin tin form. The potential within each contig­uous sphere is spherically averaged, and volume averaged (a constant Vmtz^ within the intersphere region. Within this approximation the atomic t matrices, in the angular momentum basis, are diagonal and the wavefunctions in the intersphere region can be expanded in terms of spherical harmonics and linear combinations of spherical Bessel (j ) and Hankel (h^) functions. At valence and conduction band energies partial, wave expansions about each atomic site converge rapidly, typically 1=2 is sufficient.

The cluster can be chosen either to have an atom at the centre (used in calculations for the (111) surface) or not (used in calculations of the (100) surface), see Fig.2.1. The Green function for an isolated atom, G ,a ismodified by adding the corrections due to scattering from the surrounding atoms in the cluster. To get a formula' for G for clusters with no central atom the atomic t matrices for the central atom are set to zero. This leads to a single centre expansion G+ from which £>(r,E) is obtained.

2.2.1 Derivation of G+

If both r and r’ are outside the central atommuffin-tin, and if r > r 1 then we can write G* as— — a

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G*(r,r',E) = -4iK E Y*(r'){j1(Kr')+t1hJ(Kr')} h*(Kr)YL(r){2.3}

note that the correct analytic continuation of the spheri-* mcal harmonic is "1 Y^-m(r). However, since the

argument is the angle of a real vector r, complex conjuga­tion yields the same result as the analytic continuation. The outgoing waves scatter back and forth between the central atom and the rest of the cluster. The resulting geometric progression can be summed exactly to give

G+(r,r',E) = -4iK E Y*(r') {jx(Kr' ) }

[l-tR]^„{SL„L ,hl, (Kr)+RL„L , jx, (Kr)}YL,(r) {2.4}

for equal arguments this can be written as

G+(r,r ,E) = Gt(r,r ,E) - 4iK E Y*(r)$.(r) elSla — — j_i x

[R(l-tR)_1]LL, e6l,$1,(r)YL ,(r) {2.5}

This is the same as the formula for the cluster Green function as used in the calculation of X-ray adsorption spectra (Durham et al 1982). is the regular radialwavefunction for energy E, and t is the atomic t matrix related to the logarithmic derivatives of at the sphere boundary, 6^ by

t1 = i [e2l6l -1] {2.6}

K, the intersphere electron propagation vector given by

K = V [2(E-Vmtz}J {2.7}

2.2.2 Calculation of the cluster reflection matrix

To calculate the cluster reflection matrix we appeal to an algorithm for stacking shells of atoms together, similar to that of layer doubling developed for LEED

Iu>-JI

Figure 2.2 Illustration of the four shell scattering matrices. Inward and outward arrows denote the partial waves j,(Kr)Y,(r) and hj(Kr)Y,(r) respectively with the coupling between different partial waves shown.

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calculations (Pendry 1974 p.. 141), but in an angularmomentum instead of plane wave basis. The cluster is divided into shells of atoms for which the reflection and transmission matrices are calculated. These shell matri­ces are then combined, into composite shells, thus reduc­ing the problem of a large matrix inversion into many smaller ones. A further reduction in matrix size is obtained by exploiting any rotational symmetry of the cluster about the surface normal. Rotations of spherical harmonics are easily performed about the z-axis, changing only by the multiplicative phase factor e1 . Thus if the Hamiltonian has an n-fold rotational axis it is easy tosee that the matrix element R, ,, . is zero unless m-m'lm, l mis a multiple of n. This makes the calculation manageable, since the time involved in matrix inversion scales like the dimension of the matrix cubed. Each layer is characterised by four scattering matrices T00,T01,TI0,T11, which are shown schematically in Fig.2.2. If the cluster were just composed of one shell then R would be equal to T°*, however an iterative algorithm allows many shells to be stacked together; this proceeds by combining the outermost shell , shell N, with the next shell in, shell N-l. A composite out-in reflection matrix is obtained by including all the multiple scattering between the shells. Again the resulting geometric pro­gression can be summed exactly to give

R? 1w,N-l = T.PIN-l (1+T^ )T^(1-T^ T ^ V l jiN 1N-11N )-1(l+T^1) {2.8}

Equation {2.8} can be iterated by adding on the N-2 shelland replacing T^by R^N-i to get ^ N - l N-2* The Process is repeated until all the shells in the cluster have been included. The cluster is made large enough to ensure convergence in real space, and typically three shells are sufficient with a total of about thirty atoms.

2.2.3 The model

The theory presented here does not calculate

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self-consistent electronic structure, therefore this must be borne in mind when choosing systems to study. Systems which involve large amounts of charge transfer, such as alkali metal coadsorption, in which a significant fraction of an electron is transferred to the substrate (for example see Heskett et al 1985, 1986), or carbon monoxide chemisorption (for example see Blyholder 1964), where charge transfer is implicit in forming the metal-molecule bonds, are therefore poorly represented by this approach. However, calculations based on electronegative adsorbates, such as sulphur, which involve only a little charge rear­rangement are amenable to this method. Experimental measurements of workfunction changes due to adsorbing jsulphur on nickel (Hardegree et al 1986) and theoretical slab calculations (Feibelman and Hamann 1984, 1985) bear this out.

Whilst this is a drawback of this scheme, the advan­tages gained, on the assumption that a fairly good approx­imation to the self-consistent potential can be found, are that the calculations are manageable both in CPU time and memory ( a typical calculation for thirty energy points in the band takes lmin on a Cray 1-S), and that large clus­ters of atoms can be handled. In the second approach, the MS-Xa self consistent cluster calculations are restricted to typically ten atom clusters. Using these large clus­ters, although we cannot study directly the influence of catalyst poisons on chemisorbed carbon monoxide, we are able to reach large lateral distances between the adsorbate and adsorption site. Thus, we are able to study, in general terms, the range of surface electronic perturbations and the strength of the effect at the adsorption site. In conjunction, the MS-Xa scheme is used to give a more detailed look at local interactions and modifications to the surface bonding due to these changes in surface electronic structure. Coverage effects are of great importance, since much of the experimental evidence is based on measurements of catalytic properties and activity as a function of adsorbate coverage (for a review see Goodman 1984a, 1984b). Studying these effects using layer or slab techniques (Feibelman and Hamann 1984, 1985)

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soon becomes very difficult since unit cells having many atoms are cumbersome and time consuming to deal with. In the cluster model we have described this is not a problem since we only exploit rotational symmetries of the cluster and require no long range periodicity.

2.2.4 Inputs to the calculation

Each atomic scattering potential is characterised by a set of energy dependent phaseshifts 6^(E), which are calculated using the program MUFPOT, developed at the Daresbury Laboratory. The metal substrate potentials were chosen to be the self-consistent bulk band structure potentials of Moruzzi, Janak and Williams (1978); these were used as input to MUFPOT to calculate the phaseshifts for the metal substrate. Phaseshifts for theelectronegative elements used in this work (carbon, chlorine, phosphorus, and sulphur) were constructed by overlapping the neutral atomic charge densities, derived from the dementi and Roetti (1974) parameterised wavefunctions, for bulk crystals of the substrate and the impurity. The resulting charge density is then spherical­ly averaged about each atom, from which a spherically symmetric potential is found. Phase-shifts are calculated via the logarithmic derivatives of a numerically integrat­ed regular solution to the radial Schrodinger equation. The exchange-correlation potential V (r) is calculated

a Q»according to Slater's statistical exchange form (Slater 1972),

Vxa(r) = -3<*C(3/4ti) p(r)]1/3. {2.9}

In the MS-Xa calculations a is taken from Schwartz's (1972) tabulated values, chosen to make the total energy of the isolated atom exactly equal the Hartree-Fock value; however, since our calculation is non self-consistent we allow the a parameter to be adjustable within the range 2/3 (the value for the uniform electron gas) and 1.0. The value of a is chosen to give a neutral atom through the

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Energy (eV)

8 8* 4 2 Ef

Figure 2.3 Nickel 3d phaseshifts calculated from (a) The Moruzzi Janak Williams (1978) bulk band structure poten­tial (-----),(c) MUFPOT, a = 0.8 (-x-x-), (c) MUFPOT, a =0.73 (-•-•-) , and (d) MUFPOT, a = 0.70 (---- )

Energy(eV)

Figure 2.4 P phaseshifts for C, P, S, Cl, with the values from Table 1.

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Friedel sum rule (1958). This relates the number of electrons that a scattering potential draws into itself, when placed in a free electron host, to the phaseshifts evaluated at the Fermi level, (5^(E^)).

n = 2/ti 2 (21+1) 61(Ef) {2.10}

Equation {2.10} is applied to the partially occupied p-states (1=1) of the adsorbate, and the number of p electrons (n ) is adjusted to be as close as possible toxrthat of the neutral isolated atom. The number of p electrons, n , is given byXT

np = 6/ ti 61(Ef) {2.11}

Increasing the value of a makes the atomic potential deeper, since the exchange-correlation potential given by equation {2.9} is negative. This moves atomic resonances to lower energy, effecting most those orbitals which are spatially localised and as such benefit the most from changes in the exchange-correlation potential. We show, in Fig. 2.3, the influence of varying a for a nickel 3-d phaseshift generated by overlapping neutral nickel atomic charge densities, as previously described. Good qualita­tive agreement is obtained in both the shape and in the position of the resonance between the phaseshift calculat­ed from the Moruzzi, Janak and Williams potential and from the prescription of varying a, when a is «0.73. It is interesting to note that this value of alpha is larger than Schwartz's value of =0.71 for nickel. This is to be expected, since self-consistency tends to lower the energy levels and the positions of the resonances, therefore agreement can be obtained by artificially increasing the exchange-correlation potential.

Fig. 2.4 shows the p phase shifts for carbon, chlo­rine, phosphorus and sulphur calculated according to this scheme, with Table 2.1 showing the values of a used and the corresponding occupancies of the p level.

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Table 2.1 a parameters, and corresponding atomic p-level occupancies for C, Cl, P and S.

Element a n„_____________________________________________________ EC .75 2.0Cl 1.0 4.4P 1.0 2.9S 1.0 3.9

Another input to the calculation, derived from many body corrections to the electron states, is the imaginary part of the self-energy E^. This arises from the Coulomb interaction between the electrons and gives these states a finite lifetime. In metals, for the electron energies in the valence/conduction band, this is determined by the excitation of virtual electron-hole pairs, and is energy dependent going to zero at the Fermi level. These life­times are observed in photoemission experiments as a broadening of structures and are typically leV in magni­tude. We have chosen E^ to be lev and energy independent in our calculations. The advantages of this are that the discrete cluster energy eigenstates are turned into a continuum density of states, and the absorptive nature of E^ ensures convergence of the cluster in real space after two or three shells of atoms. E^ is chosen to be leV even at the Fermi level, since this ensures convergence; although this is somewhat artificial, the mean free path of the electron is long enough to observe the surface electronic perturbations induced by impurities. Phaseshifts are then recalculated ";for the adsorbates, using the a values from Table 2.1, and for the substrate at complex energy.

2.3 The Multiple Scattering-Xa approach

The MS-Xa calculation, is also based on multiple scattering solutions to the Schrodinger equation, however it differs from the LDOS cluster calculation in both self-consistency and boundary conditions. The MS-Xa approach is a self-consistent molecular orbital

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Figure 2.5 The MS-Xa cluster for the gas phase CO mole­cule illustrating the partitioning of space between intrasphere (I), intersphere (II) and extramolecular regions (III).

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calculation of the bound states of a cluster chosen to model the catalyst surface and molecular adsorption site. The inclusion of self-consistency allows the study of systems excluded from the LDOS cluster calculations, and we use this approach to look in detail at the carbon monoxide metal bonding, the action of poisons and promoters on the electronic structure of the catalyst and on the chemisorption of carbon monoxide. Calculations of the poisoned surface bear out that neglecting self-consistency is not too bad an approximation for this system. The disadvantage of this method is that only small clusters can be chosen, typically ten atoms, so the range effects probed by the LDOS cluster technique are not easily accessible to this scheme.

The details of the method have been described fully elsewhere (Johnson and Smith 1972; Slater and Johnson 1972? Johnson 1973) and so we will just review the central features, and approximations. Space is divided into three regions I, II and III as illustrated for the free carbon monoxide molecule in Fig. 2.5.(I) An atomic region, inside the atomic muffin tins centered on each atom, within which the potential is spherically averaged.(II) An intersphere region, between each atomic sphere and the outer sphere, within which the potential is volume averaged.(III) An extramolecular region, which surrounds the molecule, within which the potential, is again spherically averaged.

Excluding the extramolecular region the division is analogous to the muffin tin approximation used in band structure calculations. Within this model of the molecule, the radial Schrodinger equation is solved numerically within each atomic sphere. The product 'of this and spherical harmonics is matched smoothly at each sphere boundary onto a multi-centre partial wave expansion in the intersphere region. The secular equation resulting from this matching condition is satisfied only for discrete values of energy, the energy eigenstates of the cluster. Using the full symmetry of the cluster, energy

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levels are found independently for each irreducible representation of the point group. For systems with low symmetry, such as surfaces, the high density of levels and the time involved in finding zeros of the secular determinant, due to its large dimension, makes the study of large clusters difficult. Once all the cluster eigenvalues are known, charge densities and occupancies of these spin orbitals can be found. The charge density is starting point for the next cycle in the self consistent calculation. Slater's statistical exchange V (r) (SlaterAU»1972) is used in this method as well as the LDOS cluster approach, the a parameters taken from Schwartz (1972) in each sphere, and a weighted average in the intersphere region. A weighted average of the new potential generated from the spin orbitals and the old potential is constructed and used as input to the next iteration. This procedure is repeated until convergence in the potential is obtained, this is known as the self-consistent field (SCF) approximation.

Previous studies with the MS-Xa method by Balazs et al (1982) established a correlation between the relative rates of certain reductive elimination reactions and bond strengths, as evidenced by the occupancy of antibonding orbitals. Although we have not performed a total energy analysis of our systems, Danese and Connolly (1974) have demonstrated the feasibility of such calculations within the MS-Xa framework, even for diatomic molecules, by including non-muffin-tin corrections to the charge densi­ty. Moreover, Case et al (1980) have used MS-Xa wavefunctions for accurate calculations of a variety of one-electron properties; including dipole and quadrupole moments, diamagnetic susceptibility, and nuclear guadrupole coupling constants. These results indicate that the MS-Xa wavefunctions are a suitable basis for the analysis of carbon monoxide-adsorbate interactions on transition metal surfaces.

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CHAPTER 3

The Layer KKR method

3.1 Introduction

Over the last ten years considerable effort has been put into generalising the schemes of calculation of bulk electronic structure to that of surfaces or interfaces. These methods fall naturally into cluster and layer methods.

Cluster approaches, such as the LDOS or MS-Xa calcu­lations, need relatively little modification to be able to calculate surface electronic structure. The cluster chosen simply models the surface of interest; this approach was reviewed in chapter 2. Extended layer or slab calcula­tions, however, are intrinsically more difficult than the bulk counterparts since translational symmetry is present only in two dimensions. The surface Bloch states are therefore characterised by a two dimensional wavevector, ki i , in the Brillouin Zone of a two dimensional reciprocal lattice. Bulk states with k in the full three dimensional Brillouin Zone are projected into this two dimensional Brillouin Zone.

In this section we will discuss layer and slab type schemes of calculation. One of the first surface calcu­lations that was self-consistent was the Pseudopotential approach of Appelbaum and Hamann (1976). The Schpdinger equation is expanded in a Bloch wave;basis set leading to a set of coupled differential equations whose solutions are matched to the bulk Bloch states of the infinite crystal and evanescent waves in the vacuum region. This method has been successfully applied to s-p bonded metals, such as sodium, many ideal and reconstructed semiconductor surfaces and to the chemisorption of simple adsorbates.

The LAPW approach, which has been successfully applied to bulk structures, has been adapted to solve surface electronic structure by Freeman and coworkers (for example see Krakauer et al 1979), Feibelman- and Hamann (1984, 1985). The LAPW method can be applied to

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transition metal surfaces, and chemisorption on them, an important step forward since these systems are often catalytically active. A detailed knowledge of the surface electronic structure is the first step to understanding the molecular details of many surface reactions. The major drawback, however, is that the surface is represent­ed by a slab of layers of atoms; the number of which, and the complexity of the surface unit cell which can be handled computationally, is limited by both time and storage requirements. Moreover, one of the disadvantages of the cluster approach is not overcome, namely a discrete set of energy levels. . The finite size of the cluster results in discrete values of kz and hence discrete energy eigenvalues of the slab, representative of the density of states. This can be circumvented by Gaussian or Lorentzian broadening of the levels to give a continuum of states. In particular surface states residing on each surface can penetrate far enough into the slab to inter­fere with each other. Inglesfield (1981) in his matching Green function technique overcomes the latter problem of discrete levels, by embedding the surface slab onto a semi infinite bulk, whose properties are calculated from self-consistent bulk potentials, thus a continuum density of states is obtained. A comparison of the calculated density of states for Al(100) using this method (Benesh and Inglesfield 1984) produced a single surface state compared to the two eigenvalues obtained from a slab LAPW calculation.

Recently the LMTO method, which has also been suc­cessfully applied to bulk structures, has been recast into a TB-LMTO formalism which is more suitable to surface, interface calculations (Lambrecht and Anderson 1986; Fujiwara 1986), though as yet this procedure is not self-consistent.

