1. The Fascinating Helium Dario Bressanini Crit05, Dresden 2005 http://scienze-como.uninsubria.it/bressanini Universita’ dell’Insubria, Como, Italy

2. 2 The Beginning First discovered in the Sun by Pierre Janssen and Norman Lockyer in 1868 First discovered in the Sun by Pierre Janssen and Norman Lockyer in 1868 First liquefied by Kamerlingh Onnes in 1908 First liquefied by Kamerlingh Onnes in 1908 First calculations by Egil Hylleraas and John Slater in 1928 First calculations by Egil Hylleraas and John Slater in 1928

3. 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

4. 4 Plan of the Talk Nodes of the Helium Atom:  (R)=0 Nodes of the Helium Atom:  (R)=0 Stability of mixed 3 He m 4 He n clusters Stability of mixed 3 He m 4 He n clusters Geometry of 4 He 3 (if time permits) Geometry of 4 He 3 (if time permits)

5. 5 Nodes Why study Nodes of wave functions? Why study Nodes of wave functions?  They are very interesting mathematical  Very little is known about them  They have practical relevance especially in Quantum Monte Carlo Simulations Nodes are region of N-dimensional space where  (R)=0

6. 6 Nodes are relevant Levinson Theorem: Levinson Theorem:  the number of nodes of the zero-energy scattering wave function gives the number of bound states Fractional quantum Hall effect Fractional quantum Hall effect Quantum Chaos Quantum Chaos Integrable system Chaotic system

7. 7 Nodes and QMC +- If we knew the exact nodes of , we could exactly simulate the system by QMC methods If we knew the exact nodes of , we could exactly simulate the system by QMC methods We restrict random walk to a positive region bounded by ( approximate ) nodes. We restrict random walk to a positive region bounded by ( approximate ) nodes.

8. 8 Common misconception on nodes Nodes are not fixed by antisymmetry alone, only a 3N-3 sub-dimensional subset Nodes are not fixed by antisymmetry alone, only a 3N-3 sub-dimensional subset

9. 9 Common misconception on nodes They have (almost) nothing to do with Orbital Nodes. They have (almost) nothing to do with Orbital Nodes.  It is (sometimes) possible to use nodeless orbitals

10. 10 Common misconceptions on nodes A common misconception is that on a node, two like-electrons are always close. This is not true A common misconception is that on a node, two like-electrons are always close. This is not true 21 12

11. 11 Common misconceptions on nodes Nodal theorem is NOT VALID in N-Dimensions Nodal theorem is NOT VALID in N-Dimensions  Higher energy states does not mean more nodes ( Courant and Hilbert )  It is only an upper bound

12. 12 Common misconceptions on nodes Not even for the same symmetry species Not even for the same symmetry species Courant counterexample

13. 13 Tiling Theorem (Ceperley) Impossible for ground state The Tiling Theorem does not say how many nodal regions we should expect Nodal regions must have the same shape

14. 14 The Helium triplet First 3 S state of He is one of very few systems where we know the exact node First 3 S state of He is one of very few systems where we know the exact node For S states we can write For S states we can write Which means that the node is Which means that the node is For the Pauli Principle For the Pauli Principle

15. 15 The Helium triplet node Independent of r 12 Independent of r 12 The node is more symmetric than the wave function itself The node is more symmetric than the wave function itself It is a polynomial in r 1 and r 2 It is a polynomial in r 1 and r 2 Present in all 3 S states of two-electron atoms Present in all 3 S states of two-electron atoms r1r1 r2r2 r 12 r1r1 r2r2

16. 16 Helium 1s2p 3 P o node independent from r 12 (numerical proof) node independent from r 12 (numerical proof) The Wave function (J.B.Anderson 1987) is

17. 17 Although, the node does not depend on   (or does very weakly) Although, the node does not depend on   (or does very weakly) Other He states: 1s2s 2 1 S r1r1   r2r2 Surface contour plot of the node A very good approximation of the node is A very good approximation of the node is

18. 18 Casual similarity ? First unstable antisymmetric stretch orbit along with the symmetric Wannier orbit r 1 = r 2 and various equipotential lines

19. 19 The second triplet has similar properties The second triplet has similar properties Other He states: 2 3 S "Almost"

