diffusion monte carlo simulations of crystalline feo under pressure · 2007. 9. 5. · diffusion...
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Diffusion Monte Carlo simulations of crystalline FeO under pressure 1/36
Diffusion Monte Carlo simulations ofcrystalline FeO under pressure
J. Kolorenc and L. Mitas
North Carolina State University
July 20, 2007
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Diffusion Monte Carlo simulations of crystalline FeO under pressure 2/36
Why?FeO
magnesiowustite, MgxFe1−xO, is(believed to be?) one of the mostabundant minerals in the lowerEarth mantle
FeO is a subset of MgxFe1−xO
quantum Monte Carlo
conventional band-structuremethods unreliable for materialswith 3d electrons
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Diffusion Monte Carlo simulations of crystalline FeO under pressure 3/36
The method
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Diffusion Monte Carlo simulations of crystalline FeO under pressure 4/36
Diffusion quantum Monte Carlo (DMC)
stochastic implementation of projector on the ground state
e−Hτ |ΨT 〉τ→∞−−−−−−−→ e−E0τ |Ψ0〉 ,
modified diffusion in 3N-dim space, Ψ(1, . . . ,N) acts asa probability distribution
fermionic Ψ changes sign (antisymmetry w.r.t. particleexchanges) −→ fixed-node approximation
sign Ψ(1, . . . ,N)
= signΨT (1, . . . ,N)
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Diffusion Monte Carlo simulations of crystalline FeO under pressure 5/36
DMC and DFT
diffusive projection
e−Hτ |ΨT 〉τ→∞−−−−−→ e−E0τ |Ψ0〉
Hohenberg-Kohn theorem&
Kohn-Sham equations
give exact answers if we know
nodes of the wave functionthe exchange-correlation
functional
nodal quality for solids: even the simplest ansatz for nodes isseen to provide considerably better results than DFT based
methods
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Diffusion Monte Carlo simulations of crystalline FeO under pressure 6/36
Trial wave function“trial” wave function sampling efficiency
nodal structureinitial guess
ΨT (1, . . . ,N) = det[ψi (j)
]︸ ︷︷ ︸× exp[J(1, . . . ,N)
]︸ ︷︷ ︸Slater determinant
of 1-body orbitals
Jastrow many-body
correlation factor
1-body orbitals = variational “parameters”Hartree-Fock approximationPBE0x — PBE-GGA mixed with x % of exact exchange
Jastrow factor
J(1, . . . ,N) =∑
i
fe(ri ) +∑ij
fee(ri − rj) +∑i ,α
feI (ri − Rα)
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Reduction to the primitive cell?
non-interacting particles in a periodic potential
E [config1] = E [config2]
interacting particles in a periodic potential
E [config1] 6= E [config2]
No 1-electron Bloch theorem −→ large simulation cell needed
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Diffusion Monte Carlo simulations of crystalline FeO under pressure 7/36
Reduction to the primitive cell?
non-interacting particles in a periodic potential
E [config1] = E [config2]
interacting particles in a periodic potential
E [config1] 6= E [config2]
No 1-electron Bloch theorem −→ large simulation cell needed
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Diffusion Monte Carlo simulations of crystalline FeO under pressure 8/36
Periodic Coulomb interactionPeriodically repeated supercell (k = 0. . . homogeneous background)
vee(r) =∑RS
1
|r − RS |' 4π
Ω
∑k6=0
1
k2e ik·r
Both sums converge slowly −→ cure: combine them into one∑k6=0
1
k2e ik·r =
∑k6=0
1
k2e−k2/(4α2)e ik·r − lim
k→0
1
k2
(1− e−k2/(4α2)
)+∑k
1
k2
(1− e−k2/(4α2)
)e ik·r
=∑k6=0
1
k2e−k2/(4α2)e ik·r − 1
4α2
+Ω
4π
∑RS
1
|r − RS |erfc(α|r − RS |)
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Diffusion Monte Carlo simulations of crystalline FeO under pressure 9/36
Total interaction energy (Ewald)
Total e–e interaction energy per simulation cell.
