lecture11

lecture11 - Chapter 9: Molecular-dynamics Integrate...

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Chapter 9: Molecular-dynamics • Integrate equations of motion-- classical! • Discrete form of Newton’s second law • Forces from interaction potential • For a simple pair potential, we get r F i = " r # i U r r 1 , r r 2 ,..., r r N ( ) U r r 1 ,..., r r N ( ) = 1 2 u ( ij " r ij )
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Integrating equations of motion F i , x = m i dv i , x dt F i , y = m i dv i , y dt F i , z = m i dv i , z dt dv i , x dt = d 2 x i dt 2 " x i ( n + 1) # 2 x i ( n ) + x i ( n # 1) $ t 2 x i ( n + 1) = 2 x i ( n ) " x i ( n " 1) + F i , x m i # t 2 This works out to give the Verlet algorithm,
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Lennard-Jones potential for noble gas V r ( ) = 4 " # r $ % ( ) 12 * r $ % ( ) 6 + , - . / 0
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Statistical ensemble • Newtonian dynamics -----> microcanonical, canonical ensemble P ~ e -E/kT • If we know U( r 1 , …, r N ), in principle we can compute anything! ….but if it were just that easy I wouldn’t have a job! Challenges: • We usually do not know interactions accurately • Timescales, length scales usually are ~ns, nm … • Problems of interest often on scales of ~ μ s, μ m and beyond!
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Placing atoms on fcc lattice… non-primitive basis ! start with non-primitive fcc basis r1(1)=0.0d0 r2(1)=0.0d0 r3(1)=0.0d0 r1(2)=0.5d0 r2(2)=0.5d0 r3(2)=0.0d0 r1(3)=0.5d0 r2(3)=0.0d0 r3(3)=0.5d0 r1(4)=0.0d0 r2(4)=0.5d0 r3(4)=0.5d0
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Some parameters for lattice… INTEGER, PARAMETER :: nn=10 INTEGER, PARAMETER :: Prec14=SELECTED_REAL_KIND(14) INTEGER, PARAMETER :: natoms=4*nn**3 INTEGER, PARAMETER :: nx=nn,ny=nn,nz=nn REAL(KIND=Prec14), PARAMETER :: sigma=1.0d0,epsilon=1.0d0 REAL(KIND=Prec14), PARAMETER :: L=sigma*2.0d0**(2.0d0/3.0d0)*nn
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lecture11 - Chapter 9: Molecular-dynamics Integrate...

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