1

!

MOLECULAR DYNAMCS !(PLAY IT AGAIN SAM)

Another pioneer of MD…

You cannot step twice in the same river

Heraclitus (Diels 91)

2

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

4

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)(

22

23

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