I Density Functional Theory of Electronic Structure: A Short
Course for Mineralogists and Geophysicists
John P. Perdew and Adrienn Ruzsinszky
INTRODUCTION AND SUMMARY ........................................ 1
KOHN-SHAM THEORY: THE ORBITALS MAKE IT ACCURATE ................. 2
Motivation and history of Kohn-Sham theory ................... 2
Summary of Kohn-Sham theory .................................. 3
Uniform electron gas ......................................... 4
What Kohn-Sham theory promises ............................... 5
Solving the Kohn-Sham equations .............................. 6
EXCHANGE-CORRELATION ENERGY: NATURE'S GLUE ...................... 6
SEMILOCAL AND NONLOCAL APPROXIMATIONS: WHEN CAN WE STAY CLOSE
TO HOME? ........................................................ 8
Local spin density approximation ............................. 8
Gradient expansion ........................................... 8
Generalized gradient approximation ........................... 8
GGA's for solids ............................................. 9
Ladder of density functional approximations: Meta-GGA and
fully nonlocal ............................................... 9
Semilocal approximations: What, why, and when? .............. 10
PRESSURE AND TEMPERATURE IN DENSITY FUNCTIONAL THEORY:
PRESSING DOWN AND HEATING UP ................................... 12
Pressure .................................................... 12
Temperature ................................................. 14
Pressure and temperature together ........................... 15
ACKNOWLEDGMENTS ................................................ 15
REFERENCES ..................................................... 15
2 The Minnesota Density Functionals and their Applications
to Problems in Mineralogy and Geochemistry
Yan Zhao, Donald G. Truhlar
INTRODUCTION ................................................... 19
DENSITY FUNCTIONALS ............................................ 20
UNIVERSITY OF MINNESOTA FUNCTIONALS ............................ 22
VALIDATIONS AND APPLICATIONS ................................... 25
Water and aqueous chemistry ................................. 25
Atmospheric chemistry ....................................... 26
Metal oxides ................................................ 27
Silicates and siliceous minerals ............................ 27
Zeolites .................................................... 28
Mineral nanoparticles ....................................... 30
CONCLUDING REMARKS ............................................. 31
ACKNOWLEDGMENTS ................................................ 31
REFERENCES ..................................................... 31
3 Density-Functional Perturbation Theory for Quasi-Harmonic
Calculations
Stefano Baroni, Paolo Giannozzi, Eyvaz Isaev
INTRODUCTION ................................................... 39
THERMAL PROPERTIES AND THE QUASI-HARMONIC APPROXIMATION ........ 40
AB INITIO PHONONS .............................................. 42
Lattice dynamics from electronic-structure theory ........... 42
Density-functional perturbation theory ...................... 44
Interatomic force constants and phonon band interpolation ... 45
COMPUTER CODES ................................................. 47
Quantum ESPRESSO ............................................ 47
The QHA code ................................................ 47
APPLICATIONS ................................................... 48
Semiconductors and insulators ............................... 48
Simple metals ............................................... 49
Hydrides .................................................... 49
Intermetallics .............................................. 50
Surfaces .................................................... 50
Earth materials ............................................. 51
CONCLUSIONS .................................................... 52
ACKNOWLEDGMENTS ................................................ 53
REFERENCES ..................................................... 53
4 Thermodynamic Properties and Phase Relations in Mantle
Minerals Investigated by First Principles Quasiharmonic
Theory
Renata M. Wentzcovitch, Yonggang G. Yu, Zhongqing Wu
INTRODUCTION ................................................... 59
THE QUASIHARMONIC APPROXIMATION (QHA) .......................... 62
THERMODYNAMIC PROPERTIES OF MANTLE PHASES ...................... 63
MgO ......................................................... 64
MgSiO3-perovskite ........................................... 68
MgSiO3 post-perovskite ...................................... 71
SiO2 stishovite ............................................. 71
Mg2SiO4 forsterite (α-phase) ................................ 72
Mg2SiO4 wadsleyite (β-phase) ................................ 72
Mg2SiO4 ringwoodite (γ-phase) ............................... 73
Low- and high-pressure MgSiO3 chnoenstatite ................. 73
MgSiO3 ilmenite ............................................. 73
MgSiO3 majorite ............................................. 74
CaO ......................................................... 75
CaSiO3 perovskite ........................................... 75
PHASE RELATIONS IN SILICATES AND OXIDES ........................ 75
Phase transitions in Mg2SiO4 ................................ 75
Mantle density discontinuities caused by phase transitions
in Mg2SiO4 .................................................. 78
Low-pressure to high-pressure MgSiO3 chnoenstatite
transition .................................................. 82
Post-perovskite transition in MgSiO3 ........................ 82
ANHARMONIC FREE ENERGY ......................................... 84
SUMMARY ........................................................ 89
APPENDIX ....................................................... 90
ACKNOWLEDGMENT ................................................. 91
REFERENCES ..................................................... 91
