Dario Dario BressaniniBressanini

QMCI Sardagna QMCI Sardagna ((TrentoTrento) ) 20082008

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

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

Some considerations on nodes Some considerations on nodes and trial wave functionsand 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

7

The influence on the nodes The influence on the nodes of of

QMC currently relies on QMC currently relies on TT(R)(R) and its nodes and its nodes

((indirectlyindirectly))

How are the nodes How are the nodes TT(R)(R) of influenced by: of influenced by: The single particle basis setThe single particle basis set The generation of the orbitals (HF, CAS, MCSCF, NO, The generation of the orbitals (HF, CAS, MCSCF, NO,

…)…) The number and type of configurations in the multidet. The number and type of configurations in the multidet.

ExpansionExpansion

The functional form of The functional form of TT(R)(R) ??

8

Improving Improving

Current Quantum Monte Carlo research Current Quantum Monte Carlo research focusesfocuses on on Optimizing the energy Optimizing the energy Adding more determinants (large number of Adding more determinants (large number of

parameters)parameters) Exploring new trial wave function forms Exploring new trial wave function forms

(moderately large number of parameters)(moderately large number of parameters)» Pfaffians, Geminals, Backflow ...Pfaffians, Geminals, Backflow ...

Node are improved (but not always) only Node are improved (but not always) only indirectlyindirectly

9

Adding more Adding more determinantsdeterminants

Use a large Slater basisUse a large Slater basis Try to reach HF Try to reach HF nodesnodes convergence convergence

Orbitals from MCSCF are goodOrbitals from MCSCF are good 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

10

LiLi22

-14.99-14.995454 E (N.R.L.)E (N.R.L.) E (N.R.L.)E (N.R.L.)

-14.99-14.995252(1)(1)

-14.99-14.993939(2)(2)

-14.99-14.993333(1)(1)

-14.99-14.993333(2)(2)

-14.99-14.991414(2)(2)

-14.99-14.992323(2)(2)

E (hartree)E (hartree) CSFCSF22 g 22 g

222 9...43 ggg 222 9...43 ggg 22 11 uyux 22 11 uyux

224 uyux nn 224 uyux nn 222 211 uuyux 222 211 uuyux

2222 3211 guuyux 2222 3211 guuyux

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

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

(1(1gg22 1 1uu

22 omitted) omitted)

11

DimersDimers

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

12

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:Tails:Often neglectedOften neglected

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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

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Asymptotic behavior of Asymptotic behavior of

)()()( 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

15

Asymptotic behavior of Asymptotic behavior of

),...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. An determinants. An exponential wallexponential wall

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GVB for atomsGVB for atoms

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GVB for atomsGVB for atoms

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GVB for atomsGVB for atoms

21

GVB for atomsGVB for atoms

23

GVB for moleculesGVB for molecules

Correct Correct asymptotic asymptotic structurestructure

Is there a Is there a nodal nodal error error componentcomponent in in HF wave HF wave function coming function coming from incorrect from incorrect dissociation?dissociation?

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GVB for moleculesGVB for molecules

Localized orbitalsLocalized orbitals

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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)

-14.9933(2)-14.9933(2)

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

DMCDMC

CI 3 det large basisCI 3 det large basisCI 3 det large basisCI 3 det large basis

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 irrelevant on the nodes,but irrelevant on the nodes,

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GVB in QMCGVB in QMC

ConclusionsConclusions The quality of the wave function improves, giving The quality of the wave function improves, giving

better VMC energies …better VMC energies … … … but the nodes are not changed, giving the but the nodes are not changed, giving the

same QMC energiessame QMC energies Bressanini and Morosi Bressanini and Morosi J. Chem. Phys. J. Chem. Phys. 129129, ,

054103 (2008) 054103 (2008)

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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))

28

Conventional wisdom on Conventional wisdom on

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))

29

Conventional wisdom on Conventional wisdom on

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

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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?

After all, the whole quantum chemistry is After all, the whole quantum chemistry is built on it!built on it!

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 ? ?

31

The curse of the The curse of the

QMC currently relies on QMC currently 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)) “discredited” the wave function as a “discredited” the wave function as a non non legitimate conceptlegitimate 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

32

Numbers and insightNumbers and insight

““The more accurate the The more accurate the calculations became, the more the calculations became, the more the concepts tended to vanish into concepts tended to vanish into thin air “thin air “(Robert Mulliken)(Robert Mulliken)

There is no shortage of accurate calculations There is no shortage of accurate calculations for few-electron systemsfor few-electron systems

−−2.90372437703411959831115924519440444662.903724377034119598311159245194404446696905379690537 a.u.a.u. Helium atom (Nakashima Helium atom (Nakashima and Nakatsuji JCP 2007)and Nakatsuji JCP 2007)

However…However…

We need new,We need new, and and differentdifferent, ,

ideasideas

A little A little intermezzo intermezzo (for the (for the students)students)

35

We need new, and We need new, and differentdifferent, , ideasideas

Research is the process of going up alleys to see if Research is the process of going up alleys to see if they are blind.  they are blind. 

Marston BatesMarston Bates

Research is the process of going up alleys to see if Research is the process of going up alleys to see if they are blind.  they are blind. 

