1

Quantum Monte Carlo And

Weak Physical Interactions

Ian Snook

RMIT University, Applied Physics, School of Applied Sciences, SET Portfolio,

GPO Box 2476V, Melbourne, Victoria, Australia, 3001

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COLLEAGUES OR THOSE WHO I BLAME FOR ERRORS/MISTAKES/MISCONCEPTIONS

Dr Nicole Benedek Mr Ryan Springall Postdoctoral Fellow PhD Student RMITImperial College

Professor Richard Needs Dr Mike Towler

TCM Group, Cavendish Lab., Cambridge

Ass. Prof. Salvy Russo RMIT

Dr Manolo PerPostDoc RMIT

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A LITTLE PERSPECTIVE

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VICTORIA – SHOWING WINE REGIONS

AND MELBOURNE

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MELBOURNE AND RMIT APPLIED PHYSICS

APPLIEDPHYSICS RMIT

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MIKE OPENS NEW SPECIALTY SHOP IN MELBOURNE

MIKE ADRESSES LARGE POLITICAL RALLY IN FEDERATION SQUARE

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MIKE HUNTS FOR FOOD IN THE AUSSIE BUSH

MIKE AT LAST FINDS HIS TRUE VOCATION

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WHY BOTHER WITH van der Waals INTERACTIONS?THE ANSWER IS CONTAINED IN "PNAS September 17, 2002, 99 (19) 12252-12256.

Gecko

Tanu Suryadi Kustadi (Nanyang Technical University, Singapore)“Wow for evolution. Weight by numbers. The ability of flies, beetles, skinks and geckos to stick to surfaces is in no way chemical, with bond strength being determined by the polarisabilty of the surface ( perturbation theory show the attractive potential between 2 atoms/molecules to be a function of the atomic polarisability of the 2 atoms) and the number of setae in contact with the surface, not being due to suction or capillary forces[PNAS September 17,

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WHAT ELSE?(THERE MUST BE MORE TO LIFE THAN GEKOS!)

Interaction of Chemically Saturated Objects

Biological Molecules

Physical Adsorption

Adhesion

Surface-Surface Forces

Initial Stages of Interactions with Surfaces vdW

VdW Universal at All Separations – Usually No

Screening

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WHY DO I WANT TO DO THIS?Or WHAT’S AN INNOCENT YOUNG AUSSIE

DOIN’ ERE AMONGST YOU LOT?

My Main Interests Are:

• Statistical Mechanics of Condensed Matter

• Surfaces

• Interaction with Surfaces – Adsorption

• Surface-Surface Interactions –Adhesion

• Colloids and Nano-structures

• Wine

• Food

Need at least a good Interaction Potential Energy Curve

It would be nice if this were non-empirical

N.B. CP-MD is Not Very Accurate for Many SystemsDs for H2O Liquid Off by Orders of Magnitude and vdW

Wrong

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HOW STRONG IS WEAK ?

• These Interactions are a lot Weaker than Covalent, Ionic or Metallic Bonds

• Well Depth He2 = 10 K = 0.0008au

• Well Depth Ar2 = 160K

• H2O – H2O De = 5 kcalmol-1 = 0.004 au

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Quantum Chemical Methods CI MPn CC

Problems:

The Interaction Energy E is Obtained as a Difference EAB = EAB

– (EA + EB)

EAB (EA + EB) Variational Principle Does Not Apply to

E

Slow Convergence (e.g. MP4 is not “exact” for He2 or Ne2

CC needs SD(T) for good results and SDT(Q) for “Exact” Results for He2)

Nm where m = 4-9??

Needs very Large Basis Set for the Non-HF Part(e.g. MP4 for Ne2 s,p,d Needed for HF but MP4

s,p,d,f,g…. and/or Bond Functions*)

Basis Set Superposition Error (BSSE) LargeNeed Counter Poise Correction (CPC) which Doubles the Calculation

Bond Functions Suffer from “Molecular Bias”*

G. Grochola, T. Petersen, S.P. Russo, I.K. Snook, Molecular Physics, 100 3867-3872 (2003)

WHAT METHODS ARE AVAILABLE?

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DFT

Currently Available Approximate Functionals

Do Not Describe van der Waals Interactions

Correctly

They are Somewhat Semi-empirical in Nature

The Results for H-Bonds are Very Functional

Dependent*

BSSE is Still Important Even with Large

Numerical Basis Sets Which are Available for

H-bonds*

* N.A. Benedek, I.K. Snook, K. Latham and I. Yarovsky, J. Chem. Phys., 122, 144102 (2005)

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Symmetry Adapted Perturbation Theory (SAPT)

H = H(0) + V = HA + HB + HAB

EAB = EAB – (EA + EB) = E(1) + E(2) + E(3) + ….

