Stochastic Molecular Dynamics Derived
from the Time-independent Schrödinger
Equation
Anders Szepessy
Schrödinger observables approximated by:
• Ehrenfest dynamics O(M −1)
• Born-Oppenheimer dynamics O(M −1/2)
• Langevin and Smoluchowski dynamics O(M −1 + gap−1)
Which Molecular Dynamics?
M Ẍ = −V 0(X) + ?
• What is V ?
• Are there fluctuations? How model temperature T ?
• Is there dissipation?
The Usual Derivation and the New
Usual start from time-dependent Schrödinger equation:
• self consistent field equation: wave function is product of nuclei
and electron function
• Ehrenfest dynamics: nuclei wave function becomes point measure
• Born-Oppenheimer approximation: electron wave function is the
ground state.
The Usual Derivation and the New
Usual start from time-dependent Schrödinger equation:
• self consistent field equation: wave function is product of nuclei
and electron function
• Ehrenfest dynamics: nuclei wave function becomes point measure
• Born-Oppenheimer approximation: electron wave function is the
ground state.
New start from time-independent Schrödinger equation:
• use WKB-Ansatz
• introduce time by momentum from characteristics of Eikonal eq.
• introduce integrating factor from divergence of momentum
• stability from consistency with Schrödinger equation.
Difference Between the Two
+ experimental evidence of time-independent Schrödinger
+ nuclei paths behave classically without separation and small support
+ long time stability
+ stochastic perturbation of ground state leads to Langevin
+ colliding characteristic paths
– only equilibrium situations
Dynamics from the Time-independent Schrödinger
Schrödinger: H(x, X)Φ(x, X) = EΦ(x, X)
N
1 X
H(x, X) = V (x, X) −
∆X n ,
2M n=1
J
1X
∆xj +
V (x, X) = −
2 j=1
−
N X
J
X
n=1 j=1
X
1≤k<j≤J
Zn
+
|xj − X n|
M 1
1
|xk − xj |
X
1≤n<m≤N
ZnZm
|X n − X m|
The WKB-Ansatz
Φ(x, X) = ψ(x, X)eiM
1/2 θ(X)
implies
iM 1/2 θ(X)
0 = (H − E)ψe
|θ0|2
= (
+ Vn − E) ψ
| 2 {z
}
=0
i
1 00
0
0
− 1/2 (ψ ◦ θ + ψθ ) + (V − Vn)ψ
2
M
1 00 iM 1/2θ
−
ψ e
,
2M
ψ·Vψ
Vn :=
,
ψ·ψ
Z
v · w :=
R3J
v ∗(x, X, t)w(x, X, t) dx
The Time-dependence
The Time-dependence1
dXt dψ(x, Xt)
ψ ◦θ =ψ ◦
=
dt
dt
|{z}
0
0
0
=:p
1
Mott 1931; Briggs and Rost 2001
The Time-dependence2
dXt dψ(x, Xt)
ψ ◦θ =ψ ◦
=
dt
dt
|{z}
0
0
0
=:p
d
θ = div p = log |p|
dt
yields the Eikonal and transport equation for ϕ := |p|1/2ψ
00
0 = (H − E)Φ
|θ0|2
+ Vn − E) ψ
= (
| 2 {z
}
=0
1/2
i
1
+ − 1/2 ϕ̇ + (V − Vn)ϕ −
|p|1/2∆X (ϕ|p|−1/2) |p|−1/2 eiM θ .
