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First-principles calculatiton combined with multicanonical ensemble method
Yoshihide YOSHIMOTOMaterials Design and Characterization Laboratory
Institute for Solid State PhysicsUniversity of Tokyo
Yoshimoto, JCP 125, 184103 (2006)
Multi-order Multi-thermal ensembleand
Thermodynamic downfolding
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Contents
• Introduction
• multi-order multi-thermal ensemble by structure factor
• thermodynamic downfolding
• results for Xtal↔Liq transition of Si
• results for Xtal↔Liq transition of MgO
Yoshimoto, JCP 125, 184103 (2006)
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DSC Cycle
developing materials
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DSC CycleDesign, Synthesis, Characterization (MDCL, ISSP)
developing materials
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DSC CycleDesign, Synthesis, Characterization (MDCL, ISSP)
D
S C
developing materials
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DSC CycleDesign, Synthesis, Characterization (MDCL, ISSP)
D
S Cmajor part of
first-principles works
minor part offirst-principles works
? ? ?
developing materials
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DSC CycleDesign, Synthesis, Characterization (MDCL, ISSP)
D
S Cmajor part of
first-principles works
minor part offirst-principles works
? ? ?
crystallization asa basic synthesis developing
materials
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Purpose of development
✦ Capture the transition directly
➡ By multicanonical method
• Without simulations of co-existing state
✓ keep simulated system small
• Without ladder ofthermodynamic integration/adiabatic switching
✓ No need for a reference system
✦ Optimized model potential based on multicanonical ensemble and ab-initio calc.
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multicanonical ensembleA generalized ensemble
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multicanonical ensembleBerg and Neuhaus, PRL 68, 9 (1992) Lee, PRL 71, 211(1993) e!!E !W (E)!1 = e!S(E)
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multicanonical ensemble
• Wide range of E is visited with equal prob.
➡overcome free energy barriers rapidly
➡capture the transition in principle
• produces entropy, S(E) [as weight]
• Re-weighting of the simulation run produces any thermal average at wide range of T
➡Simul. run represents the total thermodynamics of the system
Berg and Neuhaus, PRL 68, 9 (1992) Lee, PRL 71, 211(1993)
Wang and LandauPRL, 86, 2050 (2001)
e!!E !W (E)!1 = e!S(E)
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multicanonical ensemble
• Wide range of E is visited with equal prob.
➡overcome free energy barriers rapidly
➡capture the transition in principle
• produces entropy, S(E) [as weight]
• Re-weighting of the simulation run produces any thermal average at wide range of T
➡Simul. run represents the total thermodynamics of the system
Berg and Neuhaus, PRL 68, 9 (1992) Lee, PRL 71, 211(1993)
Wang and LandauPRL, 86, 2050 (2001)
e!!E !W (E)!1 = e!S(E)
P (E) !W (E)e!!E
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multicanonical ensemble
• Wide range of E is visited with equal prob.
➡overcome free energy barriers rapidly
➡capture the transition in principle
• produces entropy, S(E) [as weight]
• Re-weighting of the simulation run produces any thermal average at wide range of T
➡Simul. run represents the total thermodynamics of the system
Berg and Neuhaus, PRL 68, 9 (1992) Lee, PRL 71, 211(1993)
Wang and LandauPRL, 86, 2050 (2001)
e!!E !W (E)!1 = e!S(E)
P (E) !W (E)W (E)!1 = 1
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multicanonical ensemble
• Wide range of E is visited with equal prob.
➡overcome free energy barriers rapidly
➡capture the transition in principle
• produces entropy, S(E) [as weight]
• Re-weighting of the simulation run produces any thermal average at wide range of T
➡Simul. run represents the total thermodynamics of the system
Berg and Neuhaus, PRL 68, 9 (1992) Lee, PRL 71, 211(1993)
Wang and LandauPRL, 86, 2050 (2001)
e!!E !W (E)!1 = e!S(E)
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multicanonical ensemble
• Wide range of E is visited with equal prob.
➡overcome free energy barriers rapidly
➡capture the transition in principle
• produces entropy, S(E) [as weight]
• Re-weighting of the simulation run produces any thermal average at wide range of T
➡Simul. run represents the total thermodynamics of the system
Berg and Neuhaus, PRL 68, 9 (1992) Lee, PRL 71, 211(1993)
Wang and LandauPRL, 86, 2050 (2001)
e!!E !W (E)!1 = e!S(E)
!A" =!
i Ai!i 1
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multicanonical ensemble
• Wide range of E is visited with equal prob.
➡overcome free energy barriers rapidly
➡capture the transition in principle
• produces entropy, S(E) [as weight]
• Re-weighting of the simulation run produces any thermal average at wide range of T
➡Simul. run represents the total thermodynamics of the system
Berg and Neuhaus, PRL 68, 9 (1992) Lee, PRL 71, 211(1993)
Wang and LandauPRL, 86, 2050 (2001)
e!!E !W (E)!1 = e!S(E)
!A" =!
m AmW (Em)e!!Em
!m W (Em)e!!Em
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multicanonical ensemble
• Wide range of E is visited with equal prob.
