Transcript CECAM Workshop on 'van der Waals forces and Density

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
1
COLLEAGUES OR THOSE WHO I BLAME FOR
ERRORS/MISTAKES/MISCONCEPTIONS
Dr Nicole Benedek
Postdoctoral Fellow
Imperial College
Professor Richard Needs
Mr Ryan Springall
PhD Student RMIT
Dr Mike Towler
TCM Group, Cavendish Lab., Cambridge

Ass. Prof. Salvy Russo
RMIT
Dr Manolo Per
PostDoc RMIT
2
A LITTLE PERSPECTIVE
3
VICTORIA
– SHOWING WINE REGIONS
AND MELBOURNE
4
MELBOURNE AND RMIT APPLIED
PHYSICS

APPLIED
PHYSICS
RMIT
5
MIKE OPENS NEW SPECIALTY SHOP IN
MELBOURNE
MIKE ADRESSES LARGE POLITICAL RALLY
IN FEDERATION SQUARE
6
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
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suction or capillary forces
[PNAS September 17,
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 Systems
Ds 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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WHAT METHODS ARE AVAILABLE?
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) Large
Need 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)
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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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DFT
= DEFINITELY FASHIONABLE THEORY
ADVERTISING PITCH*
AN ENDORSMENT FROM THE NOBEL PRIZE
WINNING WALTER KOHN (“PROOF BY INSULT”):
"WE CONCLUDE THAT TRADITIONAL WAVEFUNCTION 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 by
Professor Kohn
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WHAT ABOUT DFT IN PRACTICE?
Neon-Dimer Binding energy(BSSE Corrected)
0.0016
Binding energy (Hartree)
0.0012
0.0008
UHF
LDA
B3LYP
EXP
0.0004
0
-0.0004
2.5
3
3.5
4
4.5
5
Seperation (Ang)
Ne2 _ HF, LDA, B3LYP and EXPERIMENT
Ne-Ne LDA Binding energy curve(BSSE Corrected)
0.000050
0.000000
Binding energy(Ha)
-0.000050
-0.000100
-0.000150
-0.000200
-0.000250
2
3
4
5
6
7
8
9
Separation (Ang)
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

DFT using DMOL3 and
GAUSSIAN03



GTO and Numerical Basis Sets
BLYP, HCTH and PBE Functionals
CPC with GTO’s Non with NBS
N.A. Benedek, I.K. Snook, K. Latham and I. Yarovsky, J. Chem. Phys.,
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122 144102 (2005)
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 GET
FROM THE CASINO
AND BE GRATEFUL!
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He2 VMC – Old Jastrow
He2 DMC Old Jastrow
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New Jastrow


Accurate but Costly
VMC Not Good

E as Small Differences
VMC Binding energy
0.00250000000
0.00150000000
Exact
0.00100000000
VMC Binding energy
0.00050000000
0.00000000000
1.8
2.3
2.8
3.3
3.8
-0.00050000000
Bond length(Ang)
Binding energy for various values of the time step
0.00030
0.00025
0.00020
Binding energy (Ha)
Binding energy(Ha)
0.00200000000
Extrapolated
0.00015
0.0025
0.01
0.00010
Extrapolated
0.00005
0.00000
2
2.2
2.4
2.6
2.8
3
3.2
-0.00005
Bond Length (Ang)
3.4
3.6
3.8
4
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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 Correlated
Gaussians
Komasa4
-10.947
-11.050.10
(r12 )MR -CI
Gdanitz5
-10.95
-10.990.02
MR-CI
van der Bovenkamp and
Duijneveldt6
-11.000.03
-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,
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7 J. Chem. Phys, 110 11141 (1999)
7 T. Van Mourik and T.H. Dunning Jr., J. Chem. Phys., 111 9248 (1999)
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. Needs
J. Chem. Phys, Accepted for publication 26/7/2006
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CAN WE RE-INCARNATE DFT
TO 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-correlation
energy density by
_
exc(r) = ½ n(r) nxc(r,r') dr'

r - r'
so that
Exc[n] =  exc(r) dr
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RELATION TO PDF
_
 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.
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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
31
Acknowledgment
RMIT Supercomputer
Centre
32
• Good Bye and Thank You
• Kindly Direct Any Awkward
Questions to Mike Towler
and/or Richard Needs
33
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, 34
Phys. Rev. B, 57, 8972 (1998)
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)
35