5 First Principles Quasiharmonic Thermoelasticity of Mantle
Minerals
Renata M. Wentzcovitch, Zhongqing Wu, Pierre Carrier
INTRODUCTION ................................................... 99
THEORETICAL BACKGROUND ........................................ 100
ELASTICITY OF LOWER MANTLE PHASES ............................. 104
MgO ........................................................ 105
MgSiO3-perovskite .......................................... 109
MgSiO3-post-perovskite ..................................... 113
SELF-CONSISTENT QHA ........................................... 118
SUMMARY ....................................................... 122
ACKNOWLEDGMENTS ............................................... 124
REFERENCES .................................................... 124
6 An Overview of Quantum Monte Carlo Methods
David M. Ceperley
MOTIVATION .................................................... 129
RANDOM WALK AND MARKOV CHAINS ................................. 129
VARIATIONAL MONTE CARLO ....................................... 130
DIFFUSION MONTE CARLO ......................................... 131
PATH INTEGRAL MC .............................................. 132
COUPLED ELECTRON ION MC ....................................... 133
AUXILLARY FIELD MC ............................................ 134
PROSPECTIVES .................................................. 134
REFERENCES .................................................... 134
7 Quantum Monte Carlo Studies of Transition Metal Oxides
Lubos Mitas, Jindřich Kolorenč
ABSTRACT ...................................................... 137
INTRODUCTION .................................................. 137
MnO CALCULATIONS .............................................. 140
FeO CALCULATIONS .............................................. 141
CONCLUSION .................................................... 144
ACKNOWLEDGMENTS ............................................... 144
REFERENCES .................................................... 144
8 Accurate and Efficient Calculations on Strongly
Correlated Minerals with the LDA+U Method: Review and
Perspectives
Matteo Cococcioni
ABSTRACT ...................................................... 147
INTRODUCTION .................................................. 147
THE LDA+U FUNCTIONAL .......................................... 148
ENERGY DERIVATIVES ............................................ 151
The Hubbard forces ......................................... 151
The Hubbard stresses ....................................... 152
CALCULATION OF U .............................................. 155
Li-ion batteries cathodes .................................. 155
Water-solvated transition-metal ions ....................... 156
Chemical reactions on transition-metal complexes ........... 157
TRANSITION-METAL-CONTAINING MINERALS .......................... 158
EXTENDED LDA+U+V APPROACH ..................................... 161
CONCLUSIONS AND OUTLOOK ....................................... 163
ACKNOWLEDGMENTS ............................................... 164
REFERENCES .................................................... 164
9 Spin-State Crossover of Iron in Lower-Mantle Minerals:
Results of DFT+t/ Investigations
Han Hsu, Koichiro Umemoto, Zhongqing Wu, Renata M.
Wentzcovitch
INTRODUCTION .................................................. 169
THE SPIN-PAIRING PHENOMENON ................................... 170
THEORETICAL APPROACH .......................................... 171
Spin- and volume-dependent Hubbard U ....................... 171
Thermodynamic treatment of the mixed spin state ............ 172
The vibrational free energy: the Vibrational Virtual
Crystal Model (VVCM) ....................................... 173
Computational details ...................................... 174
SPIN-STATE CROSSOVER IN FERROPERICLASE ........................ 175
Static LDA+U calculation ................................... 176
VVCM for ferropericlase .................................... 179
Thermodynamic properties of ferropericlase ................. 181
SPIN-STATE CROSSOVER IN FERROSILICATE PEROVSKITE .............. 186
Quadrupole splitting of ferrous iron in perovskite ......... 187
Dependence of the spin crossover pressure on the site and
orbital degeneracies ....................................... 189
Dependence of transition pressure on the concentration
and distribution of iron ................................... 191
SPIN-STATE CROSSOVER IN POST-PEROVSKITE ....................... 194
SUMMARY ....................................................... 195
ACKNOWLEDGMENT ................................................ 195
REFERENCES .................................................... 195
Simulating Diffusion
Michael W. Ammann, John P. Brodholt, David P. Dobson
INTRODUCTION .................................................. 201
BASIC METHODS ................................................. 201
Ab initio vs. empirical potentials ......................... 201
Predicting diffusion coefficients in fluids and melts ...... 202
Predicting diffusion coefficients in crystalline phases .... 202
Defect calculations: Mott-Littleton, super-cells and
embedded clusters .......................................... 204
The climbing image nudged elastic band method .............. 206
LDAvs. GGA ................................................. 208
RESULTS ON MANTLE PHASES ...................................... 208
Results on MgO ............................................. 208
Results on MgSiO3 perovskite ............................... 213
Other investigated silicon diffusion mechanisms ............ 218
IMPLICATIONS FOR THE EARTH'S LOWER MANTLE ..................... 219
Viscosity of the lower mantle .............................. 219
CONCLUSIONS ................................................... 220
ACKNOWLEDGMENTS ............................................... 220
