3.320: Lecture 11 (Mar 17 2005) MOLECULAR DYNAMICS(PLAY IT AGAIN SAM)Mar 17 2005 3.320 Atomistic Modeling of Materials -- Gerbrand Ceder and Nicola Marzari3.320: Lecture 13 (Mar 17 2005) MOLECULAR DYNAMICSMOLECULAR DYNAMICS(PLAY IT AGAIN SAM)(PLAY IT AGAIN SAM)Mar 17 2005 3.320 Atomistic Modeling of Materials -- Gerbrand Ceder and Nicola MarzariA Particle Is a Particle Is a Particle)(trr)(22rFdtrdmrr=)(tvrMar 17 2005 3.320 Atomistic Modeling of Materials -- Gerbrand Ceder and Nicola MarzariN coupled equations),,(122NiiirrFdtrdmrLrrr=• The force depends on positions only (not velocities)• The total energy of the system is conserved (microcanonical evolution)Mar 17 2005 3.320 Atomistic Modeling of Materials -- Gerbrand Ceder and Nicola MarzariConservation of the total energyMar 17 2005 3.320 Atomistic Modeling of Materials -- Gerbrand Ceder and Nicola MarzariOperational Definition• We follow the evolution of a system composed of many classical particles• Each particle interacts simultaneously with every other particle (usually – but can also have ‘hard spheres’ contact interaction), and can experience an additional external potential•It’s a many-body problem – albeit with a simpler informational content than in the case of the electrons (why ?)Mar 17 2005 3.320 Atomistic Modeling of Materials -- Gerbrand Ceder and Nicola MarzariSome history• MANIAC operational at Los Alamos in 1952• Metropolis, Rosenbluth, Rosenbluth, Teller, and Teller (1953): Metropolis Monte Carlo method• Alder and Wainwright (Livermore 1956): dynamics of hard spheres• Vineyard (Brookhaven 1959-60): radiation damage in copper• Rahman (Argonne 1964): liquid argon• Car and Parrinello (Sissa 1985): ab-initio MDMar 17 2005 3.320 Atomistic Modeling of Materials -- Gerbrand Ceder and Nicola MarzariPhase Space• If we have N particles, we need to specify positions and velocities for all of them (6Nvariables) to uniquely identify the dynamical system• One point in a 6N dimensional space (the phase space) represents our dynamical systemMar 17 2005 3.320 Atomistic Modeling of Materials -- Gerbrand Ceder and Nicola MarzariPhase Space EvolutionMar 17 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 17 2005 3.320 Atomistic Modeling of Materials -- Gerbrand Ceder and Nicola MarzariLimitations• Time scales• Length scales (PBC help a lot)• Accuracy of forces • Classical nucleiMar 17 2005 3.320 Atomistic Modeling of Materials -- Gerbrand Ceder and Nicola MarzariThermodynamical averages• Under hypothesis of ergodicity, we can assume that the temporal average along a trajectory is equal to the ensemble-average over the phase space∫∫−−=pdrdEpdrdEAArrrr)exp()exp(ββ∫=TdttATA0)(1Mar 17 2005 3.320 Atomistic Modeling of Materials -- Gerbrand Ceder and Nicola MarzariAre you ergodic ?Mar 17 2005 3.320 Atomistic Modeling of Materials -- Gerbrand Ceder and Nicola MarzariThermodynamical averages• Let’s start with the simple case: straightforward integration of the equations of motion (i.e. microcanonical: N, V and E are constant)• The trajectory in the phase space spans states belonging to the microcanonical ensemble• A long trajectory generates an excellent sample of microstatesMar 17 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 17 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 17 2005 3.320 Atomistic Modeling of Materials -- Gerbrand Ceder and Nicola MarzariMaxwell-Boltzmann distribution⎟⎟⎠⎞⎜⎜⎝⎛−⎟⎟⎠⎞⎜⎜⎝⎛∝TkmvvTkmvnBB2exp2)(2223πmTkvmTkvBrmsB3,2==Oxygen at room T: 105cm/sSPEEDNUMBER OF MOLECULES0oC1000oC2000oCFigure by MIT OCW.Mar 17 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 17 2005 3.320 Atomistic Modeling of Materials -- Gerbrand Ceder and Nicola MarzariIntegrators• (Simple) VerletMar 17 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