3.2 Discussion of the Layer KKR technique

The layer KKR formalism has been successfully applied

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SURFACE BARRIER

SURFACE LAYERS

BULK LAYERS

BULK LAYERS

INTERFACE LAYERS

BULK LAYERS

Figure 3.1 The layers of atoms used in (a) a surface(b) an interface calculation.

and

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to the calculation of LEED (Pendry 1974) and Photoemission spectra (Hopkinson et al 1980). In this section we present a self-consistent layer KKR formalism applicable to the calculation of surface or interface electronic structure. One of the earliest attempts to extend the bulk KKR (Korringa 1947; Kohn and Rostocker 1954) ' method to deal with surfaces was presented by Kar and Soven (1975). Their formalism treated a slab of a few layers with a vacuum region on either side; the model was tested for a single copper monolayer with results compared to a tight binding calculation. Maca and Scheffler (1985) and Wachutka (1986) have both formulated a layer KKR calcula­tion for surfaces, though neither of these approaches is as yet self-consistent. In the layer KKR method described here, we make the same approximation to the potential as in the bulk KKR scheme, namely a muffin tin form of the potential for each unique atom. It is this approximation to the potential, which allows us to find rapidly conver­gent analytic expansions of the Green function in both plane and spherical wave bases in the intersphere region, and in terms of numerically integrated solutions to the radial Schrodinger equation inside the atomic spheres. The coupling of the interface or surface region to a semi-infinite bulk region gives a continuum density of states similar to the embedding technique used by Inglesfield (1981).

In a surface calculation, we divide space into three regions (I) the surface barrier, (II) the surface layers, and (III) bulk layers. The interface differs from the surface calculation, in that the barrier is replaced by bulk-like layers. This is shown in Fig. 3.1.

The solution of the electronic structure of either the surface or interface system is based on the self-consistent-field approach, and uses a local approxi­mation to the exchange-correlation potential, Slater’s so called Xa statistical exchange given by equation {3.1}.

1/3Vxa = “3aC(3/4n)p(r)] (3.1}

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The a parameter in equation {3.1} is chosen from cl, - ,nrtabulated by Schwartz (1972). This is calculated so that the total energy of the isolated atom, in this approxima­tion, exactly equals the Hartree-Fock value. Once the charge density p(r) has been found for the system, then a new potential, using a solution of Poisson's equation for the Coulomb potential and equation {3.1} for the exchange-correlation potential, can be calculated. This is then averaged according to the muffin tin prescription and used as input to the next self-consistency loop. A converged, or self-consistent, solution is obtained when the input and output potentials from this method differ by less than a prescribed amount (typically 1-10%).

Both surface and interface systems are broken down into a sequence of layers, each of which may be different. In the next section, we will show how the scattering properties of these layers are used to build up both bulk and surface regions. This breakdown gives sufficient flexibility to allow a study of the properties of surfac­es, interfaces and layered compounds while it retains the simplification brought about by the two dimensional symmetry and Bloch's theorem.

The key ingredient in the calculation is a suitably tabulated charge density, which is calculated in the usual way via the total Green's function for the system G+ evaluated for equal arguments.

p(r,E) = - 1 / ti Im G+ (r,r,E) {3.2}

the charge density is obtained from the energy resolved local density of states as

efp(r) = J p (r,E) dE {3.3}

— 00

The Green's function is calculated from the Dyson equation

G = *4* -f- 4-Go + Go T Go {3.4}

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in a spherical or plane wave basis. We will show that to close the self consistency loop, within the Ewald approx­imation (Slater and DeCiccio 1963), only a spherical wave expansion of the Green function is needed. More sophisti­cated approaches to the solution of Poisson’s equation, as suggested by Weinert (1981), require the plane wave expansion of the Green function as well. Therefore a spherical wave expansion will be derived in the next section and the details of the self-consistency loop discussed. A plane wave representation of the Green function will also be useful for the output of local density of states in the surface or interface region, the equivalent of the orbital contour plots of the MS-Xa scheme, hence an expansion of the z dependent Fourier coefficients of the Green function will be given.

3.2 Spherical wave expansion of G+

As mentioned in the previous section the central feature in solving the SCF problem requires an expression for G+ in a spherical wave basis about each unique atom. The partial wave expansion of G+ is obtained in terms of a product of a matrix (describing the atomic geometry and scattering) and spherical harmonics multiplied by either linear combinations of spherical Bessel (jn) and Hankel functions (h^) functions in the intersphere region or numerical solutions to the radial Schrodinger equation inside the atomic spheres. The full T matrix, which appears in Dyson's equation {3.4}, can be written in a multi-centre form in terms of the atomic t matrices for each atom and matrix elements of the free space propagator G* (Durham et al 1982),

TijLL' tl"l6ij6LL'- GLL-"I 0.5}

where i and j refer to different atoms, and

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.13’LL' = 4ti( 1-6 . . ) 13

1 - I » »» i *1 hl" (Krij )YLn(li. / Y Y Y L L" L' dfl

0 . 6}

are the multi-centre matrix elements of the free space propagator. An expansion of this equation yields the well known multiple scattering series (Lloyd and Smith 1972).

T = E t1 + E t1G 1 t^ + E t V jtjGjktk + {3.7}

Each term in the series represents a scattering path in the crystal. Equation {3.5} is valid even when the series given by equation {3.7} does not converge; however, the derivation of subsequent formulae is easier if aresummation of this series is made. It is important tostress that all multiple scattering can be included exactly and therefore the equations hold even when a perturbation series in scattering events does not. This is to be expected whenever there is a singularity in the inversion of equation {3.5}, corresponding to bound states or shape resonances of the system. A condition for this, in terms of the T matrix defined by equation {3.5} is,

det [ T^, ] = 0 {3.8}

which is precisely the real space KKR secular equation, whose solutions are indeed the bound states.

The resummation of equation {3.7} proceeds in three stages. In the first step all the scattering paths within a layer are calculated, the Green function G^a , found in this way, is represented in the angular momentum basis. The resulting matrix elements for Glay for high angular momentum are very small. For most materials at valence and conduction band energies 1=2 or 3 is sufficient .to characterise this atomic scattering, since scattering into higher angular momentum channels is negligible. The atomic t-matrix for the spherically symmetric muffin-tin potential is diagonal and can be written in terms of the logarithmic derivatives of the radial wavefunction at the sphere boundary in the usual way

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= i ( e2l6l-l) {3.9}

The starting point for calculating G^ay is the atomicGreen function G^, given in appendix b, centered aboutatom (3 in the unit cell at the origin of the layer,denoted by the vector R Q—op

Ga t (£ o P ' ^ P' E) = " 4 i K S YL < £ X P( r >'K) V r < 'K>V £<>{3.10}

with r^ the lesser/greater of r^ = r-R^ and r ’ = r'-R^ $1 , are the regular and irregular solutions to the radial Schrodinger equation respectively, and fK2 the energy with respect to the muffin tin zero. Corrections due to the multiple scattering within a layer lead to G^ay, which after utilising the two dimensional symmetry, depends on the Bloch wavevector kyy. Details of this step are covered in appendix b, and we will only quote the results of this analysis here, for both a multi-centre expansion

Glay^—<'—>,E) = ‘ 4 i K / N s YL(£<)$ip(r<'K) ^ 1 ^

eik//-Rj d - X ) ^ /L,a e-i6^ ^ , a (r>o,K)YL ,(r>a ) {3.11}

and single site expansion of G^ay-

Glay(—<'->'E 3 Gat(—<'—> y l <£<>410<r<'K)

gifiptP-1 [(l-X)-1-l]Lp<L,pel6 , $1((r>,K)YL, (r>) {3.12}

where N is the number of unit cells, X is the intralayer scattering matrix (Pendry 1974 p. 132), which depends .on k i f , Rj denote the two dimensional lattice vectors, and the summation in this and subsequent formalae is restricted to the first Brillouin Zone. Corrections to the layer Green function are added in step 2, detailed in appendix b. We show, that once equation {3.11} is trans­formed into a plane wave basis, the scattering between

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surrounding layers can be accounted for. This is done by allowing the plane waves emanating from the layer toscatter back and forth between the surrounding layers. The terms thus generated can be summed exactly to include all the possible scattering paths; At each ^/// thetwo-dimensional symmetry ensures that the scattering is only to momenta k^+g (where g is a two dimensionalreciprocal lattice vector), most of the waves between the layers, in the valence and conduction band energies are evanescent since the wave-vector in the z-direction, ±V(K2-(k^+g) 2 ), yields an exponential decay. Thus the matrices that characterise interlayer scattering need not include many reciprocal lattice vectors, higher order beams contribute little to the scattering. The calcula­tion of the bulk reflection matrices is the only step which is done perturbatively; this is done by layerdoubling (Pendry 1974 p. 141). The individual layers in the bulk repeat unit are combined into a composite layer which in turn is added to itself, whereupon the product of each layer doubling operation is used as the input to the next, and the process repeated until convergence is achieved. Convergence of this procedure is only ensured at complex energy, where there is flux adsorption, and no singularities in the T matrix. The larger the optical potential, the faster the rate of convergence. This optical potential can be chosen in two ways. In a SCF calculation only the total charge density is required, andis obtained from equation {3.3}. From the analytical

+properties of G , discussed m the Section on the energy integration, and by the use of Cauchy’s theorem the contour can be deformed into a semi-circular contour in the upper half plane, where the finite imaginary part to the energy ensures convergence. If the LDOS is required, then the choice of the optical potential is chosen .to model the energy dependent self-energy, which could be taken either from the experimentally observable energy dependent broadening of photoemission spectra, or from theoretical calculations (Liebsch 1979). The physical significance of the self energy has been discussed in chapter 2.

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In the final stage the plane waves between each layer are expanded once again in terms of spherical waves on each atom in the layer. The spherical waves are allowed to scatter in the layer for the last time, in a similar manner to stage 1. After this last stage all multiple-scattering paths have been included and the formulae are equivalent to a direct summation of equation {3.8}. The resulting expression is spherically averaged at equal arguments (r’=r) and given by

G P ( r o 0 ' r o | 3 ' E ) = - H K n r T 3 S / d k / ; { * l p < r 0 p , K ) * } p < r o p , K ) +

®lp(roP'K) T10 ®lp(roP'K)}= 2i6f3 P-1Tlp e 1 U1

{3.13}

{3.14}

The expression for is given in appendix b, and becauseof the complexity will not be reproduced here in the text.The summation on k// has been rewritten in terms of anintegral in the first Brillouin Zone, with Q the area of the real space unit cell. The similarity between equa­tions {3.13} and {3.12} can easily be understood since the scattering and geometry of the surrounding atoms act purely as a boundary condition of G+, and arecharacterised by FLP' whilst the discontinuity of G isgiven by the atomic part

3.4 Plane wave expansion of G*

The starting point is a plane wave expansion for the free space Green function, which is then corrected for scattering from surrounding layers. Since the details of the derivation are given in appendix b, in the text .we will only quote the main results and discuss the method of calculation.

Defining a Bloch Green function by, and for conve­nience dropping the energy label,

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S]t//(r,r') = 2 G(r,r'-R. ) e“lk//-Rj {3.15}

with

G(r,r' ) = £2/ ( 2ti) 2 / dk// Gk//(r,r') {3.16}

where the integral is over the first Brillouin Zone, we can write equation {3.15} in terms of z and z' dependent matrix elements via

Gk//(r,r' > = 1/n E ei(k/?g)'R Gk//(gg, (z,z' )e' i ( k / ) g ' ) -R '{3.17}

where r = (R,z). The matrix elements of the free space Green function, G?7/ ,(z,z'), are given by

Gk//,gg'(z'z,) = -2i/Kgz eiKgzlz_z'l {3.18}

with K , the wave vector in the z-direction given by

Kgz = V(K2-(k/7+g)2) {3.19}

The waves, given by equation {3.17}, are allowed to scatter back and forth between the layers to include all multiple scattering. At equal arguments G(r,r) can be obtained from equation {3.14}. A more convenient repre­sentation of this Green function is as a Fourier series which has z-dependent coefficients

G(r,r) = E G (z) elg,R {3.20}y

The expression for the Fourier coefficients is derived in appendix b, and given by

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Ggl(z) = 2/(2nP f d k / / £ {e"iKgzZ 6gg, + eikgzZ RS(gg ->

[ (1_Rfc>Rs) 1]gg"{6g",gl-g elK9l-g2Z+ Rb,g"gl-g e l k '31 - 9 z z }

(-i/Kgl-gz> {3.21}

The origin of z is assumed to be in the bulk layer,hence R. is the bulk reflection matrix and R is the b ssurface reflection matrix referred to the origin of the bulk layer.

3.5 Core levels

The calculation of the core charge densities and eigenvalues is performed in an identical manner to the MS-Xa scheme. The Schrodinger equation is numerically integrated for trial energy values until a wavefunction is found which is both exponentially decaying and has the same logarithmic derivative at the sphere boundary for inwards and outwards integration.

3.6 Energy and k , , integrations

When the expression for G ( k ^ , E ) has been found, the energy and kj j integrations need to be performed. The kj j

integration over the first Brillouin Zone can be written as

Gp(roB'roB'E) = ^ d k / / r(k//) {3.22}

where

r ( k / / ) = s { ® i p ^ r o p , K ) 4 i p ( r o 3 , K ) + 4>i p ( r o ( 3 , K ) T i p

[flp ■ *ip<roP'K>> ' {3.23}

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Figure 3.2 Integration contours for charge calculation.

density

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The integration is evaluated by the "Method of Special Points", using those points generated, within the two dimensional Brillouin Zone, according to Cunninghams's algorithm (1971). These are chosen in such a way as to maximise accuracy with the minimum number of points. The periodic function r(k^) is expanded as a Fourier series in terms of rings of lattice vectors related by symmetry, ie

f(k//) = fQ + S fm S eik//‘R {3.24}

The exact value of the integral is fQ, and the points are chosen to make the summation in equation {3.24} as small as possible.

Once the k ^ integration has been completed then the charge density is found by

p(r) = -1/n Im J G(r,r,E) dEE .min

{3.25}

where Emin is chosen to lie below the bottom of the valence band, but above the lowest core state shown schematically in Fig. 3.2. Evaluating the energy integral along contour 1 is an inefficient process, since the density of states is not a smooth function on the real axis. However by deforming the contour into a semi-circle in the upper half plane (contour 2), where the density of states is a smooth function, integration becomes much easier since sampling need only be done at intervals of approximately the imaginary value of the energy. Since all the poles of the Green function are in the lower half plane, application of Cauchy's theorem shows that the value of the integral around either contour is the same. Hence we can write

p(r) = -1/ti J2 Im G(r,r,E) dE {3.26}

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Figure 3.3 Integration contours for Fermi level calcula tion.

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The integration is performed by the Gauss-Chebychev method (Abromowitz and Stegun 1970, p. 889, details of the integration ordinates and weights are given in appendix b) which evaluates the integral using equi-angle points in the complex plane. Gaussian integration techniques have the important advantage of not requiring function evalua­tions at the end points of the contour, where the layer doubling algorithm is not guaranteed to converge.

3.7 Calculation of the Fermi level

The above analysis assumes that the position of the Fermi energy with respect to the muffin tin zero is known. If, however, a new system where the Fermi level is unknown is required, then the Fermi level would have to be calcu­lated. This is basically a counting problem, which satisfies the following constraint

n I dE Jp(r,S) dr3 dEE .mm

{3.27}

where n is the number of electrons. The real space integration, in our program, has been approximated by replacing the unit cell by Wigner Seitz spheres, then this integration reduces to a one dimensional radial integral

N Efn = E 4n I dE

E .i m mfwsr?p(ri,E) dr. dE {3.28}

where N is the number of atoms in the unit cell. The calculation of the energy integration in equation {3.28} proceeds in two steps. In the first step the contour integral a is performed along the semicircle, shown in Fig. 3.3, again using the Gauss-Chebychev formalism. Then, the contour integral b is evaluated with the end point varied until the correct position of the Fermi energy is found. This contour integral uses the

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Gauss-Legendre method (both of these contour integrals are discussed in appendix b). Linear interpolation is- then used to find the Fermi energy, and in the test case of bulk copper, convergence was rapid.

3.8 The self-consistency loop

The input potential to the next SCF iteration was. chosen in the following way

Vnew = (1-°>Vold +0CVnew {3.29}

a is an adjustable parameter; typically a=0.1 ensuresdamping of the SCF cycle. Without this, if were usednewdirectly as input to the next SCF cycle, oscillations, would occur because of the tendency of the system to over­shoot. Thus a serves to provide a direction in which the potential should change from iteration to iteration.

The major part in finding V is to calculate thenewCoulomb contribution from the other electrons based on the charge distribution p(r). The exchange-correlation potential is based directly on p(r) through equation {3.1}, whereas the Coulomb potential depends on integrals of the charge density. If we assume that the charge is spherically symmetric within each muffin tin sphere and a constant outside, then the solution in the ' intersphere region, once the nuclear contribution has been added, is the well known Ewald solution. This solution has the property that it averages to zero over the unit cell, and hence a muffin tin form for this potential can be found (Slater and DeCiccio 1963). Inside the atomic spheres the potential and charge density are related by a one dimen­sional Poisson equation which can easily be solved. The solution, at a radius r, consists of two parts. The first comes from the charge within the sphere of radius r, the second from the charge outside. This can be written as

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Vcoul (r)r

1/r J 4nr'2 p {r1)dr' 0

+ A {3.30}

3.9 The muffin tin zero

The muffin tin zero provides the natural zero of energy, and the prescription above yields the nuclear and coulomb potentials in a muffin tin form. The exchange- correlation potential also can easily be put into a muffin-tin form.