2
1 1’
2’
Vee =1
2
∑i 6=j
1
rij+
1
2
∑i
[vee(rii )−
1
rii
]
+1
2
∑i<j
[vee(rij)−
1
rij
]+
1
2
∑j<i
[vee(rij)−
1
rij
]
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Total interaction energy (Ewald), cont.
The same formula once more in B&W.
Vee =1
2
∑i 6=j
vee(rij)︸ ︷︷ ︸interaction of i with jand with images of j
+1
2
∑i
limrii→0
[vee(rii )−
1
rii
]︸ ︷︷ ︸
interaction of i withits periodic images
=1
2
∑i 6=j
∑RS
1
|rij − RS |erfc(α|rij − RS |)
+2π
Ω
∑k6=0
1
k2e−k2/(4α2)
∑i 6=j
e ik·rij
− 1
2N2 π
Ωα2− N
α√π
+1
2N∑RS 6=0
1
|RS |erfc(α|RS |)
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Diffusion Monte Carlo simulations of crystalline FeO under pressure 11/36
FeO, part I
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Diffusion Monte Carlo simulations of crystalline FeO under pressure 12/36
Cohesive energy
Ecoh = Eatom[TM] + Eatom[O]− 1
NTMOEsupercell [TMO]
Simulation parameters
simulation cell size: 8 FeO (176 electrons)further corrections towards infinite system (will discuss later)Ne-core pseudopotentials for Fe and Mn, He-core for O(Dirac-Fock, Troullier-Martins)
LDA HF B3LYP DMC[HF] DMC[PBE020] exp.
FeO 11.68 5.69 7.95 9.23(6) 9.47(4) 9.7MnO 10.57 5.44 7.71 9.26(4) 9.5
* all calculations at experimental lattice constant
FeO total energy (hartree) −139.6105(8) −139.6210(5)
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Cohesive energy
Ecoh = Eatom[TM] + Eatom[O]− 1
NTMOEsupercell [TMO]
Simulation parameters
simulation cell size: 8 FeO (176 electrons)further corrections towards infinite system (will discuss later)Ne-core pseudopotentials for Fe and Mn, He-core for O(Dirac-Fock, Troullier-Martins)
LDA HF B3LYP DMC[HF] DMC[PBE020] exp.
FeO 11.68 5.69 7.95 9.23(6) 9.47(4) 9.7MnO 10.57 5.44 7.71 9.26(4) 9.5
* all calculations at experimental lattice constant
FeO total energy (hartree) −139.6105(8) −139.6210(5)
det[ψHFi (j)]
det[ψPBE020i (j)]
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Diffusion Monte Carlo simulations of crystalline FeO under pressure 13/36
Competing crystal structures in FeO
Fe ↑
Fe ↓
O
B1 (NaCl) AFM-II iB8 (NiAs) AFM
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Diffusion Monte Carlo simulations of crystalline FeO under pressure 14/36
Equation of state: Experimental estimates
Experiments are not particularly conclusive so far.
ener
gy
volume
iB8
B1 (insulator)
shock-wave compressionPc ∼ 70 GPa
[Jeanloz&Ahrens (1980)]
static compression900 K: Pc ∼ 74 GPa600 K: Pc ∼ 90 GPa
[Fei&Mao (1994)]
