Melting of Iron at Earth’s core conditions from quantum Monte Carlo free energy calculations Department of Earth Sciences & Department of Physics and Astronomy,

Slides:



Advertisements
Similar presentations
Wave function approaches to non-adiabatic systems
Advertisements

A method of finding the critical point in finite density QCD
Well Defined and Accurate Semiclassical Surface Hopping Propagators and Wave Functions Michael F. Herman Department of Chemistry Tulane University New.
Molecular dynamics modeling of thermal and mechanical properties Alejandro Strachan School of Materials Engineering Purdue University
Quantum Theory of Solids
Introduction to PAW method
First Principle Electronic Structure Calculation Prof. Kim Jai Sam ( ) Lab. 공학 ( ) Students : Lee Geun Sik,
2012 Summer School on Computational Materials Science Quantum Monte Carlo: Theory and Fundamentals July 23–-27, 2012 University of Illinois at Urbana–Champaign.
Quantum Monte Carlo Simulation of Vibrational Frequency Shifts in Pure and Doped Solid para-Hydrogen Lecheng Wang, Robert J. Le Roy and Pierre- Nicholas.
Wide Range Equation of State of Water Smirnova M.S., Dremov V.V., Sapozhnikov A.T. Russian Federation Nuclear Centre – Institute of Technical Physics P.O.
1 Quantum Monte Carlo Methods Jian-Sheng Wang Dept of Computational Science, National University of Singapore.
Melting Points of Aluminum at Geological Pressures Linzey Bachmeier Divesh Bhatt Ilja Siepmann Chemistry Department University of Minnesota.
Lattice regularized diffusion Monte Carlo
Quantum Simulations of Materials Under Extreme Conditions David M. Ceperley Richard M. Martin Simone Chiesa Ed Bukhman William D. Mattson* Xinlu Cheng.
GEOMAGNETISM: a dynamo at the centre of the Earth Lecture 1 How the dynamo is powered Lecture 2 How the dynamo works Lecture 3 Interpreting the observations.
A QMC Study of the Homogeneous Electron Gas Graham Spink and Richard Needs, TCM, University of Cambridge.
International Workshop on Energy Conversion and Information Processing Devices, Nice, France 1/16 Monte Carlo phonon transport at nanoscales Karl Joulain,
Quantum Monte Carlo for Electronic Structure Paul Kent Group Meeting - Friday 6th June 2003.
FUNDAMENTALS The quantum-mechanical many-electron problem and Density Functional Theory Emilio Artacho Department of Earth Sciences University of Cambridge.
Water graphene binding energy curve from diffusion Monte Carlo Department of Earth Sciences & Department of Physics and Astronomy, Thomas Young
Network for Computational Nanotechnology (NCN) Purdue, Norfolk State, Northwestern, MIT, Molecular Foundry, UC Berkeley, Univ. of Illinois, UTEP DFT Calculations.
Computational Solid State Physics 計算物性学特論 第9回 9. Transport properties I: Diffusive transport.
Monte Carlo Methods: Basics
Introduction to (Statistical) Thermodynamics
Lectures Introduction to computational modelling and statistics1 Potential models2 Density Functional.
Practical quantum Monte Carlo calculations: QWalk
Materials Process Design and Control Laboratory ON THE DEVELOPMENT OF WEIGHTED MANY- BODY EXPANSIONS USING AB-INITIO CALCULATIONS FOR PREDICTING STABLE.
Norm-conserving pseudopotentials and basis sets in electronic structure calculations Javier Junquera Universidad de Cantabria.
The Nuts and Bolts of First-Principles Simulation Durham, 6th-13th December : DFT Plane Wave Pseudopotential versus Other Approaches CASTEP Developers’
Order(N) methods in QMC Dario Alfè 2007 Summer School on Computational Materials Science Quantum Monte Carlo: From Minerals and Materials.
Computational Solid State Physics 計算物性学特論 第8回 8. Many-body effect II: Quantum Monte Carlo method.
R. Martin - Pseudopotentials1 African School on Electronic Structure Methods and Applications Lecture by Richard M. Martin Department of Physics and Materials.
Molecular Dynamics Study of Solidification in the Aluminum-Silicon System Supervisor: Dr. Jeffrey J Hoyt Peyman Saidi Winter 2013.