3.10 The surface barrier

The formalism presented allows for the inclusion of a surface barrier, characterised by a reflection matrix. The calculations presented in this work, with this model, however, do not include such a barrier, merely a step. The barrier (Blake 1985) will be included at a later stage.

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CHAPTER 4

Range effects in catalyst poisoning

4.1 Introduction

Poisoning of many metal-catalysed chemical reactions by electronegative atoms (for example the synthesis of methane by the hydrogenation of carbon monoxide) is a well documented effect (for a review see Goodman 1984a, b). Often low coverages of these coadsorbates cause large changes in product yield and catalytic activity. In this chapter we will examine the electronic perturbations induced on the catalyst surface at the atop molecular adsorption site by a set of electronegative atoms on nickel (100), and by sulphur on nickel and rhodium (111) surfaces. The calculations are based on the Green function cluster calculation, discussed in chapter 2, from which the local density of states (LDOS) at the atop molecular adsorption site can be obtained. A full discussion of the justification of the model of correlat­ing catalyst activity with changes in local electronic structure has been given in the introduction. Hence we will only mention here that, within the Blyholder model (1964) of carbon monoxide chemisorption (cf chapter 1), decreasing the LDOS at the Fermi level will reduce the matrix elements for transitions between the 5a and Ef, and between Ef and the 2 k , thus inhibiting chemisorption. In more general terms, changes in the Fermi level LDOS reflect the ability of the surface to respond to reactants, effecting both static properties such as chemisorption, and dynamic properties such as reaction kinetics, and catalytic activity.

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4.2 Poisoning on the (100) surface

4.2.1 Review of experimental and theoretical studies for nickel (100)

In this section we consider carbon, sulphur, phos­phorus and chlorine as examples of electronegative adsorbates, and calculate changes in the LDOS at the atop molecular adsorption site on a (100) nickel surface as a function both of adsorbate geometry and lateral distance between the poisoning atom and the adsorption site. The influence of electronegative coadsorbates on reaction rates and the ability of the surface to chemisorb react­ing species has been studied extensively both under UHV conditions and in model catalytic reactors. The agree­ment between reaction rates on both industrial and ideal single-crystal catalysts for the structure-insensitive methanation reaction is encouraging, since this allows many of the surface science analytical techniques to be applied realistically to single crystal catalysts in UHV. The results of these studies (Goodman 1984a, b; Kiskinova and Goodman 1981a, b; Benziger and Madix 1980; Gland et al 1983; Goodman and Kiskinova 1981; Hardegree et al 1986) can be summarised thus:(1) decreased binding energies of both carbon monoxide and hydrogen, as observed in (a) a decrease in the Thermal Desorption Temperature for carbon monoxide and hydrogen, and (b) an increased carbon monoxide stretch frequency, observed in HREELS data;(2) decreased surface carbon coverage, indicative of a decreased carbon monoxide dissociation rate;(3) decreased adsorption capacity of the nickel (100) surface for both carbon monoxide and hydrogen;(4) reduced sticking probabilities for hydrogen and carbon monoxide;(5) reduced amounts of carbon monoxide and hydrogen adsorbed both in the most tightly bound site and in the total saturation coverage;

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(6) the appearance of weaker binding energy sites for carbon monoxide and hydrogen in the presence of sulphur; and(7) decreased methanation rates.

Based on these experimental findings, the effective­ness in poisoning was, in order of decreasing strengths: chlorine; sulphur; nitrogen; carbon and phosphorus. These results are correlated with the electronegativity of the species (Table 4.1).

Table 4.1 Pauling electronegativities for C, Cl, N, P andS.

Cl 3.0N 3.0S 2.5C 2.5P 2.1

Carbon and nitrogen show an anomalous effect, since they are not as effective as poisons as theirelectronegativity would suggest. This is attributed to the adatom size, a point discussed by MacLaren et al (1986a, b) and one which we will come back to later in our discussion.

In our calculations we observe a correlationbetween electronegativity and poisoning for adsorbates of similar size, but also find that local geometry has an important, and sometimes dominant, ~;*role in determining the strength and range of catalyst poisoning for atoms of different sizes. Previous theoretical studies, based on a calculation of the Fermi level LDOS as a measure of catalytic activity, have been performed using a Surface Linearised Augmented Plane Wave method by Feibelman and Hamann (1984, 1985). In these calculations the order of poison strength was correlated with the electronegativity of the adspecies (phosphorous, sulphur and chlorine); adsorbates that were placed at approximately the same height above the substrate.

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• p(3x3) adatom •sites

Figure 4.1 Plan view of clusters used to investigate the poison/Ni(100) system showing, cluster 1 which includes three layers of nickel atoms only (clean). The other three clusters add poisons round the central site (near), as a remote ring of poisons around the central site (far), and as a p(3x3) overlayer (remote).

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An alternative viewpoint put forward by Ntfrskov et al (1984) and Lang et al (1985), which has been discussed in chapter 1, suggests that the destabilisation of carbon monoxide arises from short range changes in the electro­static potential induced by these adsorbates. However, changes in the workfunction with coadsorbed poisons are small (0.24eV S/Ni (Hardegree et al 1986)) compared to those observed on the promoted surfaces (4 eV K/Pt (Crowell et al 1982)), hence screening and electrostatic effects should be much weaker on poisoned than promoted surfaces. We stress that the effects of surface geometry are not the same as screening effects observed by Lang et al, since the adsorbate is constructed to be electrically neutral. The effects we observed would be present even for an adsorbate with no charge transfer.

As yet there is no quantitative comparison between these theories and the experimental findings on poisoned catalyst surfaces, so in a subsequent part of this chapter we will develop a simple model, based on the results of the LDOS calculations for these coadsorbates, to produce curves of catalyst activity as a function of adatom coverage which can be directly compared to the experimental results of Kiskinova and Goodman (KG).

4.2.2 Results of p(r,E) for various systems; the role of electronegativity and surface geometry.

In the following calculations, the adsorbates carbon, sulphur, phosphorus and chlorine are placed in the four-fold hollow sites on the (100) surface, shown in Fig. 4.1. The vertical spacing (d^) between the adsorbate and the first nickel layer, has been obtained, from LEED measurements in the case of sulphur and carbon chemisorption on nickel (100) and is 1.3A and 0.1A respectively (Van Hove and Tong 1975; Onuferko, et al 1979). For the other atoms, values of d± were obtained by taking bond lengths from bulk crystal structures (Wyckoff 1973), and rescaling (by a factor of 0.82) so that the value for the nickel sulphur bond length agreed

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Figure 4.2 Calculations for the clusters shown in Fig. 4.1. The LDOS is calculated above the central nickel atom in the top layer (in the direction of the (100) bond projecting from the surface), at a height of 1.23 A. (----) clean surface, (---) 4 poisons in the near config­uration and (-.-.) 8 poisons in the far configuration. For (a) C, di = 0.1 A, (b) S, d1 = 1.3 A, (c) P, dj = 1.1 A, and (d) Cl, dx = 1.7 A.

Figure 4.3 Calculation of LDOS, for the clean Ni and 3x3 S/Ni (remote sites), at a height of 1.23A above thecentral nickel atom. (--- ) clean surface, (-.-.) p(3x3)overlayer, dx =1.3A.

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with one calculated from the experimentally determined nickel sulphur plane spacing. The LDOS is evaluated along the (100) bond above the central nickel atom, in a region where bonds between the metal and the carbon monoxide molecule would form (Andersson and Pendry 1980).

Figs. 4.2(a), (b), (c), and (d) show the influence of adsorbed carbon, sulphur, phosphorus and chlorine on the LDOS of nickel for the (100) surface. In the clean surface curves we see a large peak in the density of states just below the Fermi level corresponding to the nickel d-band. Notice the realistically broad structure due to the inclusion of the lifetime effects, discussed in chapter 2, in our calculation. Adatoms in the four nearest neighbour (NEAR) sites have a profound influence on the nickel density of states. Sulphur, chlorine and phosphorus all cause a large reduction in this density of states, with the influence of sulphur and chlorine being.: roughly comparable but slightly stronger than that caused, by phosphorus. In contrast, the influence of carbon is much weaker. This is attributed to its closer approach to the nickel surface.

Peaks at higher binding energies, typically > 5eV, come from the atomic p level resonances of the adsorbed atoms. The*orbitals involved in bonding the adsorbate to the substrate form multicentre bonds (MacLaren et al 1986d), which have significant weight in this range of energy on the central nickel atom as well as on the coadsorbates. Adatoms placed in the next nearest neighbour (FAR) positions have a much weaker influence on the density of states, which is evidence for the short range nature of the interaction. The order of poisoning strength is, however, preserved. This short range is best illustrated by Fig. 4.3, which shows the results for a p (3x3) overlayer of sulphur on nickel (100). We see that the LDOS of the poisoned catalyst has returned to that of the clean surface.

To illustrate the other controlling variable, namely d_L, we have examined the effect of artificially varying djL for phosphorus and carbon on nickel (100). Figs. 4.4(a) and 4.4(b) show the results of these calculations.

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OOCO

oOCO

Ficrure 4,4 Calculation of LDOS, for the clean Ni and poisoned surface with various dj_, at a height of 1.23Aabove the central nickel atom. (--- ) clean surface, curve1 s dj_ = 0.13A, curve 2 : d± = 0.26A, curve 3 : dj_ =0.40A, curve 4 : dj_ = 0.53A, curve 5 : dj_ = 0.79A, curve 6 : dj_ = 1.06A and curve 7 : dj_ = 1.32A. For (a) P and (b)C.

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The message conveyed by these figures is clear - the reduction of the LDOS near the Fermi level and the size of the p level resonance is strongly dependent on the spacing d^. As the adsorbate is moved up from the surface so its poisoning effect increases, with the greatest sensitivity to dj_ in the range 0.4 A to 1.3A. The explanation for this is as follows - the LDOS mea­sures all scattering paths which begin and end at the point r. Hence, if the adatom is to perturb the LDOS there must be a significant contribution from scattering paths including the adsorbate. These must involve the propagation of the electron, from the point where the LDOS is calculated, to the adatom scattering from it and subsequent propagation back again. It is well known (Pendry 1974) that the mean free paths of an electron in the nickel crystal is much shorter than in free space, especially near the Fermi level where the scattering is strong. Hence, when the adsorbate lies close into the surface, the multiple scattering paths which see the adatom need to propagate, to a greater extent, through the nickel substrate, and so the influence of these paths is greatly reduced. In contrast, when the adsorbed atom is well clear of the surface, propagation can occur mainly in free space, and hence the influence of these paths is great. Evidence of this 'shadowing' effect is seen in the recent XANES calculations (Dobler et al 1986) of an oxygen-induced reconstruction on copper (110). In the saw-tooth reconstruction, oxygen atoms move into the surface because of the absence of a row of copper atoms from the second layer. Features in the XANES spectra, characteristic of scattering paths involving the adsorbed oxygen, become much less pronounced when the oxygen is 'shadowed' by top layer copper atoms. The critical ingredient needed for the damping of the poisoning effect- is the presence of other nickel atoms in - the second layer. Calculations for clusters, with these atoms artificially removed, showed a much less dramatic sensitivity to dj_ and consequently strong poisoning even when placed close to the first nickel plane. As we shall see in chapters 5 and 6, Multiple Scattering-Xa (MS-Xa)

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cluster calculations for sulphur on nickel (100) show evidence of weak nickel-sulphur bonding with second layer nickel atoms. In the case of first row elements, such as carbon, the adsorbate can get close enough to these second layer nickel atoms to bond effectively with them, as well as to the surface nickel atoms, thus reducing the poisoning effect on the surface atoms. However, in the case of the second row elements sulphur, phosphorus and chlorine the adsorbate interaction with subsurface nickel atoms is weaker because of the greater separation. Benziger and Madix (1980) also observe that carbon destabilises the surface bonding of carbon monoxide less than coadsorbed sulphur. This size effect arises from changes in the overlap matrix element between the coadsorbed poison and the carbon monoxide molecule. The effects we observe have a different origin, and would not be observed in the calculations of Benziger and Madix, since the substrate consists of only a single layer of atoms. The effects seen at the near site are also observed in the ability of the interaction to propagate laterally across the surface. Smaller perturbations on the adjacent surface nickel atoms results in smaller perturbations on the surrounding surface nickel atoms. It is this sensitivity to distance which is observed in the experiments of Kiskinova and Goodman when they contrast the differences between carbon and nitrogen against sulphur, chlorine and phosphorus.

The similarity between carbon and phosphorus, when placed at the same dj_ , demonstrates that, other things being equal, there is a correlation between electro­negativity and the depression in the LDOS. For example, for the same dj_, in the range 0.40A to 1.3A, we see that carbon, as its electronegativity would imply, poisons more strongly than phosphorus. Comparing the systems nickel / phosphorus and nickel / sulphur, the difference in poisoning is stronger than can be attributed to the difference in d± (see Figs. 4.2 and 4.4). This extra poisoning is assigned to the electronegativity difference between sulphur and phosphorus. The value of d± is critical for explaining the differences between rows of

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the periodic table where the electronegativities may be similar but values of dx and poisoning ability vary greatly.

4.2.3 Comparison with the experiments of Kiskinova and Goodman

A major reason for performing the calculations de­scribed in the first part of this paper is to provide a framework within which to discuss the results of KG. In order to explain these observations, we must develop a method to relate the density of states to catalytic activity. We adopt a simple approach in which it is assumed that the poison acts solely to reduce the number of available adsorption sites. Sites which are poisoned are assumed to have no methanation activity, whilst those which are not poisoned are totally unaffected. We assert, that sites where the LDOS is significantly changed by the presence of the adatom are poisoned. Further, we consider that a site is not poisoned if the LDOS is negligibly effected by the presence of adsorbed atoms.

The assumption of a site either being active or not seems physically reasonable for the following reasons. Although changes occur in the LDOS across the Gp VIII transition metals, these have a relatively small influ­ence on the chemisorption of both _ hydrogen and carbon monoxide (Toyoshima and Somorjai 1979). In addition, because the adsorption sites are discrete, there is quite a large change in the influence of the poison in going from one site to the next furthest out. Hence, in the case of sulphur for example, moving adatoms from the far sites to a p (3x3) overlayer cause the change in LDOS to go from significantly depressed to negligibly depressed, whilst the distance from the adsorption site changes by only 1.1A.

The results for sulphur and phosphorus on nickel (100) (Fig. 4.2) show that the range of poisoning is about 5A, ie nearest and next nearest neighbour sites are

acti

ve

site

s ac

tive

si

tes

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0 02 04 06 08 10 12 14 16adatom coverage (monolayers)

B2Figure 4.5 Comparison of the variation of with adatom coverage (normalised to 1 when poison coverage is 0) to catalyst "activity" as calculated by the numerical simula­tion. Simulation : (— o— ) nearest neighbour poisoningonly (eg for C) and (— •— ) nearest and next nearest poisoning (eg for S). Experiment : (— A —) P, (— o —) C and (— • —) S.

adatom coverage (monolayers)Figure 4.6 Comparison of the rate of methanation with adatom coverage compared to the catalyst "activity" as calculated by the numerical simulation. Simulation : (— o— ) nearest neighbour poisoning only and ( • -■)nearest and next nearest poisoning. Experiment : (— A -) p and (- • -) s.

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poisoned. In contrast, carbon only poisons nearest neighbour sites. To model the sulphur overlayer on nickel (100) we note that, at coverages below 0.25 monolayers, a (2x2) LEED structure is observed. Thus, in this coverage range, we assume no two sulphur atoms approach closer than two nickel lattice spacings, other­wise adsorption of sulphur is random.

Fig. 4.5 shows the results of a simulation for a 100 x 100 atom nickel (100) surface with adsorbed sulphur. No simulation is necessary for carbon as it is easy to show that the number of inactive sites = 4 x carbon coverage. The first set of experimental data which we have chosen to compare with the results of the simulation is the relative saturation coverage of carbon monoxide in the most tightly bound ^ adsorption site, which corre­lates with catalyst activity. Plotting the data of KG, in Fig. 4.5, for sulphur, carbon and phosphorus, we see for sulphur, that the experimental curve lies close to the simulation, also the influence of carbon is much weaker; although the experiment would suggest it is slightly stronger than first nearest neighbour poisoning. The trend is that sulphur poisons carbon monoxide adsorp­tion more than carbon does, this shows direct evidence for our postulate of the importance of d± since both these atoms have the same electronegativity (Table 4.1). The influence of phosphorus, however, seems to be much weaker than expected. From our calculations, we expect phosphorus to poison next nearest neighbour sites, though possibly not so strongly as sulphur. One possible explanation of the weaker effect of phosphorus is postu­lated by Goodman (1984a, b). Adsorption of phosphorus does not generate ordered structures but forms islands at high coverages. Therefore, phosphorus may poison to a lesser extent than if it formed ordered structures. The repulsive constraints applied in the overlayer simulation, will then overestimate the extent of poisoning.

Goodman (1984a, b) also presents data for relative methanation rates as functions of phosphorus and sulphur coverages. Fig. 4.6 shows his results compared to those

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adatom coverage (monolayers)Figure 4.7 Comparison of the rate of methanation with adatom coverage compared to the catalyst "activity", for various ranges of the interaction, as calculated by the numerical simulation. Simulation : (— o— ) range = 1.8A(nearest neighbour poisoning), (— •— ) range = 3.9A(nearest and next nearest poisoning), (— i— ) range = 5.3A and (— *— ) range = 6.1A. Experiment : (- • -) s.

adatom coverage (monolayers)Figure 4.8 Comparison of the rate of methanation with adatom coverage compared to the catalyst "activity" as calculated by the numerical simulation for the following cases. Simulation : (— o— ) S nearest and next nearestpoisoning, (— h— ) maximum repulsion between S atoms and (——*— ) curve for an ensemble of 2 nickel atoms. Experiment : (— • —) S.

