1. Molecular Dynamics Simulations

2. Objective : To understand the properties of materials Question : How to accomplish the goal? Answer : Positions and momentums of each atom have to be determined

3. Theory of molecular dynamics What is molecular dynamics? The idea –Compute the forces acting on the atoms in a molecular system –Analyze the motions –Deduce the bulk properties of the material

4. Classical Mechanics (1) (2) (3) (4) (5) (6) (7) (8) (9) (10)

5. Verlet algorithm r(t+∆t) = r(t) + v(t)∆t + (1/2)a(t)∆t 2 (1) r(t-∆t) = r(t) – v(t)∆t + (1/2)a(t)∆t 2 (2) Summing these two equations yields r(t+∆t) = 2r(t) – r(t- ∆t) + a(t)∆t 2 (3) v(t+∆t) = v(t) + a(t)∆t + (1/2)b(t)∆t 2 (4) a(t+∆t) = a(t) + b(t)∆t (5) Plugging b(t) from (5) into (4) yields v(t+∆t) = v(t) + (1/2)[a(t) + a(t+∆t)] ∆t (6)

6. Other algorithms Leap-frog algorithm r(t+∆t) = r(t) + v(t+(1/2)∆t) ∆t v(t+(1/2)∆t) = v(t-(1/2)∆t) + a(t) ∆t Beeman’s algorithm r(t+∆t) = r(t) + v(t)∆t + (2/3)a(t)∆t 2 – (1/6)a(t-∆t)∆t 2 v(t+∆t) = v(t) + v(t)∆t + (1/3)a(t)∆t + (5/6)a(t)∆t–(1/6)a(t∆t)∆t