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 '