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of our simulation. In the case of preadsorbed sulphur, where poisoning is strong at nearest neighbour and next nearest neighbour sites (see Fig. 4.2), the experimental curve shows a more dramatic poisoning than our simulation would predict. However for preadsorbed phosphorus the poisoning is strong at nearest neighbour sites and still present to some degree at next nearest neighbour sites. We observe that if next nearest neighbour sites are assumed to be totally poisoned then the theoretical curve overestimates the experimental curve. Better agreement with experimental findings could be achieved by having partial poisoning at these far sites, or by noting that phosphorus may appear to poison to a lesser extent than expected for reasons already stated.

The discrepancy between the model and the measured methanation rate on the sulphur poisoned surface illus­trates that whilst the relative trends and strengths -in catalyst poisoning may be obtained from the LDOS ap­proach, the reduction in methane formation is underesti­mated. There are several possible reasons, some of which we will discuss here, and some in the next chapter, where we look in more detail at the sulphur-metal-carbon monoxide interactions. Firstly the influence of sulphur may extend beyond next nearest neighbour sites. Fig. 4.7 shows the curves calculated assuming an increased range of interaction, even at a range of 7A the model curve is significantly weaker in its poisoning than the experiment requires. We feel that this is not a physically reason­able range of interaction, because Fig. 4.3 shows that the influence of sulphur has completely disappeared by 5A. Another possibility is that the repulsive interac­tion between the sulphur atoms may be of longer range and Fig. 4.8 shows the effects of including this into the model, as can be seen the increase in poisoning is quite marginal.

Since neither of these calculations is sufficient to achieve agreement between theory the experiments of Goodman on sulphided nickel (100), we must conclude that the influence of sulphur on the methanation of carbon monoxide on nickel (100) is more complex. As an example

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of this we note, by comparing Figs. 4.5 and 4.6 that a site may be poisoned to methanation, although still able to adsorb carbon monoxide. One possibility is that hydrogen, as well as carbon monoxide, poisoning may be important. Our calculations, which are based on changes in the electronic structure of the catalyst surface (due to adsorbed impurities such as sulphur) would suggest a range of poisoning of the catalyst of » 5A for either carbon monoxide or hydrogen since this is the range over which the LDOS is modified. To include these effects a more detailed knowledge of the reaction kinetics and the rate determining step on the poisoned catalyst would be required. Madix et al (1983) have proposed an alterna­tive explanation, based on short ranged interactions between sulphur and carbon monoxide. Their TPD data for sulphur on nickel (100) shows that the total carbon monoxide coverage varies linearly with sulphur coverage, except in the very low sulphur coverage regime where the effect of sulphur is to inhibit the formation of a "compression” structure of carbon monoxide. The dramatic poisoning of methanation at low coverages may be related to adsorbate induced changes in the binding sites of intermediates, and not necessarily due to long range effects.

The rate determining step may involve an ensemble of active nickel atoms, the requirement for such an ensem­ble, even of only two nickel atoms, would drive the calculated curve towards the experimental result, as can be seen from Fig. 4.8. However some discrepancy will remain at very low coverages unless a very large ensemble is specified. The need for such ensembles in catalysis has been demonstrated clearly by Sinfelt (1983). It is interesting to note that Dalmon and Martin (1983) have recently invoked an ensemble of clean nickel atoms to explain their results for methanation over a Ni/SiC^ catalyst. The most likely requirement for the ensemble is to permit the hydrogenation steps in the reaction (discussed in chapter 1). Carbon monoxide dissociation is not thought to be rate determining on nickel, although it may become so on the poisoned surface.

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(111) C luster

Top-layer

Second-layer

Remote S

Far S

• Near S

Figure 4.9 Plan view of cluster used to investigate the Ni(lll) & Rh(lll) / S systems showing, cluster 1 the clean surface. The other three clusters add S atoms in near, far and remote sites around the central atom in the top layer.

Figure 4.10 Calculations for the clusters shown in Fig. 4.9. The LDOS is calculated above the central atom (in the direction of the (111) bond projecting from the surface), at a height of z. The height of the S atoms above the first plane of atoms is dj_. All curves are onthe same scale and in arbitrary units. (-----) cleansurface, (o—o—o) 3 S atoms in the near sites, (-- —) 6 Satoms in the far sites and (•—•—•) 12 S atoms in the remote sites. For (a) Ni : z = 1.3 A , dj_ = 1 . 6 A . and ( b ) Rh : z = 1.4 A , dx = 1.8 A.

LDOS

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4.3 Poisoning on the (111) surface

Recent experiments, (Trenary et al 1985) with carbon monoxide chemisorbed on sulphided nickel (111) surfaces indicate that certain sites within the range (5A) of poisoning of a given sulphur atom may still be able to chemisorb carbon monoxide weakly. This contrasts with the (100) surface where all the sites within the range of sulphur poisoning are excluded. In this section we present calculations which illustrate the nickel and rhodium (111) surfaces with and without preadsorbed sulphur.

The calculations are based on the Green function formalism, described in chapter 2, for a cluster of atoms chosen to represent the (111) surface of the catalyst. The hybridisation elements of the Green function can be obtained from the full cluster Green function as follows

G+(r,r,E) = E g£l ,(r,r,E) {4.1}

with

p(r,E) = E pLL, (r,r,E) {4.2}

therefore

P L L , (r,E) = -n"1 Im { G+LL,(r,r,E) } {4.3}

4.3.1 Results and discussion

Fig. 4.9 shows the cluster geometries used for investigating the interaction of sulphur on the (111) surfaces. Sulphur adsorption sites are labelled near, far and remote, with the molecular adsorption site (where the LDOS is evaluated) at the centre of the cluster. Fig. 4.10 illustrates this LDOS for both rhodium and nickel with the sulphur atoms placed in the above men­tioned sites. The clean surfaces show characteristic features of the bulk density of states - a high density

Figure 4.11 The partial charge densities using the same geometries as in Fig. 4.10 for Ni only. All curves are on the same scale and in arbitrary units. For (a) p , (b)

(c) Pd d , (d) Ps p , (e) ppd and (f) Pgd.

LOOS

Qtd

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of states from the d bands near the Fermi level, with a fairly high sp contribution just below the d resonance. This is more pronounced than in the bulk as we only look at a type orbitals along the (111) bond. Structures are broadened, since the calculation is done in the complex energy plane, by an imaginary part to the energy describing the lifetime effects discussed in chapter 2. Adsorption of sulphur in the near sites strongly reduces the LDOS near the Fermi level as expected. This is a direct interaction with the sulphur and depletes the charge in the region where the carbon monoxide metal bonds will form. The peak in the curve below the d band comes from the sulphur p levels and is evidence of the direct interaction of the sulphur and the surface sub­strate atoms. When sulphur is placed in the far sites there is an enhancement near the Fermi level, and a depletion lower down in the band. The calculation with sulphur in the remote sites shows that at this range its influence is rather weak and essentially the LDOS has returned to that of the clean surface.

Both nickel (111) and rhodium (111) exhibit similar trends. To illustrate these interactions we have calcu­lated the partial charge densities p^ i • Since we are evaluating the LDOS along the (111), direction only a bonds contribute to it. PLL» is a 3x3 symmetric matrix where L = (1m) runs over the values (00), (10) and (20) ie s, p and d. Figs. 4.11(a), 4.11(b) and 4.11(c) show the diagonal matrix elements pss/ p pp and p thespherical charge contributions to the LDOS. The features of the clean surface correlate well with the bulk band structure of nickel (see Fig. 4.12). The bands which follow the surface calculations best are those along the (111) direction, ie the Tl direction. The large peak in

p.g s and p occurs where the sp band hybridises with the d bands, with, as expected a large p weight higher up in energy, p ^ shows a characteristic narrow peak centred around the middle of the d bands. The influence of sulphur is the most dramatic on p and 0 • Figs.4.11(d), 4.11(e), 4.11(f) show the off diagonal matrix elements pgp, and ps{ . Considering the clean surface

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k

Figure 4.12 Band structure for Ni (Moruzzi et al

Figure 4.13 Sulphur p phase shift.

1978)

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first, all three curves follow the same trend, ie nega­tive in the lower half of the band and positive above about • =0.25H. The interpretation of the matrix elements is that of the hybridisation between the differentorbitals on the same site, ie related to the coefficients of a partial wave expansion on the central site. Thetrend shows that the hybridisation is such that theweight of the orbitals in the bonding region of the band points in to the surface, whilst the antibonding charge is adjusted to be mainly in the vacuum. Since Pgp and

hybridise charge in and out of the surface this explains why these matrix elements are larger than where the hybridisation puts charge in a direction parallel to the surface. pgp goes to zero approximately in the band gap where the states are non bonding.

When sulphur is placed on the surface then pgp, Pg^ and p ^ are strongly perturbed. For the near sulphur case, there is a peak around the sulphur atomic p level showing that the influence of the sulphur potential on the surface nickel states is attractive and therefore tends to polarise charge towards itself with the conse­quence that there is a depletion of charge at higher energy, where the sulphur potential is repulsive. In p spthere is a maximum in the curve at the same position as in and pg^ though it does not cross the axis. The sp electrons are more diffuse than d electrons and hence react similarly to the presence of sulphur. This feature is present even when sulphur is placed in the far sites, where Pg< , and P are most strongly-'affected, and appear to be almost antiphase when compared to the near sulphur case. This rehybridisation gives rise to the observed enhancement near the Fermi level. The behaviour can be understood from the sulphur p phase-shift, shown in Fig. 4.13. The energy derivative of the phase-shift, 6^(E), is proportional to the number of electrons of angular momentum 1 and energy E drawn into, or repelled by the potential. This is the differential form of the well known Friedel (1958) sum rule. Thus the influence of sulphur, as seen from the slope of the phase-shift in Fig. 4.13 is attractive (positive) below =0.25 H, and

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repulsive (negative) above ~0.25 H (a more complete discussion of these ideas can be found in Pendry (1977)).

With sulphur in the remote sites the influence on all the matrix elements is small, as expected from the short range nature of the interaction. The accuracy of the calculation is such that we can say there is little difference between this and the clean surface.

4.3.2 Conclusions and Implications for catalysis

Within the Blyholder model (1964), discussed inchapter 1, for carbon monoxide chemisorption, sulphurplaced in the near sites will reduce the sticking ofcarbon monoxide by two effects. Firstly the decrease inthe LDOS near the Fermi level will decrease the matrixelements for both the hopping between the 5a and the

*metal, and between the metal and the 2tt . Secondly,since the 5o level is close in energy to the atomic plevels of sulphur, the increased charge in this regionwill hinder the donation of charge from the 5a to themetal. The case for far sulphur is less clear, based onthese calculations, the density of states factors wouldsuggest enhanced sticking to the surface. However, the *2n extends over a fairly large region of space, since it is a diffuse orbital, hence it may see areas of reduced LDOS which would tend to inhibit the sticking of carbon monoxide. The influence of sulphur atoms adsorbed in the remote sites on carbon monoxide’ sticking will be correspondingly small.

While no reports exist of the promotion of carbon monoxide adsorption by sulphur, it is interesting to compare our results with those of Trenary et al (1985). These workers have shown that the influence of adsorbed sulphur on carbon monoxide adsorption on nickel (111) extends only to the 'near' and 'far' sites of Fig 4.10. As noted, there is a very strong suppression of the density of states at the Fermi level, by sulphur in the 'near' sites. Sulphur in the 'remote' sites has little influence but an interesting result is observed for the

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NK100) surface auffin-tin LDOS for first three layers

Fiaure 4.14 MTDOS for bulk (---- ) , top ( )second layers of nickel calculated withinlayer KKR scheme.

andthe

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' far' sites. Here, the density of states close to the Fermi level is enhanced, which within our model would imply promotion of carbon monoxide adsorption. It is of interest to note, however, that the carbon monoxide-sulphur distance in this site is intermediate between that of the 'near' and 'far' sites previously examined on nickel (100), and again indicates the short range over which adatoms act, in agreement with our overall conclusions.

Trenary et al have also observed a new, weakly held, atop carbon monoxide state which they designate CO*, and which is induced by the presence of sulphur. The carbon monoxide stretch frequency is 2105 cm , compared to 1910 cm- observed on the clean (111) surface. They suggest that, since the influence of sulphur is short-range, their CO* sites are the same as our 'near' sites, whilst the excluded sites are the same as the 'far' sites of Fig. 4.10. We believe that the situation is the reverse of that suggested by Trenary efc al. The near sites are very strongly poisoned, since the LDOS, as shown in Fig 4.10, is appreciably reduced by the presence of sulphur. Also sulphur atoms in the three-fold hollow sites would not permit occupancy of the nearby atop site and could be expected to push the carbon monoxide mole­cule away, at least to one of the two adjacent bridge sites. By contrast, the 'far' sites are not sterically hindered and our calculations show that they are close enough to the sulphur atoms to be thus influenced.

4.4 Self consistent layer calculations for Ni(lOO)

Using the layer KKR program, described in chapter 3, we have calculated the electronic structure of two surface layers of nickel coupled to a semi-infinite bulk. The surface barrier was represented by a step of height0.57H. Fig. 4.14 shows the Muffin-Tin Density of States (MTDOS), which is the LDOS integrated over the volume of the muffin-tin sphere, for both the first two layers and the bulk. The higher MTDOS at the Fermi level, compared

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Ni(100) + p(1x1) S, surface auffin-tin LOOS for first four layers

Energy (eV)

Figure 4.15 MTDOS for bulk (---- ) , top ) andsecond (...) layers of nickel and sulphur (— — — )calculated within the layer KKR scheme.

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to the bulk, arises from surface states, and as expected the d-band peak is shifted closer to vacuum at the surface as a result of the less attractive potential there than in the bulk. The second layer MTDOS is now close to the bulk curve. Compared to the LDOS calcula­tions the d bands appear much larger then the s-p bands, this is due to the degeneracy factor, since spherically averaged a, n and 6 states are included in the MTDOS instead of just a orbitals.

Results for a p(lxl) sulphur overlayer are shown in Fig. 4.15. The surface barrier was raised by .009H, the experimentally determined value for a 0.5 monolayer coverage of sulphur. The influence of sulphur on the top layer of nickel atoms is most pronounced, resulting in a depletion of states near the Fermi energy, and a shift in the position of the d-band peak to lower energies. The justification of our construction of the sulphur poten­tial for the cluster calculation is best illustrated in Table 4.2 which shows the maximum relative change in the sulphur potential, with a = 0.1. Note that within the MS-Xa scheme the sulphur potential would be regarded ..as converged at iteration 1.

Table 4.2 : Maximum relative change in the sulphurpotential.

iteration change1 0.0222 0.0213 0.0194 0.0175 0.015

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CHAPTER 5

Local interactions between CO and poisoned Ni surfaces

5.1 Introduction

Electronegative elements, such as sulphur, are known to have profound poisoning effects on certain catalysed reactions. The experimental evidence for some typical systems has been reviewed in the previous chapter. Also in the previous chapter, we examined how the geometry of the adsorbate modifies the surface electronic structure, and as a consequence of this how catalytic properties varied as a function of adsorbate coverage and position. The arguments presented, based on general features of surface electronic structure, as measured through a Local Density of States (LDOS) (MacLaren et al 1986 a-c; Feibelman and Hamann 1984, 1985), are supported in the Blyholder model (1964) of carbon monoxide adsorption discussed in the introduction. However other factors, which may also be important, certainly in determining short range effects, have been proposed involving concepts such as charge mobility (Haydock 1981; Marks and Heine 1985) and the role of electrostatics (Nprskov et al 1984; Lang et al 1985). Changes in the electrostatic potential are expected to be more important in the case of alkali metal coadsorption since, in this case, adsorption occurs in conjunction with the donation of a significant fraction of an electron from the alkali metal to the substrate. Polarisation of the chemisorbed levels of carbon monoxide, as a result of this charge flow, are, in a later chapter, shown to be important.

Heine's arguments of the correlation between surface reactivity towards reconstruction and catalytic activity are interesting, with these reconstructions and catalytic activity occuring as a result of surface charge mobility. The role of poisons would be to bond to surface charge, inhibiting locally energetically mobile charge needed for easier reaction pathways. This would be seen not only in a reduction in the LDOS around the Fermi level, but also

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Fiqure 5,1 Top view of the cluster used in the MS-Xa calculation for CO coadsorbed with S on Ni(100) and the cross-sections used for displaying wavefunctions.

TOP Ni LAYER

0 2ND Ni LAYER

3RD Ni LAYER

9 ADATOMS

Figure 5.2 Top view of the cluster used in the LDOS calculation, showing the near adsorption sites. The LDOS is calculated at a height of 1.23A above the central Ni atom, with the S atoms placed at a height of 1.3A above the top plane of nickel atoms.

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in a change in orbital character leading to a decreased surface response.

The discussion of the previous chapter was aimed at understanding any long range phenomena present, and in particular how induced perturbations propagate across the surface. Whereas, in this chapter we would like to take a closer look at the local interactions on the nickel (100) surface, including changes in both the surface nickel orbitals, and the bonding of carbon monoxide in the presence of adsorbed sulphur atoms. The suppression in the LDOS and changes in the orbital character resulting in a modification of surface charge mobility caused by sulphur adsorption, will be discussed in the light of our results and the ideas of Heine and Haydock.

