Why?

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Why?

• Three options for studying the Earths interior
• Direct observations e.g. seismics, electrical
  conductivity
• High pressure experiments, e.g. multi-anvil
  press, diamond anvil cell
• Molecular modeling, e.g. atomistic methods, ab
  initio approaches

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109
Finite element modeling and continuum methods
Length (m)
0
Mesoscale modeling
Molecular mechanics
Quantum mechanics
10-9
10-15
1
1015
Time (s)
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• Macroscopic properties are strongly dependant on
  atomic-level properties
• Molecular modeling provides a way to
• interpret field/experimental observations and
  discriminate between different competing models
  to explain macroscopic observations
• Predict properties at conditions unobtainable by
  experiment

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Techniques

• Molecular mechanics
• (a) Static - geometry optimization, defect
  energies,
• elastic properties
• (b) Molecular dynamics - transport properties,
  fluids,
• glasses
• Quantum mechanics
• (a) Static - as 1a above, but also band gaps,
  spin states
• (b) Quantum dynamics - combination of molecular
• dynamics and quantum mechanics

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Molecular Mechanics

• Based on classical mechanics
• Historically, the most widely used because it is
  less computationally intensive
• Main disadvantage - highly simplified
  representation

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Potential Energy

• An accurate description of the potential energy
  of the system is the most important requirement
  of any molecular model
• Total potential energy is given by

Nonbonded energy terms
Bonded energy terms
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Electrostatic term is from the classical
description
VDW - short-range, due to atomic interactions

• Repulsion (1/r)12 due to electronic overlap as
  atoms approach
• Attraction (1/r)6 due to fluctuations in
  electron density
• Shell model including electronic polarization -
  permits elastic,
• dielectric, diffusion and model to be derived

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Bonded energy terms
- Allows for vibration about an equilibrium
distance ro

• Important in silicates, controls angles
• in Si tetrahedral or octahedral sites
• Other geometry related terms can be included as
  needed,
• e.g. out-of-plane stretch terms for systems
  with planar
• equilibrium structures

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Choice of Potentials and Validation

• Atomistic approaches require parameters
  describing
• the interactions between each pair of atoms,
  e.g. Mg-O,
• Si-O, plus any bonded terms required by the
  system geometry
• Widely available in the literature from studies
  fitting
• simple potentials to experimental or quantum
  mechanical
• results
• Validation is a major issue, e.g.
• potentials are not always developed for the
  particular
• structure they are being applied to
• need to select potentials that adequately
  describe the
• ionic or covalent type bonding
• - pressure and temperature

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Energy or Geometry Minimization
- Convenient method (in both molecular and
quantum mechanics) for obtaining a stable
configuration for a molecule or periodic system

• Initially the energy of an initial configuration
  is calculated
• Then atoms (and cell parameters for periodic
  systems) are
• adjusted using the potential energy derivatives
  to obtain a lower
• energy structure
• This is repeated until defined energy tolerances
  between
• successive steps are achieved
• Multiple initial configurations or more advanced
  techniques
• are needed for complex systems to ensure the
  global energy
• minimum is found, not a local minimum

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MgO
Buckingham potential

• Short range terms positive and rapidly increase
  at short
• distances
• Coulombic term negative due to the opposite
  charges
• Summation of the terms gives the total energy
  and the
• energy minimum gives the optimum configuration
• Potentials from Lewis and Catlow, 1986 (J. Phys.
  C, 18,
• 1149-1161)

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Full charge
Partial charge
Mg 2 Mg 1.2 O 2-
O 1.2- MgO 1.48Å MgO 1.75Å
Experimental value 2.10Å
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Molecular mechanics methods have been widely
applied in Earth Sciences, including Minimum
energy structures Defects Minor element
incorporation Elastic properties Water However,
the method is limited as it uses a highly
simplified model of atoms and their interactions
Desirable to use more realistic models that
more accurately represent how atoms interact
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Quantum Chemistry Methods

• - Widely used in chemistry and biomedical
  applications as
• well as physics and geophysics
• More realistic representation - no
• longer restricted to the classical
• ball and spring model
• - Based on a quantum mechanical
• description of atoms, where
• electrons become very important

