The Fascinating Helium

1
Universita dellInsubria, Como, Italy
The Fascinating Helium
Dario Bressanini
http//scienze-como.uninsubria.it/bressanini
Crit05, Dresden 2005
2
The Beginning

• First discovered in the Sun by Pierre Janssen and
  Norman Lockyer in 1868

• First liquefied by Kamerlingh Onnes in 1908

• First calculations by Egil Hylleraas and John
  Slater in 1928

3
Helium studies

• Thousands of theoretical and experimental papers

have been published on Helium, in its various
forms
Small Clusters
Droplets
Bulk
Atom
4
Plan of the Talk

• Nodes of the Helium Atom Y(R)0
• Stability of mixed 3Hem4Hen clusters
• Geometry of 4He3 (if time permits)

5
Nodes
Nodes are region of N-dimensional space where
Y(R)0

• Why study Nodes of wave functions?
• They are very interesting mathematical
• Very little is known about them
• They have practical relevanceespecially
  inQuantum Monte Carlo Simulations

6
Nodes are relevant

• Levinson Theorem
• the number of nodes of the zero-energy scattering
  wave function gives the number of bound states
• Fractional quantum Hall effect
• Quantum Chaos

7
Nodes and QMC

• If we knew the exact nodes of Y, we could exactly
  simulate the system by QMC methods

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

8
Common misconception on nodes

• Nodes are not fixed by antisymmetry alone, only a
  3N-3 sub-dimensional subset

9
Common misconception on nodes

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

10
Common misconceptions on nodes

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

11
Common misconceptions on nodes

• 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
Common misconceptions on nodes

• Not even for the same symmetry species

Courant counterexample
13
Tiling Theorem (Ceperley)
Impossible for ground state
Nodal regions must have the same shape
The Tiling Theorem does not say how many nodal
regions we should expect
14
The Helium triplet

• First 3S state of He is one of very few systems
  where we know the exact node
• For S states we can write

• For the Pauli Principle

• Which means that the node is

15
The Helium triplet node

• Independent of r12
• The node is more symmetric than the wave function
  itself
• It is a polynomial in r1 and r2
• Present in all 3S states of two-electron atoms

16
Helium 1s2p 3P o
The Wave function (J.B.Anderson 1987) is

• node independent from r12 (numerical proof)

17
Other He states 1s2s 2 1S
18
Casual similarity ?
First unstable antisymmetric stretch orbit along
with the symmetric Wannier orbit r1 r2 and
various equipotential lines
19
Other He states 2 3S

• The second triplet has similar properties

"Almost"
20
He Other states

• Other states have similar properties
• Breit (1930) showed that Y(P e) (x1 y2 y1
  x2) f(r1,r2,r12)
• 2p2 3P e f( ) symmetric node (x1 y2
  y1 x2) 0
• 2p3p 1P e f( ) antisymmetric node (x1 y2
  y1 x2) (r1-r2) 0

21
He 3S a look at non-physical regions

• Consider Y(r1,r2,q12) defined in all space
• A node in a non-physical regions appears. Using a
  simple trial function...

22
He 3S a look at non-physical regions

• Consider Y(r1,r2,q12) defined in all space

• Expanding Y at second order in (0,0)
• Y (10-6 0.001 (r1r2))(r1-r2)...

23
He 3S a look at non-physical regions

• If we turn off the e-e interaction we observe the
  same feature (r1r2)(r1-r2)/2...

• There is no apparent reason why even the exact
  wave function should be
• Y c (r1r2)(r1-r2)...

• It seems the nodal structure of the exact wave
  function resembles the independent electron case

24
He Hyperspherical Approximation

• In the Hyperspherical approximation

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

1s2s 3S
1s2s 1S
1s3s 1S
1s4s 3S
They have the correct topology, and a shape close
to the exact, which is more similar to
25
Helium Nodes

• Independent from r12
• Higher symmetry than the wave function
• Some are described by polynomials in distances
  and/or coordinates
• Are these general properties of nodal surfaces ?
• Is the Helium wave function separable in some
  (unknown) coordinate system?

26
Nodal Symmetry Conjecture
WARNING Conjecture Ahead...
Symmetry of (some) nodes of Y is higher than
symmetry of Y

• Other systems apparently show this featureLi
  atom, Be Atom, He2 molecule

27
Beryllium Atom

• HF predicts 4 nodal regions Bressanini et al.
  JCP 97, 9200 (1992)
• Node (r1-r2)(r3-r4) 0

• Y factors into two determinants each one
  describing a triplet Be2. The node is the
  union of the two independent nodes.

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

28
Be Nodal Topology
29
Be nodal topology

• Now there are only two nodal regions
• It can be proved that the exact Be wave function
  has exactly two regions

Node is (r1-r2)(r3-r4) ...
See Bressanini, Ceperley and Reynolds http//scie
nze-como.uninsubria.it/bressanini/ http//archive
.ncsa.uiuc.edu/Apps/CMP/
30
Avoided crossings
Be
e- gas
31
Be model node

• Second order approx.
• Gives the right topology and the right shape
• What's next?

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
Helium

• Helium as an elementary particle. A weakly
  interacting hard sphere.
• Interatomic potential is known very accurately

• 3He (fermion antisymmetric trial function, spin
  1/2)
• 4He (boson symmetric trial function, spin zero)

Highly non-classical systems. No equilibrium
structure. ab-initio methods and normal mode
analysis useless
High resolution spectroscopy
Superfluidity
Low temperature chemistry
34
Experiment on He droplets
Toennies and Vilesov, Ann. Rev. Phys. Chem. 49,
1 (1998)

• 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

35
4Hen and 3Hen Clusters Stability

• 4He3 bound. Efimov effect?

3Hem
m ? 20 lt m lt 33 critically bound. Probably
m32(Guardiola Navarro)
36
Questions

• When is 3Hem4Hen stable?
• What is the spectrum of the3He impurities?
• Can we describe it using simple models (Harmonic
  Oscillator, Rotator,...) ?
• What is the structure of these clusters?
• What excited states do they have ?

37
3Hem4Hen Stability Chart
0 1 2 3 4 5 6 7 8 9
10 11
0 1 2 3 4 5
Terra Incognita
32
3He34He8 L0 S1/2
3He24He2 L0 S0
3He34He4 L1 S1/2
3He24He4 L1 S1
38
3He4Hen Clusters Stability
39
3He4Hen energies
n 5
Total energies (cm-1)
The p state appears at n5 The d state appears at
n9 The f state (not shown) at n19
n 9
n
40
3He4Hen energies
Spectrum similar to the rigid rotator. Different
than harmonic oscillator (sometimes used in the
literature)
41
3He4Hen Structure
3He stays on the surface. Pushed outside as L
increases
3He4He7 L 1 state
42
3He24Hen Clusters Stability

• Now put two 3He

43
Evidence of 3He24He2 Kalinin, Kornilov and
Toennies
44
3He24Hen energies relative to 4Hen
The 1P and 3P states appear for n4
The energy of 3He24Hen is roughly equal to the
4Hen energy plus the 3He orbital energies.
45
What is the shape of 4He3 ?
46
What is the shape of 4He3 ?
47
The Shape of the Trimers
Ne trimer
r(Ne-center of mass)
He trimer
r(4He-center of mass)
48
Ne3 Angular Distributions
Ne trimer
49
4He3 Angular Distributions
50
Acknowledgments.. and a suggestion

• Peter Reynolds
• Silvia Tarasco Gabriele Morosi

Take a look at your nodes