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Mar 31 2005 3.320 Atomistic Modeling of Materials -- Gerbrand Ceder and Nicola Marzari
3.320: Lecture 15 (Mar 31 2005)
FIRSTFIRST--PRINCIPLES MOLECULAR DYNAMICSPRINCIPLES MOLECULAR DYNAMICS…and let us, as nature directs, begin first with first principles.
Aristotle (Poetics, I)
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Mar 31 2005 3.320 Atomistic Modeling of Materials -- Gerbrand Ceder and Nicola Marzari
Simulated Annealing
-3 -2-2
-1 0 1 2 3
12
10
8
6
4
2
0
Figure by MIT OCW.
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Mar 31 2005 3.320 Atomistic Modeling of Materials -- Gerbrand Ceder and Nicola Marzari
Micro- to macro- : diffusion coefficient
• From Fick to Einstein:
),(),( 2 trcDt
trc∇=
∂∂
∫∫ ∇=∂∂ ),(),( 222 trcrrdDtrcrrdt
rr
dDtrt
2)(2 =∂∂
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Mar 31 2005 3.320 Atomistic Modeling of Materials -- Gerbrand Ceder and Nicola Marzari
Mean Square Displacements
∑=
∆=∆N
ii tr
Ntr
1
22 )(1)(
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Mar 31 2005 3.320 Atomistic Modeling of Materials -- Gerbrand Ceder and Nicola Marzari
Mean Square Displacements
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Mar 31 2005 3.320 Atomistic Modeling of Materials -- Gerbrand Ceder and Nicola Marzari
Velocity Autocorrelation Function
∑=
∆=∆N
ii tx
Ntx
1
22 )(1)( ∫ ′′=∆t
x tvtdtx0
)()(
∫∫∫ ′′′′′′=⎟⎟⎠
⎞⎜⎜⎝
⎛′′=∆
t
xx
tt
x tvtvtdtdtvtdtx00
2
0
2 )()()()(
∫∫′
′′′′′′=t
xx
t
tvtvtdtd00
)()(2
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Mar 31 2005 3.320 Atomistic Modeling of Materials -- Gerbrand Ceder and Nicola Marzari
Green-Kubo relations2
0
( )2 lim lim 2 ( ) ( )
t
t t x x
x tD dt v t v t
t
′
′→∞ →∞
∂′′ ′ ′′= =
∂ ∫
)0()()()( xxxx vttvtvtv ′′−′=′′′
0 0
lim ( ) (0) ( ) (0)t
t x x x xD dt v t t v d v vτ τ′ ∞
′→∞ ′′ ′ ′′= − =∫ ∫
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Mar 31 2005 3.320 Atomistic Modeling of Materials -- Gerbrand Ceder and Nicola Marzari
Velocity Autocorrelation Function
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Mar 31 2005 3.320 Atomistic Modeling of Materials -- Gerbrand Ceder and Nicola Marzari
More Green-Kubo
• Other transport coefficients:– Shear viscosity, from the stress– Electrical conductivity, from the charge current– IR adsorption, from the polarization
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Mar 31 2005 3.320 Atomistic Modeling of Materials -- Gerbrand Ceder and Nicola Marzari
Dynamics, Lagrangian style
• First construct L=T-V• Then, the equations of motion are given by
• Why ? We can use generalized coordinates. Also, we only need to think at the two scalar functions T and V
0=∂∂
−⎟⎟⎠
⎞⎜⎜⎝
⎛
∂∂
jj qL
qL
dtd
& (the dot is a time derivative)
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Mar 31 2005 3.320 Atomistic Modeling of Materials -- Gerbrand Ceder and Nicola Marzari
Newton’s second law, too
• 1-d, 1 particle: T=1/2 mv2, V=V(x)
0=∂∂
−⎟⎟⎠
⎞⎜⎜⎝
⎛
∂∂
jj qL
qL
dtd
&
212 0
mxd Vdt x x
⎛ ⎞⎛ ⎞∂ ⎜ ⎟⎜ ⎟ ∂⎝ ⎠⎜ ⎟ + =∂ ∂⎜ ⎟
⎜ ⎟⎝ ⎠
&
&( )d Vmx
dt x∂
= −∂
&
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Mar 31 2005 3.320 Atomistic Modeling of Materials -- Gerbrand Ceder and Nicola Marzari
Hamiltonian
• We could use it to derive Hamiltonian dynamics (twice the number of differential equations, but all first order). We introduce a Legendre transformation
ii q
Lp&∂∂
= ∑ −=i
ii tqqLpqtpqH ),,(),,( &&
ii p
Hq∂∂
=&i
i qHp∂∂
=− &
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Mar 31 2005 3.320 Atomistic Modeling of Materials -- Gerbrand Ceder and Nicola Marzari
Thermostats, barostats…
• We might want to sample a constant-temperature ensemble, or constants pressure– Stochastic approach– Extended system– Constraint method
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Mar 31 2005 3.320 Atomistic Modeling of Materials -- Gerbrand Ceder and Nicola Marzari
Nose’ extended Lagrangian
sNsQVrsmLi
iiNOSE ln)13(21
21 222
β+
−+−=∑ &&
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Mar 31 2005 3.320 Atomistic Modeling of Materials -- Gerbrand Ceder and Nicola Marzari
Ergodicity issues
Very harmonic solids (e.g. 1 harmonic oscillator !)
