Table Of ContentCHEMICAL BONDING IN SOLIDS
Contents
Symbols and Abbreviations, xv
1. Molecules, 3
1.1 The H2 Molecule: Molecular Orbital Approach, 3
1.2 The H2 Molecule: Localized Approach, 7
1.3 Energy Levels of HHe, 9
1.4 Energy Levels of Linear Conjugated Molecules, 10
1.5 Energy Levels of Cyclic Polyenes, 12
1.6 Energy Differences and Moments, 20
1.7 The Jahn-Teller Effects, 24
2. From molecules to solids, 29
2.1 The Solid as a Giant Molecule, 29
2.2 Some Properties of Solids from the Band Picture, 38
2.3 Two-Atom Cells, 45
2.4 The Peierls Distortion, 48
2.5 Other One-Dimensional Systems, 52
2.6 Second-Order Peierls Distortions, 61
3. More details concerning energy bands, 66
3.1 The Brillouin Zone, 66
3.2 The Fermi Surface, 76
3.3 Symmetry considerations, 82
4. The electronic structure of solids, 89
4.1 Oxides with the NaCl, Ti02, and M0O2 Structures, 89
4.2 The Diamond and Zincblende Structures, 96
4.3 “Localization” of “Delocalized” Orbitals in Solids, 102
4.4 The Structure of NbO, 109
4.5 Chemical Bonding in Ionic Compounds, 115
4.6 The Transition Metals, 117
4.7 The Free- and Nearly-Free-Electron Models, 127
4.8, Compounds between Transition Metals and Main Group Elements, 132
4.9 The Nickel Arsenide and Related Structures, 134
4.10 Molecular Metals, 139
4.11 Division into Electronic Types, 142
Xll CONTENTS CONTENTS xiii
5. Metals and insulators, 144 8.6 The Coloring Problem, 291
8.7 Structural Stability and Band Gap, 298
5.1 The Importance of Structure and Composition, 144
5.2 The Structures of Calcium and Zinc, 156
5.3 Geometrical Instabilities, 157 References, 301
5.4 Importance of Electron-Electron Interactions, 163
Index, 311
5.5 Transition Metal and Rare Earth Oxides, 170
5.6 Effect of Doping, 176
5.7 Superconductivity in the Cgo Series, 179
5.8 High-Temperature Superconductors, 182
6. The structures of solids and Pauling’s rules, 191
6.1 General Description of Ion Packings, 191
6.2 The First Rule, 193
6.3 The Second Rule, 200
6.4 The Third Rule, 207
6.5 The Fifth Rule, 210
6.6 The Description of Solids in Terms of Pair Potentials, 211
6.7 More about the Orbital Description of Silicates, 216
7. The structures of some covalent solids, 219
7.1 Electron Counting, 219
7.2 Change of Structure with Electron Count, 222
7.3 Structures of Some AXz Solids, 225
7.4 Structures Derived from Simple Cubic or Rocksalt, 228
7.5 The Stability of the Rocksalt and Zincblende Structures, 234
7.6 The Structures of the Spinels, 239
7.7 Distortions of the Cadmium Halide Structure: Jahn-Teller
Considerations, 242 *
7.8 Distortions of the Cadmium Halide Structure: Trigonal Prismatic
Coordination, 251
7.9 Distortions of the Cadmium Halide Structure: t2g Block
Instabilities, 254
7.10 The Rutile versus Cadmium Halide versus Pyrite Structures, 264
7.11 Second-Order Structural Changes, 266
8. More about structures, 270
8.1 The Structures of the Elements in Terms of Moments, 270
8.2 The Structures of Some Main Group Intermetallic Compounds, 280
8.3 The Hume-Rothery Rules,, 285
8.4 Pseudopotential Theory, 287
8.5 The Structures of the First Row Elements, 290
Xll CONTENTS CONTENTS xiii
5. Metals and insulators, 144 8.6 The Coloring Problem, 291
8.7 Structural Stability and Band Gap, 298
5.1 The Importance of Structure and Composition, 144
5.2 The Structures of Calcium and Zinc, 156
5.3 Geometrical Instabilities, 157 References, 301
5.4 Importance of Electron-Electron Interactions, 163
Index, 311
5.5 Transition Metal and Rare Earth Oxides, 170
5.6 Effect of Doping, 176
5.7 Superconductivity in the Cgo Series, 179
5.8 High-Temperature Superconductors, 182
6. The structures of solids and Pauling’s rules, 191
6.1 General Description of Ion Packings, 191
6.2 The First Rule, 193
6.3 The Second Rule, 200
6.4 The Third Rule, 207
6.5 The Fifth Rule, 210
6.6 The Description of Solids in Terms of Pair Potentials, 211
6.7 More about the Orbital Description of Silicates, 216
7. The structures of some covalent solids, 219
7.1 Electron Counting, 219
7.2 Change of Structure with Electron Count, 222
7.3 Structures of Some AXz Solids, 225
7.4 Structures Derived from Simple Cubic or Rocksalt, 228
7.5 The Stability of the Rocksalt and Zincblende Structures, 234
7.6 The Structures of the Spinels, 239
7.7 Distortions of the Cadmium Halide Structure: Jahn-Teller
Considerations, 242 *
7.8 Distortions of the Cadmium Halide Structure: Trigonal Prismatic
Coordination, 251
7.9 Distortions of the Cadmium Halide Structure: t2g Block
Instabilities, 254
7.10 The Rutile versus Cadmium Halide versus Pyrite Structures, 264
7.11 Second-Order Structural Changes, 266
8. More about structures, 270
8.1 The Structures of the Elements in Terms of Moments, 270
8.2 The Structures of Some Main Group Intermetallic Compounds, 280
8.3 The Hume-Rothery Rules,, 285
8.4 Pseudopotential Theory, 287