The approach we adopt is to use the Multiple Scattering-Xa (MS-Xa) scheme, discussed in chapter 2, which is able, since the method is- a self-consistent model, to examine systems with significant charge trans­fer, such as carbon monoxide chemisorption. We also compare results for the clean and the sulphur covered surface obtained using both cluster methods, and show that the level of qualitative agreement is quite good.

5.2 Results and comparisons between the two clusterapproaches.

The results presented in this section are a compari­son between the two cluster approaches used to study catalytic surfaces and the influence of coadsorbates. The first calculation is for the clean nickel surface in which the MS-Xa energy eigenvalues of a nine atom nickel clus­ter, shown in Fig. 5.1, are compared to the Local Density of States (LDOS) evaluated above the central nickel atom of the cluster shown in Fig. 5.2. The results of the comparison are shown in Fig. 5.3(a). To facilitate a direct comparison to the LDOS, as well as the MS-Xa levels for the whole cluster, we have also plotted those whose orbital weight is >5% on the central nickel atom. This will be more representative of a LDOS for the MS-Xa

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Ni

Ni/S

Figure 5.3 Comparison of the LDOS and MS-Xa levels, for the entire cluster and those with weight on the central Ni atom, (a) Clean Ni. (b) Ni with four coadsorbed S atoms.

♦

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calculation, since there is some spatial projection included. Once the Fermi levels have been aligned there is qualitative agreement between the density of MS-Xa levels on the central nickel atom and the calculated LDOS. Both show a peak below the Fermi level, corresponding to the nickel d states.

In the case of sulphur atoms adsorbed in the nearest four fold hollow sites, shown in Fig. 5.3(b), we again see agreement between the number of MS-Xa levels on the central nickel atom and the LDOS. The striking feature of these results is that whilst there is still a similar number of levels for the cluster as a whole, compared to the clean surface, there are correspondingly fewer levels with weight on the central nickel atom. This agrees with the conclusion of the LDOS calculation, and is due to the direct interaction between the sulphur atoms and the surface nickel electrons, resulting in a depletion of charge from a region where bonding to the carbon monoxide molecule will form, and a compensating increase on the sulphur atoms. The effect at the Fermi level is smaller than deeper down in the band, the region of energy near to that of the sulphur p states, though again depletes the bonding region of electrons. The marked depression, observed in both sets of calculations, is correlated to the decrease in sticking probability and the reduction in the binding energy of carbon monoxide on nickel observed in the presence of coadsorbed sulphur in the many experi­mental studies for this system (Goodman 1984a, b; Kiskinova and Goodman 1981a, b; Goodman and Kiskinova 1981). As previously discussed, both in the Blyholder (1964) and Newns-Anderson (Newns 1969, Doyen and Ertl 1974) models of molecular chemisorption, decreases in the density of states around the adsorption site near to the Fermi energy can account for these experimental observa­tions. The similarities between the LDOS and the MS-Xa calculations again confirm the validity of the approxima­tions used in the former calculation, and show that for systems with small amounts of charge transfer the non-self-consistent calculations reproduce the trends seen in the fully self-consistent MS-Xa calculation.

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NiCOA1 and E symmetry

All levels >5 ’.on centra ink

--- 2n' _____

2» *

1*

5o

4 a

+ 0.2 -

E, -

XCtis -0.2 bpc

“0.4 -

-0.6-

-0 JB

NiSCO A1andE symmetry

All levels > 5 \ on central ni

2i*

.1* 5a

Figure 5.4 MS-Xa orbital energy levels of A. (a) and E(ti) symmetry for the entire cluster and those with weight on the central Ni atom for CO adsorbed on clean and sulphided Ni(100).

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5.3 The effects of sulphur on carbon monoxide chemisorption on nickel (100).

In our example reaction, the methanation of carbon monoxide by hydrogen,

CO + 3H2 -> CH4 + H20

Dissociation of carbon monoxide will be the next stepafter chemisorption. In this stage, electron transfer

*from the bonding 5a to the antibonding 2n orbitalsprovides the mechanism of dissociation, since in the gas

*phase the antibonding 2n level is unoccupied. Therefore,in the MS-Xa calculations for carbon monoxide on nickelwith and without coadsorbed sulphur, we will focus primar-

*ily on the 5a and 2n chemisorbed levels. For the sake of completeness, we will include the In in our discussions, since the role of the In is known to be important on the promoted surface (Heskett et al 1985, 1986), a point wewill come back to in a later chapter. The reason for this

■kis that perturbations in the 2 k , through the constraint of orthogonality, will modify in a compensating manner the In.

Fig. 5.4 shows all the MS-Xa levels for these systems belonging to the (a) and E(n) Irreducible Representa­tions of the C4v point group, and those with weight on thecentral nickel atom. Labelled on this figure are the 4a, *5a, In and 2n molecular chemisorbed levels. In both *cases, the 2n level has most of its weight above theFermi level, and hence is mainly unoccupied. The interac-*tion of the free carbon monoxide 2n level with the substrate will induce a broadened resonance. This mani­fests itself in the calculation by several cluster orbitals, of the appropriate symmetry, having the sa‘me carbon-oxygen-nickel bonding character. Deeper down in energy are the 4a, 5a and In levels. The energy differ­ences between the 4a, In and 5a and the Fermi level are given in Table 5.1.

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Fiaure 5.5 Wavefunction contour plots (with the appropri­ate cross sections in parentheses), using the contour values ±0.081, ±0.027, ±0.009, ±0.003, ±0.001. Solid and broken lines represent positive and negative contour values respectively, (a) Ni/CO 5o (I), (b) Ni/S/CO 5o (I),(c) Ni/CO 5o (II), (d) Ni/S/CO 5a (II). Atomic positions are marked by small dark circles.

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Table 5.1

Energy differences (6e) between the chemisorbed 4a, In and 5a and the Fermi level for Ni/CO and Ni/S/CO.

Ni/CO Ni/S/COlevel 6e (eV) 6e (eV)4a -9.301 -10.747In -5.864 -7.6395a -7.65 -7.72

In the presence of coadsorbed sulphur the splitting between the In and the 5a is small compared to the clean surface. Whilst this is suggestive of a weakened carbon monoxide metal bond, it is not conclusive since both these levels are occupied. More evidence comes from the orbital contour plots for these levels which will form the basis of our discussion. Broden et al (1976) have proposed a correlation between transition metals which dissociatively chemisorb carbon monoxide and the energy splitting between the 4a and the In ^4a-ijx ' °^servec in photoemission experiments for the chemisorbed molecule. This energy difference varies in the range 2.5 eV to 3.5eV with some dissociation occuring for values greater than about 3.2 eV. Although we have not performed a transition state calculation (Slater and Johnson 1972), a procedure known to give more accurate values of excitation energies, we find, from our calculations, that for carbon monoxide adsorbed on a clean nickel surface that ^ =3.4 eV compared to 3.1 eV on the sulphided surface. This result provides more evidence of a weaker carbon monoxide metal bond in the presence of sulphur.

In Fig. 5.5 we show orbital contour plots of the 5a wavefunction for the two cross-sections illustrated in the cluster geometry of Fig. 5.1. Cross-section I is in the plane through the adsorbates, while cross-section II cuts the other surface nickel atoms. In the clean surface, bonding charge occurs between the carbon and oxygen atoms and the substrate atoms in the first and second layers of nickel atoms of the cluster. In this orbital, the

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dominant interaction is between the s-d 2 hybrid orbitalon the central nickel atom and the molecule, with weakerd „ and d„ orbitals on the other nickel atoms. The 5o yz xzorbital, in the presence of sulphur atoms, is significant­ly perturbed. The nature of the interaction is repulsive on the carbon monoxide molecule and results in a rehybridisation from s-d to p-d on the central nickel atom. This is a consequence of the rearrangement of charge in this orbital from the bonding region between the molecule and the substrate, through the substrate and towards the sulphur atoms. The p-d rehybridisation is evident from the contour plots where the relative phases of the p and d orbital components redistribute charge from between the metal and the, molecule in towards the sub­strate and out towards the sulphur atoms. Quantitative values of the changes through a charge and angular momen­tum breakdown on each atomic sphere are given in Table5.2. This clearly shows an increase in the p contribution with a decrease in the s contribution on the central nickel atom induced by the coadsorbed sulphur. Also there is increased s weight on the other nickel atoms in the presence of sulphur, though the importance of this is small since there is little weight on these atoms.

Table 5.2

Orbital weights and angular momentum, decomposition for the 5a orbital on each atomic sphere for the clean surface, with the corresponding values for the sulphided surface in parenthesis.

atom weight s P dC 24 (ID 25 (31) 75 (69) 0 (0)0 10 (3) 2 (0) 98 (100) 0 (0)S (8) (6) (94) (0)Central Ni 34 (31) 40 (25) 11 (52) 49 (24)1st Ni layer 1 (5) 57 (74) 26 (21) 17 (6)2nd Ni layer 2 (13) 51 (65) 25 (20) 24 (14)

The sulphur p states induce the nodal line seen in Fig. 5.5, as a result of the direct overlap with the d

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Ficrure 5.6 Wavefunction contour plots using the notation of Fig. 5.5 for (a) Ni/CO In (I), (b) Ni/S/CO In (I), (c) Ni/CO In (II), (d) Ni/S/CO In (II).

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orbitals on the substrate nickel atoms. The rearrangement induced in the 5a is fairly strong and not just localised to the region around the sulphur atoms, since evidence of the perturbation is seen in both cross-sections of this orbital. The influence of sulphur is expected to be dramatic on this orbital, since the 5a and the sulphur p states are close in energy.

The In levels for the clean and poisoned system,shown in Fig. 5.6, demonstrate that there is some bondingof the molecule from this level. The orbital overlap iswith d„„ and d„„ on the central nickel atom, with the xz yz Tmixing between this orbital and the molecular In in phase. This orbital is more localised on the molecule than the 5a, as is the case for free carbon monoxide. The influ­ence of sulphur is again destabilising with the sulphur p-states interacting directly with this level, inducing modifications to the surface orbitals. The phase is changed on the central nickel atom d orbital, thus forming a .weak anti-bonding interaction, as opposed to weak bonding interaction, between this atom and the molecule on the clean surface. The perturbation induced by sulphur is also observable on the other surface nickel atoms where there is an induced rehybrisation to increased s weight on them. The importance of the bonding between the sulphur atoms and the subsurface nickel atoms, can also been seen in these orbital contour plots, at the expense of nickel-nickel and nickel-carbon monoxide bonding in the cluster. Quantitative values of the changes through a charge and angular momentum breakdown on each atomic sphere are given in Table 5.3.

Figrure 5.7 Wavefunction contgur plots using the ngtation of Fig. 5.5 {or (a) Ni/CO 2n (I)* (b) Ni/S/CO 2 k (I), (c) Ni/CO 2 n (II), (d) Ni/S/CO 2ti (II).

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Table 5.3

Orbital weights and angular momentum decomposition for the Ik orbital on each atomic sphere for the clean surface, with the corresponding values for the sulphided surface in parenthesis.

atom weight s P dC 11 (15) 0 (0) 100 (100) 0 (0)0 41 (32) 0 (0) 100 (100) 0 (0)S (8) (16) (84) (0)Central, Ni 5 (1) 0 (0) 5 (5) 95 (95)1st Ni layer 0 (1) 31 (63) 22 (22) 46 (15)2nd Ni layer 0 (0) 11 (34) 15 (19) 74 (47)

*The 2r levels are shown in Fig. 5.7. Changes in the*

2ti , on the other hand, are not as pronounced since these states are relatively far away in energy from the perturb­ing sulphur p-levels. The influence of the sulphur atoms is seen to be localised around the hollow sites, and afairly weak repulsive interaction between the sulphur s-p*states and the molecular 2n . In both clean and sulphidedsurfaces the carbon-oxygen-nickel interaction isantibonding in nature, as expected. There is littleevidence of compensating effect, arising from the

*orthogonality constraint, between the In and 2n orbitals• kwith sulphur adsorption since perturbations of the 2n are

small because of the large energy separation between itand the sulphur p states. This is not the case withlithium coadsorption, which will be illustrated in a later. * chapter, where the strong perturbations seen in the 2ndoes induce this compensating effect on the In. Theoverall influence of sulphur on carbon monoxidechemisorption is shown schematically in Fig. 5.8.

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Ni/S/CO

5<t

Figure 5.8 Schematic illustration of the Ni-S-CO interac­tions showing (a) the charge flow and (b) the resulting rehybridisation induced by coadsorbed S on the CO 5c and the central Ni orbitals.

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Figure 5.9 Wavefunction contour plots using the notation of Fig. 5.5 for orbitals of symmetry A. (of the C. point group) on the clean Ni surface, for (a,b) representative orbitals near the Fermi level and (c,d) representative orbitals close in energy to the 5a, in cross-section I.

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Table 5.4

Orbital weights and angular momentum decomposition for the *2n orbital on each atomic sphere for the clean surface, with the corresponding values for the sulphided surface in parenthesis.

atom weight s P dC 11 (30) 0 (0) 100 (100) 0 (0)O 8 (5) 0 (0) 100 (100) 0 (0)S (22) (42) (58) (0)Central Ni 8 (0) 0 (0) 5 (82) 95 (18)1st Ni layer 8 (35) 0 (41) 46 (6) 54 (53)2nd Ni layer 16 (3) 57 (2) 16 (69) 26 (30)

5.4 The influence of sulphur on surface charge mobility

To address the question of charge mobility on clean and poisoned surfaces, we focus on the MS-Xa orbitals that most strongly influence the interaction of the 5a level of carbon monoxide with nickel (100). These are (1) orbitals near the Fermi level, dictating the incipient response of the system to an external perturbation, and (2) orbitals near the carbon monoxide 5a, which make appreciable contributions to the associated orbitals of the coupled carbon monoxide substrate system. Examples of each type of orbital for clean nickel (100) and with adsorbed sulphur are shown in Figs. 5.9 and 5.10. To complete the picture, and to show how changes in the substrate orbitals due to coadsorbed sulphur affect the chemisorption of carbon monoxide, we have also displayed the 5a orbital for the gas phase and chemisorbed carbon monoxide in Fig. 5.11.

On the clean nickel surface, we note that for states deep in the bands (Figs. 5.9(c,d)) bonding interactions among the atoms inhibit charge flow into the vacuum. Nearer the Fermi level, however, the repulsive interaction in the generally antibonding states favours charge flow away from the surface' (Figs. 5.9(a,b)), which is facili­tated by hybridisation with s and p states (Figs.

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Figure 5.10 Wavefunction contour plots using the notation of Fig. 5.5 for orbitals of symmetry A- (of the C. point group) on the sulphided Ni surface, for (a,b) representa­tive orbitals near the Fermi level and (c,d) representa­tive orbitals close in energy to the 5a, in cross-section I.

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5.9(a,b)) (MacLaren et al 1986c). This mechanism is closely related to that invoked by Marks and Heine (1985) and Heine and Marks (1986) to explain the types of surface reconstruction observed for transition- and noble-metal surfaces.

The distribution and mobility of nickel surface charge near the Fermi level are expected to be dramatical­ly affected by the presence of an adsorbed electronegative species. In Figs. 5.10(a)-5.10(d) we show the orbitals corresponding to those in Figs. 5.9(a)-5.9(d), respective­ly. As well as decreasing the number density of MS-Xa levels with weight on the central nickel atom (Fig. 5.3), a comparison of Figs. 5.9(a,b) and Figs. 5.10(a,b) shows that sulphur also induces a massive depletion of the mobile charge from the central nickel atom within each orbital. The sulphur-sulphur interaction is attractive, in the form of a n-bond involving the central nickel, but the a-antibonding interaction with the remaining surface atoms is manifestly repulsive, inhibiting charge flow from the surface nickel atoms normal to the surface (see below).

The nature and extent of the sulphur-induced charge redistribution is evident by comparing Figs. 5.9(d) and 5.10(d), showing the formation of a sulphur-sulphur c-bond mediated by the central nickel atom. Indeed, comparing the 3d charge density on the central nickel atom in Figs. 5.9(a) and 5.10(d) demonstrates directly that charge has been drawn from the top of the nickel d-bands (Fig. 5.9(a)), where the sulphur interaction is repulsive, and mixed into orbitals (Fig. 5.10(d)) where the sulphur p levels provide an attractive potential. Further evidence of nickel-sulphur bonding is found in Fig. 5.10(c), showing the formation of a a bond between sulphur and the second-layer nickel atoms.

Having described the effect of sulphur on the surface electronic structure of nickel (100), we turn our attention to carbon monoxide chemisorption. In Fig. 5.11 we show the gas phase 5a orbital and the corresponding orbital for carbon monoxide adsorbed on a clean and poisoned nickel (100) surface. We note first of all that

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Figure 5.11 Wavefunction contour plots using the notation of Fig. 5.5, in cross section I, of the CO 5a (I) orbital in (a) the gas phase and (b) the clean Ni surface and (c) the sulphided Ni surface.

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the broadening of the carbon monoxide molecular levels in the presence of the nickel levels is manifested by the orbital in Fig. 5.11(b) being an approximately equal superposition of the molecular level in Fig. 5.11(a) 'and the nickel orbitals in Figs. 5.9(c) and 5.9(d). In contrast, the mixing between the carbon monoxide 5a level and the nickel states on the poisoned surface is restrict­ed predominantly to the orbital shown in Fig. 5.10(c). Evidently, the depletion of charge from the neighbourhood of the Fermi level weakens the interaction with the substrate, thus inhibiting the broadening of the carbon monoxide molecular levels.

