Computational Methods in Lanthanide and Actinide Chemistry

Michael Dolg, Institute for Theoretical Chemistry, University of Cologne, Germany. Professor Dolg works in the field of relativistic ab initio pseudopotentials, both their development and their applications. He performed the first wavefunction-based relativistic and correlated ab initio calculations on lanthanide compounds, in 1989, and in 1994 he extended his studies to actinides. He is currently working on various topics in lanthanide and actinide computational chemistry and is one of the leading scientists in this field.

Contributors xiii Preface xvii

1 Relativistic Configuration Interaction Calculations for Lanthanide and Actinide Anions 1
Donald R. Beck, Steven M. O’Malley and Lin Pan

1.1 Introduction 1

1.2 Bound Rare Earth Anion States 2

1.3 Lanthanide and Actinide Anion Survey 3

1.3.1 Prior Results and Motivation for the Survey 3

1.3.2 Techniques for Basis Set Construction and Analysis 6

1.3.3 Discussion of Results 9

1.4 Resonance and Photodetachment Cross Section of Anions 12

1.4.1 The Configuration Interaction in the Continuum Formalism 13

1.4.2 Calculation of the Final State Wavefunctions 15

2 Study of Actinides by Relativistic Coupled Cluster Methods 23
Ephraim Eliav and Uzi Kaldor

2.1 Introduction 23

2.2 Methodology 25

2.2.1 The Relativistic Hamiltonian 25

2.2.2 Fock-Space Coupled Cluster Approach 25

2.2.3 The Intermediate Hamiltonian CC method 27

2.3 Applications to Actinides 30

2.3.1 Actinium and Its Homologues: Interplay of Relativity and Correlation 31

2.3.2 Thorium and Eka-thorium: Different Level Structure 35

2.3.3 Rn-like actinide ions 39

2.3.4 Electronic Spectrum of Superheavy Elements Nobelium (Z=102) and Lawrencium (Z=103) 42

2.3.5 The Levels of U4+ and U5+: Dynamic Correlation and Breit Interaction 45

2.3.6 Relativistic Coupled Cluster Approach to Actinide Molecules 48

2.4 Summary and Conclusion 49

3 Relativistic All-Electron Approaches to the Study of f Element Chemistry 55
Trond Saue and Lucas Visscher

3.1 Introduction 55

3.2 Relativistic Hamiltonians 59

3.2.1 General Aspects 59

3.2.2 Four-Component Hamiltonians 61

3.2.3 Two-Component Hamiltonians 65

3.2.4 Numerical Example 69

3.3 Choice of Basis Sets 71

3.4 Electronic Structure Methods 73

3.4.1 Coupled Cluster Approaches 75

3.4.2 Multi-Reference Perturbation Theory 80

3.4.3 (Time-Dependent) Density Functional Theory 82

3.5 Conclusions and Outlook 83

4 Low-Lying Excited States of Lanthanide Diatomics Studied by Four-Component Relativistic Configuration Interaction Methods 89
Hiroshi Tatewaki, Shigeyoshi Yamamoto and Hiroko Moriyama

4.1 Introduction 89

4.2 Method of Calculation 90

4.2.1 Quaternion Symmetry 90

4.2.2 Basis Set and HFR/DC Method 91

4.2.3 GOSCI and RASCI Methods 91

4.3 Ground State 92

4.3.1 CeO Ground State 92

4.3.2 CeF Ground State 97

4.3.3 Discussion of Bonding in CeO and CeF 101

4.3.4 GdF Ground State 102

4.3.5 Summary of the Chemical Bonds, of CeO, CeF, GdF 104

4.4 Excited States 106

4.4.1 CeO Excited States 106

4.4.2 CeF Excited States 108

4.4.3 GdF Excited States 108

4.5 Conclusion 116

5 The Complete-Active-Space Self-Consistent-Field Approach and Its Application to Molecular Complexes of the f-Elements 121
Andrew Kerridge

