MD Thermodynamics Lecture 1 3/6/18 1
Molecular dynamics The force depends on positions only (not velocities) Total energy is conserved (micro canonical evolution) Newton s equations of motion (second order diff. eq.) for all particles are integrated to predict time evolution of the system i N N i i i r r r V r r F dt r d m ),, ( ),, ( 1 1 ) ( ), ( ) ( ), ( t t t t t t i i i i v r v r
More accurate integration: Verlet algorithm Expand Taylor series to higher order: Adding the two expressions gives Odd power terms vanish and leading error is only 4 th order 3
MD workflow Requires atomic position and velocities at the beginning Step through discrete time by integrating Newton s equations Repeated calculation of atomic forces at each step 4
Maxwell Boltzmann statistics The temperature and the distribution of atomic velocities in a system are related through the Maxwell Boltzmann equation n( v) m k 3 mv v exp BT kbt v rms 3kBT m H O at room T: 500 m/s Initial velocities are generated from the Gaussian distribution of v x, v y, and v z using a random number generator Total momentum should be 0 5
Limitations of accuracy Classical nuclei Energy and force models Time scales Length scales 6
Classical nuclei Maxwell Boltzmann statistics, no zero point vibrational motion Specific heat is affected by statistics (even HO at 300K) Also, light atoms at low temperature can tunnel classical quantum specific heat of a solid Einstein classical 7
Ab initio molecular dynamics How to describe bond breaking, reactions in MD? Born Oppenheimer ab initio molecular dynamics At each point on the trajectory, the KS DFT wavefunctions are minimized to their ground state Then forces are computed using Hellman Feynman theorem If not in an exact ground state, can get large noise in forces which integrator is best to use? mr dvˆ F dr i i i i 8
Length scales Fluctuations at phase transitions, long range effects, mechanics Periodic boundaries can help D. S. Ivanov and L. V. Zhigilei, Phys. Rev. Lett. 91, 105701, 003. A. Movahedi Rad, R. Alizadeh, J Mod Phys Vol.5 No.8 (014) 9
Time scales Atomic vibrations versus conformation change Coarse graining, statistics of configurations rare hops 10
Sampling the potential energy landscape The choice of initial conditions (and temperature) determines the total energy of the system This determines the region of the potential energy landscape that will be sampled 11
Quantity averaging Ergodic hypothesis: time average of a thermodynamic quantity along a long trajectory is equal to the ensemble average over the phase space (large number of independent systems) A A e d d E( rp, ) (, rp) r p 1 e E( rp, ) drp d T T 0 At () dt Role of MD equilibration phase is to generate low energy (high probability) configurations are visited more frequently Not all regions of space may be accessible Need to explicitly sample distinct initial configurations 1
Temperature in MD Equipartition energy theorem relates temperature to the average kinetic energy of the system Instantaneous temperature is related to the instantaneous velocities KE N miv i1 i 1 N f k B T ( t) 3 Nk B T ( t) Temperature is an average over degrees of freedom To obtain macroscopic temperature, can also do a time average Tt 1 N t N t i1 Tt i 13
Energy and temperature fluctuations Simplest MD is performed in constant energy NVE (microcanonical) ensemble Kinetic energy (and T) will fluctuate, equal and opposite to potential energy Fluctuation of T(t) : Tt kt NC B () For large systems, fluctuation of T is negligible Total energy should fluctuate much less (at least orders of magnitude less than KE) V total kinetic potential 14
Energy and temperature fluctuations Hard to specify the temperature from initial conditions Temperature can drift, depending on dynamics and time step Need ability to regulate the temperature of the system : canonical ensemble In the canonical ensemble (NVT), temperature is constant by definition, but total energy fluctuates E 1 E H H NkBT0 C V 0 E N For large systems, canonical ensemble (NVT) is equivalent to microcanonical ensemble (NVE) 15
Thermostats Temperature cannot be held fixed in a NVE MD simulation Algorithm needs to be modified T must be given as an input and monitored during simulation Thermostat is a feedback control algorithm acting on the average kinetic energy of the system It must periodically act on the velocities Q: Which integrator can be used? 16
Control of system variables Strong coupling methods scale system variable to give exact preset derived value Weak coupling methods scale system variable in direction of desired derived value Stochastic methods constrain a system variable to preset distribution function Extended system dynamics extend degrees of freedom to include temperature or pressure terms 17
Velocity rescaling Temperature and velocity are related as T v Scale the velocities to reach desired temperature T o Tt Desired temperature Instantaneous temperature Can be easily used in NVE simulations (which integrator?) Disadvantage: Strong coupling affects the natural dynamics 18
