Diffusion Monte Carlo simulations of crystalline FeO under pressure · 2007. 9. 5. · Diffusion...

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

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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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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 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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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 |)

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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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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)]

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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.

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.

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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

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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

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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

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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 〉

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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

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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

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Diffusion Monte Carlo simulations of crystalline FeO under pressure 27/36

Back to FeO

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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

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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

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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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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)

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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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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

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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)

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