5.1 Introduction 121

5.1.1 Treatment of Relativistic Effects 123

5.1.2 Basis Sets 123

5.2 Identifying and Incorporating Electron Correlation 124

5.2.1 The Hartree Product Wavefunction 124

5.2.2 Slater Determinants and Fermi Correlation 124

5.2.3 Coulomb Correlation 126

5.3 Configuration Interaction and the Multiconfigurational Wavefunction 127

5.3.1 The Configuration Interaction Approach 127

5.3.2 CI and the Dissociation of H2 128

5.3.3 Static Correlation and Crystal Field Splitting 130

5.3.4 Size Inconsistency and Coupled Cluster Theory 131

5.3.5 Computational Expense of CI and the Need for Truncation 132

5.4 CASSCF and Related Approaches 133

5.4.1 The Natural Orbitals 133

5.4.2 Optimisation of the CASSCF Wavefunction 133

5.4.3 Variants and Generalisations of CASSCF 137

5.5 Selection of Active Spaces 138

5.5.1 Chemical Intuition and Björn’s Rules 138

5.5.2 Natural Orbital Occupations 139

5.5.3 RAS Probing 139

5.6 Dynamical Correlation 139

5.6.1 Multireference Configuration Interaction 140

5.6.2 Multireference Second Order Perturbation Theory 140

5.7 Applications 141

5.7.1 Bonding in Actinide Dimers 141

5.7.2 Covalent Interactions in the U-O Bond of Uranyl 142

5.7.3 Covalency and Oxidation State in f-Element Metallocenes 143

5.8 Concluding Remarks 144

6 Relativistic Pseudopotentials and Their Applications 147
Xiaoyan Cao and Anna Weigand

6.1 Introduction 147

6.2 Valence-only Model Hamiltonian 149

6.2.1 Pseudopotentials 150

6.2.2 Approximations 151

6.2.3 Choice of the Core 153

6.3 Pseudopotential Adjustment 155

6.3.1 Energy-Consistent Pseudopotentials 155

6.3.2 Shape-Consistent Pseudopotentials 158

6.4 Valence Basis Sets for Pseudopotentials 161

6.5 Selected Applications 162

6.5.1 DFT Calculated M–X (M = Ln, An; X = O, S, I) Bond Lengths 163

6.5.2 Lanthanide(III) and Actinide(III) Hydration 166

6.5.3 Lanthanide(III) and Actinide(III) Separation 170

6.6 Conclusions and Outlook 172

7 Error-Balanced Segmented Contracted Gaussian Basis Sets: A Concept and Its Extension to the Lanthanides 181
Florian Weigend

7.1 Introduction 181

7.2 Core and Valence Shells: General and Segmented Contraction Scheme 182

7.3 Polarization Functions and Error Balancing 185

7.4 Considerations for Lanthanides 187

8 Gaussian Basis Sets for Lanthanide and Actinide Elements: Strategies for Their Development and Use 195
Kirk A. Peterson and Kenneth G. Dyall

8.1 Introduction 195

8.2 Basis Set Design 196

8.2.1 General Considerations 196

8.2.2 Basis Sets for the f Block 197

8.3 Overview of Existing Basis Sets for Lanthanides and Actinide Elements 204

8.3.1 All-Electron Treatments 204

8.3.2 Effective Core Potential Treatments 205

8.4 Systematically Convergent Basis Sets for the f Block 206

8.4.1 All-Electron 207

8.4.2 Pseudopotential-Based 208

8.5 Basis Set Convergence in Molecular Calculations 210

8.6 Conclusions 213

9 4f, 5d, 6s, and Impurity-Trapped Exciton States of Lanthanides in Solids 217
Zoila Barandiarán and Luis Seijo

9.1 Introduction 217

9.2 Methods 220

9.2.1 Embedded-Cluster Methods 221

9.2.2 Combined Use of Periodic Boundary Condition Methods and Embedded Cluster Methods 227

9.2.3 Absorption and Emission Spectra 227

9.3 Applications 228

9.3.1 Bond Lengths 228

9.3.2 Energy Gaps 231

9.3.3 Impurity-Trapped Excitons 232

9.3.4 Solid-State-Lighting Phosphors 234

10 Judd-Ofelt Theory — The Golden (and the Only One) Theoretical Tool of f-Electron Spectroscopy 241
Lidia Smentek

10.1 Introduction 241

10.2 Non-relativistic Approach 245

10.2.1 Standard Judd-Ofelt Theory and Its Original Formulation of 1962 248

10.2.2 Challenges of ab initio Calculations 251

10.2.3 Problems with the Interpretation of the f -Spectra 255

10.3 Third-Order Contributions 257

10.3.1 Third-Order Electron Correlation Effective Operators 259

10.4 Relativistic Approach 260

10.5 Parameterizations of the f -Spectra 262

11 Applied Computational Actinide Chemistry 269
André Severo Pereira Gomes, Florent Réal, Bernd Schimmelpfennig, Ulf Wahlgren and Valérie Vallet

