Download Atomistic and Ab Initio Modelling in the Geosciences

Why?
Three options for studying the Earth’s interior
1. Direct observations e.g. seismics, electrical
conductivity
2. High pressure experiments, e.g. multi-anvil
press, diamond anvil cell
3. Molecular modeling, e.g. atomistic methods,
ab initio approaches
Length (m)
109
Finite element
modeling and
continuum methods
0
Mesoscale
modeling
Molecular
mechanics
10-9
Quantum
mechanics
10-15
1
Time (s)
1015
• 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
Techniques
1. Molecular mechanics
(a) Static - geometry optimization, defect energies,
elastic properties…
(b) Molecular dynamics - transport properties, fluids,
glasses
2. Quantum mechanics
(a) Static - as 1a above, but also band gaps, spin states
(b) Quantum dynamics - combination of molecular
dynamics and quantum mechanics
Molecular Mechanics
• Based on classical mechanics
– Historically, the most widely used because it is less
computationally intensive
– Main disadvantage - highly simplified representation
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:
ETotal  ECoul  EVDW  E BondStretch  E AngleBend  ETorsion
Nonbonded energy
terms
Bonded energy terms
Electrostatic term is from the classical description
ECoul
qiq j
e2


4o i j rij
VDW - short-range, due to atomic interactions

EVDW
 12  6 
Ro
Ro 
  Do

2






 

r
r
 ij  
i j
 ij 
- 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
Bonded energy terms:
E BondStretch  k1(r  ro ) 2
- Allows for vibration about an equilibrium distance ro

E AngleBend  k2 (  o )2
- 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
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
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
MgO
Buckingham potential:
E MgO  k
qMgqO
rMgO
rMgO  C
 exp
 6
   rMgO
- 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)
Full charge
Mg: 2+
O: 2MgO: 1.48Å
Partial charge
Mg: 1.2+
O: 1.2MgO: 1.75Å
Experimental
value = 2.10Å
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
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
Basic molecular mechanics
Mg2+
or
MM with shells
Quantum mechanics, electrons are included
d
p
s
Mg1,2+
Time independent
Schrödinger eqn:
 h 2
eie j 
1 2

 8 2  m    r 
  E
i
ij 

i
i j

Where E = Total energy of the system
 = wavefunction
h=Planck’s 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
Four classes of Quantum Chemistry Methods
1. 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.
2. Ab initio correlated methods
- Extension of HF correcting for local distortion of an orbital
in the vicinity of another electron
3. Density functional methods (DFT)
- Method of choice
4. 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
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:
T=kinetic
E  T  U (r)  E xc (r) U=electrostatic
Exc=exchange-correlation
and its derivatives (pressure, elastic constants etc.)

Leads to a set of single-particle, Schrödinger-like, Kohn-Sham Eqns:
2

  VKS i  ii
Where i is the wave function of a single electron
i is the corresponding eigenvalue
and the effective potential is
N
VKS (r)  
i1
2Zi

r  Ri
nuclei


2(r')
dr'  VXC (r)
r  r'
electrons
exchange
correlation
- The Kohn-Sham equations are exact.
- However, limited understanding of exchange-correlation
energies means only approximate solutions are currently
possible
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.
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 isn’t always obvious, for
example the 3p electrons in iron must be treated as
valence electrons as they deform substantially at pressures
corresponding to the Earth’s core
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
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
- 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
Forsterite
DFT calculation using the pseudopotential approximation and GGA
(Jochym et al., 2004. Comp. Mat. Sci., 29, 414-418)
Perovskite…
Tsuchiya et al., 2004
(EPSL, 244, 241-248)
Stackhouse et al., 2005