Effective Crystal Field Potential (eBook)

Z. Gajek, J. Mulak (Herausgeber)

eBook Download: PDF

2000 | 1. Auflage
316 Seiten
Elsevier Science (Verlag)
978-0-08-053071-0 (ISBN)

Infos

As it results from the very nature of things, the spherical symmetry of the surrounding of a site in a crystal lattice or an atom in a molecule can never occur. Therefore, the eigenfunctions and eigenvalues of any bound ion or atom have to differ from those of spherically symmetric respective free ions. In this way, the most simplified concept of the crystal field effect or ligand field effect in the case of individual molecules can be introduced.
The conventional notion of the crystal field potential is narrowed to its non-spherical part only through ignoring the dominating spherical part which produces only a uniform energy shift of gravity centres of the free ion terms. It is well understood that the non-spherical part of the effective potential 'seen' by open-shell electrons localized on a metal ion plays an essential role in most observed properties. Light adsorption, electron paramagnetic resonance, inelastic neutron scattering and basic characteristics derived from magnetic and thermal measurements, are only examples of a much wider class of experimental results dependent on it. The influence is discerned in all kinds of materials containing unpaired localized electrons: ionic crystals, semiconductors and metallic compounds including materials as intriguing as high-Tc superconductors, or heavy fermion systems. It is evident from the above that we deal with a widespread effect relative to all free ion terms except those which can stand the lowered symmetry, e.g. S-terms.
Despite the universality of the phenomenon, the available handbooks on solid state physics pay only marginal attention to it, merely making mention of its occurrence. Present understanding of the origins of the crystal field potential differs essentially from the pioneering electrostatic picture postulated in the twenties. The considerable development of the theory that has been put forward since then can be traced in many regular articles scattered throughout the literature. The last two decades have left their impression as well but, to the authors' best knowledge, this period has not been closed with a more extended review. This has also motivated us to compile the main achievements in the field in the form of a book.

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As it results from the very nature of things, the spherical symmetry of the surrounding of a site in a crystal lattice or an atom in a molecule can never occur. Therefore, the eigenfunctions and eigenvalues of any bound ion or atom have to differ from those of spherically symmetric respective free ions. In this way, the most simplified concept of the crystal field effect or ligand field effect in the case of individual molecules can be introduced. The conventional notion of the crystal field potential is narrowed to its non-spherical part only through ignoring the dominating spherical part which produces only a uniform energy shift of gravity centres of the free ion terms. It is well understood that the non-spherical part of the effective potential "e;seen"e; by open-shell electrons localized on a metal ion plays an essential role in most observed properties. Light adsorption, electron paramagnetic resonance, inelastic neutron scattering and basic characteristics derived from magnetic and thermal measurements, are only examples of a much wider class of experimental results dependent on it. The influence is discerned in all kinds of materials containing unpaired localized electrons: ionic crystals, semiconductors and metallic compounds including materials as intriguing as high-Tc superconductors, or heavy fermion systems. It is evident from the above that we deal with a widespread effect relative to all free ion terms except those which can stand the lowered symmetry, e.g. S-terms. Despite the universality of the phenomenon, the available handbooks on solid state physics pay only marginal attention to it, merely making mention of its occurrence. Present understanding of the origins of the crystal field potential differs essentially from the pioneering electrostatic picture postulated in the twenties. The considerable development of the theory that has been put forward since then can be traced in many regular articles scattered throughout the literature. The last two decades have left their impression as well but, to the authors' best knowledge, this period has not been closed with a more extended review. This has also motivated us to compile the main achievements in the field in the form of a book.

