The nodes of trial and exact The nodes of trial and exact wave functions in Quantum wave functions in Quantum Monte CarloMonte Carlo

Dario BressaniniDario Bressanini Universita’ dell’Insubria, Como, Universita’ dell’Insubria, Como, ItalyItalyhttp://www.unico.it/http://www.unico.it/~dario~dario

Peter J. ReynoldsPeter J. ReynoldsOffice of Naval ResearchOffice of Naval Research

CECAM 2002 - LyonCECAM 2002 - Lyon

Nodes and the Sign Nodes and the Sign ProblemProblem

• So far, solutions to the sign problem have So far, solutions to the sign problem have not proven to be efficient not proven to be efficient

• Fixed-node approachFixed-node approach isis efficient. If only we efficient. If only we could have the exact nodes …could have the exact nodes …

• … … or at least a or at least a systematicsystematic way to improve way to improve the nodes ...the nodes ...

• … … we could bypass the sign problemwe could bypass the sign problem

• How do we build a How do we build a with good nodes? with good nodes? CI? MCSCF? Natural Orbitals (CI? MCSCF? Natural Orbitals (LüchowLüchow) ?) ?

NodesNodes• What do we know about wave function What do we know about wave function

nodes?nodes? Very little ....Very little ....

• NOTNOT fixed by (anti)symmetry alone. fixed by (anti)symmetry alone.Only a 3N-3 subsetOnly a 3N-3 subset

• Very very few analytic examplesVery very few analytic examples

• Nodal theorem is Nodal theorem is NOT VALIDNOT VALID Higher energy states Higher energy states does notdoes not mean more mean more

nodes nodes ((Courant and Hilbert Courant and Hilbert ))

• They have They have (almost)(almost) nothing to do with nothing to do with Orbital Nodes. It is possible to use Orbital Nodes. It is possible to use nodeless orbitals.nodeless orbitals.

• Tiling theoremTiling theorem

Tiling Theorem Tiling Theorem (Ceperley)(Ceperley)

Impossible for Impossible for ground stateground state

The Tiling Theorem does not say how The Tiling Theorem does not say how many nodal regions we should expectmany nodal regions we should expect

Nodes and Nodes and ConfigurationsConfigurations

It is necessary to get a better understanding how CSF influence It is necessary to get a better understanding how CSF influence the nodesthe nodes. . Flad, Caffarel and SavinFlad, Caffarel and Savin

The The (long term)(long term) Plan of Plan of AttackAttack

• StudyStudy the nodes of exact and good the nodes of exact and good approximate trial wave functionsapproximate trial wave functions

• UnderstandUnderstand their properties their properties

• Find a way to Find a way to sistematicallysistematically improve the improve the nodes of trial functions, or...nodes of trial functions, or...

• ……findfind a way to a way to parametrizeparametrize the nodes the nodes using simple functions, and using simple functions, and optimizeoptimize the the nodes directly minimizing the Fixed-Node nodes directly minimizing the Fixed-Node energyenergy

The Helium Triplet The Helium Triplet

• First First 33SS state of He is one of very few state of He is one of very few systems where we know exact nodesystems where we know exact node

• For For SS states we can write states we can write ),,( 1221 rrr

),,(),,( 12121221 rrrrrr

• Which means that the node isWhich means that the node is

02121 rrorrr

•For the Pauli Principle For the Pauli Principle

The Helium TripletThe Helium Triplet• Independent of Independent of rr1212

• The node is The node is more more symmetricsymmetric than the wave than the wave function itselffunction itself

• It is a polynomial in It is a polynomial in rr11 and and rr22

• Present in all Present in all 33SS states of states of two-electron atomstwo-electron atoms

r1

r2

r1

2

021 rr

r1

r2

021 rr

)(),(),,( 12211221 rrrrrr

),,(211221

1221),(),,( rrrferrNrrr

• The wave function is not The wave function is not factorizablefactorizable

butbut

The Helium TripletThe Helium Triplet),,(

2112211221),(),,( rrrferrNrrr ),,(

2112211221),(),,( rrrferrNrrr

• This is This is NOTNOT trivial trivial

• NN = = rr11-r-r22 , , Antisymmetric Antisymmetric NodalNodal FunctionFunction

• ff = = unknownunknown, , totally symmetrictotally symmetric

• The HF function has the exact nodeThe HF function has the exact node

• Which of these properties are present in other Which of these properties are present in other systemssystems??

