Dario Bressanini

Dario Dario BressaniniBressanini

Qmc in the Apuan Alps III Qmc in the Apuan Alps III ((Vallico sottoVallico sotto) ) 2007 2007

http://scienze-como.uninsubria.it/http://scienze-como.uninsubria.it/bressaninibressanini

Universita’ dell’Insubria, Como, ItalyUniversita’ dell’Insubria, Como, Italy

The quest for compact and accurateThe quest for compact and accurate trial wave functions trial wave functions

2

30 years of QMC in 30 years of QMC in chemistrychemistry

3

The Early promises?The Early promises?

Solve the Schrödinger equation Solve the Schrödinger equation exactly exactly withoutwithout approximationapproximation (very strong)(very strong)

Solve the Schrödinger equation with Solve the Schrödinger equation with controlled approximationscontrolled approximations, and converge , and converge to the exact solution to the exact solution (strong)(strong)

Solve the Schrödinger equation with Solve the Schrödinger equation with some some approximationapproximation, and do better than other , and do better than other methods methods (weak)(weak)

4

Good for Helium studiesGood for Helium studies

ThousandsThousands of theoretical and experimental of theoretical and experimental paperspapers

)()(ˆ RR nnn EH )()(ˆ RR nnn EH

have been published on Helium, in its various forms:have been published on Helium, in its various forms:

AtomAtom Small ClustersSmall Clusters DropletsDroplets BulkBulk

5

Good for vibrational Good for vibrational problemsproblems

6

For electronic structure?For electronic structure?

Sign ProblemSign Problem

Fixed Nodal error problemFixed Nodal error problem

8

What to do?What to do?

Should we be happy with the “cancellation Should we be happy with the “cancellation of error”, and pursue it?of error”, and pursue it?

If so:If so: Is there the risk, in this case, that QMC Is there the risk, in this case, that QMC

becomes becomes Yet Another Computational ToolYet Another Computational Tool, and , and not particularly efficient nor reliable?not particularly efficient nor reliable?

If not, and pursue If not, and pursue orthodox QMCorthodox QMC (no (no

pseudopotentials, no cancellation of errors, …)pseudopotentials, no cancellation of errors, …) , can we , can we avoid theavoid the curse of curse of TT ? ?

9

The curse of The curse of

QMC currently heavily relies on QMC currently heavily relies on TT(R)(R)

Walter Kohn in its Nobel lecture Walter Kohn in its Nobel lecture (R.M.P. (R.M.P. 7171, 1253 , 1253

(1999))(1999)) tried to “discredit” the wave function as a tried to “discredit” the wave function as a non legitimate conceptnon legitimate concept when when NN (number of (number of electrons) is largeelectrons) is large

1033 ppM N 1033 ppM N

pp = parameters per variable = parameters per variable

MM = total parameters needed = total parameters needed

For For MM=10=1099 and and pp=3=3 NN=6=6

The Exponential WallThe Exponential WallThe Exponential WallThe Exponential Wall

10

The curse of The curse of

Current research focusses onCurrent research focusses on optimizing the energy for moderately large optimizing the energy for moderately large

expansions (good results)expansions (good results) Exploring new trial wave function forms, with a Exploring new trial wave function forms, with a

moderately large number of parameters (good moderately large number of parameters (good results)results)

Is it hopeless to ask for both Is it hopeless to ask for both accurateaccurate and and compactcompact wave functions? wave functions?

12

LiLi2 2 J. Chem. Phys. J. Chem. Phys. 123123, 204109 (2005), 204109 (2005)

-14.9954 E (N.R.L.)E (N.R.L.)

