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