Chemical Sensors: Simulation and Modeling Volume 3: Solid-State Devices

This is the third of a new five-volume comprehensive reference work that provides computer simulation and modeling techniques in various fields of chemical sensing and the important ap-plications for chemical sensing such as bulk and surface diffusion, adsorption, surface reactions, sintering, conductivity, mass transport, and interphase interactions. In this third volume, you will find background and guidance on:

• Phenomenologicalandmolecularmodelingofprocesseswhichcontrolsensingcharacteristicsand parameters of various solid-state chemical sensors, including surface acoustic wave, metal-insulator-semiconductor (MIS), microcantilever, thermoelectric-based devices, and sensor arrays intended for “electronic nose” design

• Modelingofnanomaterials andnanosystems that showpromise for solid-state chemicalsensor design

Chemical sensors are integral to the automation of myriad industrial processes and everyday monitoring of such activities as public safety, engine performance, medical therapeutics, and many more. This five-volume reference work covering simulation and modeling will serve astheperfectcomplementtoMomentumPress’s6-volumereferencework, Chemical Sensors: Fundamentals of Sensing Materials and Chemical Sensors: Comprehensive Sensor Technologies, which present detailed information related to materials, technologies, construction, and application of various devices for chemical sensing.

About the editorGhenadii KorotcenkovreceivedhisPh.D.inPhysicsandTechnologyofSemiconductorMa-terialsandDevicesin1976,andhisHabilitateDegree(Dr.Sci.)inPhysicsandMathematicsofSemiconductorsandDielectricsin1990.Currently,Dr.KorotcenkovisaresearchProfessorattheGwangjuInstituteofScienceandTechnology,RepublicofKorea.Dr.Korotcenkovistheauthor or editor of eleven books and special issues, eleven invited review papers, seventeen book chapters,andmorethan190peer-reviewedarticles.Hisresearchactivitieshavebeenhonoredwith the Award of the Supreme Council of Science and Advanced Technology of the Republic ofMoldova(2004)andThePrizeofthePresidentsoftheUkrainian,BelarusandMoldovanAcademiesofSciences(2003),amongmanyothers.

ISBN: 978-1-60650-315-7

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CHEMICAL SENSORS VoLuME 3: soLId-stAtE dEVICEsEdited by Ghenadii Korotcenkov, Ph.d., dr. sci.

A volume in the Sensors Technology Series Edited by Joe WatsonPublishedbyMomentumPress®

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solid-state devices






ISBN: 978-1-60650-315-7

9 781606 503157

90000


CH

EMIC

AL S

EN

SO

RS

KO

RO

tc

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vO

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solid-state devices






ISBN: 978-1-60650-315-7

9 781606 503157

90000


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EN

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RS

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Contents

Preface xiiiabout the editor xviicontributors xix

1 Molecular Modeling: aPPlication to hydrogen interaction with carbon-suPPorted transition Metal systeMs 1

Samir H. MushrifGilles H. PeslherbeAlejandro D. Rey

1 Introduction 1

2 Molecular Modeling Methods 72.1 Molecular Mechanics 72.2 Electronic Structure Theory 112.3 Density Functional Theory 142.4 Plane-Wave Pseudo-Potential Methods 192.5 Optimization Techniques 22

3 Modeling Hydrogen Interaction with Doped Transition Metal Carbon Materials Using Car-Parrinello Molecular Dynamics and Metadynamics 30

3.1 Dissociative Chemisorption 333.2 Spillover and Migration of Hydrogen 35

4 Summary 42

References 43

2 surface Modification of diaMond for cheMical sensor aPPlications: siMulation and Modeling 51

Karin Larsson

1 Introduction 51

vi • contents

2 Factors Influencing Surface Reactivity 52

3 Diamond as a Sensor Material 523.1 Background 523.2 Electrochemical Properties of Diamond Surfaces 54

4 Theory and Methodology 554.1 Density Functional Theory 554.2 Force-Field Methods 62

5 Diamond Surface Chemistry 635.1 Electron Transfer from an H-Terminated Diamond (100)

Surface to an Atmospheric Water Adlayer; a Quantum Mechanical Study 63

5.2 Effect of Partial Termination with Oxygen-Containing Species on the Electron-Transfer Processes 66

5.3 The Energetic Possibility to Completely Oxygen-Terminate a Diamond Surface 70

5.4 Effect on Electron-Transfer Processes of Complete Termination with Oxygen-Containing Species 76

5.5 Biosensing 785.6 Simulation of the Pluronic F108 Adsorption Layer on

F-, H-, O-, and OH-Terminated NCD Surfaces 80

References 81

3 general aPProach to design and Modeling of nanostructure- Modified seMiconductor and nanowire interfaces for sensor and Microreactor aPPlications 87

J. L. GoleW. Laminack

1 Introduction: The IHSAB Model for Porous Silicon Sensors and Microreactors 87

2 The Interface on Extrinsic Semiconductors 89

3 The IHSAB Concept as the Basis for Nanostructure-Directed Physisorption (Electron Transduction) at Sensor Interfaces 94

4 The Extrinsic Semiconductor Framework 97

5 Physisorption (Electron Transduction) and the Response of a Nanostructure-Modified Sensor Platform 100

6 The Underlying IHSAB Principle 114

7 Application to Nanowire Configurations 116

8 Application to Additional Semiconductors 119

contents • vii

9 Time-Varying Operation and False-Positives; Sensing in an Unsaturated Mode 119

10 Sensor Rejuvenation 122

11 Summary of Sensor Attributes 123

12 Extension to Phytocatalysis-Enhanced System 123

13 Mixed Gas Format 125

14 Comparison to Alternative Technologies 126

15 Chemisorption and the Analog of the HSAB Principle 127

16 Physisorption (Electron Transduction) versus Chemisorption 129

17 Outlook 130

Acknowledgments 132

References 132

4 detection MechanisMs and Physico-cheMical Models of solid-state huMidity sensors 137

V. K. Khanna

1 Introduction 137

2 Humidity-Sensitive Materials 138

3 Resistive and Capacitive Humidity-Sensing Configurations, and Other Structures 139

4 Equivalent Circuit Modeling of Humidity Sensors 141

5 General Approaches to the Formulation of Humidity Sensor Models 143

6 Theories of Adsorption of Water on the Surfaces of Solids 1436.1 Hydroxylation of the Surface by Chemisorption of Water 1436.2 Mono- and Multilayer Physisorption and Brunauer-

Emmett-Teller (BET) Theory 1446.3 Capillary Condensation of Water Vapor 146

7 Modeling the Kinetics of Diffusion of Water in Solids 146

8 Surface Conduction Mechanisms on Solids and Humidity- Induced Surface Conductivity Modulation 147

9 Dielectric Properties of Solids Containing Adsorbed Water 1489.1 The Modified Clausius-Mosotti Equation in the Presence

of Moisture 1489.2 Maxwell-Wagner Effect in Heterogeneous Binary Systems 148

viii • contents

9.3 Sillars’s Theory for Spheroidal Particles Sparsely Distributed in an Insulator 149

10 Fleming’s Approach: Surface Electrostatic Field Model 150

11 Theory of the Porous Alumina Humidity Sensor, and Simulation of Its Capacitance and Resistance Characteristics 152

11.1 Microstructure of Porous Anodic Alumina 15211.2 Water Vapor Adsorption on Porous Alumina 15511.3 Adsorption Isotherm on Porous Alumina 15611.4 Surface Conduction Mechanisms on Porous Alumina and

Their Correlation with Surface Conductivity Variation with Humidity 157

11.5 Statistical Distribution of Humidity-Dependent Surface Conductivity of Alumina Among Pores 159

11.6 Response of Dielectric Properties of Alumina to Humidity Changes 160

11.7 Influence of Pore Shape Parameter (l) on Capacitance and Resistance Variation 166

12 Dynamic Behavior and Transient Response Modeling of Humidity Sensors 167

12.1 The Tetelin-Pellet Model 16712.2 Designing a Short-Response-Time Humidity Sensor Structure 169

13 Modeling the Diffusion Kinetics of Cylindrical Film and Cylindrical Body Structures for Enhanced–Speed Humidity Sensing 170

14 Effect of Ionic Doping on Humidity Sensor Performance 17314.1 Anionic Doping in Al2O3 Humidity Sensors 17314.2 Alternative Doping Techniques 175

15 Modeling the Drift and Ageing of Humidity Sensors 175

16 Artificial Neural Network (ANN)–Based Behavioral Modeling of Humidity Sensors 177

17 Modeling Other Types of Humidity Sensors 17917.1 Microgravimetric Humidity Sensors: The Sauerbrey Equation 17917.2 Surface Acoustic Wave (SAW) Delay-Line Humidity Sensors

Using Velocity and Attenuation Changes 18017.3 Microcantilever Stress-Based Humidity Sensors:

Stoney’s formula 18117.4 Field-Effect Humidity Sensors 182

18 Discussion of Humidity Sensor Models 184

19 Conclusions and Outlook 184

Dedication 186

contents • ix

Acknowledgments 186

References 186

5 the sensing MechanisM and resPonse siMulation of the Mis hydrogen sensor 191

Linfeng Zhang

1 Introduction 191

2 Sensors and Their Sensing Mechanisms 1922.1 Metal–Semiconductor Sensors 1922.2 Metal–Semiconductor–Metal Sensors 1942.3 Metal–Insulator–Semiconductor Sensors 197

3 Gas Diffusion 200

4 Kinetics of Surface and Interface Adsorption 205

5 Simulations 2095.1 MS Sensors 2095.2 MIS Sensors 212

6 Conclusions 219

Appendix 219

References 228

6 Modeling and signal Processing strategies for Microacoustic cheMical sensors 231

G. FischerauerF. Thalmayr

1 Sensing Principles of Microacoustic Chemical Sensors 2311.1 Introduction 2311.2 Microacoustic Chemical Sensors 233

2 Simulation and Modeling of Acoustic Wave Propagation, Excitation, and Detection 234

2.1 Analytical Solution to the Undisturbed Wave Propagation Problem 235

2.2 Analytical Solution to the Wave Excitation and Detection Problem 239

2.3 Finite-Element Method 2422.4 Equivalent-Circuit Models 245

3 Sensor Steady-State Response 2483.1 Perturbation Approaches 2483.2 Temperature Effects 253

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4 Sensor Dynamics 2544.1 Linear Model 2554.2 State-Space Description 262

5 Sensor Signal Processing 2665.1 Suppression of Temperature Effects 2665.2 Signal Processing Based on Linear Analytical Model 2685.3 Wiener Deconvolution 2695.4 Kalman Filter 2745.5 Discussion of State-Space–Based Signal Processing 277

6 Summary 281

7 Nomenclature 281

References 286

7 hierarchical siMulation of carbon nanotube array–based cheMical sensors with acoustic PickuP 289

V. BarkalineA. Chashynski

1 Introduction 289

2 Simulation Levels of Nanodesign 291

3 Prototype of Hierarchical Simulation System for Nanodesign 294

4 Continual Simulation of SAW Propagation in a Layered Medium 297

5 Structure of Carbon Nanotubes and Adsoption Properties of CNT Arrays 312

5.1 Atomic Structure of Single- and Multiwalled Nanotubes 3135.2 Quantum Mechanical Study of the Adsorption of Simple

Gases on Carbon Nanotubes 3155.3 Molecular Mechanics of Physical Adsorption of the

Individual Molecules on the CNT 324

6 Simulation of a Carbon Nanotube Array–Based Chemical Sensor with an Acoustic Pickup 332

6.1 Molecular Dynamics Calculation of the Elastic Moduli of Individual Carbon Nanotubes 335

6.2 Molecular Dynamics Study of Distribution of Adsorbed Molecules in CNT Array Pores and Calculation of Acoustic Parameters of CNT Arrays 339

6.3 SAW Phase Velocity Change Due to Molecular Adsorption on CNT Arrays in SAW-Based Chemical Sensors 341

6.4 Simulation of Adsorption on the “Swelling” CNT Array 343

contents • xi

7 Conclusion 343

References 346

8 Microcantilever-based cheMical sensors 349S. MartinG. Louarn

1 Introduction 349

2 Natural Frequencies and Normal Modes of Vibration 352

3 Experimental Procedure 352

4 Natural Frequencies of Free Rectangular Cantilevers 3534.1 Analytical Calculations 3544.2 Simulation with Finite-Element Method 3564.3 Experimental and Modelling Results on a Rectangular

Beam 361

5 Natural Frequencies of V-Shaped Microcantilevers 361

6 Natural Frequencies of V-Shaped Coated Cantilevers 366

7 Conclusion and Prospects 368

8 Acknowledgments 368

References 368

9 Modeling of MicroMachined therMoelectric gas sensors 371S. UdinaM. CarmonaC. Calaza

1 Principles of MTGS Modeling 3711.1 Introduction to the Theory of Heat Transfer 3721.2 Key Thermal Contributions and Parameters Involved in

Sensor Operation and Modeling 3741.3 Influence of the Packaging 379

2 Modeling and Simulation Methods 3792.1 Complexity Model Levels 3792.2 Analytical Models 3802.3 Finite-Element Method 3842.4 Thermal Conductivity of Gases 388

3 Application to Thermoelectric Gas Sensors 3903.1 Case Study 3903.2 Analytical Model 392

xii • contents

3.3 Static FEM 3963.4 Dynamic FEM 4023.5 Device Optimization 405

Acknowledgments 407

Nomenclature 407

References 408

10 Modeling, siMulation, and inforMation Processing for develoPMent of a PolyMeric electronic nose systeM 411

R. D. S. Yadava

1 Introduction 411

2 Sensor Array Approach 4142.1 System Characteristics 4162.2 Sensing Platform and System Design 417

3 Sensor Transient Approach 419

4 Design and Modeling of SAW Sensing Platform 4224.1 Generic Sensor Model 4224.2 Designing a SAW Platform for Mass Sensitivity 4264.3 Designing a SAW Platform for Multifrequency Sensing 438

5 Vapor Solvation, Diffusion, and Polymer Loading 4445.1 Solvation Model and Data Processing 4455.2 Sorption Kinetics and Transient Signal Model 449

6 Data Mining and Simulation for Polymer Selection 4536.1 Case Study of Explosive Vapor Detection 4546.2 Case Study of Body-Odor Detection 456

7 Optimizing Data Processing Methods 4627.1 Transient Signal Analysis 4627.2 Steady-State Sensor Array Response Analysis 4777.3 Enhancing Sensor Intelligence by Information Fusion 4797.4 Simultaneous Recognition and Quantitation 481

8 Conclusion 485

Acknowledgments 488

References 488

index 503

xiii

PrefaCe

This series, Chemical Sensors: Simulation and Modeling, is the perfect comple-ment to Momentum Press’s six-volume reference series, Chemical Sensors: Fundamentals of Sensing Materials and Chemical Sensors: Comprehensive Sensor Technologies, which present detailed information about materials, technologies, fabrication, and applications of various devices for chemical sensing. Chemical sensors are integral to the automation of myriad industrial processes and every-day monitoring of such activities as public safety, engine performance, medical therapeutics, and many more.