➡overcome free energy barriers rapidly
➡capture the transition in principle
• produces entropy, S(E) [as weight]
• Re-weighting of the simulation run produces any thermal average at wide range of T
➡Simul. run represents the total thermodynamics of the system
Berg and Neuhaus, PRL 68, 9 (1992) Lee, PRL 71, 211(1993)
Wang and LandauPRL, 86, 2050 (2001)
Ferrenberg and SwendsenPRL61, 2635 (1988); 63 1658 (1989)
e!!E !W (E)!1 = e!S(E)
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Limitation of multicanonical
0 5
10 15 20 25 30
0 0.5 1 1.5 2
O
t [108 a.u.]
multicanonical
Simulation by Wang-Landau algorithm to determine S(U)
no tunneling from Liq. to Xtal. of Si
Xtal.
Liq.
O: An order parameter of crystallization
Wang and Landau, PRL, 86 (2001) 2050
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• sampling probability in MD
• entropy [mult.-ca. weight]
Order param. as 2nd deg. of freedom
P (U,O)
Multi-Order Multi-Thermal: MOMT
explicitly generate random walk between ordered and disordered states
keep tunneling time in MD short
S(U,O)
two-component multicanonical: Higo et al., 1997 multibaric multithermal: Okumura et al., 2004
Wang and Landau, 2001
Crystalline order Structure factorcf. Bragg spots in diffraction experiments
potential energy
order param.
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Efficiency of MOMT
0 5
10 15 20 25 30
0 0.5 1 1.5 2
O
t [108 a.u.]
multicanonical
0 5
10 15 20 25 30
0 0.5 1 1.5 2
O
t [108 a.u.]
multi-order multi-thermal
-11
-10.5
-10
-9.5
-9
-8.5
0 0.5 1 1.5 2
U [H
t.]
t [108 a.u.]
multi-order multi-thermal
-11
-10.5
-10
-9.5
-9
-8.5
0 0.5 1 1.5 2
U [H
t.]
t [108 a.u.]
multicanonical
by Wang-Landau algorithm
short tunneling time between Liq. and Xtal. of Si
Xtal.
Liq.
Xtal.
Liq.
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multi-order multi-thermal MD run
Repeating transition between liquid and crystal
variable cell simulation
impuritiesin a crystal
amorphous
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Still too much steps for first-principles MD
0 5
10 15 20 25 30
0 0.5 1 1.5 2
O
t [108 a.u.]
multicanonical
0 5
10 15 20 25 30
0 0.5 1 1.5 2
O
t [108 a.u.]
multi-order multi-thermalXtal.
Liq.
Xtal.
Liq.
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Still too much steps for first-principles MD
• > 106 steps required (1 step ~ 100 a.u.)
0 5
10 15 20 25 30
0 0.5 1 1.5 2
O
t [108 a.u.]
multicanonical
0 5
10 15 20 25 30
0 0.5 1 1.5 2
O
t [108 a.u.]
multi-order multi-thermalXtal.
Liq.
Xtal.
Liq.
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Still too much steps for first-principles MD
• > 106 steps required (1 step ~ 100 a.u.)
←available steps for first-principles MD
~ 104 steps
0 5
10 15 20 25 30
0 0.5 1 1.5 2
O
t [108 a.u.]
multicanonical
0 5
10 15 20 25 30
0 0.5 1 1.5 2
O
t [108 a.u.]
multi-order multi-thermalXtal.
Liq.
Xtal.
Liq.
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Still too much steps for first-principles MD
• > 106 steps required (1 step ~ 100 a.u.)
←available steps for first-principles MD
~ 104 steps
• even more light weight method is required
0 5
10 15 20 25 30
0 0.5 1 1.5 2
O
t [108 a.u.]
multicanonical
0 5
10 15 20 25 30
0 0.5 1 1.5 2
O
t [108 a.u.]
multi-order multi-thermalXtal.
Liq.
Xtal.
Liq.
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Still too much steps for first-principles MD
• > 106 steps required (1 step ~ 100 a.u.)
←available steps for first-principles MD
~ 104 steps
• even more light weight method is required
downfolding of inter-atomic potential from a first-principles one to a model one
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Thermodynamic downfolding of U
• Multicanonical simul. : typical config.
• Re-weighting of produces any thermodynamic quantities at any T
➡ : a measure of a potential
• Optimize model potential via function L
• Self-consistency for
➡ depends on UM
!U(X) = UM (X)! UA(X)
Q = {Xi}Q
Q
QQ
model accurate
L =!
Xi!Q[!U(Xi) ! "!U#]2
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Iterative optimize of UM
UM
Q
opt UM by L(UM; Q, UA on Q)
MD/MCFPC
UA on Q
feed
back
embarrassingly parallel
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downfolded model potential of Si
Original EDIP, Justo et al. PRB 58 (1998) 2539
Stillinger Weber
optimized EDIP by DForiginal EDIP
Stillinger-Weber, PRB 31 (1985) 5262
UA: first-principles
UM: model
-0.5
0
0.5
1
1.5
2
2.5
UM
! U
A [H
t.]
