EPJ Web of Conferences 14, 02002 (2011) DOI: 10.1051/epjconf/20111402002 © Owned by the authors, published by EDP Sciences, 2011

“ Fundamentals of Thermodynamic Modelling of Materials ” November 15-19, 2010 INSTN – CEA Saclay, France

PROFESSOR & TOPIC

Julian GALE Curtin University, Australia

Potential Energy Surfaces [02002]

Organized by Bo SUNDMAN [email protected] Constantin MEIS [email protected]

This is an Open Access article distributed under the terms of the Creative Commons Attribution-Noncommercial License 3.0, which permits unrestricted use, distribution, and reproduction in any noncommercial medium, provided the original work is properly cited. Article available at http://www.epj-conferences.org or http://dx.doi.org/10.1051/epjconf/20111402002

Potential Energy Surfaces

Julian Gale Department of Chemistry Curtin University

NATIONAL COMPUTATIONAL INFRASTRUCTURE

Cricos Providerr Code Code 00301J 0

Scale

02002-p.2

Interatomic Potentials (Force Fields) N

N

N

N

N

N

1 1 U = ∑U i + ∑ ∑U ij + ∑ ∑ ∑U ijk + ... 2 i=1 j=1 6 i=1 j=1 k=1 i=1 • Choose expressions for energy based on physics • Fit parameters to experiment theory

U12−6 LJ

σ 12 σ 6 = 4ε − r r

U12−6 LJ

Ionic Materials & Minerals • Electrostatics

ij UCoulomb =

– Formal (or near formal) charges – Ewald/Parry sum etc

core

02002-p.3

4 πε 0 rij shell

• Polarisability • Short-range repulsion • Dispersion Buckingham potential:

qi q j e 2

Formal charges – why do they work? • Born effective charges: q

born i

∂µα = = qicδαβ − ∑ DcsDss−1q s ∂β

• Example: α-Quartz - Sanders et al model

Interatomic Potentials: Molecular Mechanics

Applicable to: • Organics and biomolecules • Water • Molecular anions (SO42-,CO32-)

02002-p.4

Energy

Probability Distribution

Energy Minimisation

T=0

T>0

Conformation space

Approach for Solids / Ordered Systems • Optimise each well defined minimum • Compute properties from second derivatives about equilibrium structure • Phase transformations – energy difference between minima • Defect properties • Kinetics - Activation barriers / Transition State Theory

02002-p.5

What properties can be calculated? • • • • • • • •

Phase transitions Structural properties Elastic constants Bulk moduli Dielectric constants Piezoelectric constants Refractive indices Reflectivity

• • • • • • • •

Phonon spectra Wave velocities Thermal expansion Thermal conductivity Neutron scattering Activation energies Wave velocities Surface energies & morphology

Silica Polymorphs

Henson et al, Chem. Mater., 6, 1647 (1994)

02002-p.6

Energy

Probability Distribution

Complex systems: Multiple minima

T=0 T>0

Conformation space

Molecular Dynamics

Energy

Newton’s eqns F = ma

Force Conformation space

Cf. Monte Carlo in next talk 02002-p.7

Initialise Coordinates (Optimise)

Initialise Velocities

Molecular dynamics

Equilibration

• Ensemble • Time step

Compute Forces

Evolve: Positions Velocities Accelerations

– Order of 1 fs (10-15 s)

• Thermostat • Barostat

Rescale velocities

Production

Compute Forces

• Run length:

Evolve: Positions Velocities Accelerations

– Force field: < 100 ns – QM: < 20 ps

Accumulate averages and trajectory

End

Simulation at Finite Temperature • Lattice dynamics (free energy minimisation) Includes zero point energy Includes vibrational quantisation Samples phonons across Brillouin zone Difficult to include anharmonicity

• Molecular dynamics Codes widely available Includes anharmonicity at high temperature Incorrect below Debye temperature Large supercells Statistical noise/ensemble parameter issues

02002-p.8

Lattice Dynamics vs Molecular Dynamics

MgSiO3 A. Oganov, J.D. Brodholt & G.D. Price, Phys. Earth Planet. Int., 122, 277 (2000)

Negative Thermal Expansion

Calculated = - 4.6 x 10-6 K Experiment = - 4.2 x 10-6 K 02002-p.9

Exploring the free energy landscape: Metadynamics Laio & Parrinello, PNAS, 99, 12562 (2002), Martonak et al. PRL, 90, 75503 (2003) See also: Local elevation method, Huber et al, J. Comput.-Aided Mol. Des., 8, 695 (1994)

Vmeta ( x,t ) = V ( x ) + VHD ( x,t )

