The Fascinating The Fascinating HeliumHelium

Dario Dario BressaniniBressanini

Crit05, Dresden 2005 Crit05, Dresden 2005

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

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

2

The BeginningThe Beginning

• First discovered in the Sun by First discovered in the Sun by Pierre JanssenPierre Janssen and and NormanNorman LockyerLockyer in in 18681868

• First liquefied by First liquefied by Kamerlingh Kamerlingh OnnesOnnes in in 19081908

• First calculations by First calculations by EgilEgil HylleraasHylleraas and and John SlaterJohn Slater in in 19281928

3

Helium studiesHelium studies• ThousandsThousands of theoretical and of theoretical and

experimental papersexperimental papers

)()(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

4

Plan of the TalkPlan of the Talk

•Nodes of the Helium Atom: Nodes of the Helium Atom: (R)=0(R)=0

•Stability of mixed Stability of mixed 33HeHemm44HeHenn clusters clusters

•Geometry of Geometry of 44HeHe33 (if time permits)(if time permits)

5

NodesNodes

• Why study Nodes of wave functions?Why study Nodes of wave functions? They are very interesting mathematicalThey are very interesting mathematical Very little is known about themVery little is known about them They have practical relevanceThey have practical relevance

especially inespecially inQuantum Monte Carlo SimulationsQuantum Monte Carlo Simulations

Nodes are region of N-dimensional space where (R)=0

6

Nodes are relevantNodes are relevant• Levinson Theorem:Levinson Theorem:

the number of nodes of the zero-energy the number of nodes of the zero-energy scattering wave function gives the number of scattering wave function gives the number of bound statesbound states

• Fractional quantum Hall effectFractional quantum Hall effect• Quantum ChaosQuantum Chaos

Integrable systemIntegrable system Chaotic systemChaotic system

7

Nodes and QMCNodes and QMC

++ --

If we If we knewknew the exact nodes of the exact nodes of , we , we couldcould exactly exactly simulatesimulate the system by QMC methods the system by QMC methods

We restrict random walk to a positive We restrict random walk to a positive region bounded by (region bounded by (approximateapproximate)) nodes. nodes.

8

Common misconception Common misconception on nodeson nodes

• Nodes are Nodes are notnot fixed by antisymmetry fixed by antisymmetry alone, only a 3N-3 sub-dimensional alone, only a 3N-3 sub-dimensional subsetsubset

9

Common misconception Common misconception on nodeson nodes

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

orbitalsorbitals

10

Common misconceptions Common misconceptions on on nodesnodes

• A common misconception is that A common misconception is that on a on a nodenode, two like-electrons are always , two like-electrons are always closeclose. This is not true. This is not true

22 11

0

0

0

11 22

11

Common misconceptions on Common misconceptions on nodesnodes

• Nodal theorem is Nodal theorem is NOT VALID in N-DimensionsNOT VALID in N-Dimensions Higher energy states Higher energy states does notdoes not mean more nodes mean more nodes ((Courant and Courant and

Hilbert Hilbert )) It is only an upper boundIt is only an upper bound

12

Common misconceptions on Common misconceptions on nodesnodes

• Not even for the same symmetry speciesNot even for the same symmetry species

0 0.5 1 1.5 2 2.5 3

0

0.5

1

1.5

2

2.5

3

Courant counterexampleCourant counterexample

14

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 the exact nodesystems where we know the 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

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The Helium triplet The Helium triplet nodenode

• IndependentIndependent of 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 of two-electron states of two-electron atomsatoms

r1

r2

r1

2

021 rr

r1

r2

021 rr

16

Helium 1s2p Helium 1s2p 33P P oo

•node independent fromnode independent from r r12 12 (numerical proof)(numerical proof)

),,(),,()( 121221221103 rrrfzrrrfzP

The Wave function The Wave function (J.B.Anderson 1987) (J.B.Anderson 1987) isis

),,()),(),(()( 1221221103 rrrrzgrzgP

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),,( 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 11SS

r1

r2

Surface contour plot of the node

• A very good approximation A very good approximation of the node isof the node is constrr 4

24

1

18

Casual similarity ?Casual similarity ?