300 K: Pc > 220 GPa? large barrier & slow kinetic ?
[Yagi,Suzuki,&Akimoto (1985)]
[Mao,Shu,Fei,Hu&Hemley (1996)]
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Diffusion Monte Carlo simulations of crystalline FeO under pressure 15/36
Equation of state: Failure of LDA/GGAen
ergy
volume
iB8
B1 (metal)
iB8 stable at all pressures
[Mazin,Fei,Downs&Cohen (1998)]
[Fang,Terakura,Sawada,Miyazaki
&Solovyev (1998)]
B1 has no gap (metal)
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Diffusion Monte Carlo simulations of crystalline FeO under pressure 16/36
Equation of state: “Correlated” band theories
Inclusion of Coulomb U stabilizes B1 phase.[Fang,Terakura,Sawada,Miyazaki&Solovyev (1998)]
ener
gy
volume
iB8
B1 (insulator)
FeO
method Pc (GPa)
PBE010 7PBE020 43exp. &&& 70
MnO
method Pc (GPa)
PBE010 117exp. ∼ 100
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Diffusion Monte Carlo simulations of crystalline FeO under pressure 17/36
Equation of state: DMC[PBE020]
0.0
0.2
0.4
0.6
0.8
1.0
1.2
1.4
15 16 17 18 19 20
E (
eV/F
eO)
V (Å3/FeO)
B1 AFMiB8 AFM
0
30
60
90
120
15 16 17 18 19 20
P (
GP
a)
57.4
method Pc
(GPa)
PBE-GGA —PBE010 7PBE020 43DMC 57± 5*
exp. &&& 70
* pure Ewald formula
geometry optimization:iB8 c/a (PBE020)B1 none
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Diffusion Monte Carlo simulations of crystalline FeO under pressure 18/36
Finite size errors
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Diffusion Monte Carlo simulations of crystalline FeO under pressure 19/36
Only 8 FeO in the simulation cell: Finite-size errorskinetic energy FSEaverage over 8 k-points (a.k.a. twists of
boundary conditions) → only ∼ 0.01 eV/FeO
away from converged Brillouin zone integral
−4
−3
−2
−1
0
1
2
3
4
(¼,¾,0)ΓF
E (
eV)
potential energy FSE (beyond Ewald)
-1.0
-0.8
-0.6
-0.4
-0.2
0.0
16 12 8
E-E
∞ (
eV/F
eO)
No. of FeO formula units
E∞ ≈ 139.6 hartree ≈ 3800 eV
E − E∞ comparable tothe scale of our physics(∼ 1 eV/FeO)
finite-size scaling atevery volume tooexpensive
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Diffusion Monte Carlo simulations of crystalline FeO under pressure 20/36
Improving kinetic energy — k-point average
Adding k-points effectively increases simulation cell size. . .
L = L0 L = 2L0
Γ Γ
. . . but not quite when interactions are in the game.
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Diffusion Monte Carlo simulations of crystalline FeO under pressure 20/36
Improving kinetic energy — k-point average
Adding k-points effectively increases simulation cell size. . .
L = L0 L = 2L0 L = 4L0
Γ Γ Γ
. . . but not quite when interactions are in the game.
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Only 8 FeO in the simulation cell: Finite-size errorskinetic energy FSEaverage over 8 k-points (a.k.a. twists of
boundary conditions) → only ∼ 0.01 eV/FeO
away from converged Brillouin zone integral
−4
−3
−2
−1
0
1
2
3
4
(¼,¾,0)ΓF
E (
eV)
potential energy FSE (beyond Ewald)
-1.0
-0.8
-0.6
-0.4
-0.2
0.0
16 12 8
E-E
∞ (
eV/F
eO)
No. of FeO formula units
E∞ ≈ 139.6 hartree ≈ 3800 eV
E − E∞ comparable tothe scale of our physics(∼ 1 eV/FeO)
finite-size scaling atevery volume tooexpensive
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Potential energy & static structure factor[after Chiesa,Ceperley,Martin&Holzmann (2006)]
Vee =1
2
∑i 6=j
1
rij=
2πN
Ω
∑k
(ρkρ−k
N− 1)
=2πN
Ω
∑k
(SN(k)− 1
)
Correction ∆FS =(limΩ→∞ Vee − Vee
)/N has two parts
∆(1)FS =
2π
Ω
∑k6=0
1
k2− 1
4π2
∫d3k
1
k2= lim
rii→0
[vee(rii )−
1
rii
]. . . this one we already know (and have in Ewald formula)
∆(2)FS =
1
4π2
∫d3k
S∞(k)
k2− 2π
Ω
∑k6=0
SN(k)
k2' 1
4π2
(2π/L)3∫0
d3kS∞(k)
k2
1
4π2
(2π/L)3∫0
d3kS∞(k)
k2
1
4π2
(2π/L)3∫0
d3kS∞(k)
k2
. . . this contribution is new
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Potential energy & static structure factor, cont.