-14.9952(1)

-14.9939(2)

-14.9933(1)

-14.9933(2)

-14.9914(2)

-14.9923(2)

E (hartree) CSF22 g

222 9...43 ggg 22 11 uyux

224 uyux nn 222 211 uuyux

2222 3211 guuyux

Not all CSF are usefulNot all CSF are useful Only 4 csf are needed to build a statistically Only 4 csf are needed to build a statistically

exact nodal surfaceexact nodal surface

(1(1gg22 1 1uu

22 omitted) omitted)

13

A tentative recipeA tentative recipe Use a large Slater basisUse a large Slater basis

But not too largeBut not too large Try to reach HF Try to reach HF nodesnodes convergence convergence

Orbitals from CAS Orbitals from CAS seemseem better than HF, or NO better than HF, or NO Not worth optimizing MOs, if the basis is large Not worth optimizing MOs, if the basis is large

enoughenough Only few configurationsOnly few configurations seem to improve the FN seem to improve the FN

energyenergy Use the right determinants...Use the right determinants...

...different Angular Momentum CSFs...different Angular Momentum CSFs

And not the bad onesAnd not the bad ones ...types already included...types already included

222234

222234

222234

31)31(ˆ

21)21(ˆ

21)21(ˆ

ssssi

pspsi

ssssi

222234

222234

222234

31)31(ˆ

21)21(ˆ

21)21(ˆ

ssssi

pspsi

ssssi

14

DimersDimers

Bressanini et al. J. Chem. Phys. 123, 204109 (2005) Bressanini et al. J. Chem. Phys. 123, 204109 (2005)

17

Convergence to the exact Convergence to the exact We must include the correct analytical structureWe must include the correct analytical structure

Cusps:Cusps:2

1)0( 1212

rr Zrr 1)0(

3-body coalescence and logarithmic terms:3-body coalescence and logarithmic terms:

QMC OKQMC OK

QMC OKQMC OK

Tails and fragments:Tails and fragments: Usually neglectedUsually neglected

18

Asymptotic behavior of Asymptotic behavior of

1221

22

21

1)11

()(2

1

rrrZH

1221

22

21

1)11

()(2

1

rrrZH

21

22

21

1)(

2

12

r

Z

r

ZH

r

21

22

21

1)(

2

12

r

Z

r

ZH

r

Example with 2-e atomsExample with 2-e atoms

IE2 IE222 1/)1(

210 )( rZr

err

22 1/)1(

210 )( rZr

err

)( 10 r )( 10 r is the solution of the 1 electron problemis the solution of the 1 electron problem

19


)()()( 1221 rJrr )()()( 1221 rJrr The usual formThe usual form

)()()(

)()()(

2011

2102

rrr

rrr

)()()(

)()()(

2011

2102

rrr

rrr

does does notnot satisfy the asymptotic conditions satisfy the asymptotic conditions

)())()()()(( 121221 rJrrrr )())()()()(( 121221 rJrrrr

A closed shell determinant has the A closed shell determinant has the wrongwrong structure structure

)( 12)( 21 rJe rra )( 12)( 21 rJe rra

20


),...2()())(1( 101

11

/21

11110

1111

NYerOrcr Nml

brar

N

r ),...2()())(1( 101

11

/21

11110

1111

NYerOrcr Nml

brar

N

r In generalIn general

Recursively, fixing the cusps, and setting the right symmetry…Recursively, fixing the cusps, and setting the right symmetry…

UNN eNfffA ))()...2()1((ˆ 21 UNN eNfffA ))()...2()1((ˆ 21

Each electron has its own orbital, Each electron has its own orbital, Multideterminant (GVB) Structure!Multideterminant (GVB) Structure!

Take 2N coupled electronsTake 2N coupled electrons )...)(( 434321212 N )...)(( 434321212 N

22NN determinants. Again an determinants. Again an exponential wallexponential wall

21

PsH – Positronium PsH – Positronium HydrideHydride

A wave function with the correct asymptotic A wave function with the correct asymptotic conditions:conditions:

Bressanini and Morosi: JCP Bressanini and Morosi: JCP 119119, 7037 (2003), 7037 (2003)

