Molecular Dynamics Simulations

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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)
(6)
(2)
(7)
(3)
(8)
(9)
(4)
(10)
(5)
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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
• Beemans 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