molecular dynamcs (play it again sam)...• ab-initio: bond-breaking and charge transfer; structure...
TRANSCRIPT
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!
MOLECULAR DYNAMCS !(PLAY IT AGAIN SAM)
Another pioneer of MD…
You cannot step twice in the same river
Heraclitus (Diels 91)
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Newton’s second law: N coupled equations
),,( 12
2
Nii
i rrFdtrdm
=
• The force depends on positions only (not velocities)
• The total energy of the system is conserved (microcanonical evolution)
Phase Space
• If we have N particles, we need to specify positions and velocities for all of them (6N variables) to uniquely identify the dynamical system
• One point in a 6N dimensional space (the phase space) represents our dynamical system
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Three Main Goals
• Ensemble averages (thermodynamics) • Real-time evolution (chemistry) • Ground-state of complex structures
(optimization) • Structure of low-symmetry systems: liquids, amorphous
solids, defects, surfaces • Ab-initio: bond-breaking and charge transfer; structure of
complex, non trivial systems (e.g. biomolecules)
Thermodynamical averages • Under hypothesis of ergodicity, we can
assume that the temporal average along a trajectory is equal to the ensemble-average over the phase space
∫=T
dttAT
A0
)(1∫∫
−
−=
pdrdE
pdrdEAA
)exp(
)exp(
β
β
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Real Time Evolution
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Simulated Annealing
The Computational Experiment
• Initialize: select positions and velocities • Integrate: compute all forces, and determine new
positions • Equilibrate: let the system reach equilibrium (i.e.
lose memory of initial conditions) • Average: accumulate quantities of interest
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Initialization
• Second order differential equations: boundary conditions require initial positions and initial velocities
• Initial positions: reasonably compatible with the structure to be studied. Avoid overlap, short distances.
• Velocities: zero in CP, random distribution according to temperature in BO. They thermalize quickly.
Maxwell-Boltzmann distribution
⎟⎟⎠
⎞⎜⎜⎝
⎛ −⎟⎟⎠
⎞⎜⎜⎝
⎛∝
Tkmvv
Tkmvn
BB 2exp
2)(
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π
mTkv
mTkv B
rmsB 3,2 ==
Oxygen at room T:
105 cm/s
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Integrators • Verlet
Verlet’s Algorithms
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Time Step
Time Step
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How to test for equilibration ?
• Drop longer and longer initial segments of your dynamical trajectory, when accumulating averages
Accumulate averages
• Potential, kinetic, total energy (conserved) • Temperature (K=3/2 N kBT) • Pressure • Caloric curve E(T): latent heat of fusion • Mean square displacements (diffusion) • Radial (pair) distribution function
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Correlation Functions
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Limitations
• Time scales • Length scales (PBC help a lot) • Accuracy of forces • Classical nuclei
Classical MD Bibliography
• Allen and Tildesley, Computer Simulations of Liquids (Oxford)
• Frenkel and Smit, Understanding Molecular Simulations (Academic)
• Ercolessi, A Molecular Dynamics Primer (http://www.fisica.uniud.it/~ercolessi/md)
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Ground states from self-consistent iterations
Hellmann-Feynman theorem
Fi = − dE
dRi
= −d Ψ H Ψ
dRi
=
= Ψ − dHdRi
Ψ = Ψ − dVdRi
Ψ
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Born-Oppenheimer Molecular Dynamics
miRi =Fi = Ψ − dV
dRi
Ψ
Total energy (approx, non-SCF)
212n n nn
nE Vε ψ ψ= = − ∇ +∑ ∑
)exp()( rGicrG
nGn
⋅=∑ψ
E = 12
c Gn 2
G∑ G2 + c G
n∗c ′Gn V (G −′G )
G ,′G
∑⎛⎝⎜
⎞⎠⎟n
∑
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Dynamical evolution of cG’s
We need the “force”
}][{ iEE ψ= Fi = −δE[{ψ i}]δψ i
iHψˆ−=
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Skiing down a valley
µ ψ i = Fi (= −Hψ )i
ψ i = Fi (= −Hψ i )
“Damped” dynamics
skiing
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SD or CG skiing
Lots of Skiing if Atoms Move
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Lots of Skiing if Atoms Move
The extended CP Lagrangian
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Equations of motion
Equations of motion (II)
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Constant(s) of motion
Econs =12µi ψ i
ψ ii∑ + 1
2MIRI2 + Ψ0 He Ψ0
I∑
Ephys =12MIRI2 + Ψ0 He Ψ0
I∑ = Econs −Te
Ve = Ψ0 He Ψ0
Te =12µi ψ i
ψ ii∑
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Kolmogorov-Arnold-Moser invariant tori
Born-Oppenheimer vs Car-Parrinello
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HF vs CP forces
A typical CP simulations
• Fixed ions, converge the electrons (very well) – Damped dynamics, or ideally conjugate gradient in the
future – Small steps at the beginning (1-3 a.u.) to allow for
iterative solution of Lagrangian orthogonality constraints. Then restart with larger steps
• Start CP dynamics. With no thermostats, initial configuration determines (in an unknown way) what will be the average temperature.
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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 Jülich 2000). Book of the same name.
• http://www.theochem.ruhr-uni-bochum.de/research/marx/cprev.en.html