∆(2)FS =
1
4π2
(2π/L)3∫0
d3kS∞(k)
k2
0.0
0.1
0.2
0.3
0.0 0.2 0.4 0.6 0.8 1.0 1.2
S(k
)
|k|
0.0
0.5
1.0
0 1 2 3 4
We need S∞(k) at k ≤ 2π/L
SN(k) does not depend much on N −→ S∞(k) ' SN(k)
k ≤ 2π/L correspond to wavelengths longer than the size ofour simulation cell, i.e., no direct access to SN(k) there−→ extrapolation needed
fortunately, exact identity fixes SN(0) = 0, so that theextrapolation is under control
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Extrapolated estimate for S(k)mixed estimateDMC with guiding wave function samples the mixeddistribution f (R) = Ψ0(R)ΨT (R)
〈Ψ0|S |ΨT 〉 =
∫d3NR Ψ0(R)ΨT (R)︸ ︷︷ ︸
f (R)
S(R)ΨT (R)
ΨT (R)︸ ︷︷ ︸SL(R)
=1
Nw
∑w
SL(Rw )
ΨT (R) known in explicit form −→−→−→ derivatives in S(R) wouldbe no problem in evaluation of SL(R)
extrapolated estimateapproximate expression for the desired matrix element
〈Ψ0|S |Ψ0〉 = 2〈Ψ0|S |ΨT 〉 − 〈ΨT |S |ΨT 〉+O((Ψ0 −ΨT )2
)|Ψ0〉 = |ΨT + ∆〉 : 〈ΨT + ∆|S |ΨT + ∆〉 = 〈ΨT |S |ΨT 〉 + 2〈∆|S |ΨT 〉 + 〈∆|S |∆〉
〈ΨT + ∆|S |ΨT 〉 = 〈ΨT |S |ΨT 〉 + 〈∆|S |ΨT 〉
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Comparison of various estimates for S(k)
0.0
0.1
0.2
0.3
0.0 0.2 0.4 0.6 0.8 1.0 1.2
S(k
)
|k|
~ k1.5
~ k1.9
VMCDMC mixed
DMC extrapolatedRMC
Reptation Monte Carlo(RMC)
provides pureestimates for local(“density-type”)quantities
quickly losesefficiency withincreasing systemsize
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Expectation values in DMC and DFT
DMC
expectation values calculated using explicitly correlatedmany-body wave function
in general, only mixed estimators 〈Ψ0|A|ΨT 〉 available; thesedepend on quality of |ΨT 〉for the total energy and all
[B, H
]= 0 we have
〈Ψ0|B|ΨT 〉 = 〈Ψ0|B|Ψ0〉DFT
quantities calculated from eigenfunctions of artificialnon-interacting Kohn-Sham system
these eigenfunctions (and eigenvalues) not guaranteed to havedirect physical content (but often seem to be close)
total energy prominent — K-S system constructed to have thesame total energy as the original interacting system
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Back to FeO
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“S(k) correction” does a good job
Finite size errors at different levels of compression−→−→−→ errors grow as electron density increases
−1.2
−1.0
−0.8
−0.6
−0.4
−0.2
0.0
0.2
16 12 8
E−
E∞
(eV
/FeO
)
No. of FeO formula units
V=20.4 Å3/FeO
pure EwaldS(k) correction −1.2
−1.0
−0.8
−0.6
−0.4
−0.2
0.0
0.2
16 12 8
∞
No. of FeO formula units
V=15.9 Å3/FeO
pure EwaldS(k) correction
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Transition pressure Pc revisited
Finite-size corrections (slightly) increase Pc .