Despite the large number of chemical sensors already on the market, selec-tion and design of a suitable sensor for a new application is a difficult task for the design engineer. Careful selection of the sensing material, sensor platform, technology of synthesis or deposition of sensitive materials, appropriate coatings and membranes, and the sampling system is very important, because those deci-sions can determine the specificity, sensitivity, response time, and stability of the final device. Selective functionalization of the sensor is also critical to achieving the required operating parameters. Therefore, in designing a chemical sensor, de-velopers have to answer the enormous questions related to properties of sensing materials and their functioning in various environments. This five-volume com-prehensive reference work analyzes approaches used for computer simulation and modeling in various fields of chemical sensing and discusses various phenomena important for chemical sensing, such as surface diffusion, adsorption, surface reactions, sintering, conductivity, mass transport, interphase inter actions, etc. In these volumes it is shown that theoretical modeling and simulation of the pro-cesses, being a basic for chemical sensor operation, can provide considerable assistance in choosing both optimal materials and optimal configurations of sensing elements for use in chemical sensors. The theoretical simulation and model ing of sensing material behavior during interactions with gases and liquid surroundings can promote understanding of the nature of effects responsible for high effectiveness of chemical sensors operation as well. Nevertheless, we have to understand that only very a few aspects of chemistry can be computed exactly.

xiv • preface

However, just as not all spectra are perfectly resolved, often a qualitative or ap-proximate computation can give useful insight into the chemistry of studied phe-nomena. For example, the modeling of surface-molecule interactions, which can lead to changes in the basic properties of sensing materials, can show how these steps are linked with the macroscopic parameters describing the sensor response. Using quantum mechanics calculations, it is possible to determine parameters of the energetic (electronic) levels of the surface, both inherent ones and those introduced by adsorbed species, adsorption complexes, the precursor state, etc. Statistical thermodynamics and kinetics can allow one to link those calculated surface parameters with surface coverage of adsorbed species corresponding to real experimental conditions (dependent on temperature, pressure, etc.). Finally, phenomenological modeling can tie together theoretically calculated characteris-tics with real sensor parameters. This modeling may include modeling of hot plat-forms, modern approaches to the study of sensing effects, modeling of processes responsible for chemical sensing, phenomenological modeling of operating char-acteristics of chemical sensors, etc.. In addition, it is necessary to recognize that in many cases researchers are in urgent need of theory, since many experimental observations, particularly in such fields as optical and electron spectroscopy, can hardly be interpreted correctly without applying detailed theoretical calculations.

Each modeling and simulation volume in the present series reviews model-ing principles and approaches particular to specific groups of materials and de-vices applied for chemical sensing. Volume 1: Microstructural Characterization and Modeling of Metal Oxides covers microstructural characterization using scanning electron microscopy (SEM), transmission electron spectroscopy (TEM), Raman spectroscopy, in-situ high-temperature SEM, and multiscale atomistic simulation and modeling of metal oxides, including surface state, stability, and metal oxide interactions with gas molecules, water, and metals. Volume 2: Conductometric-Type Sensors covers phenomenological modeling and computational design of conductometric chemical sensors based on nanostructured materials such as metal oxides, carbon nanotubes, and graphenes. This volume includes an over-view of the approaches used to quantitatively evaluate characteristics of sensitive structures in which electric charge transport depends on the interaction between the surfaces of the structures and chemical compounds in the surroundings. Volume 3: Solid-State Devices covers phenomenological and molecular model-ing of processes which control sensing characteristics and parameters of various solid-state chemical sensors, including surface acoustic wave, metal-insulator-semiconductor (MIS), microcantilever, thermoelectric-based devices, and sensor arrays intended for “electronic nose” design. Modeling of nanomaterials and nano-systems that show promise for solid-state chemical sensor design is analyzed as well. Volume 4: Optical Sensors covers approaches used for modeling and simu-lation of various types of optical sensors such as fiber optic, surface plasmon resonance, Fabry-Pérot interferometers, transmittance in the mid-infrared region,

preface • xv

luminescence-based devices, etc. Approaches used for design and optimization of optical systems aimed for both remote gas sensing and gas analysis cham-bers for the nondispersive infrared (NDIR) spectral range are discussed as well. A description of multiscale atomistic simulation of hierarchical nanostructured materials for optical chemical sensing is also included in this volume. Volume 5: Electrochemical Sensors covers modeling and simulation of electrochemical pro-cesses in both solid and liquid electrolytes, including charge separation and transport (gas diffusion, ion diffusion) in membranes, proton–electron transfers, electrode reactions, etc. Various models used to describe electrochemical sensors such as potentiometric, amperometric, conductometric, impedimetric, and ion-sensitive FET sensors are discussed as well.

I believe that this series will be of interest of all who work or plan to work in the field of chemical sensor design. The chapters in this series have been prepared by well-known persons with high qualification in their fields and therefore should be a significant and insightful source of valuable information for engineers and researchers who are either entering these fields for the first time, or who are al-ready conducting research in these areas but wish to extend their knowledge in the field of chemical sensors and computational chemistry. This series will also be interesting for university students, post-docs, and professors in material science, analytical chemistry, computational chemistry, physics of semiconductor devices, chemical engineering, etc. I believe that all of them will find useful information in these volumes.

G. Korotcenkov

xvii

about the editor

Ghenadii Korotcenkov received his Ph.D. in Physics and Technology of Semiconductor Materials and Devices in 1976, and his Habilitate Degree (Dr.Sci.) in Physics and Mathematics of Semiconductors and Dielectrics in 1990. For a long time he was a leader of the scientific Gas Sensor Group and manager of various national and international scientific and engineering projects carried out in the Laboratory of Micro- and Optoelectronics, Technical University of Moldova. Currently, Dr. Korotcenkov is a research professor at the Gwangju Institute of Science and Technology, Republic of Korea.

Specialists from the former Soviet Union know Dr. Korotcenkov’s research results in the field of study of Schottky barriers, MOS structures, native oxides, and photoreceivers based on Group III–V compounds very well. His current research interests include materials science and surface science, focused on nanostructured metal oxides and solid-state gas sensor design. Dr. Korotcenkov is the author or editor of 11 books and special issues, 11 invited review papers, 17 book chapters, and more than 190 peer-reviewed articles. He holds 18 patents, and he has presented more than 200 reports at national and international conferences.

Dr. Korotcenkov’s research activities have been honored by an Award of the Supreme Council of Science and Advanced Technology of the Republic of Moldova (2004), The Prize of the Presidents of the Ukrainian, Belarus, and Moldovan Academies of Sciences (2003), Senior Research Excellence Awards from the Technical University of Moldova (2001, 2003, 2005), a fellowship from the International Research Exchange Board (1998), and the National Youth Prize of the Republic of Moldova (1980), among others.

xix

Contributors

Samir H. Mushrif (Chapter 1)Catalysis Centre for Energy Innovation and Department of Chemical EngineeringUniversity of DelawareNewark, Delaware 19716, USA

Gilles H. Peslherbe (Chapter 1)Centre for Research in Molecular Modeling and Department of Chemistry and BiochemistryConcordia University Montréal, Québec, Canada H4B 1R6

Alejandro D. Rey (Chapter 1)Department of Chemical EngineeringMcGill University, Montréal, Québec, Canada H3A 2B2

Karin Larsson (Chapter 2)Department of Chemistry—Ångström LaboratoryUppsala University753 10 Uppsala, Sweden

James L. Gole (Chapter 3)Schools of Physics and Mechanical EngineeringGeorgia Institute of TechnologyAtlanta, Georgia 30332, USA

William Laminack (Chapter 3)Schools of Physics and Mechanical EngineeringGeorgia Institute of TechnologyAtlanta, Georgia 30332, USA

xx • contributors

Vinod Kumar Khanna (Chapter 4)MEMS & MicrosensorsCSIR—Central Electronics Engineering Research InstitutePilani-333031 (Rajasthan), India

Linfeng Zhang (Chapter 5)Department of Electrical EngineeringUniversity of BridgeportBridgeport, Connecticut 06604, USA

Gerhard Fischerauer (Chapter 6)Chair of Metrology and Control EngineeringUniversität Bayreuth95440 Bayreuth, Germany

Florian Thalmayr (Chapter 6)Micro Electro Mechanical Systems DesignSand 9 Inc.Cambridge, Massachusetts 02139, USA

Viatcheslav Barkaline (Chapter 7)System Dynamics & Material Mechanics LaboratoryBelarusian National Technical UniversityMinsk, 220013, Belarus

Aliaksandr Chashynski (Chapter 7)System Dynamics & Material Mechanics LaboratoryBelarusian National Technical UniversityMinsk, 220013, Belarus

Sophie Martin (Chapter 8)Institut des Matériaux Jean Rouxel (IMN), CNRS Université de Nantes44322 Nantes cedex 3, France

Guy Louarn (Chapter 8)Institut des Matériaux Jean Rouxel (IMN), CNRS Université de Nantes 44322 Nantes cedex 3, France

contributors • xxi

Sergi Udina (Chapter 9)Departament d’ElectrònicaUniversitat de Barcelona08028 Barcelona, Spain andArtificial Olfaction GroupInstitute for Bioengineering of Catalonia (IBEC)08028 Barcelona, Spain

Manuel Carmona (Chapter 9)Electronics Department Universitat de Barcelona08028 Barcelona, Spain

Carlos Calaza (Chapter 9)Instituto de Microelectrónica de Barcelona—Centro Nacional de Microelectrónica (IMB-CNM)Consejo Superior de Investigaciones Científicas (CSIC)08193 Barcelona, Spain

R. D. S. Yadava (Chapter 10)Sensors & Signal Processing Laboratory Department of Physics, Faculty of Science Banaras Hindu UniversityVaranasi 221005, India

1DOI: 10.5643/9781606503171/ch1

Chapter 1

Molecular Modeling

application to Hydrogen interaction witH carbon-Supported tranSition Metal SySteMS

Samir h. MushrifGilles h. peslherbealejandro D. rey

1. IntroductIon

Carbon can exhibit different types of orbital hybridization, sp, sp2, and sp3, thus giving rise to a variety of carbonaceous structures. The most precious form of carbon, diamond, consists of carbon with sp3-type hybridization, and it is ex-tremely hard due to its strong covalent bonds. A large family of functional car-bonaceous materials, particularly those currently being researched for chemical sensor appli cations, exhibits sp2-type bonding. One of the basic building blocks of these materials, including graphitic carbon, carbon fibers, porous carbons, carbon composites, and fullerenes, is the pentagonal/hexagonal carbon ring. A number of such rings connected to each other, in a plane, form a layer, sometimes re-ferred to as a graphene sheet. The delocalized electrons in these rings impart the graphene sheet with good electrical conductivity along the layer. The size of this layer, its agglomeration, interconnectivity, geometry, stacking, and the presence of

2 • CheMiCal SenSorS – SiMUlation anD MoDelinG: VolUMe 3

noncarbon elements vary, thus giving rise to different types of carbon materials (with different structural and functional properties) in the family of sp2-bonded carbons. Graphitic carbon consists of large-size layers composed of hexagonal carbons stacked together. Fullerenes are another special type of carbon materials made up entirely of carbon, and the hexagonal (and pentagonal) rings form dif-ferent shapes such as a hollow sphere, ellipsoid, or cylinder. The spherical fuller-enes are called buckminsterfullerenes (or buckyballs), and the cylindrical ones are called nanotubes. These carbon materials, nanotubes in particular, have gar-nered huge attention in recent years due to their exceptional mechanical, ther-mal, and electrical properties. Less structured carbon materials are the isotropic carbon-based materials. They also consist of sheets of carbon rings; however, the size of the sheet is too small to lead to an anisotropic structure. Their function-ality arises from their porous structure and high surface area, which allows the materials to adsorb gases and liquids. The functionality of all the carbonaceous materials is often enhanced by the addition of an external component. Active car-bons are often loaded with active metals. There are two different ways of doping the active carbons with metal. One of the methods is to impregnate the carbons using an aqueous solution of metal precursors (Augustine 1996), and the other method is to mix the metal precursor with the carbon precursor even before the preparation of the carbon material (Basova et al. 2005). It has to be noted that the functionality depends on the amount of loaded metal, and there exists an opti-mum amount above which the performance may decline (Furimsky 2008).

