Interatomic Forces in Condensed Matter

Interatomic Forces in Condensed Matter

by Mike Finnis
Pub. Date:
Oxford University Press
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Interatomic Forces in Condensed Matter

The series is intended to present graduate textbooks that will bring students to the frontiers of current research in materials modelling as it is being practiced in both academic and industrial science and engineering. Finnis (Queen's U. Belfast) demonstrates how to derive various approximate schemes of total energy and force calculation, to help clarify the several models of inter-atomic forces that abound in the literature. Annotation ©2004 Book News, Inc., Portland, OR

Product Details

ISBN-13: 9780198509776
Publisher: Oxford University Press
Publication date: 01/01/2004
Series: Oxford Series on Materials Modelling Series , #1
Pages: 304
Product dimensions: 9.50(w) x 6.70(h) x 0.50(d)

Table of Contents

IThe Framework
1Essential Quantum Mechanics3
1.1The Time-independent Schrodinger Equation3
1.1.1The Born-Oppenheimer Approximation5
1.1.2Atomic Units5
1.1.3Parts of the Total Energy6
1.2Wave-mechanics of Non-interacting Fermions7
1.2.1Mean Field Theory7
1.2.2Spacial and Spin Parts in Single-particle Wavefunctions8
1.2.3Determinant Wavefunctions9
1.3Basis Vectors and Representations11
1.3.1Bras and Kets11
1.3.2Expansion Coefficients15
1.3.3Eigenstates of Momentum and Matrix Elements of Operators16
1.3.4Matrix Elements of the Coulomb Potential17
1.3.5Momentum Space and k-space18
1.4Periodic Boundary Conditions20
1.4.1Fourier Transformation of a Wavefunction20
1.4.2Cells and Supercells21
1.4.3The Reciprocal Lattice23
1.4.4The First Brillouin Zone24
1.4.5Bloch's Theorem24
1.4.6Expansion in Plane Waves25
1.5Local Orbitals and Spherical Harmonics26
1.5.1Spherical Harmonics27
1.5.2Local Atomic-like Orbitals30
1.5.3Expansion of a Wavefunction in a Non-orthogonal Basis32
1.5.4Expansion of an Operator32
1.5.5Spherical Harmonics and Multipoles33
1.6The Variational Principle and the Schrodinger Equation37
1.6.1The Single-particle Schrodinger Equation37
1.6.2The Hartree-Fock Approximation39
1.7The Density Matrix and the Charge Density42
1.7.1The Fermi Distribution42
1.7.2Matrix Representations of the Density Operator43
1.7.3Mulliken Charges, Bond Charges and Bond Orders45
1.8The Density of States48
1.8.2The Green Function50
1.9.1Cancellation of Electrostatic Energies53
1.9.2The Kinetic Energy53
1.9.3Exchange and Correlation Energy55
1.10The Matrix Eigenvalue Problem55
1.10.1Local Orbitals or Plane-waves?56
1.10.2Approaches to Solving the Matrix Eigenvalue Problem57
1.11.1Basic Ideas61
1.11.2The Ashcroft Empty-core Pseudopotential62
2Essential Density Functional Theory64
2.1What is a Functional?64
2.2Functional Derivatives65
2.3The Thomas-Fermi Model68
2.3.1Description of the Thomas-Fermi Functional68
2.3.2The Euler-Lagrange Equation68
2.3.3The Local Density Approximation69
2.4The Kohn-Sham Equations70
2.4.1The Existence of a Density Functional70
2.4.2The Hohenberg-Kohn-Sham Functional71
2.4.3The Kohn and Sham Trick72
2.4.4Self-consistent Solution of the Kohn-Sham Equations75
2.4.5Approximating the Exchange and Correlation: The LDA77
3Exploiting the Variational Principle79
3.1The Hellmann-Feynman Theorem79
3.1.1Statement and Proof79
3.1.2The van der Waals Interaction81
3.1.3Another Example: The Image Potential82
3.2Perturbation Theory with the Density84
3.2.1Switching on an External Potential84
3.2.2First-order Perturbation Theory85
