Trial wave function Trial wave function construction and the construction and the nodes of trial and nodes of trial and exact wave functions exact wave functions ininQuantum Monte Quantum Monte CarloCarlo

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

ACS National Meeting 2003 – New YorkACS National Meeting 2003 – New York

2

Nodes and the Sign Nodes and the Sign ProblemProblem

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

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

3

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.

4

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

Nodal regions must have the same shapeNodal regions must have the same shape

6

Nodes and Nodes and ConfigurationsConfigurations

A better A better does not mean better nodes does not mean better nodesWhy? What can we do about it?Why? What can we do about it?

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

7

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 improve

the nodes of trial functionsthe nodes of trial functions FindFind a way to a way to parametrizeparametrize the nodes the nodes

using simple functions, and using simple functions, and optimizeoptimize the nodes directly minimizing the the nodes directly minimizing the Fixed-Node energyFixed-Node energy

8

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

9

The Helium triplet The Helium triplet nodenode

• Independent of Independent of rr1212

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

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

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

r1

r2

r1

2

021 rr

r1

r2

021 rr

13

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

Other He states: 1s2s 2 Other He states: 1s2s 2 11S S and and 2 2 33SS

• 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

14

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

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

15

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

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

)()( RR fExact eN )()( RR fExact eN

16

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

• Node has higher symmetry than Node has higher symmetry than

• 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 goodsurprisingly good

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

17

• We take an “almost exact” Hylleraas expansion We take an “almost exact” Hylleraas expansion 250 term250 term

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

• a DMC simulation with a DMC simulation with rr11 = r = r33 node node and good and good to to reduce the variancereduce 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 ?

18

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

• Li exact node is more symmetric than Li exact node is more symmetric than

321231312321

ˆ rrrkjilmnHy errrrrrA

• At convergence, there is a delicate At convergence, there is a delicate cancellation in order to build the nodecancellation in order to build the node

• Crude Crude has a good node (r has a good node (r11-r-r33)Exp(...))Exp(...)

• Increasing the expansion Increasing the expansion spoilsspoils the node, the node, by including by including rrijij terms terms

19

Nodal Symmetry Nodal Symmetry ConjectureConjecture

• This observation is general:This observation is general:

If the symmetry of the nodes is higher If the symmetry of the nodes is higher than the symmetry of than the symmetry of , adding terms in , adding terms in mightmight decrease the quality of the decrease the quality of the nodes nodes (which is what we often see)(which is what we often see)..

WARNING: Conjecture Ahead...WARNING: Conjecture Ahead...

Symmetry of nodes of Symmetry of nodes of is is higher than symmetry of higher than symmetry of Symmetry of nodes of Symmetry of nodes of is is higher than symmetry of higher than symmetry of

20

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 factors into two determinants each one “describing” a each one “describing” a triplettriplet BeBe+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

Plot cuts of (rPlot cuts of (r11-r-r22) ) vsvs (r (r33--rr44))

23

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

24

Be nodal topologyBe nodal topology

•Now there are only Now there are only twotwo nodal regionsnodal regions

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

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/

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

25

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 ?

26

Nodal RegionsNodal RegionsNodal RegionsNodal Regions

NeNe

LiLi

BeBe

BB

CC

LiLi22

224444444444

HF

222222222222

CI

28

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

WARNING: Conjecture Ahead...WARNING: Conjecture Ahead...

29

Be model nodeBe model node

• Second order approx.Second order approx.

• Gives the right topology Gives the right topology and the right shapeand the right shape

• What's next?What's next?

r1-r2r1-r2

r1+r2r1+r2

r3-r4r3-r4

0))((

0)())((

34124321

224

223

214

2134321

rrcrrrr

rrrrcrrrr

30

Be numbersBe numbers

• HF nodeHF node -14.6565(2)-14.6565(2)1s1s22 2s 2s22

• GVB nodeGVB node same same 1s1s' 2s2s'1s1s' 2s2s'

• Luechow & AndersonLuechow & Anderson -14.667-14.6672(2)2(2) +1s+1s22 2p 2p22

• Umrigar et al.Umrigar et al. -14.667-14.66718(3)18(3) +1s+1s22 2p 2p22

• Huang et al.Huang et al. -14.667-14.66726(1)26(1) +1s+1s22 2p 2p22 optopt

• Casula & SorellaCasula & Sorella -14.667-14.66728(2)28(2) +1s+1s22 2p 2p2 2 optopt

• ExactExact -14.667-14.66735553555• Including 1sIncluding 1s22 ns ms or 1s ns ms or 1s22 np mp configurations np mp configurations

does not improve the Fixed Node energy...does not improve the Fixed Node energy...

...Why?...Why?