REFERENCES .................................................... 221
10 Modeling Dislocations and Plasticity of Deep Earth Materials
Philippe Carrez, Patrick Cordier
CONTEXT ....................................................... 225
DISLOCATIONS AND PLASTIC DEFORMATION .......................... 225
MODELING DISLOCATIONS ......................................... 227
The Volterra dislocation ................................... 227
"Direct" atomistic calculations ............................ 227
The Peierls-Nabarro (PN) model of dislocations:
fundamentals and recent developments ....................... 229
Dislocation mobility at finite temperature ................. 233
SELECTED EXAMPLES ............................................. 235
SrTiO3 perovskite .......................................... 235
Magnesium oxide (MgO) ...................................... 239
CONCLUDING REMARKS ............................................ 247
REFERENCES .................................................... 248
11 Theoretical Methods for Calculating the Lattice Thermal
Conductivity of Minerals
Stephen Stackhouse, Lars Stixrude
ABSTRACT ...................................................... 253
INTRODUCTION .................................................. 253
FUNDAMENTAL PRINCIPLES ........................................ 254
THEORETICAL METHODS ........................................... 255
Green-Kubo method .......................................... 255
Non-equilibrium molecular dynamics ......................... 257
Transient non-equilibrium molecular dynamics ............... 261
Combined Quasiharmonic Lattice Dynamics and Molecular
Dynamics Method ............................................ 263
Anharmonic lattice dynamics method ......................... 264
DISCUSSION .................................................... 265
THE LATTICE THERMAL CONDUCTIVITY OF PERICLASE ................. 266
CONCLUSION .................................................... 267
ACKNOWLEDGMENTS ............................................... 267
REFERENCES .................................................... 268
13 Evolutionary Crystal Structure Prediction as a Method for
the Discovery of Minerals and Materials
Artem R. Oganov, Yanming Ma, Andriy O. Lyakhov, Mario
Valle, Carlo Gatti
ABSTRACT ...................................................... 271
INTRODUCTION .................................................. 272
EVOLUTIONARY ALGORITHM USPEX .................................. 272
OF THE ALGORITHM .............................................. 277
SOME APPLICATIONS OF THE METHOD ............................... 282
CaCO3 polymorphs ........................................... 282
Polymeric phase of CO2 ..................................... 284
Semiconducting and metallic phases of solid oxygen:
unusual molecular associations ............................. 285
Reactivity of noble gases: are Xe-C compounds possible at
high pressure? ............................................. 288
Boron: novel phase with a partially ionic character ........ 288
Sodium: a metal that goes transparent under pressure ....... 291
CONCLUSIONS ................................................... 293
ACKNOWLEDGMENTS ............................................... 295
REFERENCES .................................................... 296
14 Multi-Mbar Phase Transitions in Minerals
Koichiro Umemoto, Renata M. Wentzcovitch
INTRODUCTION .................................................. 299
COMPUTATIONAL BACKGROUND ...................................... 300
DISSOCIATION OF MgSiO3 PPV .................................... 300
LOW-PRESSURE ANALOG OF MgSiO3 ................................. 302
PREDICTION OF POST-PPV CRYSTALLINE PHASES ..................... 303
POST-POST-PEROVSKITE TRANSITION IN Al2O3 ...................... 305
RARE-EARTH SESQUISULFIDES ..................................... 307
M2O3 SESQUIOXIDES ............................................. 308
SUMMARY ....................................................... 310
ACKNOWLEDGMENTS ............................................... 311
REFERENCES .................................................... 311
15 Computer Simulations on Phase Transitions in Ice
Koichiro Umemoto
INTRODUCTION .................................................. 315
BACKGROUND OF COMPUTATIONAL METHOD ............................ 315
EXCHANGE-CORRELATION FUNCTIONAL ............................... 317
H2O monomer and dimer ...................................... 317
Solid ice .................................................. 318
CRYSTALLINE PHASES ............................................ 319
Phase diagram .............................................. 319
Order-disorder transition .................................. 320
Ice X: Hydrogen-bond symmetrization ........................ 322
Beyond ice X ............................................... 323
Isostructural transition in ice VIII at low pressure? ...... 323
AMORPHOUS ..................................................... 324
Amorphization of ice Ih .................................... 324
Amorphization of ice VIII .................................. 326
Prediction of amorphization of ice XI ...................... 328
Amorphization between low ↔ high density transformation .... 328
ACKNOWLEDGMENTS ............................................... 331
REFERENCES .................................................... 331
16 Iron at Earth's Core Conditions from First Principles
Calculations
Dario Alfe
ABSTRACT ...................................................... 337
INTRODUCTION .................................................. 337
STATIC PROPERTIES ............................................. 339
Crystal structures and phase transitions ................... 339
Elastic constants .......................................... 340
FINITE TEMPERATURE ............................................ 341
The Helmholtz free energy: low temperature and the
quasi-harmonic approximation ............................... 343
