AB-INITIO MOLECULAR DYNAMICS (THE...

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  • AB-INITIO MOLECULAR DYNAMICS(THE END)

    Nicola Marzari, DMSE, MIT

    DFT total energy

  • Evaluating H

  • Its really kinetic + (SCF) potential

    )(21 2 rVH +=

    =

    = GGGrGiiGrdrGG ,

    222

    21)exp(

    21)exp(

    21

    ( ) exp( ) ( ) exp( ) ( )G V r G dr iGr V r iG r V G G = =

    Total energy (non-SCF, sort of)

    212n n nn n

    E V = = + )exp()( rGicr

    G

    nGn =

    2 2

    ,

    1 ( )2

    n n nG G G

    n G G G

    E c G c c V G G

    = +

  • Dynamical evolution of cs

    We need the force

    }][{ iEE =i

    ii

    EF }][{=

    iH=

  • Skiing down a valley

    ( )i i iF H = =

    ( )i i iF H = =

    Damped dynamics

    skiing

  • SD or CG skiing

    Hellmann-Feynman theorem

    ii i

    i i

    d HdEFdR dR

    dH dVdR dR

    = = =

    = =

  • Born-Oppenheimer Molecular Dynamics

    i i i

    i

    dVm R FdR

    = =

    Lots of Skiing if Atoms Move

  • Lots of Skiing if Atoms Move

    The extended CP Lagrangian

  • Equations of motion

    Equations of motion (II)

  • Constant(s) of motion

    20 0

    20 0

    0 0

    1 1 2 21 2

    12

    cons i i i I I ei I

    phys I I e cons eI

    e e

    e i i ii

    E M R H

    E M R H E T

    V H

    T

    = + +

    = + =

    =

    =

  • Kolmogorov-Arnold-Moser invariant tori

  • Born-Oppenheimer vs Car-Parrinello

  • HF vs CP forces

    Quantum MD Bibliography

    Payne, Teter, Allan, Arias, Joannopoulos, Rev Mod Physics 64, 1045 (1992).

    Marx, Hutter, "Ab Initio Molecular Dynamics: Theory and Implementation", in "Modern Methods and Algorithms of Quantum Chemistry" (p. 301-449), Editor: J. Grotendorst, (NIC, FZ Jlich 2000)

    http://www.theochem.ruhr-uni-bochum.de/research/marx/cprev.en.html

  • Full class (videos, etc..)

    Atomistic Modeling of Materials

    http://ocw.mit.edu/OcwWeb/Materials-Science-and-Engineering/3-320Spring-2005/CourseHome/index.htm

  • Thanks !

    Amy Young Paolo Giannozzi and the www.quantum-

    espresso.org gang Todd Martinez, Richard Martin, Duane

    Johnson, David Ceperley Axel Kohlmeyer, Stefano Baroni, Guido

    Fratesi