A significantly important class of functional carbon materials in hydrogen-involving processes consists of nano-size transition metal clusters anchored on a carbon support. One of the key phenomena governing their functionality is the interaction of hydrogen with these materials. If understanding the functionality of a chemical sensor at the molecular level is the ultimate goal, detailed molecu-lar-level information about the interaction of the gas (hydrogen) with the sensor material is an important milestone in achieving this goal. The nature of hydrogen bonding (physisorption or chemisorption) and the possibility of its dissociation and migration on the material are the key factors to understand (Conner and Falconer 1995; Pajonk 2000; Teschner et al. 2008). Hence, the course of inter-action of hydrogen with transition metal–doped carbon-supported materials has received significant attention from researchers (Mitchell et al. 2003; Jewell and Davis 2006; Amorim and Keane 2008; Cheng et al. 2008; Teschner et al. 2008; Zhou et al. 2008), particularly since (1) the sp2 carbon-based materials such as nanotubes, activated carbon, and activated carbon fibers have been recognized as potential hydrogen-adsorbing materials (Dillon et al. 1997; Poirier et al. 2001; Schimmel et al. 2003, 2004; Takagi et al. 2004; Patchkovskii et al. 2005; Cabria et al. 2006; Strobel et al. 2006; Aga et al. 2007; Henwood and Carey 2007; Shevlin and Guo 2007; Cabria et al. 2008) and (2) transition metal doping has been shown to significantly modify the hydrogen adsorption characteristics of such materials

MoleCUlar MoDelinG • 3

(Lueking and Yang 2004; Cabria et al. 2005; Dag et al. 2005; Yildirim and Ciraci 2005; Yildirim et al. 2005; Lee et al. 2006; Zhou et al. 2006; Durgun et al. 2008; Gallego et al. 2008). In the absence of a transition metal, hydrogen mostly physi-sorbs on carbon materials in the molecular form (Schimmel et al. 2003, 2004; Takagi et al. 2004; Dag et al. 2005; Patchkovskii et al. 2005; Cabria et al. 2006; Strobel et al. 2006; Aga et al. 2007; Henwood and Carey 2007); it is believed that the addition of a transition metal to the carbon material alters this mode of inter-action, with the diatomic hydrogen dissociatively chemisorbing on the transition metal clusters, absorbing as metal hydride and possibly migrating on the cluster and eventually on to the carbon support (Mitchell et al. 2003; Lueking and Yang 2004; Matsura et al. 2004; Takagi et al. 2004; Lachawiec et al. 2005; Yildirim and Ciraci 2005; Yildirim et al. 2005; Shevlin and Guo 2007; Amorim and Keane 2008; Chen et al. 2008; Cheng et al. 2008; Fedorov et al. 2008; Gallego et al. 2008; Zhou et al. 2008; Bhat et al. 2009). The latter migration of hydrogen on the metal clus-ters and on to the support is also referred to as spillover. Spillover is nothing but the transport of adsorbed species from one site to another on an adsorbent which would not adsorb the species in the first place under the same conditions if it were by itself (Conner and Falconer 1995). This way, an adsorption site with higher ad-sorption capability is available for any new species to adsorb again. It should be noted that the spillover phenomenon that has received the most attention (Conner and Falconer 1995; Badenes et al. 1997; Mitchell et al. 2003; Heiz and Bullock 2004; Lueking and Yang 2004; Lachawiec et al. 2005; Banerjee et al. 2006; Li and Yang 2006; Strobel et al. 2006; Li et al. 2007; Chen et al. 2008; Cheng et al. 2008; Teschner et al. 2008; Zhou et al. 2008; Casolo et al. 2009; Celli et al. 2009) is that of H2, because hydrogen spillover is also envisioned as an important mechanistic step in the enhancement of the hydrogen adsorption capacity of carbon-based materials (Schimmel et al. 2003; Lueking and Yang 2004; Lachawiec et al. 2005; Banerjee et al. 2006; Strobel et al. 2006; Cheng et al. 2008; Celli et al. 2009). Hence, if the metal-doped carbon-supported material leads to any application, including as a chemical sensor, a clear and definitive understanding of hydrogen adsorption and the spillover process is required.

Though it is of great importance, the adsorption and spillover of hydrogen on metal-doped carbon supports are mostly studied experimentally as an entire process made up of a combination of sequential steps. The deficiency of elemen-tary studies of hydrogen interaction with these materials has cast some doubts about the interaction mechanism and the dynamics of the process, the very exis-tence of spillover, the energetics of the process, and the possibility of recombina-tion of monoatomic hydrogen to re-form a diatomic molecule following spillover. There are few key experimental investigations exclusively studying the interaction of hydrogen with metal-doped carbon-supported materials (Mitchell et al. 2003; Amorim and Keane 2008; Bhat et al. 2009; Contescu et al. 2009). Mitchel et al. (2003), using inelastic neutron scattering, identified two forms of hydrogen atoms


on a Pt- doped carbon support, viz., H atoms at the edge sites of the graphite and a weakly bound layer of mobile H atoms on the carbon surface. Using the same experimental method, Contescu et al. (2009) also identified the presence of C–H bonds on an activated carbon fiber doped with Pd and, using in-situ x-ray dif-fraction, they also demonstrated the presence of palladium hydride. Amorim and Keane (2008) also revealed the presence of spillover hydrogen (C–H bonds), che-misorbed hydrogen, and hydrides on Pd-supported activated carbon, graphite, and nanofibers. It should be noted that all the aforementioned valuable experi-mental investigations successfully probed the nature of hydrogen present in these materials; however, they do not directly divulge the mechanism of the formation of these different forms of hydrogen on these materials. The only experimental investigations, to the best of our knowledge, directed toward understanding the mechanism have been carried out by Yang and co-workers (Lueking and Yang 2004; Lachawiec et al. 2005; Li et al. 2007). They demonstrated that an improved contact between the metal particles and the carbon support enhances the spill-over process. They also demonstrated that the spillover process is not only depen-dent on the dissociation of hydrogen on the metal and its migration to the active carbon support, it is also controlled by the absorption capacity of the supported carbon material.

There have also been numerous theoretical investigations, in particular at the molecular level, to understand the interaction of hydrogen with transition metal clusters, carbon materials, and transition metal–doped carbon materials. Some of the key findings are summarized in the following.

1. Cabria et al. (2008) and Aga et al. ( 2007) found that the adsorption of hy-drogen molecules is dependent on the layer spacing (or slit pore size) of gra-phene sheets and that there exists an optimum layer spacing for maximum molecular hydrogen adsorption. As far as adsorption of atomic hydrogen on carbon is concerned, Yang et al. (2006) showed that the binding energy depends on the geometry of carbon nanotubes and its hydrogen occupancy, and Casolov et al. ( 2009) computed an energy barrier of up to 20 kJ/mol for the binding of atomic hydrogen to graphene. Chen et al. (2008) and Cheng et al. (2008) showed that the diffusion of an H atom chemisorbed on carbon nanotubes or graphene is energetically very difficult, since it requires the breaking of C–H covalent bonds.

2. Ferreira and Pavao (2008) showed that a tilted hydrogen molecule is a pre-cursor state of dissociative adsorption on a Pd surface and, in the case of interaction of hydrogen with a Pd cluster, Matsura et al. (2004) suggested that the vertices of the cluster are the initial points of interaction. Zhou et al. (2008) and Bartczak and Stawowska (2006) found that the diffusion of hydrogen atoms adsorbed on a Pd cluster is associated with a low energy barrier. Caputo and Alavi (2003) also showed that the octahedral site is more favorable when hydrogen is absorbed on Pd as hydride.


3. Theoretical investigations of the hydrogen interaction with metal atom (Ti, B, Ni, Pt, Li, Pd, Sc, V)–doped carbon systems (Yildirim and Ciraci 2005; Yildirim et al. 2005; Kiran et al. 2006; Lee et al. 2006; Zhou et al. 2006; Durgun et al. 2008) revealed increased hydrogen storage capacities (in atomic and mo-lecular form) due to metal doping. Yildirim and Ciraci (2005) and Dag et al. (2005) showed that Ti, Pt, and Pd atom–doped carbon nanotubes dissocia-tively adsorb the first H2 molecule with a negligible energy barrier and the subsequent adsorptions are in molecular form, with elongated H–H bonds. Kiran et al. (2006) also suggested that the number of molecular hydrogens adsorbed on to a carbon-supported metal atom increases for metals with lesser filled d orbitals. Guo et al. (2006) showed that the desorption energy barrier of hydrogen adsorbed on Pd-doped carbon nanotubes can be re-duced by deformation of the carbon nanotube. The computations of Fedorov et al. (2008) suggested that the migration of hydrogen from Pt to the carbon surface is associated with a significant energy barrier.

A significant number of theoretical studies related to molecular hydrogen ad-sorption on pure carbon systems have been motivated by the initially reported high hydrogen storage capacity of carbon materials (Dillon et al. 1997), and hence such studies are physically too unrealistic and simple to shed any light on hy-drogen interaction with metal-doped carbon materials. Similarly, the studies re-lated to metal atom–doped carbon materials were also carried out with the goal of searching for high hydrogen storage materials, and since they were mostly restricted to the adsorption of hydrogen (in atomic and molecular form) around the metal atom, they are not relevant to the subsequent hydrogen migration and dynamics. It should also be noted that the behavior of hydrogen with metal clus-ters is different than that with a single metal atom. The adsorption, absorption, energetics, and migration of hydrogen on metal surfaces and the adsorption, ener-getics, and migration of hydrogen on carbon surfaces were investigated separately in most of the aforementioned studies, thus again dividing the system into two parts (metal and carbons). Some of the few interesting investigations that have recently attempted to examine the entire hydrogen interaction process, involving the initial interaction of hydrogen with the metal and its migration, with a carbon-supported metal cluster are those by Cheng et al. ( 2008) and Chen et al. (2008). However, they did not model the dynamics of the process at a finite temperature. The spillover of hydrogen was modeled by bringing a presaturated (with hydrogen) Pt cluster near the carbon support arbitrarily, so as to make the hydrogen atoms in between the metal cluster and the support spill over. This may not be realistic, since the metal cluster is in contact with the (carbon) support even before it comes in contact with hydrogen, and hence spillover will never occur from the metal cluster surface that is in contact with the support.

To summarize, the experimental investigations geared toward understanding the hydrogen interaction process with metal-doped carbon-supported materials


are limited, due to the difficulty of isolating the interaction from the complex inter-play of various factors, and due to limitations on the accessible time and length scales. Theoretical investigations either assume the system to be too simple to be compared to a realistic metal-doped carbon-supported system, or the hydrogen interaction is arbitrated by a preset interaction mechanism. Additionally, very few theoretical studies of hydrogen interaction with metal-doped carbon-supported materials have attempted to model and reveal the dynamics of the interaction process at a finite temperature (Mushrif et al. 2010).

Given their importance in theoretical studies to investigate the interaction of hydrogen with carbon-supported metal systems, in this chapter we provide some insights into the different molecular modeling methods available for that purpose. We briefly discuss the physics of these methods and comment on their capabilities and limitations. More emphasis is given to some of the popular elec-tronic structure methods and, in particular, to the modeling of the dynamics of hydrogen inter action with a carbon-supported metal. In addition to introducing the reader to some of the state-of-the-art molecular modeling methods, we also provide enough references for details. An example of molecular modeling of hydro-gen interaction with a carbon-supported Pd cluster from our own work is also discussed in detail.

Figure 1.1. Modeling methods versus length and time scales.


2. Molecular ModelIng Methods

Matter is composed of molecules, and molecules are composed of individual atoms, or of positively charged nuclei and negatively charged electrons. Different molecules contain different atoms (or the same atoms in different arrangements), or they contain different nuclei and different numbers of electrons (or the same nuclei and the same number of electrons in different spatial positions). These two different ways of considering molecules give rise to the two most popular molecular modeling methods. One is referred to as the force-field method or mo-lecular mechanics, and the other is called electronic structure theory or the first-principles or quantum chemistry method. In molecular mechanics, an individual atom is treated as the basic particle and the potential energy of molecules is calculated as a parametric function of the atomic coordinates. In contrast, in electronic structure methods, the positively charged nuclei and the negatively charged electrons are the fundamental particles and the interactions between these charged particles determine the potential energy. The following subsections give a brief overview of molecular mechanics and electronic structure theory. Modeling methods usually used for simulations versus length and time scales are shown in Figure 1.1.

2.1. Molecular Mechanics

Force-field calculations are often referred to as molecular mechanics because molecules in these calculations are described by a ball-and-spring type of model in which the atoms are “balls” of different sizes and the bonds are “springs” of different lengths and strengths. Nonbonded interactions such as van der Waals interactions and electrostatic interactions are also taken into account in force-field calculations.

The energy in force-field calculations is given by a sum of different terms, where each term corresponds to a specific type of deformation in the species, as given in the following equation,

= + + + +

bonded nonbonded

MM stretch bend torsion vdW electrostaticE E

E E E E E E (1.1)

where stretchE is the energy of stretching a bond between two atoms, bendE is the energy of bending an angle formed by three bonded atoms, torsionE is the energy of twisting around a bond, and vdWE and electrostaticE are the energies accounting for van der Waals interaction and electrostatic interactions between two distant (nonbonded) atoms. Figure 1.2 shows a graphical illustration of the basic terms involved in calculating the force-field energy. The energy terms are parametric


functions of atomic coordinates. The minimum-energy structure corresponding to the most stable configuration can be determined by minimizing EMM as a function of atomic coordinates.