2M
| M
{z
}
=0
2
Mott 1931; Briggs and Rost 2001
Characteristics of the Schrödinger Equation
i
1
|p|1/2∆X (ϕ|p|−1/2)
− 1/2 ϕ̇ + V (Xt) − Vn ϕ =
2M
M
Characteristics of Eikonal equation:
Ẋt = pt
ṗt = −Vn0(Xt)
ϕ·Vϕ
Vn =
ϕ·ϕ
The Ehrenfest Approximation
Ẋt = pt
ṗt = −φt · V 0(Xt)φt
i
φ̇t = V (Xt)φt
M 1/2
is Hamiltonian system for HJ
HE := |p|2/2 + φ · V (X)φ = E
with characteristics (X, ϕr ; p, ϕi) and ϕ := 21/2M −1/4φ;
and ψ̂t := φte
R
iM 1/2 0t φs ·V (Xs )φs ds
implies
i ˙
ψ̂t = (V − Vn)ψ̂t
M 1/2
Ehrenfest Accuracy
Φ̂ := ρ̂
1/2
iM 1/2 θ̂
ψ̂e
implies
1
∆X (ρ̂1/2ψ̂)
2M
= O(M −1)
(H − E)Φ̂ =
so that
Z
Z
g(X) ρ̂ − ρ)dX = g(X) Φ̂ · Φ̂ − Φ · Φ dX = O(M −1)
Motivation for Stable Eigenstate Perturbation
Orthonormal eigenpairs {λn, Φn}, satisfying HΦn = λnΦn,
X
Φ̂ =:
αnΦn
n
yields
X
n
1
(λn − E)αnΦn = −
v,
2M
which establishes
Z
1
(λn − E)αn = −
Φn · v dX .
2M | T3N {z
}
=: v̂n
We have v̂n = 0, when λn = E, and let
|v̂n0 (E)| := lim sup
δ→0+
|v̂n(E + δ)|
,
δ
which implies
X |v̂n(λn)|2
≤
2
|λ
−
E|
n
n
X
|v̂n0 (E)|2
+
X
|v̂n|2.
n
{n:|λn −E|<1}
Assume that
X
{n:|λn −E|<1}
|v̂n0 (E)|2
+
X
|v̂n|2 = O(1).
(1)
n
Motivation for bounded |v̂ 0|:
Large number of nuclei N M
δ perturbation of
λn = |p|2/2 + Vn
yields O(δN −1) perturbation of paths
(X, p, ψ)
and also θ change negligible (with appropriate time) so eigenstate
change small
1/2
Φ = ψeiM θ
The Born-Oppenheimer Approximation
An electron eigenstate: ψ̂ = Ψn V (X)Ψn = λn(X)Ψn
implies
Ẋt = pt
ṗt = −λ0n(Xt)
(H − E)Φ̂ = O(M −1/2)
and
Z
Z
g(X) ρ̂ − ρ)dX =
g(X) Φ̂ · Φ̂ − Φ · Φ dX = O(M −1/2)
Stochastic Molecular Dynamics Approximation
Improve Born-Oppenheimer
Ehrenfest solution ψ̂n = Ψn + ψn⊥
ψn⊥(t)
=
St,0ψn⊥(0)
− iM
−1/2
Z
t
St,sΨ̇n(s)ds
0
so perturbation of ground state Ψ0 yields
• fluctuation from stochastic initial data ψ̂n, n > 0
• dissipation from residual Ψ̇0(s) = Ψ00 ◦ Ẋ
Which Initial Data for ψ̂n ?
Liouville eq. for Ehrenfest
∂tf + ∂pE HE ∂rE f − ∂rE HE ∂pE f = 0
has many time-independent solutions f = h(HE )
Which Initial Data for ψ̂n ?
Liouville eq. for Ehrenfest
∂tf + ∂pE HE ∂rE f − ∂rE HE ∂pE f = 0
has many time-independent solutions f = h(HE )
Idea: Nuclei act as heat bath for electrons
independent Ẋn imply h(HE ) = e−HE /T
(nuclei has this (unique) invariant SDE probability density)
so
Prob(electron configuration ψ̂) ∼ e−ψ̂·(V −λ0)ψ̂/T dψ̂ r dψ̂ i.
Stochastic Ehrenfest Dynamics
Let
r(X) :=
YZ
j≥0
−λ̄j |γj |2 /(T
e
2
j≥0 |γj | )
P
|γj |2 <C
dγjr
dγji
Y T
∼
( )1/2,
λ̄
j≥0 j
then observable in the Ehrenfest dynamics is
R
R
−λ0 (X)/T r(X)
−λ0 (X)/T
g(X)e
3N
g(X)e
r(X)dX
R
r(0) dX
R3N
R
.
= R
−λ
(X)/T
r(X)
0
−λ
(X)/T
r(X)dX
0
dX
3N e
R3N e
R
r(0)
The Spectral Gap Condition
A large spectral gap α :=
−1
j>0 λ̄j
P
1 yields
r(X)
= 1 + O(α),
r(0)
which implies
R
R
−λ0 (X)/T
−λ0 (X)/T
g(X)e
r(X)dX
g(X)e
dX
3N
3N
RR
RR
=
+ O(α).