20. 20 He: Other states Other states have similar properties Other states have similar properties Breit ( 1930 ) showed that  P e )= ( x 1 y 2 – y 1 x 2 ) f( r 1, r 2, r 12 ) Breit ( 1930 ) showed that  P e )= ( x 1 y 2 – y 1 x 2 ) f( r 1, r 2, r 12 )  2p 2 3 P e : f( ) symmetric node = ( x 1 y 2 – y 1 x 2 ) = 0  2 p 3 p 1 P e : f( ) antisymmetric node = ( x 1 y 2 – y 1 x 2 ) ( r 1 - r 2 ) = 0

21. 21 He 3 S: a look at non-physical regions Consider  ( r 1, r 2,  12 ) defined in all space Consider  ( r 1, r 2,  12 ) defined in all space A node in a non-physical regions appears. Using a simple trial function... A node in a non-physical regions appears. Using a simple trial function...

22. 22 He 3 S: a look at non-physical regions Consider  ( r 1, r 2,  12 ) defined in all space Consider  ( r 1, r 2,  12 ) defined in all space Expanding  at second order in (0,0) Expanding  at second order in (0,0)  = (10 -6 + 0.001 (r 1 +r 2 ))(r 1 -r 2 )+... r1r1r1r1 r2r2r2r2  12

23. 23 He 3 S: a look at non-physical regions If we turn off the e-e interaction we observe the same feature: (r 1 +r 2 )(r 1 -r 2 )/2+... If we turn off the e-e interaction we observe the same feature: (r 1 +r 2 )(r 1 -r 2 )/2+... There is no apparent reason why even the exact wave function should be There is no apparent reason why even the exact wave function should be  = c (r1+r2)(r1-r2)+...  = c (r1+r2)(r1-r2)+... It seems the nodal structure of the exact wave function resembles the independent electron case It seems the nodal structure of the exact wave function resembles the independent electron case

24. 24 He: Hyperspherical Approximation In the Hyperspherical approximation: In the Hyperspherical approximation: which means the first few S excited states have circular nodes.. which means the first few S excited states have circular nodes.. 1s2s 3 S 1s2s 1 S 1s3s 1 S 1s4s 3 S They have the correct topology, and a shape close to the exact, which is more similar to

25. 25 Helium Nodes Independent from r 12 Independent from r 12 Higher symmetry than the wave function Higher symmetry than the wave function Some are described by polynomials in distances and/or coordinates Some are described by polynomials in distances and/or coordinates Are these general properties of nodal surfaces ? Are these general properties of nodal surfaces ? Is the Helium wave function separable in some (unknown) coordinate system? Is the Helium wave function separable in some (unknown) coordinate system?

26. 26 Nodal Symmetry Conjecture Other systems apparently show this feature: Li atom, Be Atom, He 2 + molecule Other systems apparently show this feature: Li atom, Be Atom, He 2 + molecule WARNING: Conjecture Ahead... Symmetry of (some) nodes of  is higher than symmetry of 

27. 27 Beryllium Atom  HF predicts 4 nodal regions Bressanini et al. JCP 97, 9200 (1992)  Node: (r 1 -r 2 )(r 3 -r 4 ) = 0   factors into two determinants each one “describing” a triplet Be +2. The node is the union of the two independent nodes.  The HF node is wrong DMC energy -14.6576(4)DMC energy -14.6576(4) Exact energy -14.6673Exact energy -14.6673 Plot cuts of (r 1 -r 2 ) vs (r 3 -r 4 ) Plot cuts of (r 1 -r 2 ) vs (r 3 -r 4 )

28. 28 Be Nodal Topology r3-r4 r1-r2 r1+r2 r1-r2 r1+r2 r3-r4

29. 29 Be nodal topology Now there are only two nodal regions Now there are only two nodal regions It can be proved that the exact Be wave function has exactly two regions It can be proved that the exact Be wave function has exactly two regions See Bressanini, Ceperley and Reynolds http://scienze-como.uninsubria.it/bressanini/http://archive.ncsa.uiuc.edu/Apps/CMP/ Node is (r 1 -r 2 )(r 3 -r 4 ) +...