Another effect of the sulphur atoms is a slight in­crease of the nickel work function. Charge transfer from the Fermi level to the sulphur atoms and the subsequent sulphur-nickel bond-formation inhibits the flow of charge from the nickel toward the vacuum. This is vividly demonstrated in Figs. 5.10(a) and 5.10(b) by the appear­ance of a nodal plane between the sulphur overlayer and the substrate, as compared with Figs. 5.9(a) and 5.9(b). The sulphur atoms thus effectively screen the response and reduce the ability of nickel to chemisorb carbon monoxide, and so in general produce a substrate with substantially diminished reactivity.

5.5 Discussion

We can summarise the influence of sulphur on the chemisorption of carbon monoxide as two effects: (1) weakening carbon monoxide chemisorption by localising charge that would have been available for bonding to carbon monoxide in covalent sulphur-nickel bonds, and (2) inhibiting the incipient response of the surface by deple­ting charge near the Fermi energy (both in the number of levels and the distribution of charge within each orbit­al). As discussed in the introduction, the effect (1) is observed through a decrease in the desorption temperature and (2) an increase in the work function of sulphided nickel (100) over that of clean nickel (100) and in the

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ability of the surface to chemisorb carbon monoxide. While (1) may be understood directly in terms of the Blyholder model, (2) is more a dynamical response property of the surface, as pointed out in the preceding Section.

Although we have considered the specific case of sulphur on nickel (100), we expect our conclusions to have implications for electronegative adsorbates in general. For example, the origin of both the conclusions (1) and(2) lies not only in the electronegative nature of sul­phur, but also in the greater attraction of surface charge by the sulphur than by the carbon monoxide, as evidenced, for example, by the energies of the nickel-sulphur (3.3 eV for the atop site, 5.4 eV for the bridge site (Walsh and Goddard 1978) and nickel-carbon monoxide (1.8 eV (Andreoni and Varma 1981) bonds. This observation may be particu­larly relevant in the case of several very different adsorption sites that may be sampled by the molecule. Madix et al (1983), have indeed observed in their thermal programmed desorption data the appearance of lower-energy carbon monoxide adsorption sites on the sulphided surface, whose population increases with increasing sulphur cover­age .

Finally, we show that many of the sulphur-induced features in the surface electronic structure of the poisoned catalyst can be deduced from simple phase-shift arguments (Pendry 1977). The phase-shifts, which com­pletely characterise the potential, can be related to the number of electrons drawn into or repelled from the atom at energy E and angular momentum 1 by, n^ a d/dE [6^(E)]. A typical phase-shift for an electronegative adsorbate has positive slope below the energy of the p-level (E ) and negative slope above. Hence for states below Ep the potential is attractive, and above is repulsive.

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CHAPTER 6

Local interactions between CO and promoted Ni surfaces

6.1 Introduction

In contrast to the poisoning effect discussed in chapters 4 and 5, catalyst activity may be promoted by the coadsorption of certain atoms. Industrial catalysts often include small amounts of preadsorbed electropositive elements on the surface, whose role may be (1) to increase the reaction rate, (2) increase the selectivity towards the desired product, or (3) increase the lifetime of the catalyst.

In general, rates of molecular dissociation are dramatically increased by coadsorbed alkali metals. For example, taking potassium as a typical promoter, then increased dissociation rates have been observed for (1) carbon monoxide on Pt(lll) (Crowell et al 1982), Ni(100)(Campbell and Goodman 1982) and Fe(110) (Broden et al 1979), and (2) nitrogen on iron surfaces (Ertl et al 1979). The role and importance of surface carbon, derived from dissociated carbon monoxide, in the FTS have been discussed in detail in chapter 1. This has provided the motivation for the numerous experimental studies of chemisorbed carbon monoxide on clean and promoted surfac­es. In this chapter, we will review some of this work which will be used as a basis for understanding our calculations on promoted surfaces. Other theoretical approaches, which were reviewed in the introductory chapter, include the Effective Medium Theory (Nsfrskov et al 1984; Lang et al 1985). Electrostatic interactions, which were suggested to be important (Lang et al 1985), are shown to be of greater significance on promoted than on poisoned surfaces.

Modifications to the Blyholder model of carbonmonoxide chemisorption on promoted surfaces (cf chapter

*1) imply that increased 2 n occupancy should occur on promoted surfaces, as a result of increased "back-

1

donation" to this orbital and charge transfer from the promoter to the substrate. Evidence of the latter is seen in the dramatic decrease in the workfunction accom­panying adsorption of the promoter (for example 4 eV for potassium on Pt(lll) (Crowell et al 1982)). Moleculardissociation rates should also be increased, since*filling of the anti-bonding 2ti provides the mechanism for dissociation (this has been observed for many sys-

kterns, see above). Indirect evidence of increased 2rt occupancy on promoted surfaces comes from HREELS results for the carbon monoxide stretch frequency, and the increased heats of adsorption (for example a 5 kCal/mol increase is observed on promoted Pt(lll) (Crowell et al 1982)). Decreased stretch frequencies have been measured on many promoted surfaces (Crowell et al 1982; Weimmer et al 1985; Heskett et al 1985, 1986).

Direct evidence of the positions of molecular levels with respect the Fermi level comes from photoemission, inverse photoemission (BIS) and metastable quenching spectroscopy (MQS). Photoemission from promot­ed surfaces (for example see Heskett et al 1985, 1986;Eberhardt et al 1985) shows 4a, In and 5a levels all shifted to higher binding energies. Emission from the In is altered on the promoted surface, indicative of the strong perturbation of the n levels induced by the promoter. The authors argue, from the constraint oforbital orthogonality and their photoemission data, that

*the In as well as the 5a and 2n may play an importantrole on promoted surfaces, a point we will discuss later.MQS (Lee et al 1985) uses Penning Ionisation to producean energy distribution of electrons characteristic of the*molecular levels. This technique can probe the 2n as well as the 5a level, and provides direct evidence of

kincreased 2n occupancy on the promoted surface.The effects of alkali metals on methanation and

FTS have been examined by Campbell and Goodman (1982) and by Kiskinova (1981). The conclusions of their work, and the studies on carbon monoxide chemisorption on promoted surfaces can be summarised as follows:

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Figure 6.1 Top view of the cluster used in the MS-Xa calculations for carbon monoxide coadsorbed with^ poisons and promoters on nickel (100) and the cross-sections used for displaying the wavefunctions in Figs. 6.2, 6.5, 6.6 and 6.7.

Figure 6.2 Wavefunction contour plots the molecular and chemisorbed carbon monoxide levels 5o (a,b), In (c,d) and 2ti (e,f), in cross-section II of figure 5. Contour values are ±0.081, ±0.027, ±0.009, ±0.003 and ±0.001. Solid and broken lines represent positive and negative values of the wavefunction, respectively. Atomic posi­tions are marked by the small dark circles.

i

i

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(1) the carbon monoxide binding energy is increased, as seen in the increase in the Thermal Desorption Temper­ature of carbon monoxide;(2) the carbon monoxide stretch frequency is lowered,indicative of increased back-donation from the catalyst *to the 2n ;(3) the dissociation rate over many transition metal catalysts is increased; and(4) the methanation rate is slightly decreased, and the yield of higher weight hydrocarbons is increased, consistent with increased surface carbide concentration.

6.2 Results for alkali metal promoters.

In this section we will discuss in detail the influence of the alkali metals on adsorbed carbon monox­ide on nickel (100), and make comparisons with some of the experimental and theoretical results for carbon monoxide on promoted surfaces. All of the calculations in this section are based on the MS-Xa scheme for the cluster shown in Fig. 6.1. The lithium atoms are placed in the four-fold hollow sites with the vertical spacing from the first nickel plane determined from a lithium- nickel bond length obtained by scaling the sodium-nickel bond length determined by LEED for sodium on nickel (100) (Demuth et al 1975), which gives a nickel-lithium layer spacing of 1.86A. Carbon monoxide is known, from LEED studies (Andersson and Pendry 1980) >' to sit on the atop site with the nickel-carbon bond length of 1.71A, with the carbon monoxide bond length the same as in the gas phase, 1.15A.

To illustrate the carbon monoxide-metal bondingfor the clean surface we plot in Fig. 6.2 the molecular

*orbitals 5a, In, 2n for the gas phase carbon monoxidemolecule and chemisorbed carbon monoxide on nickel (100).The interaction is not strong as can be seen from thesmall changes observed in the orbitals. Charge transferoccurs from the 5a and In to the metal and from the metal *to the 2n .

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Figure 6,3 Direct comparison of the _ MS-Xa orbital energies and the partial density of states /in the carbon muffin tin calculated by Wimmer et al (1985) for Ni/CO and Ni/K/CO. Long and short vertical bars indicate positions of MS-Xa energy levels with weight w > 10%, and 1% < w < 10% in the carbon sphere respectively. After alignment of the Fermi levels the positions and the widths of the FLAPW 4o, 5o, In and 2n agree remarkably well with the MS-Xa calculation.

- 1 2 0 -

The first study presented here for the promoted surface is for the system Ni/K/CO. The geometry used is slightly different to that used in the cluster shown in Fig 6.1, in that the potassium atoms are placed on the atop sites surrounding the carbon monoxide molecule, with a nickel-potassium bond length of 3.14A. This arrange­ment was chosen to facilitate a direct comparison to the FLAPW slab calculation of Wimmer et al, (1985) (Fig. 6.3). We see that for fairly large changes in the electronic structure observed in the promoted system, the cluster chosen accurately reproduces the slab calcula­tions. This point is important for three reasons:(1) the extended nature of the surface is not neces­sary to reproduce the gross features of chemisorption of carbon monoxide on the promoted surface;(2) cluster calculations are particularly convenient for studying the molecular adsorption site because of the flexibility afforded by not being limited to long range periodicity; and(3) the MS-Xa scheme is numerically fast, and thus allows the study of low symmetry clusters (eg in the final calculation discussed below, the symmetry of the cluster is lowered from to Cy ).

The results for Ni/K/CO and Ni/CO are compared with those of Wimmer et al by aligning the Fermi levels and marking the positions of the MS-Xa levels for the chemisorbed carbon monoxide molecule against the partial density of states in the carbon muffin tin (Fig. 6.3). In both the clean and the promoted surfaces the position of the 4a, 5a, and In agree quite well with their posi­tions in the partial density of states. The position and

*width of the broadened 2n level seen in the slab calcu­lation agrees well with the set of levels characteristic *of the 2n resonance seen in the MS-Xa results. The potassium covered surface shifts all the chemisorbed levels to lower energy with respect to the Fermi level, due to an electrostatic interaction caused by charge transfer from the potassium atoms to the substrate. All the metal states are raised, including the Fermi level. This effectively drives the molecular levels down with

Ener

gy iR

yl

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Figure 6.4 MS-Xa orbital energy levels of Al(a) and E(n) symmetry for the entire cluster and those with apprecia­ble weight on the central nickel atom for carbon monoxide adsorbed on a clean nickel (100) surface, and with coadsorbed lithium in the geometry of Figure 6.1. To facilitate direct comparison, the Fermi levels of all theiv" systems have been aligned.

- 1 2 2 -

Figure 6.5 Wavefunction contour plots using the notation of Fig. 6.3 for (a) Ni/CO 5a in cross-section I, (b) Ni/Li/CO 5a in cross-section I, (c) Ni/CO 5a in cross- section II and (d) Ni/Li/CO 5a in cross-section II.

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respect to the conduction electron states and in particu­lar with respect to the Fermi level. Some evidence ofthis purely electrostatic effect is seen in the position*of the carbon monoxide 2n resonance relative to the la, in the carbon K edge NEXAFS results, which remains the same for clean and promoted surfaces on several metals

m(Stohr et al 1984; Sette et al 1985; Babershke 1986),.

The systems Ni/CO and Ni/Li/CO will form the basis of the rest of the discussion of the promoters. Fig. 6.4 shows the MS-Xa levels, of the A^(a) and E(n) irreduc­ible representations of the C^v point group, with the orbital weight on the central nickel atom being > 5%.Lithium increases the density of levels near the Fermi level, since the partially occupied lithium s-p states are in this energy range. There is again an electrostat­ic interaction due to the polarisation of the lithium states as in the Ni/K/CO system , and a direct interac­tion between the lithium atoms and the chemisorbed carbon

*monoxide, causing the broadening of the 2 n level. Thedirect interaction can be seen more clearly from Figs.6.5, 6.6 and 6.7, which show the contour plots of the *5a, Iti and 2 k levels for these two systems. Fig 6.5shows cross section (I) for the 5a for the systemsNi/Li/CO and Ni/CO. Compared to the clean Ni/CO, thelithium 2s orbitals mix and interact weakly with the 5adrawing charge towards the lithium atoms, since in crosssection (II) (also seen in Fig. 6.5) there is relativelylittle change in the 5a upon lithium adsorption. Theinteraction is weak and localised; around the lithiumatoms, because of the large orbital electronegativitydifference between the lithium 2s states and the 5a.

On the promoted surface there is a strong interac- *tion between the 2 k , the Fermi level states, and thelithium s-p like states due to their proximity in energy,which leads to charge polarisation towards the metal. *The 2 n is a spatially diffuse orbital in both clean andpromoted cases and is antibonding between the carbon andoxygen and central nickel atoms, but weakly bonding tothe other nickel atoms in the top layer. Any rearrange-

*ment of the charge in the 2 k must involve a compensating

124

Figure 6.6 Wavefunction contour plots using the notation of Fig. 6.3 for (a) Ni/CO In in cross-section I, (b) Ni/Li/CO In in cross-section I, (c) Ni/CO In in cross- section II and (d) Ni/Li/CO In in cross-section II.

125

Figure 6.7 Wavefunction contour plots using the notation of Fig. 6.3* for (a) Ni/CO 2n in cross-section I, (b) Ni/Li/CO 2n in cross-section J, (c) Ni/CO 2ti in cross- section II and (d) Ni/Li/CO 2n in cross-section II.

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effect on the occupied orbitals due to screening.Although the In (Fig. 6.6) is energetically as far awayfrom the lithium states as the 5a, the effect is mostpronounced on it due to the constraint of orthogonality.In Fig. 6.6 there is some mixing of the lithium s states *and 2n in this orbital, which results in charge transfer from the metal and the carbon monoxide bond to the oxygen end of the molecule, and to the lithium atoms. This in turn results in much less orbital weight on the nickel atoms, apart from the central nickel. The interaction is stronger and less localised than the influence of lithium on the 5a, as can been seen from cross-section (II). The orbital is still noticeably perturbed from the clean Ni/CO In and shows the same general features as ini tsection (I). Correspondingly, Fig. 6.7 shows the 2n , and the compensating effects of a mixing of lithium p states and some In character. This results in charge transfer in the opposite sense to that observed in the In,with charge drawn from the lithium and the oxygen end of the carbon monoxide molecule to the carbon monoxide bond and the metal carbon monoxide region, shown schemat­ically in Fig 6.8.

The direct interaction of lithium, and the rehybridisation induced has been discussed in some detail above. Our model for the interaction of the promoters is, therefore, a local one with electrostatic and direct chemical interactions. For the reasons discussed above,

i tboth are important for partially occupying the 2n through back-donation of charge to’:the region of space between the carbon and oxygen atoms. The combined rolei tof the In and the 2n in the promoted system has been studied in some detail, and been shown to be more impor­tant than on the clean surface, a point also raised by Heskett et al (1985, 1986) in the interpretation of their UPS results for Cu/K/CO. In contrast the sulphur atoms interact with the 5a level directly, destabilising molecular adsorption, and indirectly by depleting states near the Fermi level, thus inhibiting transitions to thei t2 n . Even in the case of the promoted carbon monoxide,

i tthe degree of occupancy of the carbon monoxide 2n does

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Ni Li CO

Figure 6.8 Schematic illustration of the effects of Figures 6.6 and 6.7 showing the influence- of lithium on the u bonding. Arrows indicate the direction of* charge flow from the promoter to the metal for the 2n and a compensating flow from the substrate to the lithium in the In due to screening and orthogonalisation.

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not appear large enough to suggest molecular dissociation in the upright configuration.

Since dissociation of carbon monoxide, like that of nitrogen, is an activated process, it is important to consider possible channels of energy exchange with the surface. For example, the frustrated rotational mode can couple to surface phonons, a point we will address in the next section. An alternative reaction pathway would involve tipping the carbon monoxide away from the verti­cal, as a route to carbon monoxide dissociation. Results for the electronic structure for carbon monoxidechemisorbed at various angles of tilt show that the chemisorbed levels of carbon monoxide remain relatively unaltered in energy as the molecule is tipped to 45°, from which we conclude that the carbon monoxidechemisorption bond is spatially local, and that the charge density, particularly near the Fermi level, is fairly mobile for this type of perturbation. The energy barrier to carbon monoxide tipping is, therefore, expect­ed to be small, with the frustrated rotational mode having a comparatively low frequency and a large ampli­tude. The value for the frequency of 411 cm”* for carbon monoxide bonded atop on nickel (Richardson and Bradshaw 1979) indeed confirms these observations. Calculations of the total energy as a function of this tilt angle were performed by Allison and Goddard (1982), who concluded that the minimum in the total energy was very shallow and centered for carbon monoxide in the upright configura­tion. The influence of sulphur and lithium on this mode will also be the subject of future work.