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Basic molecular mechanics
or MM with shells Quantum mechanics,
electrons are included
Mg2
Mg1,2
s
d
p
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Time independent Schrödinger eqn
Where E Total energy of the system ?
wavefunction hPlancks constant m the
mass ?2 Laplacian operator e charge on the
particles at separation rij
- Only has an exact solution for systems with one
electron - Approximations needed for the
many-electron systems of interest
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Four classes of Quantum Chemistry Methods

• Ab initio Hartree-Fock (HF)
• - Electrons are treated individually assuming the
  distribution of
• other electrons is frozen and treating their
  average distribution
• as part of the potential. Iterative process
  used to determined the
• steady state.
• Ab initio correlated methods
• - Extension of HF correcting for local
  distortion of an orbital
• in the vicinity of another electron
• Density functional methods (DFT)
• - Method of choice
• Semi-empirical methods
• - Involve empirical input to obtain approx.
  solutions of the
• Schrödinger Eqn. Less computationally
  intensive than 1-3,
• but success of DFT means this approach is
  less common
• these days

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Density Functional Theory

• In principle an exact method of dealing with the
  many-electron
• problem
• Based on the proof that the ground-state
  properties of
• a material are a unique function of the charge
  density ?(r)
• - Including the total energy

Tkinetic Uelectrostatic Excexchange-correlation
and its derivatives (pressure, elastic constants
etc.)
Leads to a set of single-particle,
Schrödinger-like, Kohn-Sham Eqns
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Where ?i is the wave function of a single
electron ?i is the corresponding
eigenvalue and the effective
potential is
exchange correlation
nuclei
electrons
- The Kohn-Sham equations are exact. - However,
limited understanding of exchange-correlation
energies means only approximate solutions are
currently possible
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Approximations in DFT
1. Exchange-correlation potential
Known exactly for only simple systems Common
approximations a. Local Density Approximation
(LDA) - assumes a uniform electron gas. Quite
successful in many applications, but shows some
failures significant in geophysics. For
example, it fails to predict the correct
ground state of iron. b. Generalized-Gradient
Appoximation (GGA) - Utilizes both the
electron density and its gradient. As good as LDA
and sometimes better. This correctly predicts
the ground state of iron.
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2. Frozen-Core Approximation

• In general only the valence electrons
  participate in bonding
• Within the frozen-core approximation the charge
  density of
• the core electrons is just that of the free
  atom
• Solve for only the valence electrons
• Choice of electrons to include isnt always
  obvious, for
• example the 3p electrons in iron must be
  treated as
• valence electrons as they deform substantially
  at pressures
• corresponding to the Earths core

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• 3. Pseudopotential Approximation
• Potential is chosen in such a way that the
  valence wave
• function in the free atom is the same as the
  all-electron
• solution beyond some cutoff, but nodeless
  within this
• radius
• Advantages
• spatial variations are much less rapid than for
  the bare
• Coulomb potential of the nucleus
• need solve only for the peudo-wave function of
  the valence
• electrons
• Construction is based on all-electron results but
  is nonunique
• Demonstrating transferability is important

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Advantages of Quantum Chemistry Approaches

• Realistic model (mostly) of atoms and their
  interactions
• Use a few approximations, but close to first
  principles models
• Electronic properties such as spin states
  accessible for study
• (potentially important in the lower mantle)

• However, some downsides
• Computationally intensive
• Questions regarding the applicability of the
  approximations
• to high pressure and temperature systems
• Scale issues
• - Lower mantle is 2000km thick.
• - A large molecular mechanics model of
  perovskite
• uses 360 atoms 30 Angstroms (1Å
  1x10-10m)
• - A large quantum mechanical model - 100
  atoms

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- Experiments suggested Al increases the amount
of Fe3 in perovskite - Molecular modeling
was carried out to investigate how Al and/or
Fe3 is incorporated, e.g. FeMg AlSi, 2FeMg
VMg
1. From molecular mechanics (Richmond and
Brodholt, 1998) Throughout lower mantle AlMg
AlSi
2. Then from quantum mechanics (Brodholt,
2000) Top of the lower mantle AlSi VO Higher
pressures AlMg AlSi
3. Large-scale quantum mechanics (Yamamoto et
al., 2003) Throughout the lower mantle AlMg
AlSi
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Forsterite
DFT calculation using the pseudopotential
approximation and GGA
(Jochym et al., 2004. Comp. Mat. Sci., 29,
414-418)
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Perovskite
Tsuchiya et al., 2004 (EPSL, 244, 241-248)
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Stackhouse et al., 2005