• Are these be Are these be general propertiesgeneral properties of the nodal of the nodal surfaces ?surfaces ?

• For a generic system, what can we say about For a generic system, what can we say about NN ? ?)()( RR fExact eN )()( RR fExact eN

Helium Singlet 1s2s 2 Helium Singlet 1s2s 2 11SS

• A very good approximation A very good approximation of the node isof the node is

constrr 42

41

• The second triplet has The second triplet has similar propertiessimilar properties

constrr 52

51r1

r2

Surface contour plot of the node

),,( 1221 rr• Although , the node does Although , the node does notnot depend on depend on (or does (or does veryvery weakly) weakly)

He: Other statesHe: Other states

• 1s2s 1s2s 33S S : (: (rr11--rr22) f() f(rr11,,rr22,,rr1212))

• 1s2p 1s2p 11P P o o : node independent from: node independent from r r1212 (J.B.Anderson)(J.B.Anderson)

• 22pp22 33P P ee : : = ( = (xx11 yy22 – – yy11 xx22) f() f(rr11,,rr22,,rr1212))

• 22pp33pp 11P P ee : : = ( = (xx11 yy22 – – yy11 xx22) () (rr11--rr22) f() f(rr11,,rr22,,rr1212))

• 1s2s 1s2s 11S S : node independent from : node independent from r r1212

Similar to Similar to 042

41 constrr

0))(( 215

25

1 rrconstrr

•1s3s 1s3s 33S S : node independent from : node independent from r r1212

Similar toSimilar to

Helium NodesHelium Nodes

• Independent from Independent from rr1212

• More “symmetric” than the wave functionMore “symmetric” than the wave function

• Some are described by polynomials in Some are described by polynomials in distances and/or coordinatesdistances and/or coordinates

• The same node is present in different statesThe same node is present in different states(as if Helium were separable)(as if Helium were separable)

• The HF The HF , sometimes, has the correct node, , sometimes, has the correct node, or a node with the correct or a node with the correct (higher)(higher) symmetry symmetry

Lithium Atom Ground Lithium Atom Ground StateState

)(1)(2)(1)(2)(1)(2)(1)(1 21331321 rsrsrsrsrsrsrsrsRHF

• The The RHFRHF node is node is rr1 1 = = rr33

if two like-spin electrons are at the same if two like-spin electrons are at the same distance from the nucleus then distance from the nucleus then =0=0

• This is the same node we found in the This is the same node we found in the HeHe 33SS

• Again, node has higher symmetryAgain, node has higher symmetry

• How good is the RHF node?How good is the RHF node?

RHFRHF is not very good, however its node is is not very good, however its node is surprisingly good surprisingly good ((might it be the exact one?might it be the exact one?))

DMC(DMC(RHF RHF ) = -7.47803(5)) = -7.47803(5) a.u.a.u. Lüchow & Anderson JCP 1996Lüchow & Anderson JCP 1996

ExactExact = -7.47806032 = -7.47806032 a.u.a.u. Drake, Hylleraas expansionDrake, Hylleraas expansion

• We take an “almost exact” Hylleraas expansion We take an “almost exact” Hylleraas expansion 250 terms250 terms

• Energy Energy HyHy = = -7.478059-7.478059 a.u. a.u.Exact = Exact = -7.4780603-7.4780603 a.u. a.u.

321231312321

ˆ rrrkjilmnHy errrrrrA

How different is its node from r1 = r3 ??

LiLi atom: Study of atom: Study of Exact NodeExact Node

r3r1

r2

• The node The node seemsseems to be to berr11 = r = r33, taking different , taking different cuts, independent from cuts, independent from rr22 or or rrijij

LiLi atom: Study of atom: Study of Exact NodeExact Node

Numerically, we found only Numerically, we found only very small very small deviationsdeviations from the HF node, or from the HF node, or artifacts artifacts of the linear expansionof the linear expansion

0.7 0.75 0.8 0.85 0.9 0.95 10.7

0.75

0.8

0.85

0.9

0.95

1

• a DMC simulation with a DMC simulation with rr11 = r = r33 node node and good and good to reduce the variance to reduce the variance givesgives

• DMCDMC -7.478061(3) a.u.a.u. ExactExact -7.4780603 a.u.a.u.