R-S Perturbation Theory

Positive Features

Gets E Directly

In Principle Exact for E

Terms in E can be given a Physical Interpretation

Negative Features

At least Third Order for Very Accurate Result

Don’t know anything about Convergence

The higher order terms become complicated

The energy expressions are not unique

Over-complete bases are used to get better results

Strictly need the exact ground state wavefunctions of the separated systems or Double Pert. Theory

Restricted to small systems – He2 , (H2O)2

In practice many different methods must be used to evaluate different parts of E

- usually supermolecule plus SAPT

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ADVERTISING PITCH*

AN ENDORSMENT FROM THE NOBEL PRIZE WINNING WALTER KOHN (“PROOF BY INSULT”):

"WE CONCLUDE THAT TRADITIONAL WAVE-FUNCTION METHODS, WHICH PROVIDE THE 'REQUIRED' CHEMICAL ACCURACY, ARE GENERALLY LIMITED TO MOLECULES WITH A SMALL TOTAL NUMBER OF CHEMICALLY ACTIVE ELECTRONS, N < O(10)“

AND

"IN GENERAL THE MANY-ELECTRON WAVEFUNCTION (r1,…….,rN) FOR A SYSTEM OF N ELECTRONS IS NOT A LEGITIMATE SCIENTIFIC CONCEPT, WHEN N N0, WHERE N0 103."TAKEN FROM, W.KOHN, REV. MOD. PHYS., 71 1257 (1998)

• The management of this workshop does not

necessarily endorse the products advertised byProfessor Kohn

DFT = DEFINITELY FASHIONABLE THEORY

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WHAT ABOUT DFT IN PRACTICE?

Neon-Dimer Binding energy(BSSE Corrected)

-0.0004

0

0.0004

0.0008

0.0012

0.0016

2.5 3 3.5 4 4.5 5

Seperation (Ang)

Bin

din

g e

ner

gy

(Har

tree

)

UHF

LDA

B3LYP

EXP

Ne2 _ HF, LDA, B3LYP and EXPERIMENT

Ne-Ne LDA Binding energy curve(BSSE Corrected)

-0.000250

-0.000200

-0.000150

-0.000100

-0.000050

0.000000

0.000050

2 3 4 5 6 7 8 9

Separation (Ang)

Bin

din

g e

ner

gy(

Ha)

He Ne Basis set 6-311++(2d,2p) Using Crystal. LDA shows binding, UHF and B3LYP do not. LDA has spurious long range behaviour

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He2 B3LYP

He2 "ROLL-YOUR-OWN FUNCTIONAL" e.g. 100% HF EXCHANGE and 85% LYP CORRELATION

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GRAPHITE

Comparison of scale difference in inter-planar (green) and intra-planar (red) calculations

RHF PBE BLYP B3LYP Exp

a [Å] - 2.4718 2.481 2.4815 2.46

c [Å] 6.774 6.307 6.4744 6.4676 6.71

C11 + C12

[GPa]12.01† 13.01 12.92 12.9 13.3

C33 [GPa] - 1.6 1.78 1.9 0.41

All calculations by Ryan Springall

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H-BONDS

N.A. Benedek, I.K. Snook, K. Latham and I. Yarovsky, J. Chem. Phys., 122 144102 (2005)

DFT using DMOL3 and GAUSSIAN03

GTO and Numerical Basis Sets

BLYP, HCTH and PBE Functionals

CPC with GTO’s Non with NBS

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DFT= DEAD F#@$’ING THEORY??

• OBITUARY: DENSITY FUNCTIONAL THEORY (1927-1993)", PETER M.W. GILL, AUST. J. CHEM. 54 661-662 (2001)

• TO QUOTE

" SHE WAS MISUNDERSTOOD AND ABUSED,

HELD IN NAÏVE AWE BY SOME,

AND IN CONTEMPT BY OTHERS,

CAPABLE OF STUNNING SUCCESSES

AND DISMAL FAILURES.

HER SIMPLICITY WAS SEDUCTIVE

BUT HER FLAWS RAN DEEP AND,

IN THE END, HER FALL WAS INEVITABLE“

RIP

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WHAT NOW?

TRY TO RE-HABILITATE WAVEFUNCTIONS

AND RID THEM OF THEIR EXCESSES (Nm !)

TRY TO RE-INCARNATE DFT – HOW?

BECOME A GAMBLER

ENTER THE CASINO!

– A PARTICULARLY AUSTRALIAN RESPONSE TO

LIFE'S PROBLEMS I FEAR

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QUANTUM MONTE CARLO (QMC) METHODS

VARIATIONAL QUANTUM MONTE CARLO (VMC)

AND DIFFUSION MONTE CARLO (DMC)

= eJ D

D IS A SINGLE DETERMINANT

J THE JASTROW FACTOR (due to BIJL)

WHAT IS THE FORM OF J?

TAKE WHAT YOU CAN GETFROM THE CASINO AND BE GRATEFUL!