30. 30 Avoided crossings Be e - gas

31. 31 Be model node Second order approx. Second order approx. Gives the right topology and the right shape Gives the right topology and the right shape What's next? What's next? r1-r2 r1+r2 r3-r4

32. 32 A (Nodal) song... He deals the cards to find the answers the secret geometry of chance the hidden law of a probable outcome the numbers lead a dance Sting: Shape of my heart

33. 33 Helium Helium as an elementary particle. A weakly interacting hard sphere. Helium as an elementary particle. A weakly interacting hard sphere. Interatomic potential is known very accurately Interatomic potential is known very accurately  3 He (fermion: antisymmetric trial function, spin 1/2)  4 He (boson: symmetric trial function, spin zero) Highly non-classical systems. No equilibrium structure. ab-initio methods and normal mode analysis useless Superfluidity High resolution spectroscopy Low temperature chemistry

34. 34 Adiabatic expansion cools helium to below the critical point, forming droplets. Adiabatic expansion cools helium to below the critical point, forming droplets. The droplets are sent through a scattering chamber to pick up impurities, and are detected with a mass spectrometer The droplets are sent through a scattering chamber to pick up impurities, and are detected with a mass spectrometer Toennies and Vilesov, Ann. Rev. Phys. Chem. 49, 1 (1998) Experiment on He droplets

35. 35 4 He n and 3 He n Clusters Stability 4 He 3 bound. Efimov effect? 4 He 3 bound. Efimov effect? Liquid: stable 4 He 2 dimer exists 4 He n All clusters bound Liquid: stable 3 He 2 dimer unbound 3 He m m = ? 20 < m < 33 critically bound. Probably m=32 (Guardiola & Navarro)

36. 36 Questions When is 3 He m 4 He n stable? When is 3 He m 4 He n stable? What is the spectrum of the 3 He impurities? What is the spectrum of the 3 He impurities? Can we describe it using simple models (Harmonic Oscillator, Rotator,...) ? Can we describe it using simple models (Harmonic Oscillator, Rotator,...) ? What is the structure of these clusters? What is the structure of these clusters? What excited states do they have ? What excited states do they have ?

37. 37 3 He m 4 He n Stability Chart 32 4 He n 4 He n 3 He m 3 He m 0 1 2 3 4 5 6 7 8 9 10 11 0 1 2 3 4 5 6 7 8 9 10 11 012345 3 He 3 4 He 8 L=0 S=1/2 3 He 2 4 He 4 L=1 S=1 3 He 2 4 He 2 L=0 S=0 3 He 3 4 He 4 L=1 S=1/2 Terra Incognita Bound L=0 Unbound Unknown L=1 S=1/2 L=1 S=1 Bound

38. 38 3 He 4 He n Clusters Stability 3 He 4 He dimer unbound 3 He 4 He 2 Trimer bound 3 He 4 He n All clusters up bound Bonding interaction Non-bonding interaction

39. 39 3 He 4 He n : energies The p state appears at n=5 The d state appears at n=9 The f state (not shown) at n=19 n = 5 n = 9 Total energies (cm -1 ) n

40. 40 3 He 4 He n : energies Total energies (cm -1 ) 3 He 4 He 30 L (angular momentum) Spectrum similar to the rigid rotator. Different than harmonic oscillator (sometimes used in the literature)

41. 41 3 He 4 He n : Structure 3 He 4 He 7 : L = 1 state 3 He stays on the surface. Pushed outside as L increases 4 He 3 He

42. 42 3 He 2 4 He n Clusters Stability Now put two 3 He Now put two 3 He 3 He 2 4 He n All clusters up bound 3 He 2 4 He Trimer unbound 3 He 2 4 He 2 Tetramer bound 5 out of 6 unbound pairs 4 He 4 E = -0.3886(1) cm -1 3 He 4 He 3 E = -0.2062(1) cm -1 3 He 2 4 He 2 E = -0.071(1) cm -1

43. 43 Evidence of 3 He 2 4 He 2 Kalinin, Kornilov and Toennies

44. 44 3 He 2 4 He n : energies relative to 4 He n l = 0 ______ l = 1 ______ l = 0 ______ l = 1 ______ l = 0 ______ 1S1S1S1S 3P3P3P3P 1P1P1P1P The 1 P and 3 P The 1 P and 3 P states appear for n=4 The energy of 3 He 2 4 He n is roughly equal to the 4 He n energy plus the 3 He orbital energies.

45. What is the shape of 4 He 3 ?

46. 46 What is the shape of 4 He 3 ?

47. 47 The Shape of the Trimers Ne trimer He trimer  ( 4 He-center of mass)  (Ne-center of mass)

48. 48 Ne 3 Angular Distributions       Ne trimer

49. 49      4 He 3 Angular Distributions

50. 50 Acknowledgments.. and a suggestion Peter Reynolds Silvia Tarasco Gabriele Morosi Take a look at your nodes