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
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