−λ0 (X)/T r(X)dX
−λ0 (X)/T dX
e
e
R3N
R3N
Langevin and Smoluchowski Dynamics
The stochastic Langevin dynamics
dXt = ptdt
√
dpt = −∂X λ0(Xt)dt − Kptdt + 2T KdWt
and the Smoluchowski dynamics
dXs = −∂X λ0(Xs)ds +
√
2T dWs
has the unique invariant probability density
e−(p◦p/2+λ0(X))/T dp dX
R
−(p◦p/2+λ0 (X))/T dp dX
e
6N
R
respectively
e−λ0(X)/T dX
R
−λ0 (X)/T dX
R3N e
Stochastic Approximation Theorem
Langevin and Smoluchowski dynamics approximate Schrödinger observables with error
O(M −1 + α)
provided the assumption in EhrenfestP
approximation holds together
with the spectral gap condition α = j>0 λ̄−1
j 1.
Conditions for Colliding Characteristics Paths (Caustics)
At a point X of two colliding characteristic paths, (X, p−, z −)
and (X, p+, z +), we need:
• the phase θ is continuous, i.e. z − = z +,
• a stable ϕ,
i
|p|1/2 X
−1/2
∆
(|p|
ϕ̇
=
(V
−
V
)ϕ
−
ϕ),
j
n
X
1/2
2M j
M
so take
|p+| = |p−|,
• θ is max-norm stable towards perturbations of the initial data,
which implies the irreversible viscosity solution θ.
SPDE from Smoluchowski MD with Erik von Schwerin
Energy conservation:
∂t(cv T + m) = div(k∇T )
Phase field for m = g(φ):
V
k0∂tφ = div(k1∇φ) − V 0(φ) + k2T + noise
Why noise?
m
SPDE from Smoluchowski MD with Erik von Schwerin
Energy conservation:
∂t(cv T + m) = div(k∇T )
Phase field for m = g(φ):
V
k0∂tφ = div(k1∇φ) − V 0(φ) + k2T + noise
Why noise?
m
Which Noise and Phase Field Equation?
1. Stochastic Smoluchowski molecular dynamics
2. Quantitative atomistic definition of phase field
3. Numerical computation of coarse-grained model functions
1. Stochastic Molecular Dynamics
Energy:
X |vi|2
+ V (X1, . . . , XN )
|
{z
}
2
m
| i {z }
cv T
Smoluchowski dynamics in diffusion time scale (γ = kB T ):
p
t
t
dX = −∂V (X )dt + 2γ dW t
2. Molecular Potential Energy
Liquid
Subcooled Liquid
m
Latent heat, L
m
Superheated Solid
Solid
TM
T
Pair interactions
1 XX
V (X) =
Ṽ (Xi − Xj )
2 i
j6=i
Localized average (as SPH)
1 XX
m(X, x) :=
Ṽ (Xi − Xj )η(x − Xi)
2 i
j6=i
{z
}
|
observable
3. Coarse-Grained Stochastic MD
Want coarse-grained approximation m̄
X
t
t
bk (m̄t)dW̄kt
dm̄ = a(m̄ )dt +
k
such that
T
T
min E g m(X , ·) − g(m̄ ) .
a,b
1. Ito implies
t
t
dm(X , ·) = α(X )dt +
X
βj (X t)dWjt.
j
2. Kolmogorov equation for ū(n, t) := E[g(m̄T ) | m̄t = n]
T
T
E g m(X , ·) − g(m̄ )
Z T
X
X
0
00
= E[
hū , α − ai + hū ,
βj ⊗ βj −
bk ⊗ bk i dt]
0
j
k
3. Expansion in α − a
Z T
1
a= E
α dt],
T
0
Z
X
1 TX
b k ⊗ bk = E
βj ⊗ βj dt].
T
0
j
k
Coarse-Grained Variables
Density Phase Field : ρloc(*;x)
Potential Energy Phase Field : m(*;x)
0.6
1.31
0.6
T d2m/dx2 (x)
m(x)
0.4
0.4
1.3
0.2
0.2
1.29
0
0
1.28
−0.2
−0.2
−0.4
1.27
−0.4
−0.6
1.26
−0.6
−0.8
1.25
−0.8
−1
−1.2
−40
−20
0
20
40
60
x
80
100
120
140
1.24
−40
−20
0
20
1
1. α = γ∂xxm + ∂xA1 + A0
RT
2. a = 0 αdt/T → 0.