Is r1 = r3 the exact node of Lithium ?

Beryllium Atom Beryllium Atom Ground StateGround State

),,,,,,,,,,,()4,3,2,1()( 444333222111 zyxzyxzyxzyx R ),,,,,,,,,,,()4,3,2,1()( 444333222111 zyxzyxzyxzyx R 12 D12 D

After spin assignmentAfter spin assignment

9 D9 D

Node Node (R) = 0(R) = 0 8-dimensional hypersurface8-dimensional hypersurface

)(2)(1)(2)(1 3221 rsrsrsrsRHF )(2)(1)(2)(1 3221 rsrsrsrsRHF

)(2)(1)(2)(1)(2)(1)(2)(1 34431221 rsrsrsrsrsrsrsrsRHF )(2)(1)(2)(1)(2)(1)(2)(1 34431221 rsrsrsrsrsrsrsrsRHF

Factor external anglesFactor external angles

Beryllium AtomBeryllium Atom

HF predicts HF predicts 44 nodal regions nodal regions Bressanini et al. Bressanini et al. JCP JCP 9797, 9200 (1992), 9200 (1992)

Node: (rNode: (r11-r-r22)(r)(r33-r-r44) = 0) = 0

factors into two determinants each one factors into two determinants each one “describing” a “describing” a triplettriplet Be Be+2+2. The node is the union of . The node is the union of the two independent nodes.the two independent nodes.

)(2)(1)(2)(1 4321 rsrsrsrsRHF )(2)(1)(2)(1 4321 rsrsrsrsRHF

The HF node is The HF node is wrongwrong•DMC energy DMC energy --14.6576(4)14.6576(4)•Exact energy Exact energy -14.6673-14.6673

Hartree-Fock NodesHartree-Fock Nodes

• HFHF has always, has always, at leastat least, 4 nodal regions , 4 nodal regions

for 4 or more electronsfor 4 or more electrons

• It It mightmight have N have N! N! N! Regions! Regions

• Ne atom: 5! 5! = 14400 possible regionsNe atom: 5! 5! = 14400 possible regions

• LiLi2 2 molecule: 3! 3! = 36 regionsmolecule: 3! 3! = 36 regions

)(,...,1,...2,1 ijHF rJNNNN )(,...,1,...2,1 ijHF rJNNNN

How Many ?How Many ?

Clustering AlgorithmClustering Algorithm

22

44

Nodal RegionsNodal Regions

44

44

44

44

622 221 pssNeNe

ss 21 2LiLi22 21 ssBeBe

pss 221 22BB

222 221 pssCC

LiLi22

g

1

HF

Be: beyond Restricted Be: beyond Restricted Hartree-FockHartree-Fock

• Hartree-Fock Hartree-Fock is is notnot the most general the most general single particle approximationsingle particle approximation

• Try a GVB wave function Try a GVB wave function (each electron in its (each electron in its own orbital)own orbital)

....)(2)(1)(2)(1

)(2)(1)(2)(1ˆ

4321

4321

termsrsrsrsrs

rsrsrsrsA

....)(2)(1)(2)(1

)(2)(1)(2)(1ˆ

4321

4321

termsrsrsrsrs

rsrsrsrsA

GVBGVB(Be) = sum of 4 Determinants(Be) = sum of 4 Determinants

Be: beyond Hartree-Be: beyond Hartree-FockFock

• GVBGVB(Be) = sum of 4 Determinants(Be) = sum of 4 Determinants

• VMC energy improvesVMC energy improves

• 22(H) improves(H) improves

• ……but but still the same nodestill the same node (r1-r2)(r3-r4) = 0

))()(()()()(ˆ44332211 rfrfrfrfA ))()(()()()(ˆ44332211 rfrfrfrfA

(r1-r2)(r3-r4) = 0 for for anyany ff11, , ff22, , ff33, , ff44

Be: CI expansionBe: CI expansion

• What happens to the HF What happens to the HF node in a CI expansion?node in a CI expansion?