8.5 The Structures of the First Row Elements, 290
Symbols and Abbreviations
gj, atomic energies of s, p, and d orbitals
Ha, diagonal entry in the Hamiltonian matrix, usually set equal to e,, gp, or
Hij, off-diagonal entry in the Hamiltonian matrix, often related to the overlap
integral S and to Ha and Hjj by the Wolfsberg-Helmholz relationship
a, Hiickel label for Hu
j?, Hiickel label for Hy
M", Schafii symbol for describing nets; each node of the nei is a part of n M-gons
E, electronic energy
Ej, total electronic energy
Ep, Fermi level
k, wave vector
kp, wave vector corresponding to Ep
p{E), electronic density of states
fF, bandwidth
r, coordination number
Pnt wth moment of electronic density of states
COOP, crystal orbital overlap population
MOOP, molecular orbital overlap population
DOS, electronic density of states
^ij, pair potential between atoms ij
□, vacancy
second-order energy parameter in the angular overlap model for X = cr,iz, or S
interactions
/a> fourth-order energy parameter in the angular overlap model for X = a, n, or d
interactions
wave functions for electronic state
i
Symbols and Abbreviations
gj, atomic energies of s, p, and d orbitals
Ha, diagonal entry in the Hamiltonian matrix, usually set equal to e,, gp, or
Hij, off-diagonal entry in the Hamiltonian matrix, often related to the overlap
integral S and to Ha and Hjj by the Wolfsberg-Helmholz relationship
a, Hiickel label for Hu
j?, Hiickel label for Hy
M", Schafii symbol for describing nets; each node of the nei is a part of n M-gons
E, electronic energy
Ej, total electronic energy
Ep, Fermi level
k, wave vector
kp, wave vector corresponding to Ep
p{E), electronic density of states
fF, bandwidth
r, coordination number
Pnt wth moment of electronic density of states
COOP, crystal orbital overlap population
MOOP, molecular orbital overlap population
DOS, electronic density of states
^ij, pair potential between atoms ij
□, vacancy
second-order energy parameter in the angular overlap model for X = cr,iz, or S
interactions
/a> fourth-order energy parameter in the angular overlap model for X = a, n, or d
interactions
wave functions for electronic state
i
CHEMICAL BONDING IN SOLIDS
1 □ Molecules
The discussion starts off in this book by looking at the electronic structure of
molecules of various types. It provides a very useful basis from which to move to the
solid State in that many of the theoretical tools needed for later chapters have their
origin here. In addition, many solid-state systems, including the Zintl phases and van
der Waals solids, are at the zeroth level of approximation “molecular” systems, too,
although diamond would be described as a “giant” molecule. Some of the examples
discussed in this chapter might appear to be rather idiosyncratic, but their utility
will become apparent in the next chapter when specific electronic concepts are
transferred from the finite to the infinite regime. The orbital model that is used (Ref.
5) has had wide success in studies of molecular problems and is ideally poised to be
able to provide a broadly applicable model for the solid state. Symmetry (Ref. 1(X),
166, 209, 330, 382) will play a major role in the development of molecular and
solid-state energy levels. This chapter begins with the simplest molecule of all, namely
H2, and progresses to larger systems.
1-1 The H2 molecule: molecular orbital approach
Within the molecular orbital framework (Refs. 5, 165, 209, 236, 259, 337) an
expression for the molecular orbitals of this diatomic molecule may be written as
f = a<l)i+ 602 (1.1)
where 0^ 2 are the two atomic hydrogen Is orbitals. The electron density function
associated with such an orbital is then
+ 2ab<f>i<f>2 (1.2)
By symmetry the electron density must be equal on both hydrogen atoms, a result
that implies a = +6. If S is the overlap integral between the two hydrogen Is orbitals.
then the bonding, if/f, (oTg) and antibonding, fa, (Cu ) orbitals may be written
(1.3)
Within the framework of Hiickel theory (Refs. 165, 183, 335, 393), if the Coulomb
integral is defined as a = ^ the interaction
1 □ Molecules
The discussion starts off in this book by looking at the electronic structure of
molecules of various types. It provides a very useful basis from which to move to the
solid State in that many of the theoretical tools needed for later chapters have their
origin here. In addition, many solid-state systems, including the Zintl phases and van
der Waals solids, are at the zeroth level of approximation “molecular” systems, too,
although diamond would be described as a “giant” molecule. Some of the examples
discussed in this chapter might appear to be rather idiosyncratic, but their utility
will become apparent in the next chapter when specific electronic concepts are
transferred from the finite to the infinite regime. The orbital model that is used (Ref.
