3 Helium studies Thousands of theoretical and experimental papers Thousands of theoretical and experimental papers have been published on Helium, in its various forms: Atom Small Clusters DropletsBulk
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• 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)
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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
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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
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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.
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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
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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
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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
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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
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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
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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
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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"
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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
• 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?
Symmetry of Symmetry of (some)(some) nodes of nodes of is higher than symmetry of is higher than symmetry of
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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
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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
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44HeHenn and and 33HeHenn Cluster Clusterss StabilityStability