First unstable antisymmetric stretch orbit along with the symmetric Wannier orbit r1 = r2 and various equipotential lines

19

• The second triplet has similar propertiesThe second triplet has similar properties

Other He states: 2 Other He states: 2 33SS

constrr 52

51"Almost""Almost"

20

He: Other statesHe: Other states

•Other states have similar propertiesOther states have similar properties•Breit Breit ((19301930) ) showed thatshowed that

P P ee)= ()= (xx11 yy22 – – yy11 xx22) f() f(rr11,,rr22,,rr1212)) 2p2p22 33P P ee : f( ) symmetric : f( ) symmetric

node = (node = (xx11 yy22 – – yy11 xx22) = 0) = 0 22pp33p p 11P P ee : f( ) antisymmetric : f( ) antisymmetric

node = (node = (xx11 yy22 – – yy11 xx22) () (rr11--rr22) = 0) = 0

24

He: Hyperspherical He: Hyperspherical ApproximationApproximation

• In the Hyperspherical approximation:In the Hyperspherical approximation:2

22

1),()(),( rrRRRFR

• which means the first few which means the first few SS excited states excited states have circular nodes..have circular nodes..

1s2s 1s2s 33SS 1s2s 1s2s 11SS 1s3s 1s3s 11SS 1s4s 1s4s 33SSThey have the correct topology, and a shape They have the correct topology, and a shape closeclose to the to the exact, which is more similar toexact, which is more similar to Constrr kk 21

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Helium NodesHelium Nodes

• Independent from Independent from rr1212

• Higher symmetry than the wave functionHigher symmetry than the wave function• Some are described by polynomials in Some are described by polynomials in

distances and/or coordinatesdistances and/or coordinates• Are these Are these general propertiesgeneral properties of nodal of nodal

surfaces ?surfaces ?• Is the Helium wave function Is the Helium wave function separableseparable in in

some (some (unknownunknown) coordinate system?) coordinate system?

)()( RR fExact eN

26

Nodal Symmetry Nodal Symmetry ConjectureConjecture

• Other systems apparently show this Other systems apparently show this feature:feature:

Li atom, Be Atom, HeLi atom, Be Atom, He22++ molecule molecule

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

Symmetry of Symmetry of (some)(some) nodes of nodes of is higher than symmetry of is higher than symmetry of

28

Be Nodal TopologyBe Nodal Topology

0HF

r3-r4r3-r4

r1-r2r1-r2

r1+r2r1+r2

0CI

r1-r2r1-r2

r1+r2r1+r2

r3-r4r3-r4

2222 2121 pscss

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A (Nodal) song...A (Nodal) song...

He deals the cards to find the answersHe deals the cards to find the answersthe secret the secret geometry of chancegeometry of chancethe the hidden lawhidden law of a probable outcome of a probable outcomethe numbers lead a dancethe numbers lead a dance

Sting: Shape of my heartSting: Shape of my heart

33

HeliumHelium Helium as an elementary particle. A weakly interacting Helium as an elementary particle. A weakly interacting

hard sphere.hard sphere. Interatomic potential is known very accuratelyInteratomic potential is known very accurately 33He (fermion: antisymmetric trial function, spin 1/2) He (fermion: antisymmetric trial function, spin 1/2) 44He (boson: symmetric trial function, spin zero)He (boson: symmetric trial function, spin zero)

Highly non-classical systems. No equilibrium structure.Highly non-classical systems. No equilibrium structure.ab-initio methods and normal mode ab-initio methods and normal mode analysisanalysis useless useless

SuperfluiditySuperfluidity

High resolution spectroscopyHigh resolution spectroscopy

Low temperature chemistryLow temperature chemistry

35

44HeHenn and and 33HeHenn Cluster Clusterss StabilityStability

44HeHe33 bound. Efimov effect? bound. Efimov effect?

Liquid: stableLiquid: stable

44HeHe22 dimer existsdimer exists

44HeHenn

All clusters All clusters boundbound

Liquid: stableLiquid: stable

33HeHe22

dimer dimer unboundunbound

33HeHemm

m = ?m = ? 20 < 20 < mm < 33 < 33critically bound. critically bound. Probably m=32Probably m=32(Guardiola & Navarro)(Guardiola & Navarro)

36

QuestionsQuestions•When is When is 33HeHemm

44HeHenn stable? stable?•What is the spectrum of theWhat is the spectrum of the

33He impurities?He impurities?•Can we describe it using simple Can we describe it using simple

models (Harmonic Oscillator, models (Harmonic Oscillator, Rotator,...) ?Rotator,...) ?