Plot cuts of (rPlot cuts of (r11-r-r22) ) vsvs (r (r33-r-r44))

msnssscn

nCI 11 msnssscn

nCI 11

• Still the Still the same topologysame topology and the and the same nodesame node(same in lithium)(same in lithium)

In DMC, CI is not necessarily better than HFIn DMC, CI is not necessarily better than HFIn DMC, CI is not necessarily better than HFIn DMC, CI is not necessarily better than HF

• Of course Of course , one would first use, one would first use 2222 2121 pscss 2222 2121 pscss

Be: CI expansionBe: CI expansion

• What happens to the HF node in a What happens to the HF node in a goodgood CI expansion? CI expansion?

Node is Node is (r(r11-r-r22)(r)(r33-r-r44)) + ... + ...

Plot cuts of (rPlot cuts of (r11-r-r22) ) vsvs (r (r33-r-r44))

• In 9-D space, the direct product structure In 9-D space, the direct product structure “opens up”“opens up”

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

Be nodal topologyBe nodal topology

• The clustering algorithm The clustering algorithm confirms that now there confirms that now there are only are only twotwo nodal regions nodal regions

• It can be proved that the It can be proved that the exactexact Be wave function Be wave function has exactly two regionshas exactly two regions

Node is (r1-r2) (r3-r4) + ??? Node is (r1-r2) (r3-r4) + ???

See See Bressanini, Ceperley and ReynoldsBressanini, Ceperley and Reynoldshttp://www.unico.it/~dario/http://www.unico.it/~dario/http://archive.ncsa.uiuc.edu/Apps/CMP/http://archive.ncsa.uiuc.edu/Apps/CMP/

Fitting NodesFitting Nodes

• We would like a simple analytical We would like a simple analytical approximation of the nodeapproximation of the node

• Ultimate goal: parametrize the node with few Ultimate goal: parametrize the node with few parameters, and parameters, and directlydirectly optimize them optimize them

• Useful also for Useful also for diagnosticdiagnostic. To see if the node . To see if the node changes, and how, by changing basis, or changes, and how, by changing basis, or expansion, or functional form.expansion, or functional form.

• How can we model the How can we model the implicitimplicit surface surface =0 ?=0 ? The studied The studied (simple)(simple) systems suggest polynomials systems suggest polynomials

of distances (plus x, y and z for Lof distances (plus x, y and z for L0 states)0 states)

Fitting NodesFitting Nodes

Model Node:Model Node: 0)(),( i

ii Pcf RcR 0)(),( i

ii Pcf RcR

)(RiP )(RiP Polynomials of Polynomials of rrii and and rrijij (or other simple (or other simple functions)functions) with the correct spin-space with the correct spin-space symmetry.symmetry.

Node

dlf ),()(min 22 cRcc

•To find To find cc::

NodeNode

ff((Node,Node,cc))

line integralline integral

Fitting NodesFitting Nodes

• Collect points on the nodal Collect points on the nodal surface during a DMC or VMC surface during a DMC or VMC walkwalk

Linear Fit:Linear Fit: i

iic

Pc 2)(min R i

iic

Pc 2)(min R

• Minimize least square deviation Minimize least square deviation 22

• Discard null solution Discard null solution ccii = 0 = 0

Be model nodeBe model node

• Second order approx.Second order approx.

• Gives the right Gives the right topology and the right topology and the right shapeshape

r1-r2r1-r2

r1+r2r1+r2

r3-r4r3-r4

0))((

0)())((

34124321

224

223

214

2134321

rrcrrrr

rrrrcrrrr

Be Node: Be Node: considerationsconsiderations

• HF and GVB give the wrong topologyHF and GVB give the wrong topology

• In CI, it seems useless to include 1sIn CI, it seems useless to include 1s22 ns ms CSFs ns ms CSFs

• 1s1s222s2s22 + 1s + 1s222p2p22 give already the right topologygive already the right topology

• Exact Exact has only 2 nodal volumeshas only 2 nodal volumes

• The nodes of the individual CSFs belong to The nodes of the individual CSFs belong to higherhigher symmetry groups than the exact symmetry groups than the exact

• The nodes of 1sThe nodes of 1s222s2s22 + 1s + 1s222p2p22 belong to belong to differentdifferent

symmetry groupssymmetry groups

0

0))((

3412

4321

rr

rrrr

222234

222234

21)21(ˆ

21)21(ˆ

pspsi

ssssi

2222

34

222234

21)21(ˆ

21)21(ˆ

pspsi

ssssi

CSFs Nodal CSFs Nodal ConjectureConjecture

Include configurations built Include configurations built with orbitals of different with orbitals of different angular momentum and angular momentum and

symmetrysymmetry

Include configurations built Include configurations built with orbitals of different with orbitals of different angular momentum and angular momentum and

symmetrysymmetry

• If we consider If we consider the nodesthe nodes of the individual of the individual CSFs as a CSFs as a basisbasis for the node of the exact for the node of the exact , , IFIF we can generalize from Be, it seems we can generalize from Be, it seems necessary necessary (maybe not sufficient)(maybe not sufficient) to to

??

