3.320: Lecture 12 (Mar 29 2005) MOLECULAR DYNAMICSNewton’s coupled equationsThe Computational ExperimentInitializationIntegrateIntegratorsGear Predictor-CorrectorLyapunov InstabilitiesTime StepThermodynamic equilibrium ?Accumulate averagesThree Main GoalsCorrelation FunctionsCold catalysis of methane ?Diffusion of AdatomsBibliographyMar 29 2005 3.320 Atomistic Modeling of Materials -- Gerbrand Ceder and Nicola Marzari3.320: Lecture 14 (Mar 29 2005) MOLECULAR DYNAMICSMOLECULAR DYNAMICSYou cannot step twice in the same riverHeraclitus (Diels 91)Mar 29 2005 3.320 Atomistic Modeling of Materials -- Gerbrand Ceder and Nicola MarzariNewton’s coupled equations),,(122NiiirrFdtrdmrLrrr=0exp( )1()exp( )TAEdrdpAAAtdtTEdrdpββ−=⇔=−∫∫∫rrrrMar 29 2005 3.320 Atomistic Modeling of Materials -- Gerbrand Ceder and Nicola MarzariThe Computational Experiment• Initialize: select positions and velocities•Integrate:compute all forces, and determine new positions• Equilibrate: let the system reach equilibrium (i.e. lose memory of initial conditions)• Average: accumulate quantities of interestMar 29 2005 3.320 Atomistic Modeling of Materials -- Gerbrand Ceder and Nicola MarzariInitialization• Second order differential equations: boundary conditions require initial positions and initial velocities• Initial positions: reasonably compatible with the structure to be studied. Avoid overlap, short distances.• Velocities: zero, or small. Then thermalizeincreasing the temperatureMar 29 2005 3.320 Atomistic Modeling of Materials -- Gerbrand Ceder and Nicola MarzariIntegrate• Use an integrator… (Verlet, leapfrog Verlet, velocity Verlet, Gear predictor-corrector)• Robust, long-term conservation of the constant of motion, time-reversible, constant volume in phase space• Choose thermodynamic ensemble (microcanonicalNVE, or canonical NVT using a thermostat, isobaric-isothermic NPT with a barostat…)• Stochastic (Langevin), constrained (velocity rescaling), extended system (Nose-Hoover)Mar 29 2005 3.320 Atomistic Modeling of Materials -- Gerbrand Ceder and Nicola MarzariIntegrators• (Simple) VerletMar 29 2005 3.320 Atomistic Modeling of Materials -- Gerbrand Ceder and Nicola MarzariGear Predictor-CorrectorMar 29 2005 3.320 Atomistic Modeling of Materials -- Gerbrand Ceder and Nicola MarzariLyapunov InstabilitiesMar 29 2005 3.320 Atomistic Modeling of Materials -- Gerbrand Ceder and Nicola MarzariTime StepMar 29 2005 3.320 Atomistic Modeling of Materials -- Gerbrand Ceder and Nicola MarzariThermodynamic equilibrium ?• Drop longer and longer initial segments of your dynamical trajectory, when accumulating averagesMar 29 2005 3.320 Atomistic Modeling of Materials -- Gerbrand Ceder and Nicola MarzariAccumulate averages• Potential, kinetic, total energy (conserved)• Temperature (K=3/2 N kBT)•Pressure• Caloric curve E(T): latent heat of fusion• Mean square displacements (diffusion)• Radial (pair) distribution functionMar 29 2005 3.320 Atomistic Modeling of Materials -- Gerbrand Ceder and Nicola MarzariThree Main Goals• Ensemble averages (thermodynamics)• Real-time evolution (chemistry)• Ground-state of complex structures (optimization)• Structure of low-symmetry systems: liquids, amorphous solids, defects, surfaces• Ab-initio: bond-breaking and charge transfer; structure of complex, non trivial systems (e.g. biomolecules)Mar 29 2005 3.320 Atomistic Modeling of Materials -- Gerbrand Ceder and Nicola MarzariCorrelation FunctionsMar 29 2005 3.320 Atomistic Modeling of Materials -- Gerbrand Ceder and Nicola MarzariSource: Di Tolla, Francesco. "Interplay of Melting, Wetting, Overheating, Faceting, on Metal Surfaces: Theory and Simulation." Ph.D. Thesis, International School For Advanced Studies (SISSA), Trieste Italy, 1995. Courtesy of SISSA. Used with permission.Mar 29 2005 3.320 Atomistic Modeling of Materials -- Gerbrand Ceder and Nicola MarzariCold catalysis of methane ?Mar 29 2005 3.320 Atomistic Modeling of Materials -- Gerbrand Ceder and Nicola MarzariMar 29 2005 3.320 Atomistic Modeling of Materials -- Gerbrand Ceder and Nicola MarzariMar 29 2005 3.320 Atomistic Modeling of Materials -- Gerbrand Ceder and Nicola MarzariMar 29 2005 3.320 Atomistic Modeling of Materials -- Gerbrand Ceder and Nicola MarzariSource: Di Tolla, Francesco. "Interplay of Melting, Wetting, Overheating, Faceting, on Metal Surfaces: Theory and Simulation." Ph.D. Thesis, International School For Advanced Studies (SISSA), Trieste Italy, 1995. Courtesy of SISSA. Used with permission.Mar 29 2005 3.320 Atomistic Modeling of Materials -- Gerbrand Ceder and Nicola MarzariDiffusion of AdatomsMar 29 2005 3.320 Atomistic Modeling of Materials -- Gerbrand Ceder and Nicola MarzariMar 29 2005 3.320 Atomistic Modeling of Materials -- Gerbrand Ceder and Nicola MarzariBibliography• Allen and Tildesley, Computer Simulations of Liquids (Oxford)• Frenkel and Smit, Understanding Molecular Simulations (Academic)• Ercolessi, A Molecular Dynamics Primer