The Helmholtz free energy: high temperature and
thermodynamic integration .................................. 345
Melting .................................................... 348
REFERENCES .................................................... 352
17 First-Principles Molecular Dynamics Simulations of
Silicate Melts: Structural and Dynamical Properties
Bijaya B. Karki
ABSTRACT ...................................................... 355
INTRODUCTION .................................................. 355
COMPUTATIONAL CHALLENGES ...................................... 358
METHODOLOGY ................................................... 359
First-principles molecular dynamics simulations ............ 359
Input configuration and simulation schedule ................ 361
Derivation of physical properties .......................... 362
Convergence tests .......................................... 364
Visualization .............................................. 366
SIMULATION RESULTS AND DISCUSSION ............................. 371
Equations of state and derived properties .................. 371
Radial distribution functions .............................. 372
Coordination environments .................................. 374
Medium range order ......................................... 380
Diffusion coefficients and viscosities ..................... 381
CONCLUSION AND FUTURE DIRECTIONS .............................. 386
ACKNOWLEDGMENTS ............................................... 386
REFERENCES
18 Lattice Dynamics from Force-Fields as a Technique for
Mineral Physics
Julian D. Gale, Kate Wright
INTRODUCTION .................................................. 391
METHODOLOGY ................................................... 392
Interatomic potentials ..................................... 392
Lattice dynamics versus molecular dynamics ................. 394
Finding stationary points .................................. 396
Mineral properties ......................................... 397
Derivation of force-fields ................................. 398
Evolution of a force-field: Silica potentials through the
ages ....................................................... 401
APPLICATIONS .................................................. 405
CONCLUSION .................................................... 408
ACKNOWLEDGMENTS ............................................... 409
REFERENCES .................................................... 409
19 An Efficient Cluster Expansion Method for Binary Solid
Solutions: Application to the Halite-Silvite, NaCl-KCl,
System
Victor Vinograd, Björn Winkler
ABSTRACT ...................................................... 413
INTRODUCTION .................................................. 414
DECOMPOSITION OF THE EXCESS ENTHALPY INTO PAIRWISE
INTERACTIONS .................................................. 417
THE DOUBLE DEFECT METHOD (DDM) ................................ 419
THE SYSTEM NaCl-KCl .......................................... 421
Quantum mechanical calculations ............................ 421
Calculation of the excess energies based on the empirical
force-field model .......................................... 421
The enthalpy of mixing in the limit of the complete
disorder ................................................... 422
Monte Carlo simulations of the effect of the temperature
on the enthalpy of mixing .................................. 424
The phase diagram .......................................... 425
The excess vibrational free energy ......................... 425
DISCUSSION AND CONCLUSIONS .................................... 430
ACKNOWLEDGMENTS ............................................... 434
REFERENCES
20 Large Scale Simulations
Mark S. Ghiorso, Frank J. Spera
INTRODUCTION .................................................. 437
THERMODYNAMIC ANALYSIS ........................................ 440
Polyamorphism .............................................. 447
Hugoniot ................................................... 450
STRUCTURAL FEATURES ........................................... 451
Microscopic to macroscopic .................................... 454
TRANSPORT PROPERTIES .......................................... 458
Viscosity .................................................. 458
Self-diffusivity ........................................... 459
Eyring relation ............................................ 461
THE FUTURE - LSS IN THE CONTEXT OF FPMD ....................... 462
ACKNOWLEDGMENTS ............................................... 462
REFERENCES .................................................... 462
21 Thermodynamics of the Earth's Mantle
Lars Stixrude, Carolina Lithgow-Bertelloni
INTRODUCTION .................................................. 465
OUR APPROACH AND PREVIOUS WORK ................................ 466
Fundamental thermodynamic relations ........................ 467
Euler form ................................................. 467
Legendre transformations ................................... 467
Anisotropic generalization ................................. 468
THERMODYNAMIC THEORY .......................................... 468
CONSTRAINING AND TESTING THE MODEL ............................ 471
Elastic constants .......................................... 472
Griineisen parameter and q ................................. 472
Temperature dependence of the shear modulus ................ 472
SCALING ....................................................... 473
APPLICATIONS .................................................. 475
Origin of the low velocity zone ............................ 475
Origins of lateral heterogeneity ........................... 476
Influence of lithologic heterogeneity ...................... 479
CONCLUSIONS AND OUTLOOK ....................................... 481
REFERENCES .................................................... 481
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