HarmOsc_MD1_LJ - Introduction to Molecular Dynamics The...

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1 Introduction to Molecular Dynamics: The Harmonic Oscillator The LeapfrogAlgorithm The Lennard-Jones Potential BIOL 7110 / CHEM 8901 / BIOL 4105 / CHEM 4804 February 15, 2011
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2 To be covered today Overview of molecular dynamics Newton’s Equations Differential equations converted to difference equations dt vs Δ t (difference equation) The MD algorithm: numerical integration of Newton’s equations The harmonic oscillator: theory The harmonic oscillator: treatment by MD (Pseudocode) The harmonic oscillator: results of the simulation The Lennard-Jones potential
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3 Newton’s Equations: Energy, Force and Acceleration MD represents the numerical integration of Newton’s equations F i = [ ˆ k j ( E / r ij )] j = 1 3 We note that the force is just the negative gradient of the energy The trajectory of a cannonball. y x x ( t ) = v x t y ( t ) = v y t 1 2 gt 2 x (0) = 0 y (0) = 0 Analytic solution
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4 Numerical Integration of Newton’s Equations (one-dimensional case) stepwise, replacing the differential equations By the difference equations Instead of solving Newton’s equations analytically, the computer uses numerical approximations x ( dt ) = x (0) + dx dx = vdt x ( Δ t ) = x (0) + Δ x Δ x = v Δ t We solve the integral equation x ( t ) = x (0) + v ( u ) du u = 0 u = t Numerical Integration of Newton’s Equations (two-dimensional case) stepwise, replacing the differential equations By the difference equations We solve the integral equations x ( t ) = x (0) + v x ( u ) du u = 0 u = t y ( t ) = y (0) + v y ( u ) du u = 0 u = t x ( dt ) = x (0) + dx y ( dt ) = y (0) + dy x ( Δ t ) = x (0) + Δ x y ( Δ t ) = y (0) + Δ y dx = v x dt dy = v y dt Δ x = v x Δ t Δ y = v y Δ t
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5 The trajectory of a cannonball. y x Discretization of the trajectory The trajectory of a cannonball. y x The discretized trajectory closely matches the analytic solution
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6 Molecular Dynamics Input to a molecular dynamics program: A set of molecular x,y,z coordinates E = f(x 1 ,y 1 ,z 1 ,…,x N ,y N ,z N ) An energy function x’ = x + v Δ t + a( Δ t) 2 /2 + … Algorithm: numerical integration of Newton’s equations of motion Just turn the crank Numerical Integration:
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