《Simulations Moléculaires》Cours04IV

Molecular Dynamics(2)

Molecular Dynamics (2)

Molecular dynamics for continuous potentials Short history. The first md simulation for a system interacting with a continuous potential (lennard-Jones potential) was carried out by A rahman in1964 A Rahman, Phys. Rev. 136, A405,(1964) Main differences between MD with continuous poential and MD Ofhis MD(continuous potentials) MD CHS continuous change of forces discontinuous changes of forces exerted on all the particles exerted on all the particles approximate solution of motion|exact solution of motion of of equations, equations wide applications restricted applications

Molecular dynamics for continuous potentials Short history: The first MD simulation for a system interacting with a continuous potential (Lennard-Jones potential) was carried out by A. Rahman in 1964. A. Rahman, Phys. Rev. 136, A405, (1964). Main differences between MD with continuous poentials and MD of HS: MD (continuous potentials) •continuous change of forces exerted on all the particles; •approximate solution of motion of equations; •wide applications. MD (HS) •discontinuous changes of forces exerted on all the particles; •exact solution of motion of equations; •restricted applications

Trajectory generation Equation of motion: m, a2r /Ot=ma;=f m;:mass of particle i r: position of particle 1; a; acceleration of particle i f; force on particle i, f =-V,V V: potential energy Numerical solution Method of finite difference

Trajectory generation Equation of motion: mi 2ri /t 2 = miai = fi mi : mass of particle i; ri : position of particle i; ai : acceleration of particle i; fi : force on particle i, fi = -iV V: potential energy Numerical solution: Method of finite difference

Desirable qualities for a good algorithm lt should be fast and requires little memory elt should permit the use of a long time step, 8t It should satisfy the known conservation laws for the energy and momentum and be time-reversible .lt should be simple in form and easy to program

Desirable qualities for a good algorithm •It should be fast and requires little memory. •It should permit the use of a long time step, dt. •It should satisfy the known conservation laws for the energy and momentum and be time-reversible. •It should be simple in form and easy to program

Verlet’ s algorith Position r(t+6t)=2r(t)-r(t6t)+(8t2a(t) The error on position is of order of (St) 4 Taylor expansion (t+6t)=r(t)+δtv(t)+(6t)2a(t)/2+ (t-δt)=r(t)-6tv(t)+(6t)2a(t)2+ velocity vt)=[r(t+δt)-r(t-6t)]/(26t The error on velocity is of order of( St)3

Verlet’s algorithm Position: r(t+dt) = 2r(t) - r(t-dt) + (dt)2a(t) The error on position is of order of (dt)4 . Taylor expansion: r(t+dt) = r(t) + dtv(t) + (dt)2a(t)/2 + … r(t- dt) = r(t) - dtv(t) + (dt)2a(t)/2 + … Velocity: v(t) = [r(t+dt) - r(t-dt)]/(2dt) The error on velocity is of order of (dt)3

How to initialize verlet s algorithm? Problem At t=0, r(-8t) is unknown Solution to the problem r(-∞t)=r(t)-δtv(t

How to initialize Verlet ’s algorithm? Problem: At t=0, r(-dt) is unknown! Solution to the problem: r(-dt) = r(t) - dt . v(t)

Advantages and drawbacks of Verlet's algorithm Advantages Good stability, i. e, relatively large time step Good energy conservation Good time-reversibility Simplicity Drawbacks Not self-starting Position and velocity are not treated with the same precision

Advantages and drawbacks of Verlet’s algorithm Advantages: Good stability, i.e., relatively large time step dt; Good energy conservation; Good time-reversibility; Simplicity. Drawbacks: Not self-starting; Position and velocity are not treated with the same precision

How to choose time step? Simple case ot must be chosen in such a way that the total energy is well conserved and the trajectory is time reversible Complicated case (multi-time scales): When there are several time scales(e.g, mixture of particles with different masses, polymers in solvent, both hard and soft modes exist in molecular systems, etc.), 8t must be chosen according to the dynamics of the component or the mode which evolves most quickly

How to choose time step? Simple case: dt must be chosen in such a way that the total energy is well conserved and the trajectory is time reversible. Complicated case (multi-time scales): When there are several time scales (e.g., mixture of particles with different masses, polymers in solvent, both hard and soft modes exist in molecular systems, etc.), dt must be chosen according to the dynamics of the component or the mode which evolves most quickly

Reduced units Temperature: T"=kT/E Energy E=E/E Pressure P=Po/8 Ime (/mo2)12t F orce f= fo/e

Reduced units Temperature: T* = kT/e Energy: E* = E/e Pressure: P* = Ps 3 /e Time: t* = (e/ms 2 ) 1/2t Force: f * = fs/e

Constant-temperature Molecular Dynamics The basic Md algorithm generates a microcanonical ensemble Different velocity adjusting methods: 1)Andersen's Method. Reference: H.C. Andersen, J. Chem. Phys. 72, 2384, 1980 Basic idea mimicing the collisions between the molecules of the considered system with those of the thermal bath Practical implementation At a preset time interval, At, the velocity of a randomly chosen molecule is reset according to the maxwell-boltzmann distribution with t

Constant-temperature Molecular Dynamics The basic MD algorithm generates a microcanonical ensemble. Different velocity adjusting methods: 1) Andersen’s Method: Reference: H.C. Andersen, J. Chem. Phys. 72, 2384, 1980. Basic idea: mimicing the collisions between the molecules of the considered system with those of the thermal bath. Practical implementation: At a preset time interval, Dt, the velocity of a randomly chosen molecule is reset according to the Maxwell-Boltzmann distribution with T