0
2
4
6
8
10
12
0 0.5 1 1.5 2
!F
[10
-2 H
t. a
.u.-1
]
UA [Ht.]
Erro
r in
For
ce
per
atom
diff.
in p
ot. e
nerg
y
each point: sampled config.
clearly superior potential was obtained
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Targets ofThermodynamic Downfolding
+ Multiorder Multithermal
Classification by type of bonding Instances
Covalent Silicon, Diamond
Ionic MgO, SiO2
Molecular H2O
Metallic Fe, Al
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Why crystal⇔liquid transition of Silicon
• Large ΔS ~ 3kB/mol for the transition
‣ comparable to that of water
‣ typical elemental substances: ΔS ~ 1kB/mol
• Overcooled as much as 300K (m.p. 1685K)
It is hard to observe the transition without over-cooling/heating by direct MD reversibly
Challenging test case for new methods
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calculation conditions for Si
✓ isobaric-isothermal MD by HA Stern
• 64 atoms in a cubic cell
✓Order param. from 8 shortest reciprocal lattice vectors
✓Model potential: EDIP type by Justo et al
✓Accurate potential: First-principles calc. by TAPP
• density functional: PBE, pseudo-potential
• plane wave basis set: Gcut ~ 3.8
• k point sampling: 2x2x2
✓Thermodynamic downfolding: |Q| = 1000, 2 iteration
PRB 58 (1998) 2539
J. Compt. Chem. 25 (2004) 749
Bernasconi and Parrinello et al.J. Phys. Chem. Solids 56 (1995) 501
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results for Si
Present Landman Kaczmarski Sugino
CarAlfe
Gillanexp.
potential opt-EDIPStillinger Weber
EDIP LDA GGA
Tm[K] 1572 1665 1572 1350 1492 1685
ΔH(ΔU)[kJ mol-1]
43.3 31.4 36.0 34 43 48.3
ΔV/Vs 0.101 0.070 -0.0095 0.1 0.106 0.1
Sugino and Car, PRL 74 (1995) 1823
Alfe and Gillan, PRB 68 (2003) 205212
Landman et al., PRB 37 (1988) 4637
Kaczmarski et al., PRB 69 (2004) 214105
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results for Si
Present Landman Kaczmarski Sugino
CarAlfe
Gillanexp.
potential opt-EDIPStillinger Weber
EDIP LDA GGA
Tm[K] 1572 1665 1572 1350 1492 1685
ΔH(ΔU)[kJ mol-1]
43.3 31.4 36.0 34 43 48.3
ΔV/Vs 0.101 0.070 -0.0095 0.1 0.106 0.1
Sugino and Car, PRL 74 (1995) 1823
Alfe and Gillan, PRB 68 (2003) 205212
Landman et al., PRB 37 (1988) 4637
Kaczmarski et al., PRB 69 (2004) 214105
accurate potential was by GGA
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downfolded model potential of Si
Original EDIP, Justo et al. PRB 58 (1998) 2539
Stillinger Weber
optimized EDIP by DForiginal EDIP
Stillinger-Weber, PRB 31 (1985) 5262
UA: first-principles
UM: model
-0.5
0
0.5
1
1.5
2
2.5
UM
! U
A [H
t.]
0
2
4
6
8
10
12
0 0.5 1 1.5 2
!F
[10
-2 H
t. a
.u.-1
]
UA [Ht.]
Erro
r in
For
ce
per
atom
diff.
in p
ot. e
nerg
y
each point: sampled config.
clearly superior potential was obtained
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liquid(1800 K) crystal(1200 K)
0
0.5
1
1.5
2
2.5
3
3.5
0 2 4 6 8 10
g(r
)
r [a.u.]
DFFP
orig
0 0.5
1 1.5
2 2.5
3 3.5
4 4.5
5
0 2 4 6 8 10
g(r
)
r [a.u.]
DFFP
orig
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
0 0.5 1 1.5 2 2.5 3
g3(!
)
! [rad.]
rm = 5.85 a.u.
DFFP
orig
0
0.5
1
1.5
2
2.5
0 0.5 1 1.5 2 2.5 3
g3(!
)
! [rad.]
rm = 5.85 a.u.
DFFP
orig
bondangle
pair corr.
bondangle
pair corr.
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Summary (Si)• Crystal⇔Liquid transition of Silicon was
successfully simulated
• Downfolded potential gives good agreement with first-principles results
• simultaneously for both phase
‣ pair correlation function
‣ bond-angle distribution function
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Conclusion• multi-order multi-thermal ensemble by
structure factor
• thermodynamic downfolding
• a method to determine optimized interatomic model potentials
• results for Xtal↔Liq transition of Si
• results for Xtal↔Liq transition of MgO