COPYRIGHT 2008, PRINCETON UNIVERSITY PRESS

simulating matter with molecular dynamics 431far-off molecules do not contribute significantly to the motion of a molecule, andwe pick a value r cut ≃ 2.5σ beyond which we ignore the effect of the potential:u(r)={4(r −12 − r −6) , for rr cut .(16.14)Accordingly, if the simulation region is large enough for u(r >L i /2) ≃ 0, an atominteracts with only the nearest image of another atom.The only problem with the cutoff potential (16.14) is that since the derivativedu/dr is singular at r = r cut , the potential is no longer conservative and thus energyconservation is no longer ensured. However, since the forces are already very smallat r cut , the violation will also be very small.16.2 Verlet and Velocity-Verlet AlgorithmsArealistic MD simulation may require integration of the 3-D equations of motion for10 10 time steps for each of 10 3 –10 6 particles. Although we could use our standardrk4 ODE solver for this, time is saved by using a simple rule embedded in the program.The Verlet algorithm uses the central-difference approximation (Chapter 7,“Differentiation & Searching”) for the second derivative to advance the solutionsby a single time step h for all N particles simultaneously:F i [r(t),t]= d2 r idt 2≃ r i(t + h)+r i (t − h) − 2r i (t)h 2 , (16.15)⇒ r i (t + h) ≃ 2r i (t) − r i (t − h)+h 2 F i (t)+O(h 4 ), (16.16)where we have set m =1. (Improved algorithms may vary the time step dependingupon the speed of the particle.) Notice that even though the atom–atom force doesnot have an explicit time dependence, we include a t dependence in it as a way ofindicating its dependence upon the atoms’ positions at a particular time. Becausethis is really an implicit time dependence, energy remains conserved.Part of the efficiency of the Verlet algorithm (16.16) is that it solves for the positionof each particle without requiring a separate solution for the particle’s velocity.However, once we have deduced the position for various times, we can use thecentral-difference approximation for the first derivative of r i to obtain the velocity:v i (t)= dr idt ≃ r i(t + h) − r i (t − h)+ O(h 2 ). (16.17)2hNote, finally, that because the Verlet algorithm needs r from two previous steps, itis not self-starting and so we start it with the forward difference,r(t = −h) ≃ r(0) − hv(0) + h2F(0). (16.18)2−101COPYRIGHT 2008, PRINCET O N UNIVE R S I T Y P R E S SEVALUATION COPY ONLY. NOT FOR USE IN COURSES.ALLpup_06.04 — 2008/2/15 — Page 431