Cornell University

Towards a diffusion Monte Carlo study of Mott transition

in MnO under pressure

Wissam A. Al-Saidi

Cornell University

Outline

1. Brief introduction to diffusion Monte Carlo

2. Improving trial wave functions:

Long range periodic Jastrow

silicon interstitials

backflow transformations:

First row atoms

Quantum dots

3. Towards MnO . . .

Pseudopotential validation

1

Cornell University

CollaborationsCyrus Umrigar

MnO

Richard Hennig

Kevin Driver

John Wilkins

Quantum dots

Devrim Guclu

Harold Baranger

Atomic systems

Frank Petruzielo

Julien Toulouse

2

Introduction to DMC

Ground state is obtained by projection from trial wavefunction ΨT (R):

ΨGS(R) = limt−→∞

e−tHΨT (R) = e−τH e−τH . . . e−τHΨT (R)

R is a 3N dimensional vector and H is Hamiltonian of system.

3

Introduction to DMC

Ground state is obtained by projection from trial wavefunction ΨT (R):

ΨGS(R) = limt−→∞

e−tHΨT (R) = e−τH e−τH . . . e−τHΨT (R)

R is a 3N dimensional vector and H is Hamiltonian of system.

Direct approach is extremely inefficient and not stable . . . sign problem!

Fixed node approximation: Use the nodes of a trial wave function as a boundary

condition

3-a

Introduction to DMC

Ground state is obtained by projection from trial wavefunction ΨT (R):

ΨGS(R) = limt−→∞

e−tHΨT (R) = e−τH e−τH . . . e−τHΨT (R)

R is a 3N dimensional vector and H is Hamiltonian of system.

Direct approach is extremely inefficient and not stable . . . sign problem!

Fixed node approximation: Use the nodes of a trial wave function as a boundary

condition

T

Node

GS

Ψ

Ψ

Stable solution:

exact if nodes are exact

variational if nodes are ap-

proximate

Find the best trial WF! VMC optimization

3-b

Trial wavefunctions in DMC

Slater type:

ΨT (R) =ndet∑

i

ci D↑i D↓

i

ci are CI coefficients, and D↑i and D↓

i are spin-up and spin-down determinants

R = (~r1,~r2, . . . ,~rN) 3N-dimensional vector.

4

Trial wavefunctions in DMC

Slater type:

ΨT (R) =ndet∑

i

ci D↑i D↓

i

ci are CI coefficients, and D↑i and D↓

i are spin-up and spin-down determinants

R = (~r1,~r2, . . . ,~rN) 3N-dimensional vector.

Slater-Jastrow type:

ΨT (R) = eJ(R)∑

ci D↑i D↓

i

J(R) = Jee + Jen + Jeen + . . .

J(R) introduces 1-body, 2-body, 3-body, . . . correlations. Forces ΨT (R) to satisfy

cusp conditions.

J(R, α) is parametrized. α = αee, αen, αeen and CI coefficients are

optimized with VMC

Slater-Jastrow type works well for molecular systems and solids.

4-a

Trial wavefunctions in DMC (periodic systems)

Jastrow factors suitable for isolated systems are short ranged!

long-range Jastrow correlations: Prendergast, Bevan, Fahy PRB ’02

JLR(R) = J(1)LR + J

(2)LR

J(1)LR =

∑

stars

nelec∑

i

ps cos(

~k · ~ri

)

+ qs sin(

~k · ~ri

)

J(2)LR =

∑

stars

nelec∑

i,j

rs cos[

~k · (~ri − ~rj)]

rs, ps, and qs are variational parameters optimized with VMC.

5

Trial wavefunctions in DMC (periodic systems)

Jastrow factors suitable for isolated systems are short ranged!

long-range Jastrow correlations: Prendergast, Bevan, Fahy PRB ’02

JLR(R) = J(1)LR + J

(2)LR

J(1)LR =

∑

stars

nelec∑

i

ps cos(

~k · ~ri

)

+ qs sin(

~k · ~ri

)

J(2)LR =

∑

stars

nelec∑

i,j

rs cos[

~k · (~ri − ~rj)]

rs, ps, and qs are variational parameters optimized with VMC.