•What is the structure of these What is the structure of these clusters?clusters?

•What excited states do they have ?What excited states do they have ?

37

33HeHemm44HeHenn Stability Stability

ChartChart

3232

44HeHenn 33HeHemm 0 1 2 3 4 5 6 70 1 2 3 4 5 6 7 8 9 10 11 8 9 10 11001122334455

33HeHe3344HeHe88 L=0 S=1/2 L=0 S=1/2

33HeHe2244HeHe44 L=1 S=1 L=1 S=1

33HeHe2244HeHe22 L=0 S=0 L=0 S=0

33HeHe3344HeHe44 L=1 S=1/2 L=1 S=1/2

Terra IncognitaTerra Incognita

Bound L=0Bound L=0

UnboundUnbound

UnknownUnknown

L=1 S=1/2L=1 S=1/2

L=1 S=1L=1 S=1

BoundBound

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33HeHe44HeHenn: energies: energies

The p state appears at n=5The d state appears at n=9The f state (not shown) at n=19

n = 5n = 5

n = 9n = 9

Tota

l ene

rgie

s (c

m-1

)

n

40

33HeHe44HeHenn: energies: energies

0 2 4 6

-47

-46 .5

-46

-45 .5

-45

-44 .5

sp

d

f

g

1s2s

He30

Tota

l ene

rgie

s (c

m-1

)

33HeHe44HeHe3030

L (angular momentum)

Spectrum similar to the rigid rotator. Different than harmonic oscillator (sometimes used in the literature)

l = 0

l = 1

l = 2

41

33HeHe44HeHenn: Structure: Structure

33HeHe44HeHe77 : L = 1 state : L = 1 state

3He stays on the surface. Pushed outside as L increases

44HeHe 33HeHe

0 10 20 30 40

0

0.002

0.004

0.006

0.008

0.01

spdf

42

33HeHe2244HeHenn Cluster Clusterss StabilityStability

Now put two Now put two 33HeHe

33HeHe2244HeHenn

All clusters All clusters up boundup bound

33HeHe2244HeHe

Trimer Trimer unboundunbound

33HeHe2244HeHe22

Tetramer boundTetramer bound5 out of 6 unbound pairs5 out of 6 unbound pairs

44HeHe4 4 E = E = -0.-0.33888866((11)) cm cm-1-1

33HeHe44HeHe33 E = E = -0.-0.20622062((11)) cm cm-1-1

33HeHe2244HeHe22 E = E = -0.-0.071071((11)) cm cm-1-1

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Evidence of Evidence of 33HeHe2244HeHe2 2 Kalinin, Kornilov Kalinin, Kornilov

and Toenniesand Toennies

44

33HeHe2244HeHenn : energies : energies

relative to relative to 44HeHenn

0 4 8 12 16 20n

-2.5

-2

-1.5

-1

-0.5

0

E (c

m-1

)

L S 0 0 s2

1 1 sp 1 0 sp

l l = 0= 0 ____________

l l = 1= 1 ____________l l = 0= 0 ______ ______

l l = 1= 1 ____________l l = 0= 0 ______ ______

11SS

33PP

11PP

TheThe 11P and P and 33P P states appear for n=4The energy of 3He2

4Hen is roughly equal to the 4Hen energy plus the 3He orbital energies.

What is the shape of What is the shape of 44HeHe33 ? ?

47

The Shape of the TrimersThe Shape of the TrimersNe trimerNe trimer

He trimerHe trimer

((44He-He-center of masscenter of mass))

((NNe-e-center of masscenter of mass))

48

NeNe33 Angular Distributions Angular Distributions

Ne trimerNe trimer

49

44HeHe33 Angular Distributions Angular Distributions

50

Acknowledgments.. and a Acknowledgments.. and a suggestionsuggestion

Peter ReynoldsPeter ReynoldsSilvia TarascoSilvia Tarasco Gabriele MorosiGabriele Morosi

Take a look at Take a look at youryour nodes nodes