Assigning numerical values to different parameters in the energy terms de-scribed above is very important in force-field calculations. Parameterization of the force field is usually achieved by reproducing the structure, relative energies, vibrational spectra of molecules obtained from electronic structure calculations for small molecules, and/or experimental data. It is required that the parameters which are fitted in a given force field be transferable among different molecules and environments. A compromise between accuracy and generality needs to be sought. Different force fields have been developed over the years; some of the main differences in these force fields are the functional forms of the energy, the number of additional energy interaction terms (other than the basic ones described above), and the information used to fit the parameters in the force field. Force fields based on simple (harmonic) functional forms are often called “harmonic” or “Class I” force fields, and those containing more complicated functional forms, additional terms, and which are sometimes heavily parameterized using electronic structure theory calculations are called “Class II” type (Jensen 2007). Depending on these factors, different force fields have been designed for different types of molecules; Table 1.1 lists a few of them.

Force-field methods are widely used in the computer modeling community, and their ability to provide an understanding of atomic and molecular structure for different (and large) systems, at a modest computational cost, has contributed greatly to scientific research in the last two decades. These methods are very popular for investigating systems such as small organic molecules, large bio-molecules such as proteins and DNA, polymers, etc. They are also several orders of magnitude faster than electronic structure methods. However, there are certain

Figure 1.2. illustration of interaction terms in molecular mechanics. (adapted from Jensen 2007.)


limitations associated with the force-field/molecular mechanics methods (Remler and Madden 1990; Payne et al. 1992; Marx and Hutter 2000; Deeth 2001; Norrby and Brandt 2001; Jensen and Norrby 2003; Comba and Kerscher 2009; Deeth et al. 2009). Some of them are discussed below.

1. Out-of-ordinary/unusual situations (Remler and Madden 1990; Payne et al. 1992; Marx and Hutter 2000; Jensen 2007). Force-field methods are based on various approximate functional forms and parameters. Since the para meters are determined using experimental data, these methods are em-pirical. Therefore, force-field methods perform extremely well when a lot of information about the system under investigation (or related systems) al-ready exists. For molecules that are “exotic” or a little “unusual,” for which there is little or no known information, force-field methods may perform poorly. In brief, the interpolative force-field methods may lead to serious errors when used for extrapolation.

2. Diverse types of molecules (Remler and Madden 1990; Payne et al. 1992; Marx and Hutter 2000; Jensen 2007). Parameterization of a force field needs a balance between generality and accuracy. The generality/transferability of a force field (from one type of molecular system to another) can be improved by including diverse types of molecules in the parameterization training data set, but for a given functional form, including additional data may not help. On the other hand, changing the functional form or using additional terms may correct the errors inherent in the simpler forms. Most of the force fields are restricted to specific types of molecules for which they have been specifi-cally parameterized.

3. Chemical changes (Marx and Hutter 2000; Duin 2002). When performing force-field calculations, the input consists of (a) the types of atoms, (b) the inter actions between those atoms (bonded or nonbonded), and (c) the geo metry of the system. The first two factors are crucial in assigning an

Force Field developers systems class

MM2, MM3, MM4 Allinger and co-workers Organics/general II hydrocarbons AMBER Kollman and co-workers General/proteins IUFF Goddard and co-workers General Between I and IICHARMM Karplus and co-workers Proteins IGROMOS University of Groningen Proteins, nucleic acids, and ETH Zurich and carbohydrates ICFF, TRIPOS Commercial General II

Source: Peslherbe 2006.

Table 1.1. A list of common force fields in molecular modeling


appropriate functional form to each interaction in the system. Force-field calculations are appropriate when every atom type and its type of inter-action do not change with changes in atomic coordinates. However, during the course of a chemical reaction, covalent bonds are formed and broken. Hence, a chemical reaction leads to different types of interactions in reac-tants and products for the same atom. The electronic structure of the sys-tem also changes significantly, thereby changing the type of an atom (e.g., a carbon that was sp3 before a chemical reaction may become sp2 or sp after reaction and vice versa). As a result, the force field initially chosen for the reactants may no longer be appropriate for the products. Hence force-field methods fail to model a system in which chemical reactions occur. For in-stance, the harmonic description of the stretching energy in Class I force fields makes it impossible to describe bond dissociation.

4. Metal systems (Comba 1993, 1994, 1999; Comba and Hambley 1995; Comba and Zimmer 1996; Deeth 2001; Norrby and Brandt 2001; Jensen 2007; Comba and Kerscher 2009; Deeth et al. 2009). Force-field methods are believed to be difficult, if not impossible, to apply to metal compounds and complexes, and especially transition metal systems. The bonding in metals is much more var-ied than in organic systems. In the case of metal–ligand complexes, the metal forms a coordination bond with ligands, while in pure metallic systems, the bonding may vary with the size of the metal cluster. Since electronic effects cannot be taken into account explicitly in force-field calculations, they need to be taken into account implicitly. The key reasons for the less successful implementation of force-field methods when it comes to modeling metal (in-cluding transition metals) complexes and compounds are as follows.

• Varied coordination numbers and geometries. In metal complexes, the coordination number of a metal is the number of ligand atoms to which the metal is bonded, and in the case of metal clusters, it is defined as the number of nearest-neighbor atoms. Transition metals may possess coor-dination numbers ranging from 1 to 12. There is also more than one way to organize ligand atoms around the central metal species, giving rise to mul-tiple isomers. In the case of pure metal clusters, multiple structures, very close in energy, exist. The geometry may also differ significantly, depend-ing on the physical state of the system (solid phase/solution/gas phase). Unlike organic compounds, metal-coordinated compounds possess much wider structural flexibility and hence a variety of structural motifs can be observed (Comba and Hambley 1995; Comba and Zimmer 1996; Cundari 1998). This leads to difficulties in defining the energy functional forms to describe these systems. Force-field methods have been successfully applied to quite a few specific systems (Hay 1993; Comba 1994; Zimmer 1995; Comba 1999; Landis et al. 2007), and a few generalized approaches (Rappe et al. 1992; Rappe and Casewit 1997; Gajewski et al. 1998) have


been developed to tackle the problem (Comba and Zimmer 1996; Comba et al. 1997). However, whenever force-field methods need to be applied to a new system, very frequently a significant modification of the force field is required, and its predictive ability remains questionable (Norrby and Brandt 2001; Comba and Kerscher 2009).

• Varied oxidation states and electronic structures. Another problem in using force-field methods is that transition metals exhibit multiple oxida-tion numbers and electronic states (e.g., Pd has oxidation states of 0, 1, 2, and 4), and a separate parameterization needs to be performed (simi-lar to carbon with sp, sp2, and sp3 hybridization). The problem is further magnified by the multitude of transition metal complexes, thus mak-ing the parameterization even more cumbersome (Bernhardt and Comba 1992; Comba 1993; Boeyens and Comba 2001; Comba and Kerscher 2009). Also, in pure metallic systems, the nature of the bonding and the electronic structure change with the number of atoms in the metal clus-ter. For example, the ground-state spin multiplicities of Pd2, Pd11, and Pd12 clusters are 3, 7, and 5, respectively (Nava et al. 2003).

• d-shell electrons (Deeth 2001; Deeth et al. 2009). In the case of tran-sition metal systems, the effects due to d-orbital electrons pose further problems in using force-field methods. The structural, spectroscopic, and magnetic properties of transition metal complexes depend significantly on the d-orbital electrons. Some issues are Jahn-Teller distortion, s-d orbital mixing, etc. Some efforts have been directed toward tackling these problems in force-field methods, and the POS (points on a sphere) model and the LFMM (ligand field augmented molecular mechanics) model have garnered some attention to that effect (Deeth et al. 2009). However, these modifications are very specific and need significant code writing, since they are difficult to implement in general, standard, force-field software, and these approaches require a lot of parameterization. Another situation that may hamper the use of these methods is when the system under investigation contains both transition metal complexes and some usual organic molecules, making the choice of force field delicate.

2.2. electronic structure theory

To model a phenomenon where bonds are broken and formed, or where a sys-tem contains novel transition metal clusters and complexes and organic species, at the atomic level, electronic structure theory can be used. Since it explicitly takes into account the electronic structure, it also offers the additional advantage of probing and predicting the bonding and electronic structure changes in the system. A nucleus is much heavier than electrons, and this large mass differ-


ence results in much slower motion than that of the electrons. Nuclei also exhibit smaller quantum effects and are usually treated as classical particles, while elec-trons are treated as quantum particles orbiting around the nuclei in a so-called orbital. In the Born-Oppenheimer approximation, the light electrons adjust in-stantaneously to any change in the position of the heavy nuclei. In quantum me-chanics, associated with any particle (e.g., an electron) is a wavefunction Ψ which determines everything about the state of the particle. The Schrödinger equation (Schrodinger 1926) is the most important equation in quantum mechanics, and it describes the evolution of the wavefunction of a particle in a given potential. Its time- independent form is given as

( ) ( )Ψ Ψ=H r E r (1.2)

where E is the total energy of the system and H is the Hamiltonian operator, de-fined as

( )=- Ñ +

22

2H U r

m (1.3)

where is the reduced Planck constant, m is the mass of the particle at position r, and U (r ) is the potential energy.

The Schrödinger equation cannot be solved analytically for a system contain-ing more than one electron. An approximate numerical solution needs to be deter-mined even for the helium atom, which contains two electrons and a nucleus. Any realistic system will contain multiple electrons. Atoms form molecules (by forming electron-sharing and non-electron-sharing bonds), and their electronic structure is affected by bonding. For example, when two H atoms form a covalent bond to make an H2 molecule, the electronic structure of the atoms is different than that of individual hydrogen atoms. According to valence bond theory, individual orbit-als of two atoms overlap and the shared electrons localize in the overlap region (the bond ) between the two atoms. However, modern electronic structure theory tends to rely on the molecular orbital theory. According to this theory, electrons are not assigned to individual bonds and are considered to arrange themselves all over the molecule, under the influence of the nuclei. Similarly to orbitals in an atom, every molecule is considered to have a set of molecular orbitals, and these molecular orbitals (wavefunctions) are usually expressed as a linear combination of atomic orbitals (LCAO) (Jensen 2007):

χ ϕ ϕ ϕ ϕ= + + + +1 1 2 2 3 3 n nc c c c (1.4)

where χ is the molecular orbital, ji is an atomic orbital, and ci is the coefficient associated with the contribution of the atomic orbital ji to the molecular orbital.


To ensure that the wavefunction is antisymmetric with respect to the interchange of any two electrons, following the Pauli principle, the wavefunction is expressed as a Slater determinant. If there are N electrons in the system with spin orbitals χ χ χ1 2, , , N then the total wavefunction is given as

( )

( ) ( ) ( )( ) ( ) ( )

( ) ( ) ( )

( ) ( ) ( )

χ χ χχ χ χ

Ψ

χ χ χ

χ χ χ

=

1 2

1 2

1 2

1 2

1 1 12 2 211,2, ,

!

= 1 2

N

N

N

N

NN

N N N

N (1.5)

Atomic orbitals are described by a set of basis functions, or basis set. A basis function can be of any type—exponential, Gaussian, polynomial, cube function, plane wave, to name a few. Though these basis functions need not be an analyti-cal solution of an atomic Schrödinger equation, they should properly describe the physics of the problem and these functions should vanish at large distances be-tween the nuclei and the electron. Usually, the basis functions are taken as func-tions describing (Slater-type orbitals) or approximating (Gaussian-type orbitals) the H-atom orbitals obtained by solving the Schrödinger equation analytically. The minimum number of basis functions that is needed to describe a system is that which can just accommodate the number of electrons present in the system. For example, two sets of s functions (1s and 2s) and a set of p functions (2px, 2py, and 2pz) are required to describe the first-row elements in the Periodic Table. Increased accuracy can be obtained using a larger number of basis functions.

Electronic structure calculations aim at characterizing the molecular orbitals and the total energy of a given molecular system. Once the type and number of basis functions has been chosen, the molecular orbitals are formed as a linear combination of atomic orbitals, which are themselves expressed in terms of basis functions. Since there is no “exact” wavefunction, one needs to determine the co-efficients of Eq. (1.4) that will give the best possible solution. A procedure that is reminiscent of the variational principle is used to calculate these coefficients and ultimately the wavefunction. Therefore, the energy calculated from this approxi-mate wavefunction will provide an upper limit to the “true” energy of the system (Jensen 2007). This implies that the closer the approximate wavefunction is to the actual solution, the lower will be the energy of the system and the closer it will be to the true ground-state energy. Hence, to obtain the best possible wavefunction, a set of coefficients that will result in the minimum energy of the system needs to be determined, i.e., the electronic energy of the system is minimized with respect to the coefficients. The detailed numerical procedure is given by Jensen (2007).


Assuming that a sufficiently large basis set is used, a possible source of error is due to the approximate treatment of electron–electron interactions in electronic structure theory. The zeroth-order method to solve the Schrödinger equation, called the Hartree-Fock method, assumes that each electron evolves in the av-erage field of all other electrons, and the motion of the electrons is said not to be explicitly correlated. In other words, in the Hartree-Fock method, an electron does not experience an interaction with each and every other electron as indi-vidual point charges. For instance, the method does not allow electrons to cross each other. Hence, methods that improve the treatment of electron correlation have been designed. One of the ways to include electron correlation effects in electronic structure theory is to express the molecular wavefunction as a linear combination of additional (Slater) determinants. Addition of determinants in the mathematical formulation can also be interpreted physically as the addition of electronically excited states, or allowing partial occupancy of unoccupied/virtual molecular orbitals.

