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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.
Preface xvii
1 Relativistic Configuration Interaction Calculations for Lanthanide and Actinide Anions 1Donald 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 23Ephraim 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 55Trond 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 89Hiroshi 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 121Andrew 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 147Xiaoyan 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 181Florian 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 195Kirk 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 217Zoila 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 241Lidia 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 269André 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 299Jochen 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 343Christos 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 375Enrique 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 401Pekka 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 425Michael 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(¿8-cyclooctatetraenyl)cerium 440
16.3.2 Bis(¿8-pentalene)cerium 443
16.4 Conclusions 446
Index 451
Color plates appear between pages 342 and 343
Donald R. Beck1,* Steven M. O'Malley2 and Lin Pan3
1Department of Physics, Michigan Technological University
2Atmospheric and Environmental Research
3Physics Department, Cedarville University
Lanthanide and actinide atoms and ions are of considerable technological importance. In condensed matter, they may be centers of lasing activity, or act as high temperature superconductors. Because the f-electrons remain quite localized in going from the atomic to the condensed state, a lot of knowledge gained from atoms is transferable to the condensed state. As atoms, they are constituents of high intensity lamps, may provide good candidates for parity non-conservation studies, and provide possible anti-proton laser cooling using bound-to-bound transitions in anions such as La- [1].
In this chapter we will concentrate on our anion work [2-4], which has identified 114 bound states in the lanthanides and 41 bound states in the actinides, over half of which are new predictions. In two anions, Ce- and La-, bound opposite parity states were found, making a total of 3 [Os- was previously known]. Bound-to-bound transitions have been observed in Ce- [5] and may have been observed in La- [6]. We have also worked on many properties of lanthanide and actinide atoms and positive ions. A complete list of publications can be found elsewhere [7].
In 1994, we began our first calculations on the electron affinities of the rare earths [8]. These are the most difficult atoms to treat, due to the open f -subshells, followed by the transition metal atoms with their open d-subshells. At that time, some accelerator mass spectrometry (AMS) measurements of the lanthanides existed [9, 10] which were rough. Larger values might be due to multiple bound states, states were uncharacterized as to dominant configuration, etc.
Local density calculations done in the 1980s had suggested anions were formed by 4f attachments to the incomplete 4f subshell. Pioneering computational work done by Vosko [11] in the early 1990s on the seemingly simple Lu- and La- anions using a combination of Dirac-Fock and local density results suggested instead that the attachment process in forming the anions involved p, not f, electrons.
Our 1994 calculation on a possible Tm anion was consistent with this, in that it showed 4f attachment was not a viable attachment process. Our calculations are done using a Relativistic Configuration Interaction (RCI) methodology [12], which does a Dirac-Hartree-Fock (DHF) calculation [13] for the reference function (s) (dominant configurations). The important correlation configurations (e.g., single and pair valence excitations from the reference configuration[s]) are then added in, using the DHF radials and relativistic screened hydrogenic function (called virtuals), whose effective charge (Z*) is found by minimization of the energy matrix, to which the Breit contributions may be added, if desired.
Experience gained in the mid-1990s suggested that good candidates for bound anion states might be found by combining observed ground and excited state neutral spectra with the computational knowledge that closing an s-subshell might lower the energy ~1.0 eV or adding a 6p-electron to a neutral atom state (7p in the actinides) might lower the energy ~0.25 eV. The variety of energetically low-lying configurations in the observed spectrum of La and Ce suggests a potential for a large number of bound anion states, which has now been computationally confirmed.
As an example of the process, a Tm-4f146s2 anion state might be bound if there were a 4f146s1 state observed in the neutral atom that was less than 1 eV above the ground state. The use of excited states with s/p attachment also has the computationally attractive feature that it avoids, to a good level of approximation, having to compute correlation effects for d and/or f electrons. An s attachment to an excited state can be equivalent to a d attachment to the ground state. The angular momentum expansions for such pair excitations converge slowly, and a lot of energy is associated with (nearly) closed d and/or f subshells. Clearly, it is best to reduce such problems if usable experimental results exist.