λ ( x ( t )) − λ ( x ( t ' )) 2 VHD ( x,t ) = ∑ h exp − 2w 2 t '

“ Fundamentals of Thermodynamic Modelling of Materials ” November 15-19, 2010 INSTN – CEA Saclay, France

PROFESSOR & TOPIC

Julian GALE Curtin University, Australia

Potential Energy Surfaces [02002]

Organized by Bo SUNDMAN [email protected] Constantin MEIS [email protected]

This is an Open Access article distributed under the terms of the Creative Commons Attribution-Noncommercial License 3.0, which permits unrestricted use, distribution, and reproduction in any noncommercial medium, provided the original work is properly cited. Article available at http://www.epj-conferences.org or http://dx.doi.org/10.1051/epjconf/20111402002

Potential Energy Surfaces

Julian Gale Department of Chemistry Curtin University

NATIONAL COMPUTATIONAL INFRASTRUCTURE

Cricos Providerr Code Code 00301J 0

Scale

02002-p.2

Interatomic Potentials (Force Fields) N

N

N

N

N

N

1 1 U = ∑U i + ∑ ∑U ij + ∑ ∑ ∑U ijk + ... 2 i=1 j=1 6 i=1 j=1 k=1 i=1 • Choose expressions for energy based on physics • Fit parameters to experiment theory

U12−6 LJ

σ 12 σ 6 = 4ε − r r

U12−6 LJ

Ionic Materials & Minerals • Electrostatics

ij UCoulomb =

– Formal (or near formal) charges – Ewald/Parry sum etc

core

02002-p.3

4 πε 0 rij shell

• Polarisability • Short-range repulsion • Dispersion Buckingham potential:

qi q j e 2

Formal charges – why do they work? • Born effective charges: q

born i

∂µα = = qicδαβ − ∑ DcsDss−1q s ∂β

• Example: α-Quartz - Sanders et al model

Interatomic Potentials: Molecular Mechanics

Applicable to: • Organics and biomolecules • Water • Molecular anions (SO42-,CO32-)

02002-p.4

Energy

Probability Distribution

Energy Minimisation

T=0

T>0

Conformation space

Approach for Solids / Ordered Systems • Optimise each well defined minimum • Compute properties from second derivatives about equilibrium structure • Phase transformations – energy difference between minima • Defect properties • Kinetics - Activation barriers / Transition State Theory

02002-p.5

What properties can be calculated? • • • • • • • •

Phase transitions Structural properties Elastic constants Bulk moduli Dielectric constants Piezoelectric constants Refractive indices Reflectivity

• • • • • • • •

Phonon spectra Wave velocities Thermal expansion Thermal conductivity Neutron scattering Activation energies Wave velocities Surface energies & morphology

Silica Polymorphs

Henson et al, Chem. Mater., 6, 1647 (1994)

02002-p.6

Energy

Probability Distribution

Complex systems: Multiple minima

T=0 T>0

Conformation space

Molecular Dynamics

Energy

Newton’s eqns F = ma

Force Conformation space

Cf. Monte Carlo in next talk 02002-p.7

Initialise Coordinates (Optimise)

Initialise Velocities

Molecular dynamics

Equilibration

• Ensemble • Time step

Compute Forces

Evolve: Positions Velocities Accelerations

– Order of 1 fs (10-15 s)

• Thermostat • Barostat

Rescale velocities

Production

Compute Forces

• Run length:

Evolve: Positions Velocities Accelerations

– Force field: < 100 ns – QM: < 20 ps

Accumulate averages and trajectory

End

Simulation at Finite Temperature • Lattice dynamics (free energy minimisation) Includes zero point energy Includes vibrational quantisation Samples phonons across Brillouin zone Difficult to include anharmonicity

• Molecular dynamics Codes widely available Includes anharmonicity at high temperature Incorrect below Debye temperature Large supercells Statistical noise/ensemble parameter issues

02002-p.8

Lattice Dynamics vs Molecular Dynamics

MgSiO3 A. Oganov, J.D. Brodholt & G.D. Price, Phys. Earth Planet. Int., 122, 277 (2000)

Negative Thermal Expansion

Calculated = - 4.6 x 10-6 K Experiment = - 4.2 x 10-6 K 02002-p.9

Exploring the free energy landscape: Metadynamics Laio & Parrinello, PNAS, 99, 12562 (2002), Martonak et al. PRL, 90, 75503 (2003) See also: Local elevation method, Huber et al, J. Comput.-Aided Mol. Des., 8, 695 (1994)

Vmeta ( x,t ) = V ( x ) + VHD ( x,t )

λ ( x ( t )) − λ ( x ( t ' )) 2 VHD ( x,t ) = ∑ h exp − 2w 2 t '