31

Be Node: Be Node: considerationsconsiderations

• ... ... (I believe)(I believe) they give the same contribution to they give the same contribution to the node expansionthe node expansion

• ex: 1sex: 1s222s2s22 and 1s and 1s223s3s22 have the same node have the same node

• ex: 2pex: 2pxx22, 2p, 2pxx3p3pxx and 3p and 3pxx

22 have the same structure have the same structure

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

• The nodes of "useful" CSFs belong to The nodes of "useful" CSFs belong to higher higher andand differentdifferent symmetry groups than the exact symmetry groups than the exact

)()( 221121 rfrfxx

32

The effect of d The effect of d orbitalsorbitals

0 0.002 0.004 0.006Time step

-14.6674

-14.6673

-14.6672

-14.6671

-14.6670

-14.6669

-14.6668

En

erg

y

1s2 2s2 = -14.6565(2)

Exact

+ 1s2 2p2

+ 1s2 3d2

33

Be numbersBe numbers

• HFHF -14.6565(2)-14.6565(2) 1s1s22 2s 2s22

• GVB nodeGVB node same same 1s1s' 2s2s'1s1s' 2s2s'

• Luechow & AndersonLuechow & Anderson -14.667-14.6672(2)2(2) +1s+1s22 2p 2p22

• Umrigar et al.Umrigar et al. -14.667-14.66718(3)18(3) +1s+1s22 2p 2p22

• Huang et al.Huang et al. -14.667-14.66726(1)26(1) +1s+1s22 2p 2p2 2

optopt

• Casula & SorellaCasula & Sorella -14.667-14.66728(2)28(2) +1s+1s22 2p 2p2 2 optopt

• Bressanini et al.Bressanini et al. -14.667-14.66733(7)33(7) +1s+1s22 3d 3d22

• ExactExact -14.667-14.66735553555

34

CSF nodal conjectureCSF nodal conjecture

WARNING: Conjecture Ahead...WARNING: Conjecture Ahead...

IfIf the basis is sufficiently the basis is sufficiently large, large, onlyonly configurations configurations

built with orbitals of built with orbitals of differentdifferent angular momentum angular momentum and and symmetrysymmetry contribute to contribute to

the shape of the nodesthe shape of the nodes

IfIf the basis is sufficiently the basis is sufficiently large, large, onlyonly configurations configurations

built with orbitals of built with orbitals of differentdifferent angular momentum angular momentum and and symmetrysymmetry contribute to contribute to

the shape of the nodesthe shape of the nodes

This explains why single excitations are not usefulThis explains why single excitations are not useful

35

Carbon Atom: Carbon Atom: TopologyTopology

222 221 pss 222 221 pss 4 Nodal Regions4 Nodal RegionsHFHF

42222 21221 pscpss 42222 21221 pscpss 2 Nodal Regions2 Nodal RegionsCICI

GVBGVBGVBGVB

4 Nodal Regions4 Nodal Regions4 Nodal Regions4 Nodal Regions

tsDeterminan4

222211ˆ ppssssA tsDeterminan4

222211ˆ ppssssA

Adding determinants might not Adding determinants might not be sufficient to change the be sufficient to change the topologytopology

36

Carbon Atom: EnergyCarbon Atom: Energy• CSFsCSFs Det.Det. EnergyEnergy

• 1 1 1s1s222s2s2 2 2p2p22 11 -37.8303(4)-37.8303(4)

• 2 2 + 1s+ 1s22 2p 2p44 22 -37.8342(4)-37.8342(4)

• 5 5 + 1s+ 1s22 2s 2s 2p2p223d3d 1818 -37.8399(1)-37.8399(1)

• 83 1s83 1s22 + 4 electrons in + 4 electrons in 2s 2p 3s 3p 3d2s 2p 3s 3p 3d shell shell422422 -37.8387(4)-37.8387(4)

adding f orbitalsadding f orbitals

• 77 (4f(4f22 + 2p + 2p334f)4f) 3434 -37.8407(1)-37.8407(1)

ExactExact -37.8450 -37.8450

Where is the missing energy? (g, core, optim..)Where is the missing energy? (g, core, optim..)

38

LiLi22 molecule, large molecule, large basisbasis

Adding CFS with a larger basis ... (1Adding CFS with a larger basis ... (1gg22 1 1uu

22 omitted) omitted)

+8+8 -14.9914(1) -14.9914(1) 96.7(1)96.7(1)2gnσ

• ++ -14.9933(1) -14.9933(1) 98.3(1)98.3(1)22 11 uyux

+4+4 -14.9933(1) -14.9933(1) 98.3(1)98.3(1)22uyux nn

• ++ -14.9952(1) -14.9952(1)99.8(1)99.8(1)

22 32 gzuz pp

22 g• HFHF -14.9919(1) -14.9919(1)97.2(1)97.2(1)

%CE%CE

Estimated n.r. limitEstimated n.r. limit -14.9954-14.9954

• GVB 8 detsGVB 8 dets -14.9907(6) -14.9907(6) 96.2(6)96.2(6)

39

OO22

• Small basisSmall basis

1 Det.1 Det. -150.268(1)-150.268(1) Filippi & UmrigarFilippi & Umrigar

7 Det.7 Det. -150.277(1)-150.277(1) ..........................................

• Large basisLarge basis

1 Det.1 Det. -150.2850(6)-150.2850(6) Tarasco, work in progressTarasco, work in progress

2 Det.2 Det. -150.2873(7)-150.2873(7) ....................................................................

ExactExact -150.3268-150.3268

40

ConclusionsConclusions• Exact or good nodes Exact or good nodes (at least for simple systems)(at least for simple systems)

seem toseem to depend on few variablesdepend on few variables have higher symmetry than have higher symmetry than itself itself resemble simple functionsresemble simple functions

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

• Use large basis, until HF nodes are convergedUse large basis, until HF nodes are converged

• Include "different" CSFsInclude "different" CSFs

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

41

Acknowledgments.. and a Acknowledgments.. and a suggestionsuggestion

Silvia Tarasco Silvia Tarasco Peter ReynoldsPeter Reynolds

Gabriele Morosi Carlos BungeGabriele Morosi Carlos Bunge

Take a look at Take a look at youryour nodes nodes