3.320: lecture 15 (mar 3 first-principles molecular dynamics · mar 31 2005 3.320 atomistic...
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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)
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.
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 =∂∂
Mar 31 2005 3.320 Atomistic Modeling of Materials -- Gerbrand Ceder and Nicola Marzari
Mean Square Displacements
∑=
∆=∆N
ii tr
Ntr
1
22 )(1)(
Mar 31 2005 3.320 Atomistic Modeling of Materials -- Gerbrand Ceder and Nicola Marzari
Mean Square Displacements
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
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′ ∞
′→∞ ′′ ′ ′′= − =∫ ∫
Mar 31 2005 3.320 Atomistic Modeling of Materials -- Gerbrand Ceder and Nicola Marzari
Velocity Autocorrelation Function
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
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)
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∂
= −∂
&
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∂∂
=− &
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
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+
−+−=∑ &&
Mar 31 2005 3.320 Atomistic Modeling of Materials -- Gerbrand Ceder and Nicola Marzari
Ergodicity issues
Very harmonic solids (e.g. 1 harmonic oscillator !)
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)
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.
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
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
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
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
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
Mar 31 2005 3.320 Atomistic Modeling of Materials -- Gerbrand Ceder and Nicola Marzari
We need the force
i
ii
EFδψψδ }][{
−=}][{ iEE ψ=
iHψˆ−=
Mar 31 2005 3.320 Atomistic Modeling of Materials -- Gerbrand Ceder and Nicola Marzari
Skiing down a valley
ii Hψψµ −=&&
ii Hψψ −=&
Mar 31 2005 3.320 Atomistic Modeling of Materials -- Gerbrand Ceder and Nicola Marzari
Conjugate-gradients minimization
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
Mar 31 2005 3.320 Atomistic Modeling of Materials -- Gerbrand Ceder and Nicola Marzari
Proof of Hellmann-Feynman
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
Mar 31 2005 3.320 Atomistic Modeling of Materials -- Gerbrand Ceder and Nicola Marzari
The extended Car-ParrinelloLagrangian
Mar 31 2005 3.320 Atomistic Modeling of Materials -- Gerbrand Ceder and Nicola Marzari
Equations of motion
Mar 31 2005 3.320 Atomistic Modeling of Materials -- Gerbrand Ceder and Nicola Marzari
Constant of Motion
+
Mar 31 2005 3.320 Atomistic Modeling of Materials -- Gerbrand Ceder and Nicola Marzari
Born-Oppenheimer vs Car-Parrinello
Mar 31 2005 3.320 Atomistic Modeling of Materials -- Gerbrand Ceder and Nicola Marzari
BO vs CP forces
Mar 31 2005 3.320 Atomistic Modeling of Materials -- Gerbrand Ceder and Nicola Marzari
Kolmogorov-Arnold-Moser invariant tori
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