Quantum Monte Carlo for Transition Metal Oxides · Quantum Monte Carlo for Transition Metal Oxides Lucas K. Wagner Department of Physics North Carolina State University and Computational

Introduction Energetics: performance of QMC Geometry minimization Non-energy properties Conclusion

Quantum Monte Carlo for Transition MetalOxides

Lucas K. Wagner

Department of PhysicsNorth Carolina State University

andComputational Nanosciences Group

University of California, Berkeley

QMC in the Apuan Alps III

In collaboration with Lubos Mitas (NCSU)and Jeff Grossman (UCB)

Lucas K. Wagner Quantum Monte Carlo for Transition Metal Oxides


Before the feature presentationAn advertisement

www.qwalk.org : open-sourceQMC codeC++, nice, very extensibleLong list of features

www.nanohub.org : pays partof my salaryLots of useful tools andlearning material, includingQMC!




























In this talk

GoalMethod to calculate the properties of transition metal (TM)materials.D-orbitals behave very differently from S and P–how?References:

JCP 126 034105 (2007)J. Phys: Condensed Matter topical review coming very soon



Methods

DFT and DMC, mostlySmall core (Ne) ECP on TMTZ basis setQWalk (www.qwalk.org) for all QMC calculationsGAMESS and CRYSTAL for mean-fieldWill decribe newish methods as we go.



TMO molecules

Two atom molecules: small, good for benchmarksTM-O bond is present from ferroelectrics to supernovaremnantsDFT and HF describe them poorly..

Want a scalable, accurate method on these materials.



Energetics: Binding energy of TiO

Results depend on the nodesHartree-Fock nodes are poorB3LYP nodes are goodd-p hybridization is important

Hartree-Fock orbital

B3LYP orbital



Energetics: Binding energy of TiO

Results depend on the nodesHartree-Fock nodes are poorB3LYP nodes are goodd-p hybridization is important

Hartree-Fock orbital

B3LYP orbital



How’s the density in QMC?

1.2 1.4 1.6 1.8distance from Ti (bohr)

0.065

0.07

0.075

0.08

Den

sity

(ar

b. u

nit

s)

B3LYPHF

B3LYP enhanceshybridization overHFQMC(HF) alsoenhanceshybridizationQMC(B3LYP)enhanceshybridization evenmoreHF nodes limithybridization inQMC





0.065

0.07

0.075

0.08

Den

sity

(ar

b. u

nit

s)

B3LYPHFRMC(HF)






0.065

0.07

0.075

0.08

Den

sity

(ar

b. u

nit

s)

B3LYPHFRMC(HF)RMC(B3LYP)




Orbital dependence of the fixed-node approximation

0

0.1

0.2

0.3

0.4

0.5

0.6

E(D

MC

, B3L

YP

orb

itals

)-E

(DM

C, H

F or

bita

ls) (

eV)

ScO TiO VO CrO MnOσ1

σ1δ2σ1δ1 σ1δ2π1 σ1δ2π2

Each type of symmetry increases theimportance of correlation in the orbitals.

σ

δ

π



Binding Energies of TMO Molecules

4

5

6

7

8

9

Bin

din

g E

ner

gy

of

MO

(eV

)

ExperimentDMCLDAUCCSD(T)

ScO TiO VO CrO MnO

Method RMS deviationLDA 2.19UCCSD(T) 0.31DMC 0.21



BaTiO3: Ferroelectric

Cubic phase

BaTiO

Ti-O distanceEn

ergy

Polarized Polarized

Centrosymmetric



BaTiO3 challenges

Ferroelectric effect driven by the Ti-O interaction which isvery sensitive to the lattice constantLDA underestimates volume by 1%, suppressespolarization by 50%GGA overestimates volume moreDistortions, etc are ok in DFT if the volume is set toexperimental



BaTiO3: cohesive energy

Orbitals matter. More d-phybridization thanHartree-Fock.Lattice constant?

LDA density versus Hartree-fockdensity



Minimum Energy GeometryA Bayesian Approach

Idea: Use accurate (but noisy) QMC energies to findequilibrium geometry.

Theory

Given data D and model M: P(M|D) = P(D|M)P(M)P(D)

P(M): Prior distributionP(D): Normalization

P(D|M) ∝∏

i exp(−(M(xi)−D(xi))2

(2σ(xi)2D)

)



An exampleStandard fitting versus Bayesian

0.6 0.8 1 1.2 1.4x

-1

-0.95

-0.9

-0.85

y

x^2-2x + r

0.8 0.9 1 1.1 1.2Minimum

Prob

abili

ty

BayesianStandard fitting

M(x) = a + bx + cx2

pdf(min) =∫∫∫

P(M|D)δ(min − −b2c )da db dc

Standard fitting underestimates the uncertainty!



Other uses for Bayesian analysis

Predicting best next pointModel comparisonWhat is the probability that Emin of phase A is larger thanEmin of phase B?



TMO Molecules: Minimum Bond Lengths

1.58

1.6

1.62

1.64

1.66

1.68

Bo

nd

len

gth

(an

gst

rom

s)

ExperimentDMCLDAUCCSD(T)

ScO TiO VO CrO MnO

Method RMS deviationLDA 0.033UCCSD(T) 0.011DMC 0.008



BaTiO3: Cubic Volume

Same method as used on TMO molecules320 electrons, 960 dimensional integrals



Reptation Monte Carlo

RMC walker on TiO

Algorithm sketchNeed the pure distribution Φ0Φ0

Walkers are paths:s = {X0, X1, . . . , Xn−1, Xn}Sample the path distributionΨT(X0)G(X0, X1, τ) . . . G(Xn−1, Xn, τ)ΨT(Xn)

pdf(X0) = pdf(Xn) = ΨTΦ0

pdf(X n2) = Φ2

0



Performance of Slater-Jastrow wave function

3

3.5

4

4.5

5

Dip

ole

mo

men

t (D

ebye

)

LDACCSD(T)RMCExp

ScO TiO VO CrO MnO

RMC mostly overestimates the dipole moment..why?



Case study: TiO

Determinant weights optimized in VMCSignificant energy gain, change in dipole momentClose to CC number, but still higher than experiment.



TiO dipole moment in detail

3.2

3.4

3.6

3.8

4

4.2

Dip

ole

mo

men

t (D

ebye

)

LDA

CCSD

TPSSh

RMC(1det)

RMC (est limit)

Experiment

Seems closer, but quite far awayPSP error ∼ 0.1 Debye..



Conclusion

New Methods:First application of RMC to heavy atomsBayesian fitting is versatile and correct

Performance on TMO’s:Energetics, with a good trial function, are wonderfulFor dipole moments, it looks more challenging. Morecalculations would be interesting.

Physics about correlation:Enters heavily in the d-p hybridization of TMO’sIs fundamentally important: it affects the one-particledensity significantly



Conclusion






Conclusion






Thanks..

Money from NSF GRF. Help from Lubos Mitas, Jeff Grossman,and many others. Computation from NCSA and NCSU PAMScluster.


Documents

Quantum Monte Carlo for Transition Metal Oxides · Quantum Monte Carlo for Transition Metal Oxides Lucas K. Wagner Department of Physics North Carolina State University and Computational