Why?

1. Why? Three options for studying the Earth’s 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

2. 109 Finite element modeling and continuum methods Length (m) 0 Mesoscale modeling Molecular mechanics Quantum mechanics 10-9 10-15 1 1015 Time (s)

3. 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

4. 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

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

6. 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

7. 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

8. 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

9. 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

10. 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

11. 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)

13. Full charge Partial charge Mg: 2+ Mg: 1.2+ O: 2- O: 1.2- MgO: 1.48Å MgO: 1.75Å Experimental value = 2.10Å

15. 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

16. 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

17. Basic molecular mechanics or MM with shells Quantum mechanics, electrons are included Mg2+ Mg1,2+ s d p

19. Time independent Schrödinger eqn: 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

20. 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

21. 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 U=electrostatic Exc=exchange-correlation and its derivatives (pressure, elastic constants etc.) Leads to a set of single-particle, Schrödinger-like, Kohn-Sham Eqns:

22. 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

23. 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.

24. 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

25. 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

28. 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

29. - 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

30. Forsterite DFT calculation using the pseudopotential approximation and GGA (Jochym et al., 2004. Comp. Mat. Sci., 29, 414-418)

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

32. Stackhouse et al., 2005