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Molecular Dynamics Simulations
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Objective : To understand the properties of materials
Question : How to accomplish the goal?
Answer : Positions and momentums of each atom have to be
determined
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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
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Classical Mechanics
(1)
(2)
(3)
(4)
(5)
(6)
(7)
(8)
(9)
(10)
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• Verlet algorithm r(t+∆t) = r(t) + v(t)∆t + (1/2)a(t)∆t2 (1)
r(t-∆t) = r(t) – v(t)∆t + (1/2)a(t)∆t2 (2)
Summing these two equations yields
r(t+∆t) = 2r(t) – r(t- ∆t) + a(t)∆t2 (3)
v(t+∆t) = v(t) + a(t)∆t + (1/2)b(t)∆t2 (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)
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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)∆t2 – (1/6)a(t-∆t)∆t2
v(t+∆t) = v(t) + v(t)∆t + (1/3)a(t)∆t + (5/6)a(t)∆t–(1/6)a(t∆t)∆t