)()()()()ˆ1(),2,1(112 ee

rgPsrfHPe )()()()()ˆ1(),2,1(112 ee

rgPsrfHPe

22

BasisBasis

In order to build In order to build compactcompact wave functions we used wave functions we used basis functions where basis functions where the cuspthe cusp and the and the asymptotic asymptotic behaviorbehavior are decoupled are decoupled

r

brar

es

1

2

1 r

brar

es

1

2

10 r

are 0 rare

rbre rbre

r

brar

x exp

1

2

2 r

brar

x exp

1

2

2

Use one function per electron plus a simple JastrowUse one function per electron plus a simple Jastrow Can fix the cusps of the orbitals. Can fix the cusps of the orbitals. Very few Very few

parametersparameters

23

GVB for atomsGVB for atoms

24


25


26


27


28

Conventional wisdom on Conventional wisdom on

EEVMCVMC((RHFRHF) > E) > EVMCVMC((UHFUHF) > E) > EVMCVMC((GVBGVB))

Single particle approximationsSingle particle approximations

RHFRHF = = |1s|1sRR 2s 2sRR 2p 2pxx 2p 2pyy 2p 2pzz| |1s| |1sRR 2s 2sRR||

UHFUHF = = |1s|1sUU 2s 2sUU 2p 2pxx 2p 2pyy 2p 2pzz| |1s’| |1s’UU 2s’ 2s’UU||

Consider the Consider the NN atom atom

EEDMCDMC((RHFRHF) ) > ? <> ? < E EDMCDMC((UHFUHF))

29


We can build a We can build a RHFRHF with the same nodes of with the same nodes of UHFUHF

UHFUHF = = |1s|1sUU 2s 2sUU 2p 2pxx 2p 2pyy 2p 2pzz| |1s’| |1s’UU 2s’ 2s’UU||

’’RHFRHF = = |1s|1sUU 2s 2sUU 2p 2pxx 2p 2pyy 2p 2pzz| |1s| |1sUU 2s 2sUU||

EEDMCDMC((’’RHFRHF) ) == EEDMCDMC((UHFUHF))

EEVMCVMC((’’RHFRHF) ) >> E EVMCVMC((RHFRHF) ) >> E EVMCVMC((UHFUHF))

30


Node equivalent to a Node equivalent to a UHF UHF |f(r) g(r) 2p|f(r) g(r) 2p33| |1s | |1s 2s|2s|

EEDMCDMC((GVBGVB) ) == E EDMCDMC((’’’’RHFRHF))

GVBGVB = = |1s 2s 2p|1s 2s 2p33| |1s’ 2s’| - |1s’ 2s 2p| |1s’ 2s’| - |1s’ 2s 2p33| |1s 2s’| + | |1s 2s’| +

|1s’ 2s’ 2p |1s’ 2s’ 2p33| |1s 2s|- |1s 2s’ 2p| |1s 2s|- |1s 2s’ 2p33| |1s’ 2s|| |1s’ 2s|

Same NodeSame Node

32

GVB for moleculesGVB for molecules

Correct asymptotic Correct asymptotic structurestructure

Nodal error Nodal error component in HF component in HF wave function wave function coming from coming from incorrect incorrect dissociation?dissociation?

33

GVB for moleculesGVB for molecules

Localized orbitalsLocalized orbitals

34

GVB LiGVB Li22

E (N.R.L.)E (N.R.L.)

-14.9936(1)-14.9936(1)GVB CI 24 det compactGVB CI 24 det compact

-14.9632(1)-14.9632(1)CI 3 det compactCI 3 det compactCI 3 det compactCI 3 det compact

-14.9688(1)-14.9688(1)GVB 8 det compactGVB 8 det compact

-14.9523(2)-14.9523(2)HF 1 det compactHF 1 det compact

VMCVMC

Wave functionsWave functions

-14.9916(1)-14.9916(1)

-14.9915(1)-14.9915(1)

-14.9931(1)-14.9931(1)

-14.9782(1)-14.9782(1)