It has always been our position to use no more than moderately size d wavefunction expansions. Current limits are about 20,000 symmetry adapted wavefunctions built from fewer than 1 million Slater determinants, and use of two virtuals per l, per shell (n). This allows the "physics" (systematics) to be more visible and reduces the need for "large" computational resources that were frequently unavailable in the "old" days. Development of systematic "rules" is one of the main goals of our research. Some examples follow: (i) determining which correlation effects are most important for a specific property [14, 15], (ii) near conservation of f-value sums for nearly degenerate states [15, 16], (iii) similar conservation of g-value sums [16], (iv) similar conservation of magnetic dipole hyperfine constants [17, 18]. This approach does mean near maximal use of symmetry, creating extra auxiliary computer codes, and increases the necessity of automating data preparation and file manipulation. Much stricter development of this automation is one of the two factors that reduced calculation of the entire actinide row to less than the time it used to take to complete the calculation for one anion (>4 months for Nd-). Use of moderately sized wavefunctions also requires careful selection of which property-specific configurations to include and careful optimization of the virtual radial functions.
In 2008 and 2009 our group presented a series of three papers [2-4] representing an unprecedented and complete survey of the bound lanthanide and actinide anion states predicted by valence level RCI calculations. The first of these [2] was a study of all 6p attachments to 4fn 6s2 ground and excited states of the lanthanide neutral spectra (then and throughout the discussion here we use n as an occupancy of N-2 where N is the total number of valence electrons in the neutral atom configuration, including the core like 4f /5f subshells). The second paper [3] completed the lanthanide survey with 6p attachments to 4fm 5d 6s2 thresholds and 6s attachments to 4fm 5d2 6s thresholds (m = N - 3). The final paper in the series [4] included the equivalent 7s and 7p attachments to corresponding actinide neutral thresholds as well as additional states in Th- and Pa- representing 7p attachments to 5fq 6d2 7s2 thresholds (q = N - 4). The approach used to handle the complexity of these calculations represented the culmination of over three decades of group experience in developing techniques and computational tools for RCI basis set construction. The path that led to this comprehensive lanthanide and actinide anion survey was somewhat circuitous and developed originally through adjustments to increasing difficulties with each step toward more complex systems. In the following subsections we discuss some milestones leading up to the survey, the computational issues and solutions, the improved analytical tools that were needed, and a summary of results of the survey.
Throughout the 1990s and early 2000s our group had been steadily pushing our methodology towards more and more complex atomic systems. The ability to do so was partly from techniques described in Section 1.3.2 but also largely due to ever-increasing computer power. Mid-row transition metal studies had become fairly routine, e.g., binding energies of Ru- [19], Os- [20], and Tc- [21]. However, the added complexity of a near-half-full f subshell over that of a d had relegated us for the most part to the outer edges of the lanthanide and actinide rows, e.g., Ce- [22, 23], Th- [24], Pr- [25], U- [26], Pa- [27], La- [28], and Lu- [29]. During the mid-to late-1990s, we were twice enticed by the unique case of Tb to attempt to skip to the center of the lanthanide row [30]. The Tb ground state is 4f9 6s2, but the low-lying first excited state (~35 meV [31]) is of the opposite parity 4f8 5d 6s2 configuration, and the possibility of opposite parity Tb- bound anion states resulting from the same 6p attachment mechanism was a tempting prize. Unfortunately, those initial attempts at this mid-row anion were premature and Tb- would have to wait to use basis set construction techniques that we eventually developed in the mid-2000s.
As we were gradually working our way inwards from the ends of the lanthanide and actinide rows, our papers began to take on a back-and-forth dialog with the work of the experimental atomic physics group at University of Nevada, Reno (Thompson and...
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