3.320: Lecture 15 (Mar 31 2005)
FIRST-PRINCIPLES 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
12
10
8
6
4
2
0
-2
-3
-2
-1
0
1
2
Figure by MIT OCW.
Mar 31 2005
3.320 Atomistic Modeling of Materials -- Gerbrand Ceder and Nicola Marzari
3
Micro- to macro- : diffusion coefficient
• From Fick to Einstein:
∂c(r , t )
= D∇ 2 c(r , t )
∂t
r 2
r 2 2
∂
d
r
r
c
(
r
,
t
)
=
D
d
r
r
∇
c
(
r
,
t
)
∫
∂t ∫
∂ 2
r (t ) = 2dD
∂t
Mar 31 2005
3.320 Atomistic Modeling of Materials -- Gerbrand Ceder and Nicola Marzari
Mean Square Displacements
∆r (t ) 2
Mar 31 2005
1
=
N
N
2
∆
r
(
t
)
∑ i
i =1
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
∆x(t )
2
1
=
N
∆x(t )
2
∑ ∆xi (t )
∆x(t ) = ∫ dt ′ v x (t ′)
2
i =1
⎛
⎞
= ⎜⎜ ∫ dt ′v x (t ′) ⎟⎟
⎝0
⎠
t
t
N
2
0
t
t
0
0
= ∫ dt ′∫ dt ′′ v x (t ′)v x (t ′′)
t
t′
0
0
= 2 ∫ dt ′∫ dt ′′ v x (t ′)v x (t ′′)
Mar 31 2005
3.320 Atomistic Modeling of Materials -- Gerbrand Ceder and Nicola Marzari
Green-Kubo relations
2 D = limt →∞
∂ x 2 (t )
∂t
t′
= limt ′→∞ 2 ∫ dt ′′ vx (t ′)vx (t ′′)
0
v x (t ′)v x (t ′′) = v x (t ′ − t ′′)v x (0)
t′
∞
0
0
D = limt ′→∞ ∫ dt ′′ vx (t ′ − t ′′)vx (0) = ∫ dτ vx (τ )vx (0)
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
d ⎛⎜ ∂L
dt ⎜⎝ ∂q& j
⎞ ∂L
⎟−
=0
⎟ ∂q
j
⎠
(the dot is a time derivative)
• Why ? We can use generalized coordinates.
Also, we only need to think at the two
scalar functions T and V
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)
d ⎛⎜ ∂L
dt ⎜⎝ ∂q& j
⎞ ∂L
⎟−
=0
⎟ ∂q
j
⎠
⎛ ⎛1 2⎞⎞
∂ ⎜ mx& ⎟ ⎟
⎜
∂V
d
2
⎝
⎠
⎜
⎟+
=0
dt ⎜
∂x&
⎟ ∂x
⎜
⎟
⎝
⎠
Mar 31 2005
d
∂V
( mx& ) = −
∂x
dt
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
∂L
pi =
∂q&i
H (q, p, t ) = ∑ q&i pi − L(q, q& , t )
i
∂H
q&i =
∂pi
Mar 31 2005
∂H
− p& i =
∂qi
3.320 Atomistic Modeling of Materials -- Gerbrand Ceder and Nicola Marzari
Thermostats, barostats…
• We might want to sample a constanttemperature 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
LNOSE
1
1 2 (3 N + 1)
2 2
= ∑ mi s r&i − V + Qs& −
ln s
β
2
i 2
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
r r
Gi ⋅ a j = 2πδ ij
Mar 31 2005
3.320 Atomistic Modeling of Materials -- Gerbrand Ceder and Nicola Marzari
It’s really kinetic + potential
1
r
2
Hˆ = − ∇ + V (r )
2
r
r
r
n
r exp(i G ⋅ r )
ψ n (r ) = ∑
c
G
r
G
E = ∑n ε n = ∑ ψ n Hˆ ψ n
n
Mar 31 2005
3.320 Atomistic Modeling of Materials -- Gerbrand Ceder and Nicola Marzari
Kinetic energy
Ekin = ∑
n
1 2
ψn − ∇ ψn
2
r r
r
n
r exp(i G ⋅ r )
ψ n (r ) = ∑
c
G
r
G
1 2
1 2⎤
1 2
⎡
G − ∇ G′ = ∫ dr exp(−iGr ) ⎢− ∇ ⎥ exp(iG′r ) = G δ G ,G′
2
2
⎣ 2 ⎦
Ekin
Mar 31 2005
1
2
n 2
r
=∑ ∑
c
G
G
r
2
n
G
3.320 Atomistic Modeling of Materials -- Gerbrand Ceder and Nicola Marzari
Total energy (non-SCF)
E pot = ∑
r
ψ n V (r ) ψ n
n
r r
r
n
r exp(i G ⋅ r )
ψ n (r ) = ∑
c
G
r
G
G V (r ) G′ = ∫ dr exp(−iGr )V (r ) exp(iG′r ) = V (G − G′)
r r ⎞
⎛1
2
n 2
n∗ n
r
r c r V (G − G ′)
Etot = ∑ ⎜ ∑
c
G
+
c
⎟
∑
G
r
r r G G′
n ⎝2 G
G ,G ′
⎠
Mar 31 2005
3.320 Atomistic Modeling of Materials -- Gerbrand Ceder and Nicola Marzari
Dynamical evolution of c’s
r r ⎞
⎛1
2
n 2
n∗ n
r
r c r V (G − G ′)
Etot = ∑ ⎜ ∑
c
G
+
c
⎟
∑
G
r
r r G G′
n ⎝2 G
G ,G ′
⎠
Mar 31 2005
3.320 Atomistic Modeling of Materials -- Gerbrand Ceder and Nicola Marzari
We need the force
E = E[{ψ i }]
δE[{ψ i }]
Fi = −
δψ i
= − Hˆ ψ i
Mar 31 2005
3.320 Atomistic Modeling of Materials -- Gerbrand Ceder and Nicola Marzari
Skiing down a valley
µψ&&i = − Hψ i
ψ& i = − Hψ i
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
r
d Ψ Hˆ Ψ
dE
r
Fi = − r = −
=
dRi
dRi
dHˆ
dVˆ
= Ψ − r Ψ = Ψ − r Ψ
dRi
dRi
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
r&& r
dVˆ
mi Ri = Fi = Ψ − r Ψ
dRi
Mar 31 2005
3.320 Atomistic Modeling of Materials -- Gerbrand Ceder and Nicola Marzari
The extended Car-Parrinello
Lagrangian
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-unibochum.de/research/marx/cprev.en.html
Mar 31 2005
3.320 Atomistic Modeling of Materials -- Gerbrand Ceder and Nicola Marzari