5) has had wide success in studies of molecular problems and is ideally poised to be
able to provide a broadly applicable model for the solid state. Symmetry (Ref. 1(X),
166, 209, 330, 382) will play a major role in the development of molecular and
solid-state energy levels. This chapter begins with the simplest molecule of all, namely
H2, and progresses to larger systems.
1-1 The H2 molecule: molecular orbital approach
Within the molecular orbital framework (Refs. 5, 165, 209, 236, 259, 337) an
expression for the molecular orbitals of this diatomic molecule may be written as
f = a<l)i+ 602 (1.1)
where 0^ 2 are the two atomic hydrogen Is orbitals. The electron density function
associated with such an orbital is then
+ 2ab<f>i<f>2 (1.2)
By symmetry the electron density must be equal on both hydrogen atoms, a result
that implies a = +6. If S is the overlap integral between the two hydrogen Is orbitals.
then the bonding, if/f, (oTg) and antibonding, fa, (Cu ) orbitals may be written
(1.3)
Within the framework of Hiickel theory (Refs. 165, 183, 335, 393), if the Coulomb
integral is defined as a = ^ the interaction
4 CHEMICAL BONDING IN SOLIDS
MOLECULES 5
Figure 1.1
Figure 1.2 Energetic behavior of the configurations and with interatomic separation.
integral is <</>i where is some effective Hamiltonian for the problem,
then via This State of affairs is shown in Figure 1.1(b) and leads to an understanding of why
the Me2 molecule does not exist. Of the four electrons, two are stabilized on formation
Ea = {0L- PW - S) of the molecular orbitals, but two are destabilized and by a larger amount.
(1.4)
The overlap integral, and hence describing such orbital interactions generally
Et = (<x + + s)
in magnitude as the interatomic separation decreases. Thus the bonding
orbital goes down in energy and the antibonding orbital goes up. However, on
Ignoring S, then the simple Huckel result is
bringing the nuclei closer together a strong repulsion sets in. Indeed the equilibrium
interatomic separation is set by the balance of this repulsion and the stabilization
£b = a + jS
(1.5) afforded the two electrons in the bonding orbital. A minimum is shown in the lower
E„ = a-p curve of Figure 1.2, which represents the balance of these two effects. It may be
regarded as describing the energetics of the ground electronic configuration (^j,)^.
This well-known result is shown in Figure 1.1 and for H2 leads to the ground The upper curve with no minimum describes the energetics of the excited electronic
electronic configuration (^j,)^.
configuration where both the electronic term and the internuclear interaction
An alternative way to generate these energies without using symmetry per se is to
ar6 repulsive.
solve either the Huckel determinant — £| = 0 or the extended Huckel (Ref. 174)
There are two ways that these results are used in the simplest, one-electron, models
determinant \Hij — S,j£| = 0. For the former case the result is very simple of Qhemical bonding. The Huckel approach itself leaves a and ^ as parameters and
the total energy is expressed in terms of them. In the extended Huckel model the
11-.E H,2 parameter Ha = a,- is most usually associated with some measure of the atomic orbital
(1.6)
energy, g.. (A table of some useful values is given in Ref. 41.) The parameter = jS
^21 H22- £
may be estimated in two ways. It depends on the overlap between the two orbitals
or
concerned and thus usually increases with a decrease in internuclear separation. The
01 — E Wolfsberg-Helmholz approximation is generally used with p set proportional to
(1.7) cither the arithmetic or geometric means of and «2, that is, p = KS(oli 4- «2)/2 or
P a — £
The overlap integral S, a vital ingredient in the problem, is evaluated
numerically by locating Slater-type wave functions on each atomic center. For all
leading immediately to £ = a ± jS as before. Including overlap gives
t>ut the* simplest systems the determination of the energy levels and molecular wave
functions obtained by solution of the extended Huckel (Ref. 174) determinant
H,,-E f^l2 — ^12^
1^0’ ~ ^ijE\ = 0 needs to be done numerically.
(1.8)
H21 — S21E H22-E The simplest many-electron wave function that could be written for the arrange
ment with two electrons in the bonding orbital (ignoring antisymmetrization and
or
overlap in the normalization) for this ground electronic configuration is the one due
a — £ P-SE
^ Mulliken and Hund and written
0 (1.9)
=
§-SE a — £
'Tmh =
which gives
= K01 + + (I>2)i2)
£ = (a + P)!{\,+ S) and (a - P)/{\ - S) (1.10) = iC0l(l)0l(2) + <^2(1)02(2) + <Ai(1)<^2(2) + <^2(1)0i(2)] (1.11)