Berendsen thermostat Weakly coupled, reformulated velocity scaling Scale velocities at each step such that, dt dt t 1 T target T t Thermal bath is introduced with a coupling parameter Scale factor: dt T T t target 1 1 Value of has to be chosen carefully; typically = 0.1 0.4ps Large : sampling unphysical ensemble, too slow Small poor coupling to the thermostat, too aggressive Kinetic energy fluctuations do not match canonical ensemble For = dt, this is nothing but velocity scaling 19
Andresen thermostat Stochastic: influence the temperature by reassigning the velocity of a random particle (a collision ) Probability of collision The new velocity is from the Maxwell Boltzmann distribution corresponding to the desired T. Langevin thermostat: apply friction and random forces to momenta Sample a micro canonical ensemble between collisions It can be shown that overall the canonical ensemble is preserved Should not be used while computing transport and nonequilibrium properties: rates, diffusion. 0
Nosé Hoover thermostat It is possible to rigorously maintain the canonical ensemble Extended system method: introduced by Nosé and later developed by Hoover Principle: Introduce a heat bath as an integral part of the system, as an additional degree of freedom in the system This is done by modifying the Lagrangian of the system by introducing a fictitious variable s associated with a mass Q and velocity ds/dt v t sr st 1
Nosé Hoover thermostat An extra degree of freedom is introduced, which represents friction which slows down or accelerates particles until the temperature reaches target value T Combined system evolves microcanonically, but energy is exchanged between the real system and bath, sampling the true canonical statistics for the real system
Harmonic oscillator (NVE) 3
Harmonic oscillator Nosé Hoover thermostat on oscillator 4
Example: LJ fluid 5
Pressure in MD The pressure is defined as the average force on the container wall due to the physical system enclosed therein. Thermodynamically, it is defined as P H V H is the Helmholtz free energy V is the volume of the system Microscopic estimator can be expressed from Virial theorem P NT V 1 3V i r i F i 1 3V i i p m i 1 i j r ij F ij F i is the force on particle i 6
Berendsen barostat Pressure control by rescaling positions and volume r i r i 1 t ( P target P( t)) 1/3 Instantaneous pressure vs time in an MD simulation of 56 particles using the Berendsen barostat to impose an instantaneous pressure jump from 1.0 to 6.0. Each curve corresponds to a different value of the time constant τ 7
Thermostat summary Method Pro Con Berendsen Nosé Hoover stochastic (Langevin, Andersen) Fast, smooth first order approach to equilibrium, good for non equilibrium Canonical ensemble. Considered most reliable for simulation at equilibrium Canonical ensemble Non canonical, less reliable for simulation at equilibrium slow, second order approach to equilibrium, affects dynamics affects dynamics, not reliable for kinetics 8
Equilibration How to test for equilibration? Drop longer and longer initial segments of your dynamical trajectory, when accumulating averages Transient part of the trajectory can be long, depends on dynamics Equilibration Production 9
Accumulating averages Potential + kinetic = total energy (conserved) divided by N Energy conservation needs to be monitored in every simulation Temperature (from kinetic energy) T(t) 1 N N i1 1 mv i (t) v i (t) Pressure Potential energy: caloric curve E(T), latent heat of fusion Radial (pair) distribution function local structure Mean square displacements (diffusion) 30
Error bars Averages are computed as: A 1 n n i1 A i Statistical error is approximated by the variance ( ) n( n 1) Error decreases with sample size n Actually this is strictly true only for uncorrelated results With MD it s often good to average over ensemble of trajectories started from different structures ( A i i A A ) 31
Caloric curve in Pb slab (100) 1 M time steps 6400 atoms Surface melting transition 3
Radial distribution function The probability density p(r) of finding a particle at a given distance, (between r and r+dr) increases with r, since the shell volume is 4r dr Define a normalized distribution function counting the probability of finding two particles at a given distance r g( r) ( N 1) p( r) r N V d 3 rg( r) ( N 1) 33
RDF: measure of local structure RDF: ratio of the average number density (r) at a distance r from any given atom and the density at a distance r from an atom in an uniform gas at the same overall density Any deviation in g(r) from 1 reflects correlations between particles due to the intermolecular interactions 34
Water with ab initio MD (3 molecules) Can tell the size required to simulate infinite sample Does not freeze into an ordered crystal, T m is off Nuclear quantum effects are important 35