6.3 Discussion

The above analysis reveals that the influence of the surface alkali metal on the electronic structure of chemisorbed carbon monoxide can be summarised as follows: (1) a shift to larger binding energy of the carbon monoxide chemisorbed levels, due to a decrease in the

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work function caused by polarisation of charge from promoter towards the substrate;(2) a direct interaction of promoter s-p states with*the carbon end of the 2 n ; and(3) a compensating interaction with the oxygen end of the In, resulting in a weakening of the carbon oxygen bond.

As a result, the promoter makes dissociative adsorption easier on those metals where the surface carbide is stable (Joyner 1977). This conclusion is supported by Campbell and Goodman’s study of methanation on single nickel crystals (1982), where the coverage of the surface carbide was higher in the potassium promoted than the unpromoted case, other things being equal. Since the surface carbide is closely related to the polymerising species, the increase in selectivity to longer chain hydrocarbons than methane can be understood. It is less easy, however, to account for the observed overall decrease in the rate of carbon monoxide conver­sion.

There is a significant reason for arguing that the weakening of the carbon-oxygen bond is only a partial explanation of the role of the promoter. It is important to recognise that carbon monoxide dissociation is not rate determining on the unpromoted nickel metal. This is shown by the lack of structure sensitivity of the reac­tion, which has been noted in single crystal and support­ed catalyst studies (Campbell and Goodman 1982). This contrasts with clear evidence that carbon monoxide dissociation is structure sensitive, occuring to a very limited extent on the close-packed (111) plane, and much more rapidly on stepped surfaces (Eastman et al 1973; Erley and Wagner 1978).

The observations on carbon monoxide adsorption -on nickel closely parallel those of Ertl and his colleagues on the interaction of alkali metal with nitrogen on the iron single crystal planes. Dissociative adsorption of nitrogen on iron surfaces is an activated process, (Ertl 1983), and Ertl et al have shown that the presence of alkali metals lowers this activation barrier

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significantly. Recent evidence suggests that, in this case also, the role of the promoter may be complex. Kinetic analysis by Bowker et al (1985), has indicated that, under ammonia synthesis conditions, the barrier to nitrogen dissociation is insignificant compared to that involved in hydrogenation of the adsorbed nitrogenous species. The reduction in the barrier to dissociative adsorption is inadequate to explain the promotional influence of the alkali metals on ammonia synthesis on iron catalysts. Bowker et al argue that the heat of adsorption of the reactive nitrogen adspecies is much lower than that observed in the static experiments of Ertl et al, and we may speculate that the promoter has some role in this decrease of the nitrogen binding energy at high coverage. An alternative analysis of the Ertl data has been proposed by Stolze and Norskov (1985) but the situation remains to be resolved.

There are additional reasons for considering that the role of the alkali metals in carbon monoxide adsorp­tion and catalytic chemistry is complex. Evidence for the magnitude of the weakening of the carbon - oxygen bond by the promoter is provided by vibrational spectros­copy studies. The extent of this weakening, however, is very variable ranging from 120 cm""1 on the Pt(lll)/K surface (Crowell et al 1982), to over 700 cm""1 on copperpromoted by potassium (King 1986), and results from an*increased occupancy of the chemisorbed 2n on the promot­ed surface. Using effective medium theory, Holloway and Nerrskov (1984) have shown that the electrostatic part of the promoter/CO interaction can account for a weakening of ca. 120 cm”1. It is generally assumed that shifts of a greater magnitude indicate a tipping of the molecule away from the vertical or the change in the adsorption site. The formation of a squarate species, K+(CO)^, has been suggested on copper surfaces (King 1986).

One possible role of the alkali metal promotion in methanol synthesis could be to tip the adsorbed carbon monoxide species towards the surface. This could be significant if methoxide, which is known to bind to the metal through the oxygen atom, is a reaction

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intermediate. It would be particularly interesting to understand the role of the promoter in methanol synthe­sis, since the promotional effect is much greater than the change noted in ammonia synthesis on iron catalysts or in methanation (Kikuzono et al 1982). Our calcula­tions have shown, however, that little change in the electronic state of the adsorbed carbon monoxide molecule on tipping the carbon monoxide away from the vertical plane. This is perhaps not surprising given the soft nature and large amplitude of the metal-carbon monoxide wag mode. This mode will be important for the exchange of energy between the surface and the molecule, since its energy is low enough to couple to the surface phonons. The anharmonicity in the carbon monoxide metal potential allows coupling between this frustrated rotational mode and the carbon monoxide stretch frequency, observable in the spectroscopic studies of Hoffman et al (1986), and by Harris et al (1977) in a temperature-dependent lineshape in the carbon monoxide stretch mode. The coupling to the surface phonons is more effective for carbon monoxide bonded to the bridge site (Persson and Ryberg 1985)rather than to the top sites, because of the lower frequency of this adsorption geometry.

Other models for energy exchange between substrateand adsorbed molecules involve the creation ofelectron-hole pairs, a point discussed by Persson andRyberg for carbon monoxide on nickel (100) (1985), andPersson and Persson for carbon monoxide on copper (100)(1980). The electron-hole excitation will be most*effective when the 2ti resonance is close to the Fermilevel (Persson and Persson 1980), as on the promotedsurface, and consequently less important for clean andsulphur covered surfaces. The carbon monoxide rotationalmode neither shifts the position nor changes the occupan- *cy of the 2n greatly, so energy-damping via surface phonons will be more likely than the creation of electron-hole pairs.

The changes in bonding of the molecular carbon monoxide state brought about by the alkali-metal promot­er, which have been the subject of the calculations

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presented here, are insufficient to explain all of the changes noted in catalysis. Additional roles for the promoter may be to modify the stability of catalytic intermediates, to mediate the reactivity of chemisorbed hydrogen, or to change the surface diffusion parameters of reactants, intermediates, or products. It is not clear which of these is of dominant importance and we intend to explore some of these questions in subsequent work.

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APPENDIX A

Derivation of formulae in chapter 2

a.1 Calculation of the shell scattering matrices

Suppose the final scattered wavefield from atom i in the shell is given by

S b£ h]|(Kri)YL{ri) {a.l>

with r^ = r - R^, is the vector from the centre of thecluster to the atom labelled i. Suppose there are the following incident waves on atom j in this shell given by

S j^Kr ■ )YL (r .) {a.2}

Then the total incident flux on the jth atom arises from the scattered waves on every other atom in the shell plus the incident waves given by equation {a.2}. This is given by

E a£ j1(Krj)YL(rj) + E b£ g££, jL, (KTj )YL , (£j ) {a.3}

Where 2, is proportional to (1-5..), and is the matrixL iIj 1 j

element of G^, used in expanding the Hankel functions about a new origin (Durham et al 1982).

The wavefield given in equation {a.3}, once it has scattered from atom j must be of the form given by equa­tion {a.l}, so that the following relation is derived

S(aL 6ij6LL' + b: .13'LL ) t: = b; {a.4}

or

al = s bL' ^ L {a.5}

where

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HL'L = SLL'6ij t i ' 1- G31L'L {a.6}

hence

bL = s 4 ' t l -l {a.7}

where T is the matrix inverse of H.Equation {a.7}, which relates incoming to outgoing

partial waves, is the central equation for calculating T°°, T01, T10 and T11.

If there are a set of outgoing waves incident on the shell given by

E Al h£(Kr)YL(r) (a.8}

then expanding these about each site gives the following incident waves

,oiE G~£, jlf(Kri)YL ,(r..) {a.9}

using equation {a.7} the final scattered waves are

S Al G°£, T ^ l„ h1^(Krj)YL„(rj) {a.10}

if these are expanded about the cluster origin then {a.10} equals

E A rp ’ L L ^LLl 1L1I

r« a n ^ l rplj

a a

rj°GL2L' h i - (Kr)YL ,(r) r > R (max) a

r JOGL2L'

{a.11}(Kr)YL ,(r) r > R (min) a

) are the maximum (minimum) radii betweenthe cluster origin and any of the atoms in shell a. G is the alternative form of the propagator which applies to points at which outgoing waves, when expanded at a new origin, still appear as outgoing waves. From this we can identify T00 and T01. T10 and T11 can be derived in a

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similar manner by calculating the incident wave on eachatom in a shell, given the incident waves on the shell, tweThe results^summarised by:

T111LL' = S ZoiLL1

rpi j XL1L2

r j°GL2L'

T10 iLL ’ = S ZoiGLL1 Tij

L1L2Z jo GL2L'

T01iLL' = D roiGLL1

TijXL1L2

rjoGL2L'

T°°1LL' = S -oi

GLL1 rpij1 L 1 L 2

r i °GL2L' {a.12}

The expansions of spherical Bessel and Hankel functions are given by

(1) |r-r.| < r.

h*(Kr)YL(r) = E G°£, j 1, (K.|r-ri|)YL, (r-r±)

(2) |r-r.| > rA

hA(Kr)YL(r) = E G°£, hj;, (K|r-r±| )YL , (r-r^)

( 3)j1(Kr)YL(r) = E g££, jx , (K | ± |)YL, (r-ri) {a.13}

with

gLL' = ^(l-6io) S i1-1'-1"h3„(Kr.)Y*„(r.) J Y^.Y*. d£2

GLL' = 4"<l-Sio> S i1-1'_1"j!„(KrA)Y^„(r^) / Y ^ Y * , da{a.14}

a.2 A note on the matrices

The shell scattering matrix T is labelled by four indices, the angular momentum and atom labels. The angular momentum index runs over the l,m's required to

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converge atomic scattering and the atomic index runs over the unique atoms in the shell. The structure is therefore similar to that of the X matrix encountered in layer scattering (Pendry 1974 p. 132). Intershell scattering does not converge until 1=14 typically, so the propagators between the multi-centre shell T matrix and the single centre matrices T00, T01, T10 and T11 are therefore rectangular matrices.

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APPENDIX B

b.l Spherical wave expansion for G+

The atomic Green function, including spin degeneracy, for an atom at the origin is given by,

Gat(£/£'-E) = -4iK S Y*(r<)$1(r<,K)4>];(r).,K)Y*(r>) £b.l>

where K is the propagation vector with respect to the energy zero, and and are the regular and irregularradial solutions to the Schrodinger equation respectively,r^ are the greater and lesser of r and r'. Outside themuffin tin these radial solutions match smoothly ontospherical Bessel (j^) and Hankel (h^) functions as follows

«1(r,K) = e"lSl [ j1(Kr) + t ± h^(Kr) ](b. 2}

$^(r,K) = el6l h^(Kr)

If we consider r > r' and outside the muffin tin, and the atom labelled (3 to be at Ro^, then

at = S tf ( K r p)YL (rop) {b.3}

with

AL0 = -4iK *l(rV K)YL(£o|3) ei6{3 *.13-1 {b. 4}

with r Q = r-R 0.- o ( 3 ---o(3G^t appears like a set of outgoing waves leaving atom type (3 in the unit cell at the origin of the layer. These wave are then allowed to scatter from the other atoms in the layer. Pendry (1975) has developed a method for including all intralayer multiple scattering. Firstly using the identity

1/N E elk//*Rj = 6.J 30 (b. 5}

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where is a real space lattice vector and there are N unit cells in the layer. The k ^ summation is restricted to the first Brillouin Zone.- Using this, equation {b.3>, can be expressed as a Bloch sum on each atom in the layer as follows

Gat = 1/N 2 ALP fcl hi‘KrjP,YL(£jP) eik//‘R3 ^.6}

Secondly all scattering is included by applying the condition that the outgoing waves on a site are the result of scattering from this site of the incident waves formed from the scattered waves from the other sites. Pendry (1974 p. 128) has calculated this in terms of the lattice scattering X matrix, the layer equivalent to the KKR structure constants. The layer Green function can be written as

Glay = l/N S A^[l-X]-JjL,atJ,h^, (Krja)YL' (r.:-3a eik//.Rj{b.7}

A single centre expansion can be obtained about the atom (3 by expanding the waves from the other atoms about this site. An alternative approach, which we shall follow, is to regard the outgoing waves from this atom, excluding the initial waves, as arising from a set of incident waves on this site. The set of waves necessary to give this is

V N S 31 ,(Kr0&)YL ,(rop) {b.8}

Hence the sum of these and the outgoing waves gives the single site expression

Glay = 1/N 2 ALP[1-X]l S,L'P tl'hi'(KroP,YL'(£oP) +“I

1/N 2 AL(3C(1-X) "1 L(3,L'P ^1* (Kro(3)YL' (-o(3* (b. 9}

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4

(b°"}

>g

| layer i

>

0 + lb. )

>9

Figure b.l Schematic view of plane waves leaving layer i. The beam amplitudes are shown in parentheses.

I d " }>9

V

1

-+ (d }

ig

layer i

++Cd. }

19 V

4

t

K+-

Id. ) •9

Ficrure b.2 Schematic view of the final set of plane waves in the interstitial region surrounding layer i. The beam amplitudes are shown in parentheses•

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Suppose o u t s id e l a y e r i , in a p la n e wave b a s is we have th e f o l lo w in g beams on e i t h e r s id e o f th e l a y e r ,

shown s c h e m a t ic a l ly i n F i g . b . l .

E b ? + e iK§ * (r-0 i )i g

E b ? “ e i K 5 * ( r ’ C i } l g

w ith

z > c1Z

z < c i z { b .10}

Kg = k / / +g * V (K 2 “ ( k / / +<? ) 2 ) { b . 11}

Then we a sk how do th e s u rro u n d in g la y e r s m o d ify th e s e beam s. F o r s i m p l i c i t y i n su b se q u e n t d e r iv a t io n s c . ( th e

i.Uo r i g i n o f th e i la y e r ) w i l l be s e t to z e r o . The m ostg e n e r a l s o lu t io n w i l l in v o lv e fo u r s e t s o f beam a m p litu d e sf o r e ach la y e r w h ich we s h a l l c a l l dT* th e se d e s c r ib el gw aves on th e + and - s id e o f th e la y e r t r a v e l l i n g i n th e + and - d i r e c t io n s , w h ich a re shown s c h e m a t ic a lly in F i g . b .2 . A know ledge o f a l l th e s e beam a m p litu d e s s o lv e s sta g e . 2 o f the p ro b le m .

To c a lc u la t e d7* f i r s t th e b u lk i s c o u p le d to la y e r i , and th e n th e se b u lk l a y e r s a re co u p le d to th e s u r f a c e l a y e r s . O n ly th e w aves t r a v e l l i n g in the + d i r e c t io n ca n s c a t t e r from th e b u lk . L e a d in g to a w a v e f ie ld on th e + s id e o f th e la y e r com posed o f th e f o l lo w in g term s

S b Tg e i K a - r + S (P I S i + l Pt > g g - bTg' +

S <RPI S i + l Pt > g g ’ bT g ’ e * * ' * +

E < P ls b+1 P ^ R p -Sb + 1 P + )g g , b ° ; , e i K i - r + { b . 12}

w here R and T a re th e r e f l e c t i o n m a t r ic e s f o r th e la y e r s /toand S . a re th e s u r f a c e / b u lk r e f l e c t i o n m a t r ic e s w ith 1 +

r e s p e c t to an o r i g i n m la y e r i . P7 a re th e p r o p a g a to r sbetw een la y e r i and i+ 1 f o r waves t r a v e l l i n g i n th e ±s e n s e s .

T h is s e r ie s sums e x a c t l y to g iv e

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F ig u r e b .3 Schem atic view o f p lan e waves le a v in g la y e r i tow ards towards the s u r fa c e , in c lu d in g a l l m u lt ip le s c a t t e r in g between the la y e r and the b u lk . The beam a m p litu d e s are shown in p a re n th e se s.