40
60
x
1
80
100
120
140
−1
−10
0
10
20
x1
30
40
50
60
Drift
0.5
f(m)
0.4
0.3
0.3
0.2
0.2
0.1
0.1
0
−1
f(m;1.0)
1.492 f(m;2.0)
2
0.885 f(m;0.7)
0.4
Orientation 1
Orientation 2
0
−0.5
0
m
0.5
−0.1
−1
−0.8
−0.6
−0.4
−0.2
m
0
0.2
0.4
0.6
Diffusion
35
3
3
30
25
30
2.5
2.5
2
2
20
1.5
20
10
1
0
100
1.5
15
0.5
1
50
100
10
0
100
50
y1
0
0
0.5
5
50
x1
0
y1
0
0
20
40
x1
60
80
100
0
Orientation dependence
Let
qi := A0(Oi)/A0(O1)
ri := τ (Oi)/τ (O1)
Then
dm̄i =
riqi γqi−1m̄00i
1/2
+ A0(m̄i; O1) dt + ri b̄(m̄i; Oi)dW t
becomes
−1
(riqi) dm̄ =
γqi−1m̄00
−1/2 −1
qi b̄(m̄; Oi)dW t
+ A0(m̄; O1) dt + ri
Extensions
• more directions
• constant pressure
• undercooled melt
• real material
• hybrid simulations
3. Expansion in α − a
m(X t, x) ≈ m̄(x)
leads to
E
Z
T
0
t
ū m(X , x), t , α(x) − a m̄(x) dt ≈
0
Z
0
T
ū m̄(x), T , E
α(x) − a m̄(x) dt
0
2. Derivation of the Error
T
T
E g m(X , ·) − g(m̄ )
2. Derivation of the Error
T
T
E g m(X , ·) − g(m̄ )
= E[ū(mT , T ) − ū(m0, 0)]
2. Derivation of the Error
T
T
E g m(X , ·) − g(m̄ )
= E[ū(mT , T ) − ū(m0, 0)]
RT
= E[ 0 dū(mt, t)]
2. Derivation of the Error
T
T
E g m(X , ·) − g(m̄ )
= E[ū(mT , T ) − ū(m0, 0)]
RT
= E[ 0 dū(mt, t)]
hR
i
P
T
= E 0 hū0, αi + hū00, j βj ⊗ βj i + ∂tū dt
2. Derivation of the Error
T
T
E g m(X , ·) − g(m̄ )
= E[ū(mT , T ) − ū(m0, 0)]
RT
= E[ 0 dū(mt, t)]
hR
i
P
T
= E 0 hū0, αi + hū00, j βj ⊗ βj i + ∂tū dt
hR
i
P
P
T
= E 0 hū0, α − ai + hū00, j βj ⊗ βj − k bk ⊗ bk i dt
Factorization of the density
The Ehrenfest equation
|θ̂0|2
1
i
0=
+ V − E ψ̌ − 1/2 (ψ̌ 0θ̂0 + ψ̌ θ̂00)
2
2
M
yields
Z
XZ
0=
(∂X j ψ̌ ∗ψ̌ + ψ̌ ∗∂X j ψ̌)dx ∂X j θ̂ +
ψ̌ ∗ψ̌dx ∂X j X j θ̂
j
=
X
T3J
T3J
∂X j (ρ̂∂X j θ̂)
j
and
X
j
∂X j X j θ = div p =
∂p11
∂X11
=
∂p11
∂t
∂X11
∂t
ṗ11 dtd |p|2
d
= 1=
=
log |p|
2
p1
2|p|
dt
implies
˙ X̂t) =
ρ̂(
X
=
X
˙
∂X̂ j ρ̂(X̂t)X̂ j
j
∂X̂ j ρ̂(X̂t)∂X̂ j θ̂n
j
= −ρ̂(X̂t)
X
∂X̂ j X̂ j θ̂n
j
= −ρ̂(X̂t) div p̂
d
= −ρ̂(X̂t) log |p̂t|
dt
with the solution
C
,
|p̂t|
where C is a positive constant for each characteristic. Coordinates
X11 ∈ R parallel and X0 ∈ R3N −1 orthogonal to the characteristic
ρ̂(X̂t) =
direction Ẋ gives
ρ̂(X)dX = ρ̂(X0) R
dt
dX11
= ρ̂(X0)dX0 ,
|p̂11|
T
dX0
X11 (T ) dX11
0
|p̂11 |
using
|dX̂11|
dX̂11
= dt,
=
1
1
|p̂1|
|dX̂1 /dt|
and the observable
Z
Z
g(X̂, ψ̂, ρ̂)dX̂ =
T3N
0
T
Z
A(Xt)ρ̂(X̂0)dX̂0
I
dt
.
