Orbital Approach to the Electronic Structure of Solids

, by Canadell, Enric; Doublet, Marie-Liesse; Iung, Christophe

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ISBN: 9780199534937 | 0199534934
Cover: Hardcover
Copyright: 3/21/2012

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This book provides an intuitive yet sound understanding of how structure and properties of solids may be related. The natural link is provided by the band theory approach to the electronic structure of solids. The chemically insightful concept of orbital interaction and the essential machinery of band theory are used throughout the book to build links between the crystal and electronic structure of periodic systems. In such a way, it is shown how important tools for understandingproperties of solids like the density of states, the Fermi surface etc. can be qualitatively sketched and used to either understand the results of quantitative calculations or to rationalize experimental observations. Extensive use of the orbital interaction approach appears to be a very efficient wayof building bridges between physically and chemically based notions to understand the structure and properties of solids.

Elementary introduction to the transport properties of solids	p. 1
Free electron model	p. 1
One-dimensional system	p. 2
Generalisation to a three-dimensional system	p. 9
Conductivity of real solids	p. 10
Factors influencing the conductivity	p. 10
Band structure of real solids	p. 11
Metallic behaviour	p. 11
Semiconducting and insulating behaviour	p. 12
Number of carriers	p. 13
Electronic structure of molecules: use of symmetry	p. 14
Molecular orbital theory	p. 14
Born-Oppenheimer approximation	p. 15
One-electron approximation	p. 15
LCAO approximation	p. 15
Secular equations and secular determinant	p. 16
Basic features of the Hückel and extended Hückel methods	p. 17
Symmetry properties of the molecular orbitals	p. 18
A short review of the theory of symmetry point groups	p. 19
Different symmetry point groups	p. 19
Classes	p. 21
Basis for an irreducible representation	p. 22
Application to the study of the ¿ system of regular cyclobutadiene	p. 25
Decomposition of the ¿(p_z) basis	p. 26
Determination of the basis elements for different irreducible representations	p. 27
Molecular orbital diagram of the ¿ system of regular cyclobutadiene	p. 30
Transition metal complexes	p. 30
Ligands and formal oxidation-state	p. 31
The ML₆ octahedral complex	p. 33
Distortions of a complex	p. 39
Electronic structure of one-dimensional systems: basic notions	p. 44
Bloch and crystal orbitals	p. 45
Bloch orbitals	p. 46
Crystal orbitals	p. 49
Electronic structure of the model chain H_n	p. 51
Representation of the CO(¿) and CO(X) functions	p. 51
Energy of the crystal orbitals in the Hückel approach	p. 52
Band structure	p. 54
Basis for an energy level E(±k)	p. 55
Fermi level of the H_n chain	p. 57
Electronic structure of the dimerised model chain (H₂)_n′	p. 58
Formal determination of the band structure	p. 58
Qualitative determination of the band structure	p. 61
Comparison of the regular H_n and dimerised (H₂)_n′ chains	p. 63
Comparison of the band structures of the regular H_n chain generated by either a simple or a double unit cell	p. 63
Dimerisation in the H_n chain: notion of distortion in a periodic system	p. 67
First-order Peierls distortions in periodic 1D systems	p. 72
Analysis of the model system (H^0.5+)_n	p. 72
Effect of a tetramerisation on the Fermi level	p. 73
Effect of a tetramerisation on the states near the Fermi level	p. 74
Effect of a tetramerisation on the band structure	p. 76
Analysis of first-order Peierls distortion in terms of a charge density wave	p. 77
Nesting vector	p. 81
Commensurate and incommensurate distortions	p. 81
Commensurate distortion	p. 81
Incommensurate distortion	p. 83
Comparison	p. 83
Conclusions	p. 83
Application to trans-polyacetylene	p. 85
Electronic structure of ethylene	p. 86
Main aspects of the band structure for trans-polyacetylene	p. 87
Detailed analysis of the band structure of trans-polyacetylene	p. 88
Determination of the band structure of trans-polyacetylene using the fragment formalism	p. 89
Calculation of the band structure by means of the Hückel approach	p. 91
Qualitative determination of the band structure	p. 92
Band gap opening at the Fermi level in trans-polyacetylene	p. 93
Handling the symmetry in 1D compounds	p. 96
Analysis of the A_n system	p. 96
Analysis of the cyclic A_n system	p. 96
Analysis of the linear A_n system	p. 101
Notion of group of a k point	p. 104
Application to the determination of the band structure for the A_n linear system, where A is an atom	p. 104