-142-

E ( 1 - R P :s ^ +1P t ) “ J , b g , e iK £ * r +

E ((1- PlsgiPlRJ^Plsg.P+J , bg, eiKg*r {b.13}

Hence th e re n o rm a lis e d f l u x t r a v e l l i n g in th e - d i r e c t io n , in c lu d in g s c a t t e r i n g fro m th e b u lk la y e r s , can be w r it t e n as

« _ i K - . rs bige g z<0 {b.14}

w ith th e a m p litu d e bT , shown in F i g . b .3 , g iv e n by■y

bIg = h T g + S (T(1' PISi+lPtR)'lpISi+lpt>gg' b?g- <b-15>

Now th e s e r e n o rm a lis e d a m p litu d e s a re a llo w e d to s c a t t e r from th e s u r f a c e l a y e r s . The s c a t t e r in g g e n e ra te s th e f o l lo w in g term s

r i K f r x p /B+ c s D- x . - i K + . rS b e g + E , b. , e g +

S <SiPt-lS!-lPI-l>gg' bIg- +

S (Pt - l ^ - l V l SiPt-lS!-lPI-l>gg- big' ^ +{ b .1 6 }

w h ic h , once summed g iv e s

v 1 1 d ” \ — 1 u.“ iKi • r--. ,S (1_ S i P i - l S i - l P i - l , g g ' b i g ' e g •+ { b . 17}

S <(1“ PI-l^-lPI-lSi>‘lpt-lSL l PI-l>gg' big'

From t h i s we ca n i d e n t i f y d g and d g by e x p r e s s in g e q u a t io n { b .1 7 } as

r LK+.r v , — i K - . rS dig e g + E d i g e g { b .1 8 }

i d e n t i f y i n g

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d"~ = E ( 1 - S^Ppt , S? -PT 1 ) " 1 , bT , lg ' 1 l-ri-ri-rgg' ig’

diq = 2 ((1- Pt-lSI - l V l Si»"lpt-lS!-lPI-l»qq. bIq. <b’19>

In a s i m i l a r m anner the o th e r p la n e wave c o e f f i c i e n t s h— ++d. and d. ca n be fo u n d by r e p e a t in g th e a n a ly s i s m the l g l g

o p p o s it e s e n s e , i e by c o u p lin g la y e r i to th e s u r fa c e f i r s t and th e n c o u p l in g t h i s system to th e b u lk la y e r s . O n ly th e w aves t r a v e l l i n g in the - d i r e c t io n ca n s c a t t e r from th e s u r f a c e . L e a d in g to a w a v e f ie ld on th e - s id e o f th e la y e r com posed o f th e f o l lo w in g term s

2 h T g eiK5-r + s <Pi-lSL l PI - l V b Tg> eiKS-r +

2 (RPt-lSi-lPI-l V big eii r +

2 Pi-lSi-lPi-lRPi-lSi-lPi-l^gg' hV g ' + {b.20}

T h is s e r ie s sums e x a c t l y to g iv e

2 (1" RPi-lSi-lPi-l^gg' b Tg< ^ +

2 ((1- PI-lS!-lPI-Rl)'lpt-lSi-lPI-l>gg' bTg> ^ <b'21>

Hence th e r e n o r m a lis e d f l u x t r a v e l l i n g in th e + d i r e c t io n , in c lu d in g s c a t t e r i n g from the s u r f a c e l a y e r s , can be w r it t e n as

_ . + iK + r 2 b i g e « z>0 { b . 22}

w ith th e a m p litu d e b t^ g iv e n by

b!g = bS + 2 (T(i- PI-*1 *”lpi-lsi-lPI-l*gg' big-{ b .23}

Now th e se r e n o r m a lis e d a m p litu d e s a re a llo w e d to s c a t t e r from th e b u lk la y e r s . The s c a t t e r in g g e n e ra te s th e f o l lo w in g term s

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+ i K + . r -^b i K - . r2 b i g e g + E ( P i S ^ P i ) , b . g , e ~ g +

2 <S K S i + l Pt > g gK+ ^ i K + . r b i g - e g +

s (pTsb .,pts?pTsb ,,pt)1 1 l + l 1 1 i l + l i ' g gH + 4.b ig , e g + . . . {b.24}

w h ic h , once summed g iv e s

2 (1- h + e ^ ^a*r +E U SiPiSi+1Pi )gg, Dig, e g +

*-„b „ + „ S v “ l^ - ~ b + i K - . r2 ( ( 1 - P ^ P ^ ) ^ S ^ P p , b i g l e g { b . 25}

. . +- ++From t h i s we ca n i d e n t i f y d ig and d ig by e x p r e s s in ge q u a t io n { b .2 5 } as

„ ,+ - i K - . r , „ ,+ + i K + . r s d . g e g + E d . g e g { b . 26}

i d e n t i f y i n g

< g = 2 (1- SiPISi+lPi^gg' btg'

d ig 2 f'1- PISi+lPtSi>"lpISi+lPI)gg' big' { b . 27}

The w aves, a f t e r a l l i n t e r l a y e r m u lt ip le s c a t t e r in g , w h ich a re in c id e n t on la y e r i a re g iv e n by

r A +- _ i K - . r , „ j - + ^ i K + . r E d . g e g + E d . g e g { b . 28}

e x p a n d in g t h i s in t o s p h e r i c a l waves to in c lu d e , s ta g e 3, s c a t t e r i n g in th e la y e r f o r th e l a s t tim e g iv e s ,

2 <dtg eiKi-R°0 *L(Kg> + dig V£J>>

4n i 1j 1 (Kr0(3)Y L (r0 p) { b . 29}

in t h i s and su b se q u e n t fo rm u la e th e com plex c o n ju g a te o f•kth e s p h e r ic a l h a rm o n ic w i l l be w r it t e n as Y lm ( x ) , however c a l c u l a t i o n s u se th e a n a l y t i c c o n t in u a t io n , nam ely - l mY^_m( x ) . E q u a t io n { b .2 9 } ca n be w r it t e n as

-145-

2 D L{3 3l«K r o P >Y L (£ o P )

w ith

{b.30}

D . 1 , , + - iK - . R . -+ iK + .RL(3 = S 4 n i (d i g e g - oP Y j.(K g ) + d i g e g ‘ op YL (K g ))

{ b . 31}S c a t t e r in g i n th e la y e r f o r th e l a s t tim e in t r o d u c e s an ( 1 - X ) ” f a c t o r , hence i f we d e f in e th e f o l lo w in g

° L P = 2 DL ' a C1- X l V a , L P { b . 32}

th e n th e f u l l G reen f u n c t io n i s g iv e n by

G = Gl a y + 2 ° L P ” l ( K r op> + t Xh i < K r o P , l y L ( £oP> { b ' 33}

The c o e f f i c i e n t s b . “ a re r e la t e d to th e e x p a n s io n o fi gth e H a n k e l f u n c t io n s in e q u a t io n in p la n e w aves. T h is i s a w e l l known r e s u l t (P e n d ry 1974 p . 2 7 3 ) , and s u b s t i t u t in g f o r th e s e we g e t

bi g " 2 r gL(3 AL(3 ( b . 34}

w ith th e f o l lo w in g

r gLP = 2 n i - 1 /(K£2K+,) e - ^ * Rop YL <K+) t f

ALP = AL ' a ^1 _ X ^ L 'a ,L p { b . 35}

th e n i f

° L P = 2 r g L P d i g + r g Lp d i g { b . 36}

w ith

2gLP = 4 n i l ®i K 9 - R°P { b . 37}

th e n th e c o e f f i c i e n t s o f th e p la n e waves a re r e la t e d to th o se o f th e i n i t i a l o u t g o in g s p h e r ic a l waves by

-146-

d t" = E F t ' AtQ i g igLf3 L0

d I g = s F I g L p aLP

where th e m a t r ix e le m e n ts o f F a re g iv e n b y

+ “ - „ b „ + « S v - l - - « b - + * r „ - +

{b.38}

k+ «s “1 tn +

Pt>gg' [ sg'g

PI-l>g''g" Fg":

F Ig L 3 = ( ( 1 - pi - l s i - l pI - l s i ^ - l p i - l s i - l p i - l , g g ' [ 6g 'g » r g"LP

+ <T <X- p I s i + l pi R *"l p I s i + l pi 1g ' g" r g "L0 ] ( b . 39}

th e c o u p l in g betw een th e i n i t i a l and f i n a l s p h e r ic a l w aves ca n be e x p r e s s e d as

DL(3 " S E

S r gLp F i g L ’ a + r gL(3 F i g L ' a

L L ' , a p AL ' a

w ith

L 'L ,a { 3

and d e f in in g

EL ' L , a 3 = E EL ' L " a S C 1 -X JL " 5 , L 'P

We f i n a l l y a r r i v e a t th e f o l lo w in g e x p r e s s io n f o r

S'3 = 1/N E A ° p 31 , ( K r o p )Y L , ( r o p )

{ b . 40}

( b . 41}

-1 ( b . 42}

t ? ,h ? ‘ l (K r A )Y r , ( r q )+ 1/N E AL(5 d-X)Lp^L ,p w1|4*1( v — *L. v±.Qp

+ 1/N E ALp EL ’L'\a£

•+ t ^ „ h 1 ,l (KrQ p) ]Yl „ ( L q $ ) ( b . 43 }

P e r fo rm in g th e a n g u la r in t e g r a t io n to g e t an s p h e r i c a l l y a v e ra g e d G g iv e s

-147-

®P = -iK/(nM)E U j ^ K r ^ J + t P h l f K r ^ J t ^ C d - X ) - 1- ! ] ^ ^ ^

^ l (KroP,+tlhi(KroP,)

+ (^ i (Kro P )+tih i (Kro P

+ (ii<Kr0 p ,+tih i (Kr0 p

) )hi<Kro p )

) ) ( 1 - x ) L P , L ' a EL L ' , a p

^ l (Krop,+tlhi(Krop,,} { b . 44}

S u b s t i t u t in g f o r th e r a d i a l w a v e fu n c t io n s , w h ich a l lo w s th e e x p a n s io n f o r G t o be used w it h in th e m u ff in t i n s p h e re . The f i n a l e x p r e s s io n , once the sum m ation has been tu rn e d in t o an i n t e g r a l o v e r the f i r s t B r i l l o u i n Zo n e, i s

G * = - i i t t T 3 / d k / / { E * l p ( r 0 p ) * i p ( r o p )

+ ^lp^op* Tip [FLP " $ip(rop)}w ith

F L{3 S (1 “X ) L (3 ,L 'a (6 L L ' 6a(5 + E L 'L ,a ( 5 )

( b . 45}

{ b . 46}

b . 2 D e r iv a t io n o f th e p la n e wave e x p a n s io n f o r G+

In d e r iv in g a F o u r ie r s e r i e s w ith z dependent c o e f f i ­c i e n t s f o r th e G reen f u n c t io n , we s t a r t by d e f in in g a B lo c h Green f u n c t io n w it h in th e r e g io n o f c o n s ta n t p o te n ­t i a l as

Gk / / ( £ , r ' ) = 2 G ( r , r ' - R . ) e " l k / / , R j { b - 4 7 }

an e x p a n s io n o f t h i s B lo c h G reen f u n c t io n in term s o f a p la n e wave b a s is y i e l d s

-148-

Gk//(£,r') = 1/n 2 el(k/tg),R Gk//,gg , ( z , z ' Je - 1 *14/ ? 5 ' * •R ’{ b . 48}

a p p l i c a t io n o f th e S c h r o d in g e r e q u a t io n on G ( r , r - R j ) g iv e s th e f o l lo w in g

[*V2 + E] G ( r , r ' - R j ) = 6 ( r - r ' + R j ) { b . 4 9 }

- ik . Rnow i f we m u lt ip ly by e //* j , a n d sum o v e r a l l l a t t i c e s i t e s R j th e n

tiv*.+ E] G k / / (r,r') = S S(r-r'+R.) e~lk//,Rj {b.50}

e x p a n d in g th e r^ y dependence o f th e 6 f u n c t io n as

6(r-r'+Rj) = 6(z-z')/Nfl S elk//'(R_R'+Rj)e_lk//-Rj {b.51}

hence

[ i v * + E] Gk / / ( r , r ' ) = 6 ( z - z ' ) / N G S e l k / / - ( R _ R , )

S e ' l ( k / / ~ k / / ) , R j { b . 52}

The r e a l sp a ce sum m ation i s z e ro u n le s s k ' - k = g , t h e r e ­f o r e

civ* + E] Gj, ,(r,r' ) = 6(z-z ' )/Q E e^(k//+g) •(R_R' * {b.53}

a f t e r s u b s t i t u t i n g e q u a t io n { b . 4 8 } in t o e q u a tio n { b . 5 3 } , m u lt ip ly in g by 1/n e ~ i ( k / / + g ) - R e ^ k / / + g ' > ‘ R ' and i n t e - g r a t in g o v e r d R 2 and d R ' 2 we g e t th e f o l lo w in g

[ d V d z * + K | z ] Gk / / / g g , ( z , z ' ) = 2 6 ( z - z ' ) 6 g g , ( b - 54}

In th e a b se n ce o f s c a t t e r e r s th en th e se m a t r ix e le m e n ts ca n be form ed from th e l i n e a r l y in d e p e n d e n t

-149-

s o lu t io n s to hom ogeneous p a r t o f e q u a tio n ( b . 5 4 ) . The d is c o n t i n u i t y o f s lo p e a t z = z ' le a d s to the r e s u l t t h a t

Gok//,gg ( z , z ' ) - i / K g z e i K g J Z - Z ' ( b . 55}

The n e x t s ta g e i s to a l lo w t h i s form f o r G ° to be c o r ­r e c t e d f o r s c a t t e r i n g betw een th e s u rro u n d in g la y e r s . Suppose both z and z ' l i e i n th e i n t e r s t i t i a l r e g io n betw een la y e r s i and i - 1 , a l s o l e t z > z ' th e n th e waves t r a v e l l i n g to w a rd s th e b u lk , g iv e n by

e l K g z A, { b . 56}

s c a t t e r from th e b u lk to g iv e th e Green f u n c t io n in th e p re s e n c e o f th e b u lk as

r , „ _i\ _ QiK__Iz-z’I -iK _ _ „ -ilC ,Gk / / , g g , ( z , z ' “ e g z a g + e gz Rg g ' g e gz

a g = “ i / K gz { b . 57}

I f we make z<z* th e n th e w aves t r a v e l l i n g tow ards th e s u r f a c e r e g io n c a n now s c a t t e r b ack and f o r t h between s u r f a c e and b u lk to g iv e th e f u l l Green f u n c t io n

V / , g " g v

, „ b ~ s »-1

/ _ — l K || z (e g' z 6g " g " ' + ei K g " z z Rg llg „ , )

- i K z ' g e g z + R ^ , e g ' g ” i K g z z ' ) a g { b . 5 8 }

Gk / / (£ ' r ) , the e q u a l argum ent G reenf u n c t io n , from th e m a t r ix e le m e n ts in e q u a tio n { b . 5 8 } by u s in g e q u a tio n { b . 4 8 } g i v i n g

V /( r , r ) = 2/Q S ( e " l K g ,,r 6 + e lK g " r R ® , , )g - g - g g

/ *| pbls.-l / A -iK-r _b -iK+rx „(1-R R gM,g’ 5g'ge g + Rg'g e g ag ( b . 59}

once th e s p in d e g e n e ra c y h a s been in c lu d e d . F i n a l l y i n t e g r a t in g e q u a t io n { b . 4 7 } o v e r the f i r s t B r i l l o u i n Zone g iv e s G ( r , r ) as

-150-

G(r,r) = 2 / (2 n )2 /dk/y E (e'iK§"r 6gllg„, + eiK§"rR=„gl,, )

{ b . 60},, DbDs. -1 , ~ -iK-r Db o-iK+ru( ! - R R ) g „ , g ( ( 6 g l g e g + Rg , g e g ) a g

C o n s t r u c t in g a F o u r ie r s e r i e s as fo llo w s

G ( r , r ) = E G ( z ) ey

i g . R { b . 61}

th e n th e se c o e f f i c i e n t s a re g iv e n by

G ( z ) = s T 1 J d = r G ( r , r ) e - i g >y

R { b . 62}

P e rfo rm in g th e r e a l s p a c e in t e g r a t io n o f e q u a t io n { b . 5 8 } g iv e s

Ggl(z) = 2 /( 2ti) 2/dk1 1 E (e'iKgzz6ggl + eiKgzzR®g , ) { b . 6 3 }

(l-RbRS)-lg„(6g„gl_ge-iKgl-gzz + Rg,.gl.ge‘iKgl-gzZ)agl_g

b . 3 W e igh ts and o r d in a t e s f o r th e e n e rg y in t e g r a t io n

b . 3 . 1 The s e m i - c i r c u l a r c o n to u r i n t e g r a l

D e fo rm in g th e i n t e g r a l to th e s e m i - c i r c l e d is c u s s e din th e e n e rg y i n t e g r a t io n s e c t io n g iv e s , a f t e r th e s u b s t i -

10t u t io n E -E = E e (E i s th e c e n tr e o f th e s e m i - c i r c l e ) o r o

Im J ( 1 G( E) dE} = Im J ( 2G ( E ) e 10 Er id 0 } { b . 6 4 }

where E r i s th e r a d iu s o f the s e m i - c i r c l e . The G a u ss-C h e b y ch e v i n t e g r a t io n fo rm u la (A brom ow itz and S te g u n 1970, P. 889) ca n be a p p l ie d a f t e r th e change o f v a r ia b le s 0 = n - 0 . We g e t

p ( r ) = Im E ( i e - 1 0 j E r G(E. . ) w .)

where 0^ = ( 2 j - l ) / 2 n , w = n / n and E j = E (©^ )

{ b .65}

-151-

b . 3 , 2 The F e rm i e n e r g y c o n to u r i n t e g r a l

p ( r ) = Xm J { G ( E ) d E ) a + Im / { G ( E ) d E }, { b . 6 6 }

C o n to u r a i s 1/4 o f c i r c u l a r a r c betw een E . and (E r , E )mm f o ' rin th e com plex p la n e . By m aking th e s u b s t i t u t io n 0 =2 ( 11- 0 ) , th e n t h i s can be r e c a s t in a form s im i la r to e q u a t io n { b . 6 5 } , ie

0. ( r ) = E Im { i e “ *1 0 j G( E •) } { b . 6 7 }J

C o n to u r b , whose l i m i t s ca n be w r it t e n as and E 2 can a ls o be e x p re s s e d in te rm s o f a G a u s s ia n in t e g r a t io n fo rm u la (A brom ow itz and S te g u n 1970, p. 8 8 9 ) . By m aking th e s u b s t i t u t i o n E = E^ + ( E 2- E ^ ) t th e n we ca n w r it e

PhCr) = E Im { w. (E^E^ G(Ej) } { b . 68}

w ith E j = E^ + ( E 2- E ^ ) t j . The w e ig h ts and o r d in a t e s are ta k e n a s th e p o s i t i v e o r d in a t e s f o r G a u ss -L e g e n d re i n t e ­g r a t io n o f 2n p o in t s , f o r exam ple w ith n=4 (c o rre s p o n d in g to th e n=8 G a u s s -L e g e n d re in t e g r a t io n ) we have

w0 .3 6 260 . 3 1 37 00 . 2 2 23 80.101222

t0 . 18 34 30 . 52 55 30 . 79 6 660 . 96 0 28

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