Langevin Dynamics in Constant Pressure Extended Systems
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1 Langevin Dynamics in Constant Pressure Extended Systems D. Quigley and M.I.J. Probert CMMP
2 Talk Outline Molecular dynamics and ensembles. Existing methods for sampling at NPT. Langevin dynamics in constant pressure systems. Results classical and ab-initio. Conclusions. 2
3 MD and Ensembles MD models dynamical properties of a system using numerical integration of classical equations of motion. Computationally limited to simulating a small number of particles to represent a sample of the larger system. Should be subject to both temperature and density fluctuations due to exchange of energy and particles with the rest of the system. EOMs must sample phase space with correct probability requires ergodicity. Therm al Fluctuations - NVT + density (volum e) fluctuations NPT Ensem ble 3
4 Solutions Equilibrium density fluctuations (barostat) Extended system with volume/cell vectors as dynamical variable. Either Hoover (isotropic) or Parrinello-Rahman (anisotropic). Number of atoms remains constant density fluctuates Equilibrium temperature fluctuations (thermostat) Ergodicity requires chaotic behaviour. Standard method is Nosé-Hoover chains. Deterministic and simple requires chaotic behaviour to be provided by the atomic system I.e. need large N! Large N not feasible for ab-initio MD. Is there an alternative? 4
5 Langevin Dynamics (LD) Convert Hamilton s EOM for p into Langevin equation. Dam ping + stochastic buffeting Balancing the two new terms generates thermal equilibrium. Langevin thermostat correct temperature fluctuations with no ergodicity problems. BUT no equivalent method of producing density fluctuations. What would happen if we could perform LD in a pressure regulating extended system? 1. Would it give the correct E & fluctuations (NPT ensemble?) 2. What would be the advantages of doing this? 5
6 Hoover system Expansion and contraction of system shape of simulation cell constant. Have two extra phase space co-ordinates volume V and a momentum p (strain rate x fictitious volume mass W). Particle motions are coupled to the box momentum. X is the deterministic part of the internal pressure 6
7 Yes it does sample the NPT ensemble! Construct Fokker-Planck equations for the EOMs continuity equation for probability density in the extended phase space. This has the desired probability density as a solution. Equations correctly sample the NPT ensemble. Complications The Hoover equations conserve the quantity BUT, this is not a Hamiltonian. No Hamiltonian can be written down for the Hoover system. Probability of equal energy configurations may not be equal. See Tuckerman et al Europhys Lett 45 p149 (1999). Implies probability gradients imbalances Langevin dynamics. Analysis shows not a problem for Hoover system. 7
8 Parrinello-Rahman System Allows size and shape to change anisotopic fluctuations. Dynamical co-ordinate + momentum for each component of 3 cell vectors. X is deterministic part of pressure tensor. Again non-hamiltonian. In this case probability gradients prohibit Langevin dynamics of the cell variable not crucial to correct sampling. 8
9 Parameters Langevin damping co-efficient or thermostat relaxation time 1/. Fictitious mass W of the cell variable(s), corresponds to cell relaxation time. MD regime LD regime 0 identified from bulk memory function. See Kneller and Hinsen J.Chem.Phys 115(24) p11097 (2001) Choose cell paramter W based on e.g. velocity spectrum. Integration Algorithm Developed from Louiville operators for the extended system. Leads to Verlet-like algorithm reversible in zero friction limit. For details see Quigley & Probert J.Chem.Phys 2004 (In Press) matter/
10 Classical Core Softened Potentials Mapping of the Stell-Hemm er phase diagram (pair potential) Both runs seeking 600 K, 5 MPa with equal relaxation times No prior knowledge of dynamics. Langevin dynamics in the Hoover Ensem ble. Conventional Nosé-Hoover chain based constant pressure schem e. 10
11 CASTEP - Silicon (isotropic) Simulated using the optimal damping coefficient identified from a bulk sim ulation using the Tersoff potential. 8,400 time-steps ( t = 8 fs) at 293 K 1 atm. Using E xc = E LDA E cut-off = 160 ev (ultrasofts) 8 atom cell 2x2x2 MP k-point grid Volume fluctuations yield K T =103.5 ± 5.2 GPa c.f. experiment 102 GPa and equivalent Nosé-Hoover run (not converged at this run length) K T =58.7 GPa! 11
12 CASTEP - P 2 I 4 (Fully Flexible) Triclinic cell, a b c. No prior knowledge of dynamics use conservative and W from earlier work on elemental P and I (running in LD regime). Good results without prior knowledge of memory function or ideal W - 20,000 timesteps ( t = 8fs) at 250 K, 50 MPa. 12
13 Conclusions Langevin dynamics in the Hoover or Parrinello-Rahman system is a valid method for sampling the NPT ensemble. This method has been tested in a classical MD code and implemented in the plane wave DFT code CASTEP. Langevin NPT has statistical efficiency advantages over Nosé- Hoover based schemes. reduced correlation times guaranteed ergodicity on shorter timescales Further details see D. Quigley and M.I.J Probert J.Chem.Phys (2004) (in press). matter/
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