For one-body J(1)LR all stars consistent with the point group of the crystal are included:

non-inversion symmetry case

symmorphic and non-symmorphic cases

5-a

Silicon interstitials (preliminary)

-66.8

-66.79

-66.78

-66.77

-66.76

-66.75

-66.74

VM

C e

nerg

y (H

artr

ee)

H T X

With LRPJ (cosine)

With LRPJ (cosine+sine)

Short range Jastrow

DMC study using CHAMP by Batista and coworkers PRB ’06

(16 +1) atom supercell- (64 + 4) electrons. Three kinds of interstitials H, T and X

variance reduction: From σ2 = 0.8 to σ2 = 0.6 Hartree2

smaller locality error, efficient simulation, . . .

6

Trial wavefunctions in DMC

Slater type:

ΨT (R) =ndet∑

i

ci D↑i D↓

i

Slater-Jastrow type:

ΨT (R) = eJ(R)∑

ci D↑i D↓

i

J(R) = Jee + Jen + Jeen + . . .

J(R) introduces 1-body, 2-body, 3-body, . . . correlations.

For solids, Slater-Jastrow with periodic long-range correlations:

JLR(R) =∑

s

nelec∑

i

[

ps cos(

~k · ~ri

)

+ qs sin(

~k · ~ri

)]

+∑

s

nelec∑

i,j

rs cos[

~k · (~ri − ~rj)]

With a single determinant, all these WFs have same nodes!

all DMC energies should be the same

7

backflow transformations

Feynman-Cohen backflow (liquid He II): Phs. Rev. ’56

excited-state WF with an excitation of momentum ~q:

∏

i

ei~q·~riφ0 −→∏

i

ei~q·~xiφ0

where ~xi = ~ri +∑

j 6=i η(rij)~rij

e

8

backflow transformations

Feynman-Cohen backflow (liquid He II): Phs. Rev. ’56

excited-state WF with an excitation of momentum ~q:

∏

i

ei~q·~riφ0 −→∏

i

ei~q·~xiφ0

where ~xi = ~ri +∑

j 6=i η(rij)~rij

In Fermionic systems, backflow transformations modify the nodes

Applied to the Fermionic homogeneous electron gas [Schmidt PRL ’81]

Generalized to inhomogeneous systems [Rios PRE ’06]

~xi = ~ri +∑

j 6=i

η(ri, rj , rij)~rij +∑

j 6=i

ξ(ri, rj , rij)~ri

ri

rije

e−−

backflow is highly desirable in solids

implemented in champ and applied to molecular systems and quantum dots

8-a

VMC energies of first row atoms (preliminary)

Li Be B C N O F Ne

-1

-0.8

-0.6

-0.4

-0.2

0

Dev

iatio

ns f

rom

exa

ct v

alue

(eV

)

VMC-1detVMC-BF-1det only VMC with 1-det

Li Be B C N O F Ne40

50

60

70

80

90

100

vari

ance

% c

hang

e Large reduction in variance

9

Quantum dots

Quantum dots are very similar to atoms (artificial atoms)

H =∑

i

(

−1

2∇i

2 +1

2ω2r2

i

)

+∑

i 6=j

1

|ri − rj |

Highly tunable systems (vary ω)— strongly correlated physics (Wigner crystallization)

10

Quantum dots

Quantum dots are very similar to atoms (artificial atoms)

H =∑

i

(

−1

2∇i

2 +1

2ω2r2

i

)

+∑

i 6=j

1

|ri − rj |

Highly tunable systems (vary ω)— strongly correlated physics (Wigner crystallization)

Electron density:

Ω=0.0015625 Ω= 0.0005 Ω=0.0001

10-a

qdots with backflow

N = 7 and w = 0.0015625

L=0,S=1/2 L=0,S=5/2 L=1,S=3/2 L=1,S=7/2 L=3,S=7/2

1.255

1.256

1.257

1.258

1.259

1.26

1.261

1.262

1.263

1.264

1.265

1.266

1.267

1.268

Ene

rgy

(Ryd

berg

* )

VMC (SJ)DMC (SJ)

11

qdots with backflow

N = 7 and w = 0.0015625

L=0,S=1/2 L=0,S=5/2 L=1,S=3/2 L=1,S=7/2 L=3,S=7/2

1.255

1.256

1.257

1.258

1.259

1.26

1.261

1.262

1.263

1.264

1.265

1.266

1.267

1.268

Ene

rgy

(Ryd

berg

* )

VMC (SJ)DMC (SJ)VMC (BF+SJ)DMC (BF+SJ)

12

qdots with backflow

N = 7 and w = 0.0015625

L=0,S=1/2 L=0,S=5/2 L=1,S=3/2 L=1,S=7/2 L=3,S=7/2

1.255

1.256

1.257

1.258

1.259

1.26

1.261

1.262

1.263

1.264

1.265

1.266

1.267

1.268

Ene

rgy

(Ryd

berg

* )

VMC (SJ)DMC (SJ)VMC (BF+SJ)DMC (BF+SJ)

pair density g↓,↓ (N=7, w = 0.0005)

VMC DMC VMC-BF

12-a

Where are we?