Inhalt

Cover 1
Contents 10
Chapter 1. Introduction 16
Chapter 2. Parameterization of crystal field Hamiltonian 26
2.1. Operators and parameters of the crystal field Hamiltonian 27
2.2. Basic parameterizations 29
2.3. Symmetry transformations of the operators 33
2.4. The number of independent crystal field parameters 38
2.5. Standardization of the crystal field Hamiltonian 41
2.6. Final remark 44
Chapter 3. The effective crystal field potential. Chronological development of crystal field models 46
Chapter 4. Ionic complex or quasi-molecular cluster. Generalized product function 56
4.1 Concept of the generalized product function 56
4.2 The density functions and the transition density functions 58
4.3 Model of the generalized product functions 59
4.4 Crystal field effect in the product function model 64
Chapter 5. Point charge model (PCM) 68
5.1 PCM potential and its parameters 68
5.2 Simple partial PCM potentials 71
5.3 Extension of PCM–higher point multipole contribution 76
Chapter 6. One-configurational model with neglecting the non-orthogonality. The charge penetration and exchange effects 80
6.1 Classical electrostatic potential produced by the ligand charge distribution 80
6.2 The charge penetration effect and the exchange interaction in the generalized product function model 83
6.3 The weight of the penetration and exchange effects in the crystal field potential 86
6.4 Calculation of the two-centre integrals 88
6.5 Final remarks 89
Chapter 7. The exclusion model. One-configurational approach with regard to non-orthogonality of the wave functions 92
7.1 Three types of the non-orthogonality 92
7.2 The renormalization of the open-shell Hamiltonian Ha owing to the non-orthogonality of the one-electron functions 94
7.3 The contact-covalency–the main component of the crystal field potential 99
7.4 The contact-shielding 102
7.5 The contact-polarization 103
7.6 Mechanisms of the contact-shielding and contact-polarization in terms of the exchange charge notion 103
Chapter 8. Covalency contribution, i.e. the charge transfer effect 106
8.1 The one-electron excitations. Group product function for the excited state 106
8.2 The renormalization of the open-shell Hamiltonian due to the covalency effect 109
8.3 Basic approximations 111
8.4 The one-electron covalency potential Vcov 112
8.5 The one-electron covalency potential V cov in the molecular-orbital formalism 116
8.6 Remarks on the covalency mechanism 117
Chapter 9. Schielding and antishielding effect: contributions from closed electron shells 120
9.1 Phenomenological quantification of the screening effect 121
9.2 Microscopic model of the screening effect 122
9.3 General expressions for the screening factors 124
9.4 The screening factors 131
Chapter 10. Electrostatic crystal field contributions with consistent multipolar effects. Polarization 134
10.1 Expansion of the electrostatic potential of point charge system into the multipole series 134
10.2 Extended formula for the crystal field parameters including all multipole moments of the surroundings 136
10.3 The self-consistent system of permanent and induced multipole moments in crystal lattice 141
10.4 The off-axial polarization terms in local coordinate systems. 142
10.5 Typical examples of dipole and quadrupole polarization contributions to the crystal field potential 144
Chapter 11. Crystal field effect in the Stevens perturbation approach 146
11.1 The Wannier functions 147
11.2 The perturbation scheme for degenerate systems employing projection operators 148
11.3 The crystal field effect 151
Chapter 12. Specific mechanisms of metallic states contributing to the crystal field potential 158
Chapter 13. Screening the crystal field in metallic materials 162
13.1 The Fourier form of the crystal lattice potential 164
13.2 The dielectric static screening function e(q) 168
13.3 The dynamic mechanism of the screening - zero-point plasmon 174
Chapter 14. Virtual bound state contribution to the crystal field potential 178
14.1 The resonance scattering of conduction electrons by a central potential 178
14.2 The nature of the virtual bound state 181
14.3 Spin-polarization of the virtual bound state 182
14.4 Experimental manifestations of existing the virtual bound states and methods of estimating their localization degree 182
14.5 The crystal field splitting of the virtual bound state 183
14.6 The primary crystal field effect relative to the open-shell states (4f) 184
14.7 Corrections to the simple model of the virtual bound state mechanism 189
Chapter 15. Hybridization or covalent mixing between localized states and conduction band states in metallic crystals 192
15.1 The essence of the hybridization 192
15.2 Hybridization contribution to the crystal field parameters 193
15.3 The scale of the hybridization effect 197
15.4 Contribution to the crystal field potential from a split-off state from the conduction band in impurity systems 199
Chapter 16. Density functional theory approach 200
16.1 Electron density as a key variable 200
16.2 The Kohn–Sham equations 203
16.3 Local density approximation 205
16.4 Extensions 207
16.5 Exchange-correlation energy 216
16.6 Mapping DFT on effective Hamiltonian 219
16.7 Applications 221
Chapter 17. Analysis of the experimental data . Interpretation of crystal field parameters with additive models 226
17.1 Phenomenological Hamiltonian 227
17.2 Simplified crystal field models 230
17.3 Towards applications 239
Chapter 18. Lattice dynamics contribution 244
18.1 Adiabatic and harmonic approximations 245
18.2 Collective (normal) coordinates and the "quasi-molecular" model 248
18.3 The Jahn-Teller effect 250
18.4 Lattice dynamics and the crystal field effect 256
Chapter 19. Extension of the crystal field potential beyond the one-electron model 262
19.1 Two-electron correlation effect in the crystal field model 262
19.2 Parameterization of the two-electron potential 263
19.3 The term dependent crystal field 265
19.4 Spin correlated crystal field (SCCF) 267
19.5 Many-electron approach to the crystal field effect 269
Chapter 20. Appendices 272
A. Transformation from local to the global coordinate system 272
B. 3n-j symbols 274
C. Methods of orthogonalization of functions 276
References 278
Author Index 302
Subject Index 308

Produkt-Details

• Erscheint lt. Verlag: 22.6.2000
• Sprache: englisch
• Themenwelt: Naturwissenschaften ► Chemie ► Physikalische Chemie
• Themenwelt: Naturwissenschaften ► Geowissenschaften ► Mineralogie / Paläontologie
• Themenwelt: Naturwissenschaften ► Physik / Astronomie ► Festkörperphysik
• Themenwelt: Technik ► Maschinenbau
• ISBN-10: 0-08-053071-0 / 0080530710
• ISBN-13: 978-0-08-053071-0 / 9780080530710