Ψ Ψ Ψ Ψ= + + +0 HF 1 1 2 2corrrelated c c c (1.6)

where YHF is the single-determinant wavefunction obtained using the Hartree-Fock procedure. The Yi are the additional determinants and the ic are the coeffi-cients. Three main methods are commonly used to take into account electron correlation: configuration interaction (CI) (Sherrill and Schaefer 1999), coupled cluster (CC) (Bartlett 1989), and Møller-Plesset (MP) perturbation theory (Moller and Plesset 1934), and they differ mainly in the ways to calculate the coefficients in Eq. (1.6).

2.3. Density Functional theory

The electronic structure methods mentioned above are computationally intensive, and a significant amount of computational effort is needed to perform this type of calculation. An alternative method, which does take into account electron cor-relation, but at a computational cost similar to that of uncorrelated methods, is the so-called density functional theory (DFT). As the name suggests, the density of electrons (square of the wavefunction) is the key ingredient of the theory, and not the wavefunction. This method has gained tremendous popularity in the mo-lecular modeling community.

2.3.1. Origin and Formulation of Density Functional Theory

The Schrödinger equation for an N-electron system is given in atomic units by


( )

Ψ Ψ

ν

é ùê úê úê úê ú- Ñ - + =ê úê úê úê úê úë û

åå åå å

21 12

N N N NA

ij ii i j i

ZE

r r

r

(1.7)

and the ground-state electron density of the system can be calculated as

( ) ( ) ( )ρ Ψ Ψ*= ò 1 1 1 1 1 12 , ,2, , , ,2, ,r N dx d dN r x N r x N (1.8)

Equations (1.7) and (1.8) show that for a given potential ( )ν

r , it is possible to com-pute the ground-state density ( )ρ

r from the electronic wavefunction Y. However, Hohenberg and Kohn (1964) showed that there exists an inverse mapping with which it is possible to obtain an external potential, if the ground-state density is provided. In their seminal paper (Hohenberg and Kohn 1964), they also showed that this inverse mapping can be used to calculate the ground-state energy of a multielectron system using the variational principle, without having to resort to calculating the wavefunction. If the wavefunction is expressed as

( )Ψ Ψ ν= (1.9)

then, since ( )ν ν ρ= , the wavefunction can be written as

[ ]( ) [ ]Ψ Ψ ν ρ Ψ ρ= = (1.10)

where the brackets indicate a functional and not a function. Using the variational principle, the energy of the system can be calculated as

[ ] [ ]

[ ] [ ]Ψ ρ

ρ

Ψ Ψ Ψ ρ Ψ ρ

Ψ ρ ν Ψ ρ

= =

= + +

min min

min ee

E H H

T V

(1.11)

where

νæ ö÷çæ ö-Ñ ÷ç÷ç ÷ç÷ = - =ç ÷÷ çç ÷÷ç ç ÷è ø ç ÷çè ø

åå å åå2 1=

2

N N N NA

eei iji i i j

ZT V

r r

Equation (1.11) can be separated as


[ ] [ ] [ ] [ ]

( ) ( ) [ ]{ }

ρ

ρ

Ψ ρ ν Ψ ρ Ψ ρ Ψ ρ

ρ

ρ ν ρ

ì üï ïï ïï ïï ïï ï= + +í ýï ïï ïï ïé ùï ïë ûï ïî þ

= +ò

3

E min

min

eeT V

F

d r r r F

(1.12)

The energy is thus minimized with respect to the density and not the wavefunc-tion. However, in Eq. (1.12), the functional form of F(ρ) is unknown.

2.3.2. Kohn-Sham Formulation

To calculate the unknown functional F(ρ) consisting of the electronic kinetic energy part and the electron–electron interaction part, Kohn and Sham (1965) devised a scheme in which the total energy of the system is calculated using a combination of density functional theory and the orbital/wavefunction approach.

Kohn and Sham (1965) introduced a noninteracting system for which the Hamiltonian can be given as

( )ν=- Ñ +2

KS KS12

h r (1.13)

and they showed that the solution of the Schrödinger equation is given by

( ) ( ) ( ) ( )χ χ χ χ χ= KS 1 2 3 1 2 3 N N (1.14)

where the orbitals χi (usually referred to as Kohn-Sham orbitals) are obtained from the solution of

χ χ=KS KS KS KSh E (1.15)

In the Kohn-Sham formulation, the potential ( )ν

KS r is defined in such a way that the electron density calculated using the Kohn-Sham orbitals χKS is equivalent to the exact density of the electron-interacting system with the actual potential ( )ν

r . Using the Kohn-Sham orbitals, the kinetic energy of the noninteracting system can be given as

χ χ= KS KSST T (1.16)


From Eqs. (1.13–1.16), with the actual wavefunction of the system Ψ, the ground-state electronic energy of the system can be expressed as

( ) ( )

( ) ( ) ( ) ( )

[ ][ ]

[ ]

Ψ ν Ψ Ψ Ψ ρ ν Ψ Ψ

ρ ρρ ν

ρ

Ψ Ψ ρ

ρ

= + + = + +

¢¢= + +

¢-

+ - + -

ò

ò ò

3

3 3 31 2

ee ee

K

S

K s ee

XC

E T V T d r r r VT

r rT d r r r d rd r

r r

U

T T V U

E

(1.17)

where TS is the kinetic energy which can be calculated using the Kohn-Sham orbitals. [ ]ρXCE is the only unknown in the above equation, and it needs to be approximated. This term typically accounts for less than 10% of the total energy (Peslherbe 2006).

It is possible to calculate the total ground-state energy of the interacting sys-tem using the following steps:

1. Define the Kohn-Sham noninteracting system. 2. Calculate the density and kinetic energy of the system. 3. Calculate the ground-state energy.

However, this is only possible if the Kohn-Sham potential ( )ν

KS r is known. The methodology to determine the potential is as follows.

From Eqs. (1.13) and (1.15), one obtains

[ ] ( ) ( ){ }ρρ ρ ν= +ò

3KS KSmin SE T d r r r (1.18)

where

δ δ νδρ δρ

= \ + =KSKS0 0SE T

(1.19)

(δ denotes a functional derivative). From Eq. (17), the actual ground-state energy of the system is given as

[ ] ( ) ( ) [ ] [ ]{ }ρρ ρ ν ρ ρ= + + +ò

3min S XCE T d r r r U E (1.20)


where

( ) ( )ρδ δδ ν

δρ δρ δρ

¢¢= \ + + + =

¢-ò

30 0S XCrT EE r d r

r r (1.21)

Combining Eqs. (1.19) and (1.21), the Kohn-Sham potential can be obtained as

( ) ( ) ( )ρ δν ν

δρ

¢¢= + +

¢-ò

3KS

XCr Er r d r

r r (1.22)

If an approximate functional for [ ]ρXCE is known, the Kohn-Sham noninteract-ing system can be solved to obtain the Kohn-Sham orbitals χKS and the density ρ. Using this information, the total energy of the system can be calculated using Eq. (1.20). However, since the Kohn-Sham potential depends on the density ρ, the above equations need to be solved iteratively. To use the Kohn-Sham density functional theory, one needs an appropriate functional form for [ ]ρXCE , and the following section describes various approaches to obtain it.

2.3.3. Local Density and Generalized Gradient Approximation

From Eq. (1.17), the [ ]ρXCE term accounts for (1) the difference between the actual/exact electronic kinetic energy of the system and the Kohn-Sham non-interacting electronic kinetic energy ( )-K ST T and (2) the difference between the exact electron–electron interaction energy and [ ]ρU . In other words, it accounts for the exchange-correlation energy ( )= +XC X CE E E . The two common approxi-mations for the functional [ ]ρXCE are the local density approximation (LDA) and the generalized gradient approximation (GGA) (Parr and Yang 1989). In the LDA, the exchange correlation term [ ]ρXCE in Eq. (1.17) is assumed to be equal to that of the uniform electron gas:

[ ] ( ) ( ) δρ ρ ρ

π δρæ öæ ö ÷ç÷ç ÷= ÷ çç ÷÷ çç ÷çè ø è øò ò

1 34 3 3 33 3 +

4C

XCEE r d r r d r (1.23)

The GGA is an improvement over the LDA in that it implements a gradient cor-rection as

( )ρ ρ= ÑòGGA 3 g ,XCE d r (1.24)


For the GGA to be practically useful, it is again important that it be expressed as an analytical form like the LDA. Different popular parameterized forms of the GGA (Perdew and Yue 1986; Lee et al. 1988; Becke 1992, 1993; Perdew et al. 1992, 1996) have been proposed.

2.4. Plane-Wave PseuDo-Potential MethoDs

The previous section described multielectron-system electronic structure calcu-lations using Kohn-Sham density functional theory within the framework of the Born-Oppenheimer approximation. Even though density functional theory takes into account electron correlation at a computational cost equivalent to that of the Hartree-Fock method, it is computationally still a difficult task to apply this method to an extended system such as crystals or bulk soft matter. The solu-tion to this problem is to define a tractable-size system, which, when repeated periodically in all spatial directions, will represent the bulk—i.e., to apply peri-odic boundary conditions to the cell containing a set of atoms/molecules. For electronic structure calculations with periodic boundary conditions (periodic po-tential) using the Kohn-Sham implementation of density functional theory, plane-wave bases are an obvious choice to describe the Kohn-Sham orbitals due to Bloch’s theorem (see Meyer 2006 for further details).

A finite number of plane waves can be used to perform computations. Since the accuracy of the Kohn-Sham potential is dependent on the accuracy of the basis set used for the Kohn-Sham orbitals, larger basis sets result in a more ac-curate Kohn-Sham potential. Some of the advantages and disadvantages of using plane waves as basis functions are

• Plane waves are orthonormal and energy-independent. • Plane waves are not biased with respect to any particular atom. Hence, the

entire space is treated on an equal footing and it does not cause basis-set superposition error (due to the overlap of individual atoms’ basis sets).

• It does not take advantage of the vacuum space in the simulation cell by avoiding having a basis set in that region.

• Since the plane waves are independent of the positions of atoms, the Hellman-Feynman theorem can be used to compute energy gradients (Payne et al. 1992), thereby reducing the computational cost when intermolecular forces are needed. In other words, if a basis set is dependent on the nuclear coordinates, then, when calculating forces, the derivatives of the coefficients (with respect to nuclear coordinates) associated with the basis set also need to be computed. This is not the case for plane-wave bases.

• The valence wavefunctions are nodal in the core region of the atom (Pauli’s exclusion principle), and hence a large number of plane waves are needed to represent these large oscillations.


For a practical application of the plane-wave approach, a solution to the nodal structure of valence wavefunctions problem needs to be identified. The solution is to use a “frozen core” approximation, i.e., to treat the core electrons and the nucleus as a pseudo-core. The consequence is that the electron–nuclear potential will also have to be replaced by a pseudo-potential. Since this pseudo-potential eradicates the core electrons from the system, it is very important that the pseudo-potential take into account the electron–nucleus interaction (as if shielded by the core electrons) and the electron–electron interaction (the classical Columbic and exchange-correlation interaction between the valence and core electrons). Hence, the pseudo-potential is angular momentum–dependent as well. Due to this pseudo-potential, the all-electron wavefunction is also replaced by a pseudo-wavefunction. Outside a certain cutoff radius, the pseudo-potential matches the true potential of the system and the pseudo-wavefunction matches the true wave-function of the system (see Figure 1.3).

It is worth noting that the contribution of core electrons to chemical bonding is negligible, and only the valence electrons play a significant part. The core elec-trons play an important part in the calculation of the total energy, though, and this implies that the removal of core electrons will also result in lower energy dif-ferences between different configurations, thereby reducing the efforts in achiev-ing the required accuracy. As mentioned before, a smaller number of plane waves is required for the pseudo-potential approach than with the all-electron approach.

Figure 1.3. Wavefunction of the system subject to the nuclear potential and the pseudo-potential [ ]( )Ψ Ψ νé ùê úë ûae pseudo pseudo and Z r .


Hamann et al. (1979) laid down a set of conditions for a good pseudo- potential. These are

1. The all-electron and pseudo-valence eigenvalues must agree for a particular atomic configuration.

2. The all-electron and pseudo-wavefunction must agree beyond a chosen core radius rcut

3. The logarithmic derivative of both wavefunctions must agree at rcut, i.e.,

ΨΨΨ ΨΨ Ψ

= Þ =¢ ¢

cut cut cut cut

pseudoAEAE pseudo

AE pseudo

ln lnr r r r

d ddr dr

4. Although inside rcut the pseudo- and all-electron wavefunctions and the respective potentials differ, the integrated charge densities for both must agree, i.e.,

Ψ Ψ=ò òcut cut

22 3 3AE pseudo

0 0

r r

d r d r

5. The first energy derivatives for both wavefunctions must agree at rcut.

All the pseudo-potentials that satisfy condition 4 are referred to as norm-conserving pseudo-potentials, because the “norm” is strictly conserved. When a pseudo-potential is developed, the conditions listed above (equivalency of the ener gies and the first derivatives) are satisfied for the reference energy; however, with a change in the chemical environment of the atoms, the eigenstates will be at a different energy. Hence, for practical applications of the pseudo-potential ap-proach, the above equalities with the all-electron wavefunction should still hold in different chemical environments and in a wider range of energies. It was shown that the norm-conserving condition enhances this transferability (Hamann et al. 1979). The two key aspects associated with any pseudo-potential are “softness” and “transferability.” For soft pseudo-potentials, fewer plane waves are needed, more electrons are frozen in the pseudo-core, and a large rcut is employed. However, to make the pseudo-potential more transferable, fewer electrons should be frozen in the pseudo-core (more electrons treated explicitly), a small rcut is employed, and hence more plane waves are required. A balance needs to be sought between soft-ness and transferability when developing pseudo-potentials.