1. Brief review of diffusion Monte Carlo

2. Trial wave functions?

3. Long range periodic Jastrow

silicon intersteials

4. backflow transformations:

First row atoms

Quantum dots

5. Towards MnO . . .

Pseudopotentials validation

HF or GGA pseudopotentials

13

Study of anti-ferromagnetic MnO (preliminary)

AF FCC structure: Mn moments anti-parallel along adjacent sheets in the [111] plane

Experimentally:

First-order insulator-metal transition near 100 GPa

volume collapse by ≈ 5% equilbrium volume

moment collapse: 5 −→ 1 bohr magneton

Rich physics Cohen Science ’97, Kasinathan PRB ’06, Kunes Nature materials ’08

14

All-electron equation of state (preliminary)

Calculations done with GGA/PBE using WIEN2k

175 200 225 250 275 300

Volume (Bohr3)

-1233.9

-1233.9

-1233.8

Ene

rgy

(Har

tree

)

Non-magnetic

Ferromagnetic anti-ferromagnetic

Three different phases of MnO (FCC structure)

AFM phase is the ground state under ambient pressures

volume collapse in AFM phase for high pressures

15

Magnetic and volume collapse (all-electron) (preliminary )

160 170 180 190 200 210 220 230 240 250 260 270 280 290 300 310 320

Volume (bohr3)

1

2

3

4

5

Magnetic moment at Mn site

Energy

high-spin to low-spin transition (5/2 −→ 1/2) Kasinathan PRB ’06,

16

Towards a DMC of MnO

Which pseudopotentials to use?

Wide spread belief that HF pseudopotentials are better than DFT ones in QMC

HF pseudopotentials are cleaner in descreening step (DFT fixed by NLCC)

Limited experience shows that HF and GGA PsP are okay but not LDA one

OPIUM can generate HF and DFT pseudopotentials Al-Saidi, Walter and Rappe

PRB’08

small core pseudopotentials

17

Towards a DMC of MnO

Which pseudopotentials to use?

Wide spread belief that HF pseudopotentials are better than DFT ones in QMC

HF pseudopotentials are cleaner in descreening step (DFT fixed by NLCC)

Limited experience shows that HF and GGA PsP are okay but not LDA one

OPIUM can generate HF and DFT pseudopotentials Al-Saidi, Walter and Rappe

PRB’08

small core pseudopotentials

What kind of trial WF?

We want to optimize the orbitals

For high pressures, GGA trial wavefunctions will be okay

Near the equilibrium geometry, LDA+U (or HF) might be best

17-a

Non magnetic phase (preliminary)

90 100 110 120 130 140 150

Volume (Bohr3)

0

0.25

0.5

0.75

1

1.25

1.5

1.75

2

Ene

rgy

(eV

)AE PsP (GGA)PsP (HF)

GGA/PBE calculations

GGA and HF PsPs

Excellent agreement between all-electron and pseudopotential results

18

AFM magnetic phase (preliminary)

180 200 220 240 260 280 300

Volume (Bohr3)

0

1

2

3

4

5

Ene

rgy

(eV

)

AEGGA (PsP)HF (PsP)

GGA/PBE calculations

GGA and HF PsPs

pseudopotential equilibrium volume is ≈ 1 % smaller than all-electron one

Birch-Murnaghan B0 and B0′ are also in excellent agreement

pseudopotentials do not seem to give a volume collapse????

19

AFM magnetic phase (aligning equilb. positions)

200 250 300

Volume (Bohr3)

0

2

4

6E

nerg

y (e

V)

AEGGA (PsP)HF (PsP)

GGA/PBE calculations

GGA and HF PsPs

High pressures, larger deviations between pseudopotential and all-electron

change in magnetic properties of system

similar behavior has been seen by Kolorenc and Mitas PRB ’07

20

Cornell University

Summary

Most general long-range periodic Jastrow factor for solids

Backflow transformations:

atoms and molecules

quantum dots

solids . . . easily implemented

Investigating different pseudopotentials for our MnO study

Future work

orbital optimization for solids

finite-size correction scheme

al-saidi@cornell.edu

21