3.320: Lecture 15 (Mar 3 FIRST-PRINCIPLES MOLECULAR DYNAMICS · Mar 31 2005 3.320 Atomistic Modeling of Materials -- Gerbrand Ceder and Nicola Marzari 3.320: Lecture 15 (Mar 31 2005)

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