Group of the different k points	p. 105
Symmetry of the different Bloch orbitals	p. 105
Bands associated with a ¿-type overlaps	p. 107
Complete band structure	p. 108
Band structure of the hypothetical (NaCl)_n chain	p. 109
Group of the different k points	p. 110
Bands associated with a ¿-type overlaps	p. 110
Complete band structure	p. 112
Consequences of the existence of a glide plane	p. 113
Using point group symmetry properties in trans-polyacetylene	p. 113
Complete space group (non-symmorphic) of trans-polyacetylene	p. 115
Crystal orbitals of trans-polyacetylene by means of the non-symmorphic space group G = T_n ⊗ C_2h ⊗ {E, g_¿}	p. 117
Concluding remarks	p. 119
Work plan for the study of a 1D system	p. 120
Application to polyacene	p. 122
Band structure near the Fermi level	p. 123
Unit cell definition	p. 123
Symmetry analysis of the chain	p. 123
Appropriate fragment orbitals	p. 123
Crystal orbitals at the ¿ and X points	p. 124
¿-type band structure of polyacene	p. 126
Distortions in polyacene	p. 128
Disappearance of the ¿_xy symmetry plane	p. 128
Disappearance of the ¿_yz symmetry plane	p. 128
General remarks concerning Peierls distortions	p. 130
First-order Peierls distortions	p. 130
Second-order Peierls distortions	p. 131
Electronic structure of selected inorganic chains	p. 133
KCP	p. 133
Band structure of the eclipsed chain [Pt(CN)₄]^(2-¿)-	p. 134
Band structure of KCP (staggered chain)	p. 139
Conclusions	p. 142
(ML₄L′)_n chains	p. 143
Symmetry	p. 143
Choice of the fragment orbitals to generate the Bloch orbitals	p. 143
Analysis of the Bloch orbitals at the ¿ and X points	p. 144
Symmetry of the Bloch orbitals	p. 144
Band structure	p. 145
Study of the (ReCl₄N)_n chain	p. 147
Electronic structure of the (Pt(NH₂Et)₄Cl²⁺)_n chain	p. 149
Suggested studies	p. 153
Electronic structure of 2D and 3D systems	p. 157
Basic concepts	p. 157
Direct and reciprocal lattices	p. 157
Bloch and crystal orbitals	p. 159
Brillouin zone	p. 161
Symmetry and the Brillouin zone	p. 162
Analysis of the electronic structure of 2D model systems	p. 166
The square lattice 2 ∞ [H_n] system	p. 166
The square lattice 2 ∞ [A_n] system	p. 169
¿-type band structure of hexagonal graphene layers	p. 173
Density of states	p. 181
Calculation and analysis of the density of states	p. 181
Density of states	p. 181
Projected density of states	p. 183
Crystal orbital overlap population	p. 185
Combined use of DOS and COOP: electronic structure of the MPS3 layered phases	p. 186
Step-by-step determination of the density of states: the (Pt(NH₃)₄Cl)²⁺ chain	p. 188
Density of states and fragment molecular orbital interaction analysis: application to the [(C₅H₅)M] chains	p. 193
Transition metal diborides with the A1B₂ structure type: a 3D case study	p. 196
Fermi surface and low-dimensional metals	p. 203
Notion of Fermi surface	p. 204
Nesting vector and electronic instabilities in low-dimensional metals	p. 207
Monoclinic TaS₃ versus NbSe₃	p. 210
Crystal structure and electron counting	p. 211
Qualitative band structure	p. 212
Qualitative Fermi surface: differences between NbSe₃ and TaS₃	p. 214
Molybdenum bronzes	p. 215
Octahedral distortions and t_2g level splitting in MoO₆ octahedra	p. 216
MoO₅ chain with comer-sharing octahedra: counting of 2p oxygen antibonding contributions	p. 217
A_0.33MoO₃ (A = K, Rb, Cs, Tl) 2D red bronzes: metallic or insulating?	p. 219
A_0.3MoO₃ (A = K, Rb, Tl) blue bronzes: 2D solids with pseudo-ID behaviour	p. 224
Looking for ID systems where there seem to be none: the concept of hidden nesting	p. 227
Low-dimensional molecular conductors	p. 232
An archetypal molecular metal: (TMTSF)₂PF₆	p. 234
Chemically modifying the electronic structure of molecular conductors	p. 235
Structurally complex materials with simple band structures	p. 238
A case study: ID vs 2D character of the carriers in some ¿ phases of BEDT-TTF	p. 242
Electronic structure and folding: how to relate the band structure and Fermi surface of different salts of the same family	p. 247
Electron repulsion	p. 256
From the Hückel model to the Hubbard model	p. 256
The delocalised picture of H₂	p. 256
The localised picture of H₂	p. 260
From the molecule to the solid state	p. 266
Application to one-band systems	p. 269
Mean-field approaches	p. 273
The many-body problem	p. 273
The Hartree-Fock method	p. 274
Density functional theory	p. 281
Conclusion	p. 287
Solutions for exercises	p. 291
Appendix: Character tables	p. 342
Index	p. 345
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Orbital Approach to the Electronic Structure of Solids

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