11.1 Introduction 269

11.1.1 Relativistic Correlated Methods for Ground and Excited States 270

11.1.2 Spin-Orbit Effects on Heavy Elements 272

11.2 Valence Spectroscopy and Excited States 273

11.2.1 Accuracy of Electron Correlation Methods for Actinide Excited States: WFT and DFT Methods 273

11.2.2 Valence Spectra of Larger Molecular Systems 275

11.2.3 Effects of the Condensed-Phase Environment 276

11.2.4 Current Challenges for Electronic Structure Calculations of Heavy Elements 278

11.3 Core Spectroscopies 278

11.3.1 X-ray Photoelectron Spectroscopy (XPS) 279

11.3.2 X-ray Absorption Spectroscopies 280

11.4 Complex Formation and Ligand-Exchange Reactions 283

11.5 Calculations of Standard Reduction Potential and Studies of Redox Chemical Processes 286

11.6 General Conclusions 288

12 Computational Tools for Predictive Modeling of Properties in Complex Actinide Systems 299
Jochen Autschbach, Niranjan Govind, Raymond Atta-Fynn, Eric J. Bylaska, John W. Weare and Wibe A. de Jong

12.1 Introduction 299

12.2 ZORA Hamiltonian and Magnetic Property Calculations 300

12.2.1 ZORA Hamiltonian 300

12.2.2 Magnetic properties 303

12.3 X2C Hamiltonian and Molecular Properties from X2C Calculations 312

12.4 Role of Dynamics on Thermodynamic Properties 319

12.4.1 Sampling Free Energy Space with Metadynamics 319

12.4.2 Hydrolysis constants for U(IV), U(V), and U(VI) 320

12.4.3 Effects of Counter Ions on the Coordination of Cm(III) in Aqueous Solution 322

12.5 Modeling of XAS (EXAFS, XANES) Properties 325

12.5.1 EXAFS of U(IV) and U(V) Species 327

12.5.2 XANES Spectra of Actinide Complexes 330

13 Theoretical Treatment of the Redox Chemistry of Low Valent Lanthanide and Actinide Complexes 343
Christos E. Kefalidis, Ludovic Castro, Ahmed Yahia, Lionel Perrin and Laurent Maron

13.1 Introduction 343

13.2 Divalent Lanthanides 349

13.2.1 Computing the Nature of the Ground State 349

13.2.2 Single Electron Transfer Energy Determination in Divalent Lanthanide Chemistry 352

13.3 Low-Valent Actinides 356

13.3.1 Actinide(III) Reactivity 356

13.3.2 Other Oxidation State (Uranyl…) 361

13.4 Conclusions 365

14 Computational Studies of Bonding and Reactivity in Actinide Molecular Complexes 375
Enrique R. Batista, Richard L. Martin and Ping Yang

14.1 Introduction 375

14.2 Basic Considerations 376

14.2.1 Bond Energies 376

14.2.2 Effect of Scalar Relativistic Corrections 377

14.2.3 Spin-Orbit Corrections 378

14.2.4 Relativistic Effective Core Potentials (RECP) 379

14.2.5 Basis Sets 380

14.2.6 Density Functional Approximations for Use with f-Element Complexes 381

14.2.7 Example of application: Performance in Sample Situation (UF6→UF5 +F) [39, 40] 382

14.2.8 Molecular Systems with Unpaired Electrons 384

14.3 Nature of Bonding Interactions 385

14.4 Chemistry Application: Reactivity 387

14.4.1 First Example: Study of C–H Bond Activation Reaction 387

14.4.2 Study of Imido-Exchange Reaction Mechanism 395

14.5 Final Remarks 397

15 The 32-Electron Principle: A New Magic Number 401
Pekka Pyykkö, Carine Clavaguéra and Jean-Pierre Dognon

15.1 Introduction 401

15.1.1 Mononuclear, MLn systems 401

15.1.2 Metal Clusters as ‘Superatoms’ 402

15.1.3 The Present Review: An@Ln-Type Systems 404

15.2 Cases So Far Studied 404

15.2.1 The Early Years: Pb2−12 and Sn2−12 Clusters 404

15.2.2 The Validation: An@C28 (An = Th, Pa+, U2+, Pu4+) Series 410

15.2.3 The Confirmation: [U@Si20]6−-like Isoelectronic Series 413

15.3 Influence of Relativity 418

15.4 A Survey of the Current Literature on Lanthanideand Actinide-Centered Clusters 420

15.5 Concluding Remarks 421

16 Shell Structure, Relativistic and Electron Correlation Effects in f Elements and Their Importance for Cerium(III)-based Molecular Kondo Systems 425
Michael Dolg

16.1 Introduction 425

16.2 Shell Structure, Relativistic and Electron Correlation Effects 429

16.2.1 Shell Structure 430

16.2.2 Relativistic Effects 433

16.2.3 Electron Correlation Effects 437

16.3 Molecular Kondo-type Systems 439

16.3.1 Bis(η8-cyclooctatetraenyl)cerium 440

16.3.2 Bis(η8-pentalene)cerium 443

16.4 Conclusions 446

Index 451

Color plates appear between pages 342 and 343