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Mar 31 2005 3.320 Atomistic Modeling of Materials -- Gerbrand Ceder and Nicola Marzari
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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Mar 31 2005 3.320 Atomistic Modeling of Materials -- Gerbrand Ceder and Nicola Marzari
First-principles molecular dynamics
Graph removed for copyright reasons.Shows dramatic increase in number of citations per year of “CP PRL 1985” and “AIMD” beginning around 1990.
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Mar 31 2005 3.320 Atomistic Modeling of Materials -- Gerbrand Ceder and Nicola Marzari
Plane waves basis set
2i j ijG a πδ⋅ =r r
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Mar 31 2005 3.320 Atomistic Modeling of Materials -- Gerbrand Ceder and Nicola Marzari
It’s really kinetic + potential
21ˆ ( )2
H V r= − ∇ +r
)exp()( rGicrG
nGn
rrrr
r ⋅= ∑ψ
∑∑ ==n
nnn n HE ψψε ˆ
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Mar 31 2005 3.320 Atomistic Modeling of Materials -- Gerbrand Ceder and Nicola Marzari
Kinetic energy
212kin n n
nE ψ ψ= − ∇∑ )exp()( rGicr
G
nGn
rrrr
r ⋅= ∑ψ
2 212
nkin G
n G
E c G=∑ ∑ rr
∫ ′=′⎥⎦⎤
⎢⎣⎡ ∇−−=′∇− GGGrGiiGrdrGG ,
222
21)exp(
21)exp(
21 δ
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Mar 31 2005 3.320 Atomistic Modeling of Materials -- Gerbrand Ceder and Nicola Marzari
Total energy (non-SCF)
( )pot n nn
E V rψ ψ=∑ r)exp()( rGicr
G
nGn
rrrr
r ⋅= ∑ψ
∫ ′−=′−=′ )()exp()()exp()( GGVrGirViGrdrGrVG
2 2
,
1 ( )2
n n ntot G G G
n G G G
E c G c c V G G∗′
′
⎛ ⎞′= + −⎜ ⎟
⎝ ⎠∑ ∑ ∑r r r
r r r
r r
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Mar 31 2005 3.320 Atomistic Modeling of Materials -- Gerbrand Ceder and Nicola Marzari
Dynamical evolution of c’s
2 2
,
1 ( )2
n n ntot G G G
n G G G
E c G c c V G G∗′
′
⎛ ⎞′= + −⎜ ⎟
⎝ ⎠∑ ∑ ∑r r r
r r r
r r
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Mar 31 2005 3.320 Atomistic Modeling of Materials -- Gerbrand Ceder and Nicola Marzari
We need the force
i
ii
EFδψψδ }][{
−=}][{ iEE ψ=
iHψˆ−=
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Mar 31 2005 3.320 Atomistic Modeling of Materials -- Gerbrand Ceder and Nicola Marzari
Skiing down a valley
ii Hψψµ −=&&
ii Hψψ −=&
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Mar 31 2005 3.320 Atomistic Modeling of Materials -- Gerbrand Ceder and Nicola Marzari
Conjugate-gradients minimization
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Mar 31 2005 3.320 Atomistic Modeling of Materials -- Gerbrand Ceder and Nicola Marzari
Hellmann-Feynman theorem
ˆ
ˆ ˆ
ii i
i i
d HdEFdR dR
dH dVdR dR
Ψ Ψ= − = − =
= Ψ − Ψ = Ψ − Ψ
rr r
r r
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Mar 31 2005 3.320 Atomistic Modeling of Materials -- Gerbrand Ceder and Nicola Marzari
Proof of Hellmann-Feynman
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Mar 31 2005 3.320 Atomistic Modeling of Materials -- Gerbrand Ceder and Nicola Marzari
Born-Oppenheimer Molecular Dynamics
ˆi i i
i
dVm R FdR
= = Ψ − Ψr r&& r
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Mar 31 2005 3.320 Atomistic Modeling of Materials -- Gerbrand Ceder and Nicola Marzari
The extended Car-ParrinelloLagrangian
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Mar 31 2005 3.320 Atomistic Modeling of Materials -- Gerbrand Ceder and Nicola Marzari
Equations of motion
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Mar 31 2005 3.320 Atomistic Modeling of Materials -- Gerbrand Ceder and Nicola Marzari
Constant of Motion
+
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Mar 31 2005 3.320 Atomistic Modeling of Materials -- Gerbrand Ceder and Nicola Marzari
Born-Oppenheimer vs Car-Parrinello
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Mar 31 2005 3.320 Atomistic Modeling of Materials -- Gerbrand Ceder and Nicola Marzari
BO vs CP forces
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Mar 31 2005 3.320 Atomistic Modeling of Materials -- Gerbrand Ceder and Nicola Marzari
Kolmogorov-Arnold-Moser invariant tori
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Mar 31 2005 3.320 Atomistic Modeling of Materials -- Gerbrand Ceder and Nicola Marzari
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)
• http://www.theochem.ruhr-uni-bochum.de/research/marx/cprev.en.html