-14.9952(1)-14.9952(1)2222 3211 guuyux -14.9954-14.9954

DMCDMC

CI 5 det large basisCI 5 det large basisCI 5 det large basisCI 5 det large basis

Improvement in the wave functionImprovement in the wave function

but but irrelevantirrelevant on the nodes, on the nodes,

35

Different coordinatesDifferent coordinates

The usual coordinates The usual coordinates might not be the best to might not be the best to describe describe orbitalsorbitals and and wave functionswave functions

In LCAO need to use In LCAO need to use large basislarge basis

For dimers, For dimers, elliptical elliptical confocal coordinatesconfocal coordinates are are more “natural”more “natural”

AB

iBiAi

AB

iBiAi

R

rr

R

rr

AB

iBiAi

AB

iBiAi

R

rr

R

rr

36


LiLi22 ground state ground state

Compact MOs built using elliptic coordinatesCompact MOs built using elliptic coordinates

expexp

eccs

ep

es

yx 22

)1(2

2

1

221

expexp

eccs

ep

es

yx 22

)1(2

2

1

221

37

LiLi22

E (N.R.L.)E (N.R.L.)

-14.9632(1)-14.9632(1)CI 3 det compactCI 3 det compact

-14.9523(2)-14.9523(2)HF 1 det compactHF 1 det compact -14.9916(1)-14.9916(1)

-14.9931(1)-14.9931(1)

-14.9954

CI 3 det ellipticCI 3 det elliptic -14.9937(1)-14.9937(1)-14.9670(1)-14.9670(1)

HF 1 det ellipticHF 1 det elliptic -14.9543(1)-14.9543(1) -14.9916(1)-14.9916(1)

VMCVMC

Wave functionsWave functions

DMCDMC

Some improvement in the wave functionSome improvement in the wave function

but but negligiblenegligible on the nodes, on the nodes,

38

Different coordinatesDifferent coordinates It It mightmight make a make a

difference even on difference even on nodes for etheronucleinodes for etheronuclei

Consider LiHConsider LiH+3+3 the 2s the 2s state:state:

HF LCAOHF LCAO

HHLiLi

The wave function is The wave function is dominated by the 2s on Lidominated by the 2s on Li

The node (in The node (in redred) is ) is asymmetricalasymmetrical

However the exact node However the exact node must be must be symmetricsymmetric

39


This is an explicit example of a phenomenon already This is an explicit example of a phenomenon already encountered in other systems, encountered in other systems, the symmetry of the the symmetry of the nodenode is is higherhigher than the than the symmetry of the wave symmetry of the wave functionfunction

The convergence to the exact node, in The convergence to the exact node, in LCAOLCAO, is , is very very slowslow..

Using elliptical coordinates is the right way to Using elliptical coordinates is the right way to proceedproceed

HF LCAOHF LCAO

HHLiLi

Future work will explore if Future work will explore if this effect might be this effect might be important in the important in the construction of many body construction of many body nodesnodes

40

Playing directly with nodes?Playing directly with nodes?

It would be useful to be able to optimize only It would be useful to be able to optimize only those parameters that alter the nodal structurethose parameters that alter the nodal structure

A first “A first “explorationexploration” using a simple test system:” using a simple test system:

HeHe22++

21)1( cc 21)1( cc

The nodes seem to The nodes seem to be smooth and be smooth and ““simplesimple””

Can we “expand” the Can we “expand” the nodes on a basis?nodes on a basis?

41

HeHe22++: “expanding” the node: “expanding” the node

0:)( 1 cNode

BABA rrrr 3311

1:)( 2 cNode

031 zz

Exact

It is a one It is a one parameter parameter

42

““expanding” nodesexpanding” nodes

This was only a kind of “This was only a kind of “proof of conceptproof of concept”” It remains to be seen if it can be applied to It remains to be seen if it can be applied to

larger systemslarger systems Writing “simple” (algebraic?) trial nodes is not Writing “simple” (algebraic?) trial nodes is not

difficult ….difficult …. Waierstrass theoremWaierstrass theorem The goal is to have only The goal is to have only few linearfew linear parameters to parameters to

optimizeoptimize Will it work???????Will it work???????