Marston BatesMarston Bates

Different representationsDifferent representations Different dimensionsDifferent dimensions Different equationsDifferent equations Different potentialDifferent potential Radically different algorithmsRadically different algorithms Different Different somethingsomething

36

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 ?

37

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

38

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

39

Another Another (failed so far) (failed so far) exampleexample

Different dimensionality: HypernodesDifferent dimensionality: Hypernodes Given Given HH((RR) = ) = EE((RR)) build build

dimensionsNHHH 6)()( 21 RR dimensionsNHHH 6)()( 21 RR

)()()()( 2121 RRRR BFFBT )()()()( 2121 RRRR BFFBT

•Use the Use the HypernodeHypernode of of TT

• The hope was that it could be better than Fixed Node The hope was that it could be better than Fixed Node

40

HypernodesHypernodes

Exact nodeExact node

Trial nodeTrial node

Fixed NodeFixed Node

Exact nodeExact node

Trial nodeTrial node

Fixed HyperNodeFixed HyperNode

The energy is still an upper boundThe energy is still an upper bound Unfortunately, it seems to recover exactly the FN Unfortunately, it seems to recover exactly the FN

energyenergy

0)( RT 0)( RTThe intuitive idea was that the system The intuitive idea was that the system couldcould correct correct the wrong fixed nodes, by exploring regions wherethe wrong fixed nodes, by exploring regions where

41

Feynman on simulating Feynman on simulating naturenature

Nature isn’t classical, dammit, and if you Nature isn’t classical, dammit, and if you want to make a simulation of Nature, you’d want to make a simulation of Nature, you’d better make it quantum mechanical, and better make it quantum mechanical, and by golly it’s a wonderful problem, because by golly it’s a wonderful problem, because it doesn’t look so easyit doesn’t look so easy””

Richard Feynman Richard Feynman 19811981

42

NodesNodes

Should we concentrate on Should we concentrate on nodes?nodes?

Conjectures on nodesConjectures on nodes have higher symmetry than have higher symmetry than itself itself resemble simple functionsresemble simple functions the ground state has the ground state has only 2 nodal only 2 nodal

volumesvolumes HF nodes are often a god starting pointHF nodes are often a god starting point

43

How to How to directlydirectly improve improve nodes?nodes?

Fit to a functional form and optimize the Fit to a functional form and optimize the parameters parameters ((maybe for small systemsmaybe for small systems))

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

)(RR T )(RR T

44

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

0:)( 1 cNode 0:)( 1 cNode

BABA rrrr 3311 BABA rrrr 3311

1:)( 2 cNode 1:)( 2 cNode

031 zz 031 zz

Exact

It is a one It is a one parameter parameter

45

““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 …. The goal is to have only few linear parameters to The goal is to have only few linear parameters to

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

46

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? Feynman’s backflow ?) (Linear? Feynman’s backflow ?) preserving the topologypreserving the topology

Miller-Good transformationsMiller-Good transformations

47

The need for the correct The need for the correct topologytopology

Using Backflow alone, on a single Using Backflow alone, on a single determinant determinant is not sufficientis not sufficient, since the , since the topology is still wrongtopology is still wrong

More determinants are necessary (only More determinants are necessary (only two?)two?)

)()( RR TTBF )()( RR TTBF

nodestwoleastatTT 0)()(

RR nodestwoleastatTT 0)()(

RR

48

Be Nodal TopologyBe Nodal Topology

0HF 0HF

r3-r4r3-r4

r1-r2r1-r2

r1+r2r1+r2

0CI 0CI

r1-r2r1-r2

r1+r2r1+r2

r3-r4r3-r4

2222 2121 pscss 2222 2121 pscss

49

Avoided crossingsAvoided crossings

BeBe

ee-- gas gas

StadiumStadium

50

Nodal topologyNodal topology

The conjecture The conjecture (which I believe is true)(which I believe is true) claims claims that there are only two nodal volumes in that there are only two nodal volumes in the fermion ground statethe fermion ground state

See, among others:See, among others: Ceperley Ceperley J.Stat.PhysJ.Stat.Phys 6363, 1237 (1991), 1237 (1991) Bressanini and coworkers. Bressanini and coworkers. JCPJCP 9797, 9200 (1992), 9200 (1992) Bressanini, Ceperley, Reynolds, “Bressanini, Ceperley, Reynolds, “What do we know about What do we know about

wave function nodes?wave function nodes?”, in ”, in Recent Advances in Quantum Recent Advances in Quantum Monte Carlo Methods IIMonte Carlo Methods II, ed. S. Rothstein, World , ed. S. Rothstein, World Scientfic (2001)Scientfic (2001)

Mitas and coworkers Mitas and coworkers PRBPRB 7272, 075131 (2005), 075131 (2005) Mitas Mitas PRL PRL 9696, 240402 (2006), 240402 (2006)

52

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... (any help is appreciated)) a proof... (any help is appreciated)

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

53

He atom with noninteracting He atom with noninteracting electronselectrons

Sss 153 Sss 153

54

57

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 of a probableprobable outcome outcome

the the numbersnumbers lead a dance lead a dance

Sting: Shape of my heartSting: Shape of my heart

58

Take a look at your nodes!Take a look at your nodes!

Think DifferentThink Different