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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
NC STATE UNIVERSITY
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
NC STATE UNIVERSITY
Diffusion Monte Carlo simulations of crystalline FeO under pressure 3/36
The method
NC STATE UNIVERSITY
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)
NC STATE UNIVERSITY
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
NC STATE UNIVERSITY
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)
NC STATE UNIVERSITY
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
NC STATE UNIVERSITY
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
NC STATE UNIVERSITY
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 |)
NC STATE UNIVERSITY
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
]
NC STATE UNIVERSITY
Diffusion Monte Carlo simulations of crystalline FeO under pressure 10/36
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 |)
NC STATE UNIVERSITY
Diffusion Monte Carlo simulations of crystalline FeO under pressure 11/36
FeO, part I
NC STATE UNIVERSITY
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)
NC STATE UNIVERSITY
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)
det[ψHFi (j)]
det[ψPBE020i (j)]
NC STATE UNIVERSITY
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
NC STATE UNIVERSITY
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)]
NC STATE UNIVERSITY
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)
NC STATE UNIVERSITY
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
NC STATE UNIVERSITY
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
NC STATE UNIVERSITY
Diffusion Monte Carlo simulations of crystalline FeO under pressure 18/36
Finite size errors
NC STATE UNIVERSITY
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
NC STATE UNIVERSITY
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.
NC STATE UNIVERSITY
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.
NC STATE UNIVERSITY
Diffusion Monte Carlo simulations of crystalline FeO under pressure 21/36
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
NC STATE UNIVERSITY
Diffusion Monte Carlo simulations of crystalline FeO under pressure 22/36
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
NC STATE UNIVERSITY
Diffusion Monte Carlo simulations of crystalline FeO under pressure 23/36
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
NC STATE UNIVERSITY
Diffusion Monte Carlo simulations of crystalline FeO under pressure 24/36
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 〉
NC STATE UNIVERSITY
Diffusion Monte Carlo simulations of crystalline FeO under pressure 25/36
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
NC STATE UNIVERSITY
Diffusion Monte Carlo simulations of crystalline FeO under pressure 26/36
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
NC STATE UNIVERSITY
Diffusion Monte Carlo simulations of crystalline FeO under pressure 27/36
Back to FeO
NC STATE UNIVERSITY
Diffusion Monte Carlo simulations of crystalline FeO under pressure 28/36
“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
NC STATE UNIVERSITY
Diffusion Monte Carlo simulations of crystalline FeO under pressure 29/36
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
NC STATE UNIVERSITY
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
NC STATE UNIVERSITY
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
NC STATE UNIVERSITY
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)
NC STATE UNIVERSITY
Diffusion Monte Carlo simulations of crystalline FeO under pressure 33/36
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)
NC STATE UNIVERSITY
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
NC STATE UNIVERSITY
Diffusion Monte Carlo simulations of crystalline FeO under pressure 35/36
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
NC STATE UNIVERSITY
Diffusion Monte Carlo simulations of crystalline FeO under pressure 36/36
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)
NC STATE UNIVERSITY