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

Finite elementmodeling and

continuum methods

Mesoscalemodeling

Molecularmechanics

Quantum mechanics

Time (s)

Len

gth

(m)

10-15 1 101510-9

0

109

• 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 + EBondStretch + EAngleBend + ETorsion

Nonbonded energy terms

Bonded energy terms

€

ECoul =e2

4πεo

qiq jriji≠ j

∑

€

EVDW = DoRorij

⎛

⎝ ⎜ ⎜

⎞

⎠ ⎟ ⎟

12

− 2Rorij

⎛

⎝ ⎜ ⎜

⎞

⎠ ⎟ ⎟

6 ⎡

⎣

⎢ ⎢

⎤

⎦

⎥ ⎥i≠ j

∑

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

€

EBondStretch = k1(r − ro)2

€

EAngleBend = k2(θ −θo)2

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

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:

€

EMgO = kqMgqOrMgO

+ exp−rMgOρ

⎛

⎝ ⎜

⎞

⎠ ⎟−C

rMgO6

- 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 Partial charge

Mg: 2+ Mg: 1.2+O: 2- O: 1.2-MgO: 1.48Å MgO: 1.75Å

Experimental value = 2.10Å

Molecular mechanics methods have been widely applied in Earth Sciences, including:

Minimum energy structuresDefects Minor element incorporationElastic propertiesWater

However, the method is limited as it uses a highly simplifiedmodel of atoms and their interactions

Desirable to use more realistic models that more accuratelyrepresent how atoms interact

Quantum Chemistry Methods

- Widely used in chemistry and biomedical applications aswell 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 or

MM with shells

Quantum mechanics, electrons are included

Mg2+ Mg1,2+

s d p

Time independent Schrödinger eqn:

€

−h2

8π 2

1

mi

∇ 2 +eie jriji≠ j

∑i

∑ ⎛

⎝ ⎜ ⎜

⎞

⎠ ⎟ ⎟Ψ = EΨ

- Only has an exact solution for systems with one electron- Approximations needed for the many-electron systems of interest

Where E = Total energy of the system = wavefunctionh=Planck’s constantm = the mass2 = Laplacian operatore = charge on the particles at separation rij

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 electron3. Density functional methods (DFT)

- Method of choice4. 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:

€

E = T +U ρ(r)[ ] + Exc ρ (r)[ ]

and its derivatives (pressure, elastic constants etc.)

T=kineticU=electrostaticExc=exchange-correlation

Leads to a set of single-particle, Schrödinger-like, Kohn-Sham Eqns:

€

−∇2 +VKS[ ]Ψi = ε iΨi

Where i is the wave function of a single electron i is the corresponding eigenvalue and the effective potential is

€

VKS ρ(r)[ ] =2Zir − Ri

+2ρ (r')

r − r'dr'∫ +VXC ρ (r)[ ]

i=1

N

∑

nuclei 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)

Tsuchiya et al., 2004(EPSL, 244, 241-248)

Perovskite…

Stackhouse et al., 2005