pure Ewald formula → Pc = 57 ± 5GPa
S(k) correction → Pc = 65 ± 5GPa
0.9
1.0
1.1
1.2
15 16 17 18 19 20
∆ FS
(2) (
eV/F
eO)
V (Å3/FeO)
~ V−0.7
~ rs
−2.1
S(k) correctionscaling correction
0.0
0.2
0.4
0.6
0.8
1.0
1.2
1.4
15 16 17 18 19 20
E (
eV/F
eO)
V (Å3/FeO)
B1 AFMiB8 AFM
0
30
60
90
120
15 16 17 18 19 20
P (
GP
a)
65.2
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Diffusion Monte Carlo simulations of crystalline FeO under pressure 30/36
Equilibrium volume and related properties
Murnaghan equation of state fits the B1 AFM-II data nicely.
0.0
0.2
0.4
0.6
0.8
1.0
1.2
16 18 20 22 24
E (
eV/F
eO)
V (Å3/FeO)
0.00
0.05
0.10
0.15
18 19 20 21 22 23
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Diffusion Monte Carlo simulations of crystalline FeO under pressure 31/36
Equilibrium volume and related properties, cont.
E (V ) = E0 +K0V
K ′0
((V0/V )K
′0
K ′0 − 1
+ 1
)− K0V0
K ′0 − 1
K0 = −V(∂P/∂V
)T
K ′0 =
(∂K0/∂P
)T
a0 (A) K0 (GPa) K ′0
DMC, pure Ewald 4.283(7) 189(8) 5.5(7)DMC + S(k) correction 4.324(6) 170(10) 5.3(7)PBE020 4.328 182 3.7PBE010 4.327 177 3.7PBE 4.300 191 3.5LDA 4.185 224 4.0experiment 4.307–4.334 140–180 2.1–5.6
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Diffusion Monte Carlo simulations of crystalline FeO under pressure 32/36
Can we access also spectral information?
[Bowen,Adler&Auker (1975)]
2.48 1.24 0.82 (eV)
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Band gap estimate in B1 at ambient pressure
∆ = E s.cellsolid [e.s.]− E s.cell
solid [g .s.] = 2.8 ± 0.4 eV
−4
−3
−2
−1
0
1
2
3
4
(¼,¾,0)ΓF
E (
eV)
−4
−3
−2
−1
0
1
2
3
4
(¼,¾,0)ΓF
E (
eV)
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Diffusion Monte Carlo simulations of crystalline FeO under pressure 34/36
Band gap estimate in B1 at ambient pressure, cont.
DMC, hybrid-functional DFT and LDA+U compared.
0
1
2
3
4
5
6
0 10 20 30 40 50
∆ (e
V)
HF percentage in PBE0
DMC[PBE020]
PBE0 x
exp. ~ 2.2 eV
U=4.3 eV
U=6.8 eV
B3LYP
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Notes on band gaps in DMCin ∆ = E [e.s.]− E [g .s.], the intensive quantity ∆ iscalculated from extensive energies
−→−→−→ unfavorable for errorbars
single k-point quantities; although large cancellation of kineticenergy finite-size errors is likely (E [e.s.] and E [g .s.] are at thesame k-point), safe elimination of these is through a largesimulation cell
−→−→−→ unfavorable for errorbars
(the lowest) excited state must have a different symmetrythan the ground state (exc. state is then a groundstate withinthat symmetry)
−→−→−→ might not be the case in large supercell with smallnumber of symmetry operations
other methods for extracting excited-state information fromDMC available, but considerably more costly
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Final words
quantum Monte Carlo is ready to be applied to solids withcorrelated d electrons
FeO case study is very encouraginggood agreement with experimental data at both ambientconditions and elevated pressureconsistently accurate for various quantities (cohesion,Pc(B1 → iB8), equilibrium lattice constant, bulk modulus, . . . )
computer time provided by INCITE ORNL and NCSA
more good news: you can try it at home
www.qwalk.org
(Lucas Wagner, Michal Bajdich and Lubos Mitas)
the bad news: you need some 100,000+ CPU hours (for EoS)
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