Boron AtomBoron Atom

pss 221 22 pss 221 22 4 Nodal Regions4 Nodal RegionsHFHF

3222 21221 pscpss 3222 21221 pscpss 2 Nodal Regions2 Nodal RegionsCICI

GVBGVBGVBGVB

4 Nodal Regions4 Nodal Regions4 Nodal Regions4 Nodal Regions

tsDeterminan4

22211ˆ pssssA tsDeterminan4

22211ˆ pssssA

Is it possible to change the topology of the nodal Is it possible to change the topology of the nodal hypersurfaces by adding particular CSFs or do they hypersurfaces by adding particular CSFs or do they merely generate deformations? merely generate deformations? Flad, Caffarel and SavinFlad, Caffarel and Savin

Both!Both!

LiLi22 molecule molecule

222 211 gug 222 211 gug 4 Nodal Regions4 Nodal RegionsHFHF

GVBGVBGVBGVB 2 Nodal Regions2 Nodal Regions2 Nodal Regions2 Nodal Regions8 determinants8 determinantsBut energy not different than But energy not different than HFHF

2 Nodal Regions2 Nodal RegionsCICI222222 211211 uuggug c 222222 211211 uuggug c

But energy not different than But energy not different than HFHF

Nodal Topology Nodal Topology ConjectureConjecture

The HF ground state of Atomic The HF ground state of Atomic and Molecular systems has 4 and Molecular systems has 4

Nodal Regions, while the Nodal Regions, while the Exact ground state has only 2Exact ground state has only 2

The HF ground state of Atomic The HF ground state of Atomic and Molecular systems has 4 and Molecular systems has 4

Nodal Regions, while the Nodal Regions, while the Exact ground state has only 2Exact ground state has only 2

??

Node Optimization ? Node Optimization ? LiHLiH

• LiH Exact -8.0702LiH Exact -8.0702

• Simple node –8.0673(6) Simple node –8.0673(6) optimized with derivativesoptimized with derivatives

• Higher terms -8.0693(6) Higher terms -8.0693(6) optimized with derivativesoptimized with derivatives

• HoweverHowever HF nodes gives practically the exact resultHF nodes gives practically the exact result Big fluctuations, due to Big fluctuations, due to poorpoor SS((RR))

)()(LiH RR SExact eN )()(LiH RR SExact eN

)()()()( 43432121 HHLiLiHHLiLi rrcrrrrcrrN

ConclusionsConclusions• Algorithms to study topology and shape of nodesAlgorithms to study topology and shape of nodes

• ““Nodes are weird” M. Foulkes. Seattle meeting 1999Nodes are weird” M. Foulkes. Seattle meeting 1999

“...“...maybemaybe not” Bressanini, CECAM workshop not” Bressanini, CECAM workshop 20022002

• Exact or good nodes Exact or good nodes (at least for simple systems)(at least for simple systems) seem to seem to depend on few variablesdepend on few variables have higher symmetry than have higher symmetry than itself itself resemble polynomial functionsresemble polynomial functions

• Possible explanation on why HF nodes are quite good: Possible explanation on why HF nodes are quite good: they “they “naturallynaturally” have these properties” have these properties

• Hints on how to build compact MultiDet. expansionsHints on how to build compact MultiDet. expansions

• It seems possible to optimize nodes directlyIt seems possible to optimize nodes directly

• Has the ground state only 2 nodal volumes?Has the ground state only 2 nodal volumes?

AcknowledgmentsAcknowledgments

Peter ReynoldsPeter Reynolds

Gabriele MorosiGabriele Morosi

Mose’ CasalegnoMose’ Casalegno

Silvia TarascoSilvia Tarasco