44

ConclusionsConclusions

The wave function can be improved by The wave function can be improved by incorporating the known analytical incorporating the known analytical structure… structure… with a small number of with a small number of parametersparameters

… … but the nodes do not seem to improve but the nodes do not seem to improve It seems more promising to directly It seems more promising to directly

“manipulate” the nodes.“manipulate” the nodes.

45

A QMC song...A QMC song...

He deals the cards to find the answersHe deals the cards to find the answers

the sacredthe sacred geometry of chancegeometry of chance

thethe hidden lawhidden law of a probable outcomeof a probable outcome

the the numbersnumbers lead a dance lead a dance

Sting: Shape of my heartSting: Shape of my heart

46

JustJust an example an example

Try a different representationTry a different representation Is Is somesome QMC in the momentum QMC in the momentum

representationrepresentation Possible ? And if so, is it:Possible ? And if so, is it: Practical ?Practical ? Useful/Advantageus ?Useful/Advantageus ? Eventually better than Eventually better than plain vanillaplain vanilla QMC ? QMC ? Better suited for some problems/systems ?Better suited for some problems/systems ? Less plagued by the usual problems ?Less plagued by the usual problems ?

47

The other half of Quantum The other half of Quantum mechanicsmechanics

))((ˆ)( rFp ))((ˆ)( rFp The Schrodinger equation in the momentum representationThe Schrodinger equation in the momentum representation

pdpppVpm

pE )()(ˆ)2()()

2( 2/1

2

pdpppVpm

pE )()(ˆ)2()()

2( 2/1

2

SomeSome QMC (GFMC) should be possible, given the iterative form QMC (GFMC) should be possible, given the iterative form

OrOr write the imaginary time propagator in momentum space write the imaginary time propagator in momentum space

48

Better?Better? For coulomb systems:For coulomb systems:

2

2)1(ˆ)(ˆ

jiij pprFpV

2

2)1(ˆ)(ˆ

jiij pprFpV

There are There are NO cuspsNO cusps in momentum space. in momentum space. convergence should be fasterconvergence should be faster

Hydrogenic orbitals are Hydrogenic orbitals are simple rational functionssimple rational functions

222

2/15

1 )(

)8()(

Zp

Zps

222

2/15

1 )(

)8()(

Zp

Zps

54

Avoided nodal crossingAvoided nodal crossing

At a nodal crossing, At a nodal crossing, and and are zero are zero Avoided nodal crossing is the rule, not the Avoided nodal crossing is the rule, not the

exceptionexception Not (Not (yetyet) a proof...) a proof...

0

0

0

0variablesNwithN

eqsN

eq313

.3

.1

variablesNwithNeqsN

eq313

.3

.1

IfIf HFHF has 4 nodeshas 4 nodes HF HF has 2 nodes, with a properhas 2 nodes, with a proper

In the generic case there is no solution to these equationsIn the generic case there is no solution to these equations

55

He atom with noninteracting He atom with noninteracting electronselectrons

Sss 153 Sss 153

56

59

How to How to directlydirectly improve improve nodes?nodes?

““expand” the nodes and optimize the expand” the nodes and optimize the parametersparameters

IFIF the topology is correct, use a coordinate the topology is correct, use a coordinate transformationtransformation

)(RR T )(RR T

60

Coordinate transformationCoordinate transformation

Take a wave function with the correct nodal Take a wave function with the correct nodal topologytopology

)(RR T )(RR T

HF HF

Change the nodes with a coordinate Change the nodes with a coordinate transformation transformation (Linear? Backflow ?) (Linear? Backflow ?) preserving preserving the topologythe topology

Miller-Good transformationsMiller-Good transformations

Date post:	02-Feb-2016
Category:	Documents
Upload:	lara
View:	49 times
Download:	0 times

Dario Bressanini

Documents