He2 VMC – Old Jastrow

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He2 DMC Old Jastrow

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New Jastrow Accurate but Costly VMC Not Good

E as Small Differences

VMC Binding energy

-0.00050000000

0.00000000000

0.00050000000

0.00100000000

0.00150000000

0.00200000000

0.00250000000

1.8 2.3 2.8 3.3 3.8

Bond length(Ang)

Bin

ding

ene

rgy(

Ha)

Exact

VMC Binding energy

Binding energy for various values of the time step

-0.00005

0.00000

0.00005

0.00010

0.00015

0.00020

0.00025

0.00030

2 2.2 2.4 2.6 2.8 3 3.2 3.4 3.6 3.8 4

Bond Length (Ang)

Bindin

g ene

rgy (H

a)

Extrapolated

0.0025

0.01

Extrapolated

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WELL DEPTH He2

WELL DEPTH(0K)

METHOD AUTHORS

-11.01 0.10 “Exact” QMC Anderson et al1

-10.68-11.00

Explicitly Correlated Coupled Cluster

Klopper and Noga2

-11.06 0.03 SAPT to Third Order Korona et al3

-10.947-10.978

Exponentially CorrelatedGaussians

Komasa4

-10.947-11.050.10

(r12 )MR -CI Gdanitz5

-10.95-10.990.02

MR-CI van der Bovenkamp and Duijneveldt6

-11.000.03-10.99

CC(SDT) + CI(Q) van Mourik and Dunning7

-10.97 Fixed Node DMC Us

1 J.B. Anderson, C.A. Traynor and B.M. Boghosian, J. Chem. Phys. 99 345 (1993)2 W. Klopper and J. Noga, J. Chem. Phys. 103 6127 (1995)3 T. Korona, H.L.Williams, R. Bukowski, B. Jeriorski and K. Szalewicz, J. Chem. Phys. 106 5109 (1997)4 J. Komasa, J. Chem. Phys., 110 7909 (1999)5 R.J. Gdanitz, Mol. Phys. 96 1423 (1999)6 J. van de Bovenkamp and F.B. van Duijeveldt, 7 J. Chem. Phys, 110 11141 (1999)

7 T. Van Mourik and T.H. Dunning Jr., J. Chem. Phys., 111 9248 (1999)

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H2O – H2O

VMC and DMC using CASINO

Single Determinant Plus Jastrow

Large Basis Set

Extrapolate to t = 0

All Electron

Orbitals from HF and B3LYP

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Quantum Monte Carlo calculations of the dissociation energy of the water dimer, N.A. Benedek, I.K. Snook, M.D. Towler and R.J. NeedsJ. Chem. Phys, Accepted for publication 26/7/2006

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CAN WE RE-INCARNATE DFTTO MAKE IT A

“DEFINITELY FINE THEORY” ?

It would be valuable to be able to find the form

of exchange-correlation functional for some

particular van der Waals systems

Use this to aid in constructing better

approximate forms of this functional

This can to be done, in principle, by

Quantum Monte Carlo (QMC)

Calculate the exact exchange-correlation

energy density and exchange-correlation hole

function

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WHAT'S THE CONNECTION?

THE ADIABATIC CONNECTION! _

Exc[n] = ½ n(r) nxc(r,r') dr dr' r - r'

where _ 1 nxc(r,r') = nxc(r,r')d 0

nxc(r,r') is the exchange-correlation hole of a fictitious system in which the strength of the electron-electron interaction has been reduced by a factor while the external potential has been adjusted to keep the electron density at n(r).

This requires the use of a Hamiltonian operator defined by

H = T + V + V

We may also define the exchange-correlationenergy density by

_exc(r) = ½ n(r) nxc(r,r') dr' r - r'

so that

Exc[n] = exc(r) dr

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_

Now nxc(r,r') is related to the coupling-constant- _integrated pair correlation function g(r,r') by

_ _nxc(r,r') = n(r)[ g(r,r') – 1] _

If we write g in terms of its constituent spin components i.e. _ _

g(r,r') = [n(r)n(r')/ n(r)n(r')] g(r,r')

then _

g(r,r')

= [N(N-1)/ n(r)n(r')] 1

x d dx3 ……dxN (r,r',x3 ……,xN) 2

0 where N is the number of electrons, n(r) is the electronic density for spin and xi is the ith

electron's spatial and spin co-ordinates.

All this can be done by VMC.

RELATION TO PDF

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WHAT ARE WE UP TO?

He3 at Short Distances to Describe Very High Pressure He Melting

He3 Larger Separations by SAPT

PMIC Phase Diagram of He4 from First Principles

Better Jastrows for vdW

Adiabatic Connection Method

Seeing if DMC can be used with Adiabatic Connection

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AcknowledgmentRMIT Supercomputer

Centre

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• Good Bye and Thank You

• Kindly Direct Any Awkward Questions to Mike Towler and/or Richard Needs

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FOR SOLID Si

ABOVE DIAGRAM COURTESY OF DR. MIKE TOWLER, TCM GROUP, CAVENDISH LAB., CAMBRIDGE UNIVERSITY

FROM: "Exchange and correlation in silicon", R.Q. Hood, M.Y. Chou, A.J. Williamson, G. Rajagopal and R.J. Needs,

Phys. Rev. B, 57, 8972 (1998)

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STRONGLY INHOMOGENEOUS ELECTRON GAS

FROM: "Quantum Monte Carlo Analysis of Exchange and Correlation in the Strongly Inhomogeneous Electron Gas", M. Nekovee, W.M.C. Foulkes and R.J. Needs, Phys. Rev. Letters, 87, 36401 (2001)