单击此处编辑母版标题样式,单击此处编辑母版文本样式,第二级,第三级,第四级,第五级,*,Chapter 3 Molecular Dynamics Simulation,3.1 Molecular Dynamics:The Idea,What is molecular dynamics?,It is a technique to compute the equilibrium and transport properties of a classical many-body system.,Means that the nuclear motion of the constituent particles obeys the laws of classical mechanics.,Newtons law,Lagrangian equation,and Langevin equation.,This is an excellent approximation for a wide range of materials.,Chapter 3 Molecular Dynamics S,1,MD simulation is similar to real experiments,When we perform a real experiments,we proceed:,Preparing a sample of the material studied;,Connecting the sample to a measuring instrument;,Measuring the property of interest during a certain time;,If the measurements are subject to statistical noise,then the longer we average,the more accurate our,measurement becomes.,In a MD simulation,we follow,exactly the same approach,.,e.g.,a thermometer,manometer,or viscometer,etc.,MD simulation is similar to re,2,MD approach,First,prepare a sample:select a model system,consisting of,N,particles;,Second,solve Newtons equation of motion until the,properties of the system no longer change with time.,Interaction energy potential,pair potential is frequently used,Equilibrate the system,After equilibration,perform the actual measurement.,Some of the most common mistakes in MD are similar,to,the mistakes that may be made in real experiments.,The sample is not prepared correctly,the time is too short,the system undergoes an irreversible change during the experiment,or we do not measure what we think.,MD approach First,prepare a s,3,How to measure an observable quantity?,To measure an observable quantity in a MD simulation,we must first of all be able to express this observable as,a function of the positions and momenta of the particles,in the system.,Let us take temperature as an example.Making use of,the equipartition of energy over all degrees of freedom,N,f,we have:,The relative fluctuations in the temperature will be of order .As,N,f,is typically of the order of 10,2,-10,3,the statistical fluctuations are of the order of 5-10%.,Average over many fluctuations,How to measure an observable,4,In which case should we worry about quantum effects?,When we consider the the translational or rotational motion of light atoms or molecules,or vibrational motion with a frequency,such that,h,k,B,T,.,He,H,2,D,2,etc.,Of course,our course of this vast subject is incomplete.,If you need the knowledge beyond the course,you can,read the references on the coming slide.,In which case should we worry,5,References for Molecular Dynamics,MP Allen&DJ Tildesley,1987,Computer Simulation,of Liquids.,HC Berendsen&WF van Gunsteren,1984,Molecular,Dynamics Simulations:Techniques&Approached,NATO ASI Series C123.,CL Brooks et al.,1988,Proteins.A Theoretical,Perspective of Dynamics,Structure and,Thermodynamics.Advances in chemical Physics.,Volume LXXI.,JM Haile,1992,Molecular Dynamics Simulation.,Elementary Methods.,References for Molecular Dynam,6,4.2 Molecular Dynamics:A Program,The best introduction to MD is to consider a simple program.We keep it as simple as possible to illustrate some important features of MD.It is constructed as:,read in the parameters that specify the conditions of,the run(e.g.,initial temperature,number of particles,density,time step,etc.);,Initialize the system(select initial,r,i,and,v,i,);,Compute the energy and forces on all particles;,Integrate Newtons equations of motion;,measure the quantities in the present time;,After completion of the central loop,compute and print the averages of measured quantities,and stop.,Central loop,the core of the simulation,4.2 Molecular Dynamics:A Prog,7,A simple MD program,Program MD,simple MD program,call init,initialization,t=0,set time,do while(t.lt.tmax),MD loop,call force(f,en),determine the forces,call integrate(f,en),integrate equations of motion,t=t+delt,update time,call sample,sample averages,enddo,stop,Subroutine,init,force,and,integrate,will be described later.Subroutine,sample,is used to calculate averages of quantities of interest like pressure and temperature,etc.,A simple MD programProgram MD,8,4.2.1 Initiallization,To start the simulation,we should assign initial,positions and velocities to all particles in the system.,Often this is achieved by initially placing the,particles on a cubic lattice.,Avoiding positions that result in an appreciable overlap;choose positions compatible with the structure that we are aiming to simulate.,As the equilibrium properties of the system do not depend on the choice of initial conditions,all reasonable initial conditions are in principle acceptable.,4.2.1 Initiallization To start,9,Initial positions,To simulate a solid state of a model system,it is,logical to prepare the system in the crystal structure,of interest.,If we are interest in the fluid phase,we simply,prepare the system in any convenient crystal structure.,If the density close to the freezing,selectio