Kinetic Monte Carlo Triangular lattice. Diffusion Thermodynamic factor Self Diffusion Coefficient.
Phase diagram calculation based on cluster expansion and Monte Carlo methods Wei LI 05/07/2007.
8. Selected Applications. Applications of Monte Carlo Method Structural and thermodynamic properties of matter [gas, liquid, solid, polymers, (bio)-macro-
First-Principles study of Thermal and Elastic Properties of Al 2 O 3 Bin Xu and Jianjun Dong, Physics Department, Auburn University, Auburn, AL
Burkhard Militzer, Carnegie Institution of Washington: “Path Integral Monte Carlo”, 2007 Burkhard Militzer Geophysical Laboratory Carnegie Institution.
Linear and non-Linear Dielectric Response of Periodic Systems from Quantum Monte Carlo Calculations. Paolo Umari CNR CNR-INFM DEMOCRITOS
Dario Bressanini Critical stability V (Erice) Universita’ dell’Insubria, Como, Italy Boundary-condition-determined.
Time-dependent Schrodinger Equation Numerical solution of the time-independent equation is straightforward constant energy solutions do not require us.
NCN nanoHUB.org Wagner The basics of quantum Monte Carlo Lucas K. Wagner Computational Nanosciences Group University of California, Berkeley In collaboration.
Approach Toward Linear Time QMC a,c David Ceperley, a Bryan Clark, b,d Eric de Sturler, a,c Jeongnim Kim, b,e Chris Siefert University of Illinois at Urbana-Champaign,
Extending the domain of quantum mechanical simulations with HPCx: Melting Dario Alfè University College London.
Fundamentals of DFT R. Wentzcovitch U of Minnesota VLab Tutorial Hohemberg-Kohn and Kohn-Sham theorems Self-consistency cycle Extensions of DFT.
Molecular Modelling - Lecture 2 Techniques for Conformational Sampling Uses CHARMM force field Written in C++
Path Integral Quantum Monte Carlo Consider a harmonic oscillator potential a classical particle moves back and forth periodically in such a potential x(t)=
Quantum Monte Carlo on geomaterials Dario Alfè 2007 Summer School on Computational Materials Science Quantum Monte Carlo: From Minerals.
Electric field which acts on core C due to the valence electrons and the other cores. Where is a cutoff function for the electric field inside the core.
The Study Of Statistical Thermodynamics In Melting of Atomic Cluster Pooja Shrestha.
An Introduction to Monte Carlo Methods in Statistical Physics Kristen A. Fichthorn The Pennsylvania State University University Park, PA
Calculating Potential Energy Curves With Quantum Monte Carlo Andrew D Powell, Richard Dawes Department of Chemistry, Missouri University of Science and.
Monte Carlo methods (II) Simulating different ensembles
A. Ambrosetti, F. Pederiva and E. Lipparini
Materials Process Design and Control Laboratory ON THE DEVELOPMENT OF WEIGHTED MANY- BODY EXPANSIONS USING AB-INITIO CALCULATIONS FOR PREDICTING STABLE.
Advanced methods of molecular dynamics 1.Monte Carlo methods 2.Free energy calculations 3.Ab initio molecular dynamics 4.Quantum molecular dynamics 5.Trajectory.
Composition of the Earth’s core from ab-initio calculation of chemical potentials Department of Earth Sciences & Department of Physics and Astronomy, Thomas.
Comp. Mat. Science School Electrons in Materials Density Functional Theory Richard M. Martin Electron density in La 2 CuO 4 - difference from sum.
1 B3-B1 phase transition in GaAs: A Quantum Monte Carlo Study C N M Ouma 1, 2, M Z Mapelu 1, G. O. Amolo 1, N W Makau 1, and R Maezono 3, 1 Computational.
Ignacio Martin-Bragado1, Ignacio Dopico1 and Pedro Castrillo2
Introduction to Quantum Monte Carlo Methods 2! Claudio Attaccalite.
Solid State Physics Lecture 11
Concept test 15.1 Suppose at time
Production of an S(α,β) Covariance Matrix with a Monte Carlo-Generated
Quantum Model of the Atom
Concept test 15.1 Suppose at time
Prof. Sanjay. V. Khare Department of Physics and Astronomy,
Nodal surfaces in Quantum Monte Carlo: a user’s guide
Quantum Chemistry / Quantum Mechanics
Phase Changes Notes.
Presentation transcript:

Melting of Iron at Earth’s core conditions from quantum Monte Carlo free energy calculations Department of Earth Sciences & Department of Physics and Astronomy, Materials Simulation Laboratory & London Centre for Nanotechnology University College London Dario ALFÈ

Birch (1952) - “The Core is iron alloyed with a small fraction of lighter elements” Nature of light element inferred from: Cosmochemistry Meteoritics Equations of state Core composition

Temperature of the Earth’s core Exploit solid-liquid boundary Exploit core is mainly Fe Melting temperature of Fe

Thermodynamic melting

The Helmholtz free energy Solids: Low T

Phonons of Fe

The Helmholtz free energy Solids: Liquids: Low T High T

Thermodynamic integration

Example: anharmonic free energy of solid Fe at ~350 GPa

Improving the efficiency of TI F is independent on the choice of U ref, but for efficiency choose U ref such that: is minimum. For solid iron at Earth’s core conditions a good U ref is:

Improving the efficiency of TI (2) At high temperature we find c 1 = 0.2, c 2 = 0.8

F independent on choice of U ref, but for efficiency choose U ref such that is minimum. For liquid iron a good U ref is: Free energy for liquid Fe:

Liquid Fe

Size tests

Hugoniot of Fe

Alfè, Price,Gillan, Nature, 401, 462 (1999); Phys. Rev. B, 64, (2001); Phys. Rev. B, 65, (2002); J. Chem. Phys., 116, 6170 (2002 ) The melting curve of Fe

NVE ensemble: for fixed V, if E is between solid and liquid values, simulation will give coexisting solid and liquid Melting: coexistence of phases

Alfè, Price,Gillan, Nature, 401, 462 (1999); Phys. Rev. B, 64, (2001); Phys. Rev. B, 65, (2002); J. Chem. Phys., 116, 6170 (2002 ) The melting curve of Fe Free energy approach and Coexistence give same result (as they should !)

Thermodynamic integration, a perturbative approach:

Only need to run simulations with one potential (the reference potential for example).

Melting of Fe from QMC: Free energy corrections from DFT to QMC:

Thermodynamic integration, a perturbative approach:

Extracting the ground state: substitute  = it Beyond DFT, Diffusion Monte Carlo:

Imaginary time Schroedinger equation with V = 0: Diffusion equation in a 3-N dimensional space: Brownian particles (walkers)  distribution function Potential energy V --> source or sink of walkers Problems: Fixed nodes approximation: Pseudopotentials (locality approximation) DMC is ~ times more expensive than DFT

QMC on Fe, technical details CASINO code: R. J. Needs, M. D. Towler, N. D. Drummond, P. Lopez-Rios, CASINO user manual, version 2.0, University of Cambridge, DFT pseudopotential, 3s 2 3p 6 4s 1 3d 7 (16 electrons in valence) Single particle orbitals from PWSCF (plane waves), 150 Ry PW cutoff. Then expanded in B-splines. (D. Alfè and M. J. Gillan, Phys. Rev. B, 70, (R), (2004))

Blips Storing the coefficients: avc(x,y,z,ib) [old] avc(ib,x,y,z) [new] New is faster on large systems, but slower on small systems (cutoff ~ 250 electrons)

Solid (h.c.p.) Fe, finite size Ester Sola

Solid Fe, equation of state at 300 K

QMC correction to the DFT Fe melting curve

Conclusions Melting temperature of Fe at 330 GPa = K Melting point depression due to impurities ~ 800 K Probable temperature of the Earth’s core is ~ 6000 K