In addition to the conditions listed above, Hamann et al. (1979) also provided a methodology to generate the norm-conserving pseudo-potentials. Generation of a pseudo-potential begins with all-electron calculations. The atomic potential is


multiplied by a cutoff function so as to eliminate the strong attractive part. The parameters of the cutoff function are adjusted to yield eigenvalues equal to those from the all-electron calculations and the pseudo-wavefunctions which will agree with the all-electron wavefunction after the cutoff radius. The total potential is then calculated by inverting the Schrödinger equation. The pseudo-potential acting on the valence electrons is then screened by subtracting the classical Columb poten-tial and the exchange-correlation potential, to obtain the ionic pseudo-potential.

2.5. oPtiMization techniques

The electronic structure methods described in Sections 2.2 and 2.3 are used to calculate the ground-state energy of a system for a particular nuclear (atomic) configuration. The energy calculated using electronic structure theory is a func-tion of the nuclear configurations, and one needs to minimize the system energy with respect to the nuclear configuration to determine the most stable system configuration. Finding stationary points of a system, where the first derivative of the energy with respect to the nuclear configuration is zero ( )é ùÑ =ê úë û

0E R , is an im-portant aspect of molecular modeling. Finding the nearest stationary point to an arbitrary initial-guess structure can be achieved with some of the common mini-mization/optimization methods, such as Newton-Raphson or steepest-descent techniques. For example, in the Newton-Raphson method, which essentially as-sumes a locally quadratic potential energy function with respect to the nuclear coordinates, the steps involved in the optimization process are as follows:

1. Calculate ( )

initial .E R 2. Calculate numerically ( )Ñ

initialRE R and ( )

initial

,R

H R where

H is the Hessian matrix.

3. Continue the iteration scheme ( ) ( )-= - Ñ

initial initial

11 initial

R RR R H R E R until

convergence is reached ( )é ùÑ =ê úë û

0E R .

This type of energy minimization leads to the closest local minimum, and the minimized configuration may not be the “true” global minimum. Two of the most widely used methods in molecular simulations to find the global minimum more reliably are molecular dynamics and Monte Carlo techniques. The Monte Carlo method is based on making random changes to the system configuration (see Figure 1.4). A specific criterion is defined to accept or reject these random changes, thus helping the system move toward lower-energy configuration states. In molecular dynamics techniques, every atom (nucleus) is assigned a finite velo-city at the initial system configuration, and consecutive configurations are gen-erated by solving Newton’s classical equations of motion. As mentioned before, nuclei are orders of magnitude heavier than electrons and hence they can be


treated using classical Newtonian mechanics for that purpose. The velocity of the atoms determines the kinetic energy of the system, which is related to the system temperature( )= =2 31

kinetic 2 2 BE mV nk T . Thus, molecular dynamics also accounts for finite-temperature effects on the system. In molecular dynamics simulations, the forces acting on every atom of the system are calculated as the negative gradi-ent of the potential energy (the energy calculated using electronic structure theory or force-field methods) at each configuration during the molecular dynamics run.

2.5.1. Car-Parrinello Molecular Dynamics

Molecular dynamics simulations can be classified into the following two categories:

1. Standard molecular dynamics simulations using force fields. The potential energy ( )

E R and forces

dE dR are calculated using force-field methods or molecular mechanics, as described in Section 2.1.

2. First-principles molecular dynamics simulations. The potential energy and forces are calculated using electronic structure theory, as described in Sections 2.2 and 2.3.

Standard molecular dynamics simulations are obviously much less computa-tionally intensive than first-principles molecular dynamics simulations, because the latter require electronic structure calculations along the trajectories. This also means that the time and length scales that can be accessed by standard molecular dynamics are much larger than those by first-principles molecular

Figure 1.4. illustration of a one-dimensional potential energy surface of a system.


dynamics. However, as mentioned before, standard molecular dynamics tech-niques will suffer from the limitations of the force field, and are thus inadequate to model chemically complex systems (where electronic structure and bonding patterns change due to reactions, for systems containing transition metal com-pounds or giving rise to new molecular species), and hence first-principles mo-lecular dynamics techniques are sometimes necessary. In the Born-Oppenheimer variety of first-principles molecular dynamics simulations, electronic structure calculations (optimizing the wavefunction for a fixed nuclear configuration) are performed at every time step of the trajectory. The Lagrangian associated with Born-Oppenheimer molecular dynamics (without the thermostat) is

£ Ψ Ψ= -å

2BO

12 i i

i

m R H (1.25)

The above Lagrangian is appended to include a thermostat (to control the tempera-ture) when first-principles molecular dynamics simulations are performed to gen-erate a (fixed-temperature) canonical ensemble. Electronic structure calculations can be performed using Hartree-Fock, post-Hartree-Fock or Kohn-Sham density functional theory methods for all electrons, or using the plane-wave pseudo-po-tential approach for valence electrons only. Born-Oppenheimer molecular dynam-ics techniques allow the simulation of complex, diverse, and poorly characterized systems in materials science and chemistry. However, due to the high computa-tional cost, they have not gained popularity.

A breakthrough in the field of first-principles molecular dynamics was made by Car and Parrinello (1985) with the introduction of a scheme that significantly reduces the computational cost of the simulations. Unlike Born-Oppenheimer molecular dynamics, the Car-Parrinello scheme does not require optimization of the wavefunction (or density) to be performed after every time step, and it also en-sures that the electronic wavefunction stays close to the optimized value through-out the course of the molecular dynamics simulation.

Born-Oppenheimer molecular dynamics is a combined modeling approach where the electronic motion is treated purely quantum mechanically (adiabati-cally) and the nuclear motion is treated classically at every molecular dynamics step. In contrast, in the Car-Parrinello approach, the electron density of the sys-tem (optimized at the start the molecular dynamics simulations) is propagated classically along with the nuclear coordinates, making it possible to avoid an electronic structure calculation after every time step.

To that effect, Car and Parrinello formulated an extended Lagrangian as

£ µ χ χ Ψ Ψ= + - +å å

2CP

potential energykinetic energy of kinetic energy of nuclei wavefunction

1 constraints2i i i j j

i j

m R H (1.26)


where χj is the orbital/wavefunction of the jth electron in the system, µ (the unit of µ is ´ 2energy time ) is a fictitious mass associated with the electronic wave-functions, and the constraints in Eq. (1.26) can be some external constraints on the system or internal constraints such as orthonormality. In the traditional Car-Parrinello approach, the electron wavefunctions are Kohn-Sham orbitals from density functional theory, but similar schemes have been proposed for other types of electronic structure theory. It should also be noted that the kinetic energy term

µ χ χå

12 i j jj

has no relation to the physical kinetic energy of the electrons and is completely fictitious. The Euler-Lagrange equations,

£ £ £ £χ χ

¶ ¶ ¶ ¶= =

¶ ¶ ¶¶

CP CP CP CP* *and

i i ii

d ddt R dtR

(1.27)

can then be used to derive the equations of motion,

( )

( )

Ψ Ψ

δ δµ χ Ψ Ψδχ δχ

¶ ¶=- +

¶ ¶

=- +

* *

constraints

constraints

i ii i

i ii i

m R HR R

H

(1.28)

Equation (1.28) can then be solved numerically. The kinetic energy of the nu-clei (and the system) iså

2ii

m R and thus the temperature of the system µ å

212 ii

T m R . Similarly the fictitious temperature associated with the wave-function is µ χ χµå

12 i j jj

. A very small value of µ is chosen so as to keep the fictitious temperature associated with the electronic wavefunction very low. The reasons for this choice will become clear shortly.

The main concerns about the Car-Parrinello scheme are whether (1) the elec-tronic wavefunction maps the Born-Oppenheimer surface throughout the mo-lecular dynamics simulation, (2) the forces calculated using the Car-Parrinello Lagrangian are equal to the Born-Oppenheimer molecular dynamics forces, and (3) the total energy of the system is conserved in the microcanonical ensemble (Pastore et al. 1991; Payne et al. 1992).

At the beginning of a Car-Parrinello molecular dynamics simulation, the elec-tronic wavefunction is optimized for the initial nuclear configuration. When the nuclei move, their motion affects the electronic structure of the system, thereby changing the wavefunction minimizing the energy at a given instantaneous nu-clear configuration. According to the Car-Parrinello Lagrangian, the wavefunc-tion is also propagated classically according to Eq. (1.28). As mentioned before, Car-Parrinello molecular dynamics will agree with Born-Oppenheimer molecular dyna mics when the wavefunction propagated using the Car-Parrinello scheme


will result in the same electronic energy as that from the optimized wavefunc-tion. This will be the case when no energy is transferred from the nuclei to the classically propagated electronic degrees of freedom. When this adiabaticity is achieved, it is normally attributed to the time-scale difference between the very fast electronic motion and the slow nuclear motion. In some cases, however, some energy can be transferred to the electronic degrees of freedom, resulting in larger forces on the wavefunction [in addition to ( )δ δχ Ψ Ψ*

i H ], thereby deviating from the Born-Oppenheimer surface. It is shown that when a small perturbation in the minimum energy state of a system results in some additional force on the elec-tronic wavefunction, then the minimum frequency related to the dynamics of the orbital is (Pastore et al. 1991)

ω

µ

æ ö÷ç ÷µ ç ÷ç ÷çè ø

1/2gap

mine E

(1.29)

where Egap is the energy difference between the highest occupied and the lowest unoccupied orbital. The parameter Egap in Eq. (1.29) is determined by the physics of the system; however, the parameter µ is completely fictitious and hence can be set to any arbitrary value. The frequency range of the orbital dynamics thus can be switched toward the higher side by choosing a very small value of µ. If a suf-ficiently small value is chosen, then the power spectra emerging from the wave-function dynamics (fast motion, high frequency) can be completely separated from that emerging from the nuclear dynamics (slow motion, low frequency). If these two power spectra do not have any overlap in the frequency domain, then there will be no energy transfer from the nuclei to the wavefunction, the ficti-tious kinetic energy of the wavefunction µ χ χå

12 i j jj

will remain constant, and the electrons will not “heat up.” In fact, the fictitious kinetic ener gy pro-vides a measure of the correctness of the Car-Parrinello implementation. A small and constant fictitious kinetic energy ensures that the system remains on the ground-state potential energy surface, essentially making the scheme equiva-lent to Born-Oppenheimer molecular dynamics, but at a reduced computational cost. Since

µ χ χ Ψ Ψì ü ì üï ï ï ïï ï ï ïï ï ï ï+í ý í ýï ï ï ïï ï ï ïï ïî þï ïî þå å

212 i j j i i

j i

m R H (1.30)

and the fictitious kinetic energy remains constant, from a statistical mechanical point of view, it can be said that the system samples a microcanonical ensemble, where the total energy of the system is conserved.


2.5.2. Metadynamics

Even though the Car-Parrinello molecular dynamics scheme significantly reduces the computational cost of first-principles molecular dynamics simulations, it still cannot access time scales of more than a few picoseconds when attempting to simulate hundreds of atoms (even with state-of-the-art supercomputers). With the technological evolution of computer hardware, which has become increasingly faster and with better performance, researchers are hoping to increase the acces-sible lengths and time scales of first-principles molecular dynamics simulations. However, to perform a simulation of hundreds of nanoseconds (which is possible using standard force-field molecular dynamics) and to be able to simulate realistic phenomena that take place at much longer time scales than those accessed by first-principles molecular dynamics, in addition to the development of computer hardware, it also becomes necessary to implement methods that can accelerate the simulation of rare events. If the reaction of interest is associated with a large energy barrier, then the time scale for the reaction is long and thus difficult to access using costly first-principles molecular dynamics. For instance, the system could be stuck in a local minimum on the energy surface, and crossing the energy barrier to reach the global minimum might simply not be achieved on the simula-tion time scale (see Figure 1.5).

Molecular dynamics simulations can be performed at a higher temperature so as to accelerate events which are expected to take place at a longer time scale in the real system; however, this may cause undesirable and unrealistic events to also happen to the system. Several methods have been implemented in the past to overcome this difficulty, such as umbrella sampling (Torrie and Valleau 1977), nudged elastic band (Henkelman and Jonsson 2000), finite temperature string method (Weinan et al. 2005), transition path sampling (Dellago et al. 1998),

Figure 1.5. illustration of the conformational sampling issue for a system with a large energy barrier between two stable states.


milestoning (Faradjian and Elber 2004), multiple-time-scale accelerated molecu-lar dynamics (Miron and Fichthorn 2004), to name a few. The most recent of these methods is called metadynamics (Laio and Gervasio 2008), and it will be discussed in this chapter for the following reasons.

1. It encompasses several benefits of all the aforementioned methods. 2. It can accelerate rare events so as to be able to observe them in a realistic

computer simulation time. 3. It can be used to reconstruct the energy surface so as to get quantitative

information on the energy landscape and barriers. 4. It was recently coupled to the Car-Parrinello molecular dynamics scheme by

Iannuzzi et al. (2003).

The metadynamics technique, as described by Laio and Gervasio (2008), is based on the principle of filling up the energy surface with arbitrary potentials so as to “flatten” the energy surface. As illustrated on the energy profile in Figure 1.6, if the system is residing in a potential well A, from which it is taking too long to escape due to the energy barriers, then

1. Potential well A is filled up with small potentials so that the system slowly comes to a higher-energy position.

2. As soon as the middle potential well is filled up, the system can freely escape to potential well B, on the left, through the saddle point S1.

3. Potential well B is then gradually filled up until the saddle point S2 is reached, when the system may escape to potential well C.

Figure 1.6. Illustration of the metadynamics scheme (filling up the energy surface to reduce barrier heights). (adapted from laio and Gervasio 2008.)


4. Gradually, well C is filled up with potentials. 5. The system is forced to cross the energy barriers to reach the global energy

minimum by filling up the potentials, and if the magnitude and position of the potentials are tracked, then the entire energy profile can be recon-structed (Laio et al. 2005; Bussi et al. 2006).