3.2.3Second-order Perturbation Theory85
3.3The Second-order HKS Functional87
3.3.1Derivation of E[superscript (2)]87
3.3.2The Error in E[superscript (2)]90
3.4The Harris-Foulkes Functional and its Generalizations92
3.4.1The First-order Functional and the Harris-Foulkes Functional92
3.4.2Generalization of the Harris-Foulkes Functional93
4Linear response theory96
4.1Definition of the Response Function x[subscript e] (r, r')96
4.2Relationship to HKS Density Functional97
4.2.1Matrix and Vector Algebra Notation for Integrals99
4.3The Non-interacting Response Function100
4.4The Dielectric Function101
4.5The Error in the Harris-Foulkes Functional102
4.6Linear Response and the Green Function105
4.7Linear Response in Jellium107
4.8Electron-Electron Interactions in the Jellium Response111
4.9The Long Wavelength Limit of Response Functions in Jellium114
4.9.1The Thomas-Fermi Response Function114
4.9.2The Compressibility Sum Rule115
4.10Linear Response in a Perfect Crystal118
4.11Non-local Potentials120
4.11.1General Ideas120
4.11.2Non-local Perturbations in Jellium122
4.11.3Second-order Energy in Jellium125
IIModelling Atoms Within Solids
5Testing an interatomic force model129
5.1The Cohesive Energy and Crystal Structures130
5.2The Structural Energy Difference Theorem131
5.2.1Examples of the SEDT133
5.3Elastic Constants133
5.3.1Cubic Crystals136
5.3.2Some Subtleties: Pair Potentials and Cauchy Pressure141
5.4.1Lattice Dynamics in the Harmonic Approximation147
5.4.2Calculating Force Constants150
5.5Point Defects152
5.5.1Definition of Vacancy Formation Energies154
6Pairwise Potentials in Simple Metals158
6.2The Energy in Terms of Pseudopotentials161
6.2.1Structure Factors161
6.2.2The q = 0 Problem162
6.2.3The Madelung Energy [Delta]E[subscript ZZ]166
6.2.4The Total Energy and the Energy-wavenumber Characteristic169
6.3Periodic Boundary Conditions171
6.4The Effective Pairwise Interaction172
6.5Example: The Ashcroft Empty-core Potential175
6.6Asymptotic Forms of the Pair Potential178
6.7The Pseudoatom Picture180
6.7.1The Energy of a Pseudoatom and the Local Density184
7Tight Binding187
7.1.1Predicting the Past188
7.1.2Ab Initio Tight Binding189
7.1.3One-, Two- and Three-centre integrals190
7.2Non-self-consistent Tight Binding191
7.3Slater-Koster Parameters194
7.3.1The Symmetries of Two-centre Integrals194
7.3.2Distance Dependence of the Bond Integrals197
7.4The Repulsive Energy198
7.5The Tight-Binding Bond Model200
7.5.1Basic Ideas200
7.5.2Development of the Model202
7.5.3A Closer Look at the Energy in a Non-Orthogonal Basis204
7.6Hellmann-Feynman Forces207
7.6.1The Pitfall of Incomplete Bases207
7.6.2The Force on an Ion208
7.7Self-consistent Tight-Binding211
7.7.1The Self-consistent Charge Transfer Model211
7.7.2Local Charge Neutrality214
7.7.3Including Atomic Polarization215
7.8Moments of the Density of States218
7.9The Recursion Method220
7.9.1Block Recursion229
7.10Second-moment Models230
7.10.1General Ideas230
7.10.2The TBBM with a Gaussian Density of States231
7.10.3An Effective Pairwise Potential236
7.11Fourth-moment Models237
7.12Bond-order Potentials242
7.12.1A Multi-atom Expansion of the Bond Energy242
7.12.2Example: Analytic Bond Orders in a p-Bonded Trimer247
8Hybrid Schemes253
8.1Generalized Pseudopotential Theory253
8.2Effective Medium Theory257
9Ionic models263
9.2The Rigid Ion Model Derived264
9.3Beyond the Rigid Ion Model270
9.3.1The Basic Second-order Model270
9.3.2Deformable Ions270
9.3.3Variable Charge Transfer Models273

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