AB-INITIO MOLECULAR DYNAMICS(THE END)

Nicola Marzari, DMSE, MIT

DFT total energy

Evaluating Hψ

It’s really kinetic + (SCF) potential

)(21ˆ 2 rVH +∇−=

∫ ′=′⎥⎦⎤

⎢⎣⎡ ∇−−=′∇− GGGrGiiGrdrGG ,

222

21)exp(

21)exp(

21 δ

( ) exp( ) ( ) exp( ) ( )G V r G dr iGr V r iG r V G G′ ′ ′= − = −∫

Total energy (non-SCF, sort of)

212n n nn

nE Vε ψ ψ= = − ∇ +∑ ∑

)exp()( rGicrG

nGn ⋅=∑ψ

2 2

,

1 ( )2

n n nG G G

n G G G

E c G c c V G G∗′

′

⎛ ⎞′= + −⎜ ⎟

⎝ ⎠∑ ∑ ∑

Dynamical evolution of c’s

We need the “force”

}][{ iEE ψ=i

ii

EFδψψδ }][{−=

iHψˆ−=

Skiing down a valley

( )i i iF Hµψ ψ= = −

( )i i iF Hψ ψ= = −

“Damped” dynamics

skiing

SD or CG skiing

Hellmann-Feynman theorem

ˆ

ˆ ˆ

ii i

i i

d HdEFdR dR

dH dVdR dR

Ψ Ψ= − = − =

= Ψ − Ψ = Ψ − Ψ

Born-Oppenheimer Molecular Dynamics

ˆi i i

i

dVm R FdR

= = Ψ − Ψ

Lots of Skiing if Atoms Move

Lots of Skiing if Atoms Move

The extended CP Lagrangian

Equations of motion

Equations of motion (II)

Constant(s) of motion

20 0

20 0

0 0

1 1 ˆ2 21 ˆ2ˆ

12

cons i i i I I ei I

phys I I e cons eI

e e

e i i ii

E M R H

E M R H E T

V H

T

µ ψ ψ

µ ψ ψ

= + + Ψ Ψ

= + Ψ Ψ = −

= Ψ Ψ

=

∑ ∑

∑

∑

Kolmogorov-Arnold-Moser invariant tori

Born-Oppenheimer vs Car-Parrinello

HF vs CP forces

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

Full class (videos, etc..)

Atomistic Modeling of Materials

http://ocw.mit.edu/OcwWeb/Materials-Science-and-Engineering/3-320Spring-2005/CourseHome/index.htm

Thanks !

• Amy Young• Paolo Giannozzi and the www.quantum-

espresso.org gang• Todd Martinez, Richard Martin, Duane

Johnson, David Ceperley• Axel Kohlmeyer, Stefano Baroni, Guido

Fratesi