T
(2)
Dynamics from the time-dependent Schrödinger
i∂tΦ(x, X, t) = H(x, X)Φ(x, X, t)
N
1 X
H(x, X) = V (x, X) −
∆X n ,
2M n=1
J
1X
∆j+
V (x, X) = −
2 j=1 x
−
N X
J
X
n=1 j=1
J
X
=: −
1
2
X
1≤k<j≤J
j=1
1
|xk − xj |
Zn
+
|xj − X n|
∆xj + HI
M 1
X
1≤n<m≤N
ZnZm
|X n − X m|
Z
v ∗(x, X, t)w(x, X, t) dx
ZR3J Z
hv, wi :=
v ∗(x, X, t)w(x, X, t) dx dX
v · w :=
R3N
R3J
Usual derivation:
1. self consistent field equation: wave function is product of nuclei
and electron function
2. Ehrenfest dynamics: nuclei wave function becomes point measure
3. Born-Oppenheimer approximation: electron wave function is the
ground state.
1. Time-dependent self consistent field equations
Approximation Ansatz of separation
Z t
s s
s s
Φ(x, X, t) = ΨN (X, t)ΨE (x, t) exp i
hΨN ΨE , HI ΨE ΨN i ds
{z
}
|
0
H̄I
satisfies time dependent self consistent field equation3
N
X
− (2M )−1
∆Xn + ΨE · HI (X)ΨE ΨN ,
n=1
Z
∗
i∂tΨE =
ΨN (X)V (X)ΨN (X) dX ΨE .
i∂tΨN =
R3N
3
Dirac P.A.M., Proc. Cambridge Phil. Soc. 26 (1930) 376–385.
Φ solves perturbed full Schrödinger
N
X
J
1X
−1
i∂tΦ = − (2M )
∆Xn −
∆xj + ΨE · HI ΨE
2 j=1
n=1
Z
+
Ψ∗N HI ΨN dX − H̄I Φ,
R3N
and compactly supported ΨN in δ small domain leads4 to O(δ)
approximation of full Schrödinger in L2(dxdX).
4
Bornemann F.A., Nettesheim P. and Schütte C., J. Chem. Phys, 105 (1996) 1074–1083.
2. Ehrenfest dynamics from WKB
ΨN = ψeiM
1/2 θ
leads to Ehrenfest
Ẍ = −ΨE · ∂X V (X)ΨE
iM −1/2Ψ̇E = V ΨE
(X, ΨE ) approximates5 TDSCF with error O(δ 2 + M −1/2)
5
Bornemann F.A., Nettesheim P. and Schütte C., J. Chem. Phys, 105 (1996) 1074–1083.
Tully J.C., Faraday Discuss., 110 (1998) 407–419.
Marx D. and Hutter J., Ab initio molecular dynamics: Theory and implementation, Modern Methods and Algorithms of
Quantum Chemistry, J.Grotendorst(Ed.), John von Neumann Institute for Computing, Jülich, NIC Series, Vol. 1, ISBN
3-00-005618-1, pp. 301-449, 2000
3. Born-Oppenheimer approximation: ΨE = ground state
An electron eigenstate: ΨE = Ψn
V (X)Ψn = λn(X)Ψn
implies
Ẍ = −Ψn · ∂X V (X)Ψn = −∂X λn(X)
Spectral gap can be used to prove6 O(M −1/4) approximation of
Schrödinger in L2.
6
Hagedorn G.A., Commun. Math. Phys., 77 (1980) 1–19.
Panati G., Spohn H. and Teufel S., Math. Mod. Numer. Anal.