As described above, the metadynamics technique is based on filling up the energy surface by dropping potentials at small time intervals in the coordinate space of interest. Though this method can be implemented in any type of molecu-lar dynamics techniques (standard and first-principles), the mathematics rele-vant to the implementation of this method in combination with the Car-Parrinello scheme is described in this chapter, as originally given by Ianuzzi et al. (2003) and further extended by Laio et al. (Laio and Parrinello 2002, 2006; Ensing et al. 2005; Laio and Gervasio 2008). Given a vector ş of the collective variables of interest—for example, the bond distance between two hydrogen atoms if the dynamics and energy profile of hydrogen dissociation is studied, or some coordination number if a more complex phenomenon is studied—the metadynamics approach extends the Car-Parrinello Lagrangian as (Iannuzzi et al. 2003)

£ £ Ş ş ν

ş2

MTD CP cv cv cv cv cv cv cvcv cv

1ş ,2

m k R t (1.31)

where £CP is the Lagrangian defined in Eq. (1.26), the second term of the right-hand side is the kinetic energy of the collective variables, the third term is the harmonic restraining potential, and the last term is the potential that is dropped to fill the energy well in the collective variable ş space at different time intervals. Defining the kinetic and potential energy of the additional collective variables al-lows controlling their dynamics in the canonical ensemble using a suitable thermo-stat. The dynamics of the nuclear and electronic (fictitious) degrees of freedom are separated in Car-Parrinello molecular dynamics by choosing an appro priate value for the mass associated with the fictitious kinetic energy of the electronic wavefunction (as described in Section 2.5.1). Analogously, the dynamics of the collective variables are separated from that of the nuclear and fictitious electronic degrees of freedom by choosing an appropriate value for the fictitious mass mcv of the collective variables. If the fictitious mass mcv is large, then the dynamics of the collective variables will be slow and thus it can be separated from nuclear dynamics. Forces acting on the collective variables rise from the potential energy

( )é ù-ê úë û

cv cv cv cvŞ şk R , and the potential drops ν şcv ,t . Hence the dynamics of the col-

lective variables is also dependent on kcv. It has been shown that the extra term in the meta dynamics introduces an additional frequency for the motion of the col-lective variables as cv cvk m (Ensing et al. 2005). The force constant kcv is chosen


such that the collective variables are close to the actual coordinates of the system. If a small kcv is used, it will result in a large variation in the collective coordinate, even with a small potential drop. However, a very large value may make it neces-sary to employ very small time steps and may result in an excessively long com-putational time. Since kcv is selected by the above constraints, it sets limitation on the value of mcv so as to maintain the adiabaticity.

The potential used to fill the energy well at a time t is given as (Laio and Gervasio 2008)

ν

ş ş ş şş şş

22 1

cv MTD 2 4, exp exp2 2i

i i ii

t t i

t Hw w

(1.32)

where the parameter HMTD is the height of the Gaussian. The above functional form of the potential is a product of Gaussian functions, one of which has width

ş ş1i iiw . This mathematical formulation narrows the width of the poten-

tial in the direction of the trajectory, thus depositing potentials close to each other in the direction of the trajectory. The values of the height and width of the Gaussians depend on the topology of the energy landscape of the system under investigation (Ensing et al. 2005). The procedure to choose optimum values for these parameters is provided in the next section, along with further details about the implementation of the metadynamics method for a particular system.

3. ModelIng hydrogen InteractIon wIth doped transItIon Metal carbon MaterIals usIng car-parrInello Molecular dynaMIcs and MetadynaMIcs

In this section, we describe an example (Mushrif et al. 2010) in which the finite-temperature dynamics and energetics of the interaction of hydrogen with a carbon-supported Pd cluster were modeled. The dynamics were modeled using the Car-Parrinello scheme, and the energy barriers in the process were overcome, the dynamics accelerated, and the energy surface reconstructed using the meta-dynamics scheme (Laio and Parrinello 2002, 2006; Iannuzzi et al. 2003; Laio and Gervasio 2008).

A cubic simulation cell 16 Å long on each side was used for a system contain-ing a coronene molecule, a Pd4 cluster, and a hydrogen molecule (see Figure 1.7). Since the carbon support is usually made up of sp2-type carbons, a polycyclic aromatic hydrocarbon such as coronene was used to mimic the carbon support (Jeloaica and Sidis 1999; Mitchell et al. 2003). The geometry of the molecule was first optimized. Small Pd clusters prefer high-spin states, and several close-lying


energy states are possible (Nava et al. 2003). The convergence of energy calcula-tions is often difficult; as cluster size increases, the highest occupied molecular orbital (HOMO) to lowest unoccupied molecular orbital (LUMO) gap decreases, and convergence of the calculations is only possible if fractional occupation is taken into account (Nava et al. 2003). If the free-energy functional approach (Alavi et al. 1994) is implemented for energy calculations of a system with fractional oc-cupation numbers, the electronic structure at a particular nuclear configuration is completely independent from that of another—even close—configuration. This makes the implementation of the Car-Parrinello scheme impossible, and Born-Oppenheimer molecular dynamics must be used instead, thereby significantly increasing the computational cost. In order to be able to implement the Car-Parrinello scheme, a small Pd cluster of only 4 atoms was thus chosen. The low-est energy state for this cluster, according to the literature (Nava et al. 2003) and our calculations, has a multiplicity of 3. The optimized tetrahedral geometry was adapted from that of Nava et al. (2003). The coronene-supported Pd4 cluster was prepared by monitoring the energies (after geometry optimization without any symmetry constraint on the Pd4 cluster) of the system with the Pd4 tetrahedron at various distances from the center of the plane of the coronene molecule. A dis-tance of ~2 Å gave the minimum energy, and the resulting coronene-supported

Figure 1.7. Coronene-supported pd4 cluster and interacting h2 molecule. the collective variables for the metadynamics simulation are also shown.


Pd4 cluster structure was adopted. A hydrogen molecule was then placed 5 Å above the tip of the coronene-supported Pd4 cluster tetrahedron and the system geometry was reoptimized (see Figure 1.7).

All the calculations in this work were performed with the CPMD software, version 3.13.2, which provides an implementation of the Car-Parrinello molecu-lar dynamics scheme. The first-principles calculations were performed using the plane-wave pseudo-potential implementation of Kohn-Sham density functional theory (Kohn and Sham 1965). A combination of Goedecker pseudo-potential (Goedecker et al. 1996; Hartwigsen et al. 1998) with the local density approxima-tion (Kohn and Sham 1965), which has been validated in previous work (Mushrif et al. 2008), was used.

All the simulations were performed at room temperature. The fictitious elec-tron mass in the Car-Parrinello scheme was taken as 800 a.u. Short molecular dynamics runs were performed without a thermostat to obtain an approximate value around which the fictitious electronic kinetic energy oscillates, and based on this observation, a value of 0.016 a.u. was chosen. A molecular dynamics time step of 0.0964 fs was used in the simulations. Energies, including the fictitious electronic kinetic energy, were monitored to ascertain that the system did not deviate from the Born-Oppenheimer ground-state surface in the course of the molecular dynamics simulations. Trajectories were visualized using the VMD soft-ware (Humphrey et al. 1996). Since previous literature has reported the existence of energy barriers associated with the interaction of hydrogen with metal-doped carbon-supported materials, we employed the metadynamics technique to accel-erate the dynamics and to reconstruct the energy surface as a function of coor-dinates of interest. The metadynamics technique was implemented in the CPMD software, version 3.13.2.

The distance between the two hydrogen atoms in the dihydrogen molecule in-teracting with the Pd cluster-coronene system was chosen as one of the collective variables for the metadynamics, while the other collective variable was chosen as the average coordination number of both hydrogen atoms with the coronene car-bons. The coordination number is defined as follows:

( )( )

-

= -

-=

-åCarbon

60

H 121 0

1

1

nH j

j H j

d dCN

d d (1.33)

where the reference distance d0 was chosen such that the magnitude of the dif-ference between the minimum and the maximum coordination number was the largest. The tips and guidelines provided by Ensing et al. (2005), Laio et al. (2005), and Schreiner et al. (2008) were followed and implemented to estimate the para-meters associated with the metadynamics technique. The Gaussian width para-meter δş was taken as one-fourth of the fluctuation of the collective coordinate


with the smallest amplitude of oscillation. The oscillations of the collective coordi-nates were calculated by performing a sample metadynamics simulation without adding Gaussian potentials. The parameter w is calculated using the criterion

δ w cv cvmax Ş Ş ş . The height of the potential was kept fixed at ~0.27 kJ/mol. The metadynamics time step to add the Gaussian potential was adjusted in such a way that the following criterion was satisfied:

δ 1.5iş t ş t ş (1.34)

An additional criterion that the potential be also added if the time Dtadd given by the following equation had passed since the last metadynamics step was also applied:

∆ δ w cvadd 1.5

B

mt şk T

(1.35)

The dynamics of the collective variables were separated from that of nuclear and fictitious electronic degrees of freedom by choosing an appropriate value for the fictitious mass mcv of the collective variables. As suggested by Schreiner et al. (2008) and Ensing et al. (2005), the choice of kcv and mcv are made in such a way that, during a sample metadynamics run of 20 fs without addition of potentials,

cv cvŞ and ş move close to each other. The temperature of the collective variables was set to room temperature (the same as the physical temperature of the system) and was controlled in a window of ±200 K using velocity rescaling.

3.1. Dissociative cheMisorPtion

A regular Car-Parrinello molecular dynamics simulation (without implementing metadynamics) was initially performed for the system shown in Figure 1.7. Figure 1.8a displays the distance between the Pd atom at the tip of the tetrahedral Pd4 cluster and the two H atoms in the H2 molecule along the trajectory. It can be seen that the H2 molecule chemisorbs on the Pd4 cluster in less than 0.2 ps. Similarly, Figure 1.8b displays the H–H bond in the H2 molecule along the trajectory, as it attaches to the Pd4 cluster. It can be seen that the adsorption of hydrogen on the Pd4 cluster tip is simultaneously accompanied by the dissociation of the H–H bond. When hydrogen chemisorbs on the cluster at 0.16 ps, the H–H bond distance changes from ~0.75 Å to ~0.9 Å (with wide oscillations), thus resulting in dissociative chemisorption of hydrogen. The process is associated with a neg-ligible energy barrier, and the Kohn-Sham energy variation along the trajectory shows that the adsorption energy is approximately 50 kJ/mol. This value of H2


chemisorption on a coronene-supported Pd4 cluster is in agreement with that for the isolated cluster, as calculated by Zhou et al. (2008). The chemisorption energy value on an isolated Pd4 cluster was, however, reported by Matsura et al. (2004) to be ~36 kJ/mol. It is worth mentioning that Matsura et al. (2004) and Zhou et al. (2008) found minimum-energy structures of the chemisorbed H2–Pd4 system with H–H bond distances of 0.82 Å and 0.85 Å, respectively (slightly higher than the H–H bond distance in the free H2 molecule), and they referred to the chemisorbed state as a state with a “stretched” or “weakened” H–H bond. In contrast, the H–H bond distance obtained in our simulations and shown in Figure 1.8b clearly

Figure 1.8. Selected geometric parameters along a CpMD trajectory of h2 interacting with a cor-onene-supported pd4 cluster: (a) distance of two h atoms from the tip atom of the pd4 cluster; (b) distance between the two h atoms. the inset of (a) shows snapshots of the two h atoms before (as an intact h2 molecule) and after dissociative chemisorption on the pd cluster.


suggests dissociation of the H2 molecule. However, a weak, non-electron-sharing inter action between the two hydrogens could still exist at this H–H distance.

3.2. sPillover anD Migration oF hyDrogen

The CPMD simulation was run for up to 7 ps following hydrogen dissociative chemi sorption; however, no further chemical event was observed. This suggests that further migration of the dissociated H atoms involves a significant free-energy barrier. Hence, to accelerate the dynamics of the process and to compute the as-sociated free-energy barriers, metadynamics was employed, as described earlier. The starting structure of the metadynamics simulation has two H atoms dissocia-tively chemisorbed on the tip of the coronene-supported Pd4 cluster.

Figure 1.9 shows the reconstructed free-energy surface as a function of the collective variables. Snapshots of the system at key landmarks in the free-energy surface are also shown. The reconstructed free-energy surface does not appear as smooth as one might have expected, consistent with the recent literature (Zhou et al. 2008), which suggests the presence of several local minima in the migration of H atoms on a Pd cluster.

The starting structure S1, with H2 dissociatively chemisorbed on the coronene-supported Pd4 cluster, appears to be a local minimum in the free-energy surface (Figure 1.9). Analysis of the free-energy surface reveals that the system has to overcome a free-energy barrier of ~6 kJ/mol to escape from this free-energy well.

After escaping from this local minimum, the system falls into another free-energy well, S2, which corresponds to a molecular configuration with the H atoms adsorbed on the adjacent edges of the Pd4 tetrahedron. The configuration with one H atom on the edge of the tetrahedron and the other dangling on the face of the tetrahedron corresponds to a state 5 kJ/mol higher in free energy than S2 (just before the system falls into the free-energy well S2). As shown in Figure 1.9, the free-energy difference between the configurations S1 and S2, i.e., with the H atoms dissociatively chemisorbed on the tip of the Pd4 cluster and with the H atoms attached on the edges of the Pd4 cluster, is very small. For the isolated Pd4 cluster, however, the free-energy difference between the two states is significantly larger (Matsura et al. 2004). This demonstrates that the presence of the sup-port significantly alters the energetics of the hydrogen adsorption and migration process. Further migration of the H atoms from the state S2 requires the system to cross a free-energy barrier of ~3.5 kJ/mol. After crossing this barrier, the H atoms gradually move toward the carbon support, thus taking the system to a free-energy state S3, which is 5 kJ/mol lower in free energy than that with the H atoms attached to the edges of the cluster. The state S3 is the lowest free-energy state of the system. It should be noted that, unlike the isolated (unsupported) Pd4 cluster, the lowest free-energy state is not that with the H atoms attached to the


edges of the cluster, demonstrating again the key role of the carbon support. As the H atoms keep moving from the tip of the cluster toward the carbon support, the system keeps accessing lower free-energy states. It would be of great interest to simulate the dynamics and compute the energetics of the complete migration of the H atoms from the Pd cluster to the carbon support; however, the event was not observed on the practical time scale of the metadynamics simulation. Simulations were stopped when the system reached the state S3 because of the following ad-ditional reasons.

1. The addition of a H atom on a coronene molecule is highly site-specific, and no binding was reported to occur on the hollow or bridge sites (Rauls and Hornekaer 2008). The only possible binding sites are the carbon atoms of

Figure 1.9. three-dimensional free-energy surface reconstructed from the metadynamics simula-tion of h2 dissociatively chemisorbed and interacting with a coronene-supported pd4 cluster (with fixed Pd coordinates). S1, S2, and S3 are key minima on the free-energy surface; snapshots of these structures are shown as insets.


the central carbon ring and those on the edge and outer edge of the coronene molecule (where the carbon atoms already have one bound H atom) (Rauls and Hornekaer 2008). The Pd4 cluster in our simulations is situated on the central hexagonal carbon ring of the coronene molecule, and hence spillover of the H atom on the central ring carbon atom is simply not feasible due to its location right beneath the cluster. The outer edge carbons are too far for the hydrogen atoms to reach. The only remaining option is the edge carbon atom site, which may not again be in close proximity to the bottom Pd atom from which a H atom would spill over. Furthermore, binding of a H atom to the edge carbon also involves a free-energy barrier of 25 kJ/mol (in addition to the barrier that might govern the migration from the Pd cluster).

2. The relatively weak Pd–C interaction in coronene (Duca et al. 2007) may sig-nificantly increase the free-energy barrier of the spillover step (Lachawiec et al. 2005).

3. To obtain the correct relative depth of the free-energy basins in metadynamics, it is important to stop the simulation after a recrossing event, i.e., when the system crosses the barrier in the reverse direction (Laio and Parrinello 2006). In our simulation, such a recrossing and a diffuse motion of H atoms were observed after the system reached the state S3. Gervasio et al. (2005) have also demonstrated that overfilling the free-energy surface in the meta-dynamics simulation, in order to explore high-free-energy regions, may sig-nificantly alter the topology of the free-energy surface and give rise to a false energetic interpretation.

It has been reported in the literature that carbon-supported Pd clusters, upon exposure to hydrogen, form Pd-hydride even before dihydrogen chemisorbs (Amorim and Keane 2008; Bhat et al. 2009). However, we found out that even an icosahedral Pd13 cluster was not big enough to accommodate hydrogen in the absorbed hydride form, and that the computational cost of performing first-prin-ciples molecular dynamics and metadynamics simulations on a larger cluster was simply not feasible (the simulation cell size, the size of the carbon support, and the number of spin states to be considered also increase considerably with cluster size). We thus resorted to metadynamics simulations with the Pd4 cluster partially saturated with 3 H atoms chemisorbed on the edges of the cluster, in a first at-tempt to investigate the dynamics and energetics of the interaction of dihydrogen with a Pd-hydride cluster. Unlike the bare Pd4 cluster, the Pd4-H3 cluster has a singlet ground state. The starting point of the metadynamics simulation is again the system with the 2 H atoms dissociatively chemisorbed on the tip of the Pd4-H3

cluster. The system is equilibrated before initiating the metadynamics simulation. Only the 2 H atoms chemisorbed on the tip of the cluster are part of the metady-namics collective variables, while the remaining the H atoms on the edges of the cluster are simply allowed to move under the influence of the normal interaction


forces. Figure 1.10 displays the results of the simulation. In the course of their migration from the tip of the cluster, the 2 H atoms interact with the preexisting H atoms on the edges of the cluster. The separation distance between them is ap-proximately 1 Å, suggesting that their state is similar to that of the H atoms that

Figure 1.10. three-dimensional free-energy surface reconstructed from the metadynamics simula-tion of h2 dissociatively chemisorbed and interacting with a coronene-supported pd4 cluster partially saturated with 3 h atoms. S1–S6 are key minima on the free-energy surface; snapshots of these structures are shown as insets.. the h atoms which are part of the collective variables are shown in dark gray, and the others are shown in white.


are dissociatively chemisorbed on the tip of the Pd4 cluster, still with some non-electron-sharing interaction (see S1 in Figure 1.9). The state S3 with 4 H atoms chemisorbed on two tips of the Pd4 cluster and 1 H atom chemi sorbed on an edge of the cluster is lower in free energy than the state S1 with 2 H atoms chemisorbed on the tip and 3 H atoms chemisorbed on the edges. This is consistent with the previous observation from Figure 1.9 that, as the adsorbed H atoms migrate to-ward the carbon support, the system falls into lower free-energy states. The free-energy barrier associated with the migration of H atoms from the tip of the cluster in this case is ~3.5 kJ/mol. The migration of H atoms during the metadynamics simulation reveals several structures that are close in free energy, as shown in Figure 1.10. Running metadynamics further results in one of the H atoms mov-ing under the influence of metadynamics to combine with another H atom, whose motion is not influenced by metadynamics, to form a desorbed H2 molecule. As expected, the associative desorption of two H atoms in the form of a H2 molecule results in a higher free-energy state of the system. The process is thus slightly endothermic. Since the spillover of atomic hydrogen from the Pd cluster is compu-tationally not feasible with the system investigated, due to large free-energy bar-riers, the metadynamics simulation results instead in the desorption of one of the chemisorbed H atoms in the form of a H2 molecule. The small free-energy barrier associated with the associative desorption of the H2 molecule from the cluster is approximately 2.5 kJ/mol, which is much less than the barrier for the desorption of atomic hydrogen chemisorbed on a carbon support (Chen et al. 2008; Cheng et al. 2008). This small computed free-energy barrier is in agreement with experi-mental results (Amorim and Keane 2008) in that the desorption of hydrogen che-misorbed on the surface of metal clusters takes place earlier than the desorption of H atoms from the carbon support.

To investigate the transport of atomic hydrogen on an intact, partially satu-rated Pd4 cluster, the metadynamics simulation was repeated, but with the po-sition of the preexisting H atoms on the edges of the cluster kept fixed (so as to avoid associative desorption). Figure 1.11 shows the computed free-energy sur-face reconstructed from the simulation. The free-energy surfaces in Figures 1.9 and 1.11 exhibit significant similarity. The topologies of the free-energy surfaces are quite alike; however, the free-energy barrier associated with the initial migra-tion of the H atoms from the tip toward the sides of the cluster is reduced from 6 kJ/mol (as for a bare Pd4 cluster) to 2 kJ/mol. After this initial migration, the sys-tem falls into a state S2. The difference between the energies of the two states S1 and S2 is significantly higher in this case. It is merely 0.5 kJ/mol for the bare Pd4 cluster; however, it is 6.5 kJ/mol for the partially saturated Pd4 cluster. Running the metadynamics simulation further results in crossing a free-energy barrier of 3.5 kJ/mol associated with further migration of the H atoms. This free-energy barrier is exactly the same as that for the bare Pd4 cluster. The lowest free-energy state observed in this simulation is the state S5 shown in Figure 1.11. Since the


starting point S1 state is exactly the same for both simulations, with and without fixed preexisting H atoms, we can say that the states S2 and S5 which are explored in the case of frozen H coordinates are energetically more stable than any of the states that are explored in the case of freely moving preexisting H atoms (see

Figure 1.11. three-dimensional free-energy surface reconstructed from the metadynamics simula-tion of h2 dissociatively chemisorbed and interacting with a coronene-supported pd4 cluster partially saturated with 3 H atoms (the coordinates of the 3 H atoms are fixed). S1–S5 are key minima on the free-energy surface; snapshots of these structures are shown as insets. the h atoms which are part of the collective variables are shown in dark gray color, and the others are shown in white.


Figure 1.10). As before, the metadynamics simulation is also stopped at the point where a recrossing event takes place.

To summarize, an example of first-principles molecular dynamics simulations of dissociative chemisorption of H2 on the tip of a coronene-supported Pd4 cluster and first-principles metadynamics simulations of the migration of dissociatively chemisorbed H atoms on the tip of (1) a bare Pd4 cluster, (2) a Pd4 cluster partially saturated with 3 H atoms, and (3) a Pd4 cluster partially saturated with 3 immo-bile H atoms have been presented. Though the initial dissociative chemisorption is barrierless, further migration of the H atoms on the cluster involves small free-energy barriers. The barrier associated with the initial migration of the H atoms from the tip of the cluster is reduced when the cluster is partially saturated with preexisting chemisorbed H atoms. We suspect that the presence of absorbed hy-drogen in the form of hydride may have a similar effect of reducing energy barriers associated with the migration of surface H atoms. Migration of the H atoms from the tip of the cluster toward the support results in the system accessing lower free-energy states. For an isolated or unsupported Pd4 cluster, the H atoms tend to chemisorb on the edges of the cluster as the most energetically stable state (Matsura et al. 2004; Zhou et al. 2008). However, when supported on carbon, the state with the H atoms attached to the tip of the Pd cluster in contact with the carbon support is the most energetically stable configuration. The migration of the chemisorbed H atoms on the Pd4 cluster toward the carbon support is thus an energetically favorable process. However, the barrier associated with the complete spillover of atomic hydrogen from the cluster to the carbon support seems to be large. This barrier is dependent not only on the metal cluster but also on the ca-pacity of the carbon support to accept the spillover H atoms. Spillover of hydrogen, if it happens on a sub-nano-sized cluster, will involve migration of H atoms from the metal cluster to the carbon support, and freshly vacated sites on the cluster will then accommodate other incoming hydrogens. To model this effect, we par-tially saturated the cluster with H atoms. It is found that, due to a small barrier associated with the desorption of hydrogen from the partially saturated cluster, 2 H atoms recombine to form a H2 molecule which desorbs from the cluster. When quenching this associative desorption of the H atoms by keeping the position of the preexisting H atoms fixed during the simulations, we found that the topology of the free-energy surface was relatively unaffected. If the migration of H atoms on the metal cluster from its tip toward the carbon support involves small barriers, a significant barrier is associated with the complete transfer of an H atom from the Pd cluster to the carbon support.

Given the small size of the coronene molecule, physisorption of the desorbed H2 molecule on coronene was not observed in the simulations. If the physisorption free energy of H2 on carbon materials is considered to be approximately 10 kJ/mol, then, upon physisorption, the system would land into a state that is lower in free energy than the state S3 (see Figure 1.9). The free-energy surface shown


in Figure 1.10 also suggests that a small free-energy barrier is associated with the recombination of H atoms to form a desorbed H2 molecule. However, as men-tioned before, the spillover of atomic hydrogen from the Pd4 cluster is expected to involve a much larger free-energy barrier. It is possible that, for sub-nano-sized Pd model clusters, where the formation of Pd-hydride does not take place, hydrogen spillover does not occur at room temperature. When the Pd cluster is exposed to additional H2 molecules, new molecules may replace existing ones on the cluster, which are associatively desorbed and may physisorb on the support, thus enhancing the physisorption capacity of the carbon support. Experimental investigations which confirm the presence of atomic hydrogen chemisorbed on the carbon support (Amorim and Keane 2008; Bhat et al. 2009; Contescu et al. 2009) also report the presence of Pd-hydride, and we suspect that the formation of hydride may play a significant role in the complete spillover mechanism. The pres-ence of hydrides may possibly lower the free-energy barrier associated with the migration of surface H atoms from the cluster to the carbon support; alternatively, it may also change the postulated mechanism of spillover, where not the chemi-sorbed surface H atoms but the H atoms in the hydride phase would be pumped out of the cluster to adsorb on the carbon support in the atomic form (Bhat et al. 2009). The energetics associated with this process will be completely different than those for the migration of surface H atoms from the cluster.

4. suMMary

In this chapter we have (1) introduced the reader to the field of functional carbon-based materials, particularly those doped with active transition metals, which are heavily researched for applications related to hydrogen adsorption and storage, (2) briefly summarized the literature on the mechanistic details of hydrogen adsorp-tion/migration and desorption on these materials, (3) introduced the reader to various molecular modeling approaches that can be used for modeling adsorption on carbon materials and discussed their capabilities and limitations, (4) given a brief overview of various electronic structure methods which have been employed to investigate hydrogen adsorption on metal-doped carbon materials, (5) discussed some fundamentals behind the plane-wave/pseudo-potential implementation of the Kohn-Sham density functional theory, which is a widely used electronic struc-ture method to simulate materials, (6) provided the reader with some background on the Car-Parrinello molecular dynamics technique, which provides a state-of-the-art avenue to model hydrogen adsorption on carbon-supported metal clus-ters, (7) briefly discussed one of the most recent metadynamics technique used to accelerate the dynamics of processes involving rare events, and (8) discussed the mechanism and energetics of hydrogen adsorption and migration on a carbon-supported palladium cluster, as an example to demonstrate the usefulness of the


plane-wave/pseudo-potential implementation of Kohn-Sham density functional theory in conjunction with the Car-Parrinello scheme and metadynamics to inves-tigate the interaction of hydrogen with metal-doped carbon materials theoretically.

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