1. Much ado about… zeroes (of wave functions)
Universita’ dell’Insubria, Como, Italy
Much ado about… zeroes (of wave functions)
Dario Bressanini
Electronic Structure beyond DFT, Leiden 2004

2. A little advertisement
Besides nodes, I am interested in
VMC improvement
Robust optimization
Delayed rejection VMC
Mixed 3He/4He clusters, ground and excited states
Sign problem
Other QMC topics

3. Fixed Node Approximation
Restrict random walk to a positive region bounded by (approximate) nodes.
The energy is an upper bound + -
Fixed Node IS efficient, but approximation is uncontrolled
There is not (yet) a way to sistematically improve the nodes
How do we build a Y with good nodes?

4. Fixed Node Approximation
circa 1950
Rediscovered
by Anderson
and Ceperly
in the ’70s

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

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

7. Common misconceptions on nodes
A common misconception is that on a node, two like-electrons are always close. This is not true 2 1 1 2

8. 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

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

10. 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

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

12. Generalized Variational Principle
Upper bound to ground state
Higher states can be above or below
Bressanini and Reynolds, to be published

13. Nodes and Configurations
A better Y does not mean better nodes
Why? What can we do about it?
It is necessary to get a better understanding how CSF influence the nodes. Flad, Caffarel and Savin

14. The (long term) Plan of Attack
Study the nodes of exact and good approximate trial wave functions
Understand their properties
Find a way to sistematically improve the nodes of trial functions
...building them from scratch
…improving existing nodes

15. 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

16. 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 r1 r2

17. 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)
2p3p 1P e : f( ) antisymmetric node = (x1 y2 – y1 x2) (r1-r2)
1s2p 1P o : node independent from r12 (J.B.Anderson)

18. Other He states: 1s2s 2 1S and 2 3S
Although , the node does not depend on q12 (or does very weakly) r1
q12 r2
Surface contour plot of the node
A very good approximation of the node is
The second triplet has similar properties

19. Helium Nodes Independent from r12
Higher symmetry than the wave function
Some are described by polynomials in distances and/or coordinates
The HF Y, sometimes, has the correct node, or a node with the correct (higher) symmetry
Are these general properties of nodal surfaces ?

20. Lithium Atom Ground State
The RHF node is r1 = r3
if two like-spin electrons are at the same distance from the nucleus then Y =0
Node has higher symmetry than Y
How good is the RHF node?
YRHF is not very good, however its node is surprisingly good
DMC(YRHF ) = (5) a.u. Lüchow & Anderson JCP 1996
Exact = a.u. Drake, Hylleraas expansion

21. Li atom: Study of Exact Node
We take an “almost exact” Hylleraas expansion 250 term r3 r1 r2
The node seems to be r1 = r3, taking different cuts, independent from r2 or rij
a DMC simulation with r1 = r3 node and good Y to reduce the variance gives
DMC (3) a.u. Exact a.u.
Is r1 = r3 the exact node of Lithium ?

22. Li atom: Study of Exact Node
Li exact node is more symmetric than Y
At convergence, there is a delicate cancellation in order to build the node
Crude Y has a good node (r1-r3)Exp(...)
Increasing the expansion spoils the node, by including rij terms

23. Nodal Symmetry Conjecture
This observation is general: If the symmetry of the nodes is higher than the symmetry of Y, adding terms in Y might decrease the quality of the nodes (which is what we often see).
WARNING: Conjecture Ahead...
Symmetry of nodes of Y is higher than symmetry of Y

24. 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 Be+2. The node is the union of the two independent nodes.
Plot cuts of (r1-r2) vs (r3-r4)
The HF node is wrong
DMC energy (4)
Exact energy

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

26. 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

27. Hartree-Fock Nodes How Many ?
YHF has always, at least, 4 nodal regions for 4 or more electrons
It might have Na! Nb! Regions
Ne atom: 5! 5! = possible regions
Li2 molecule: 3! 3! = 36 regions
How Many ?

28. Nodal Regions
Nodal Regions 2 4 2 Ne Li Be B C
Li2

29. Nodal Topology Conjecture
WARNING: Conjecture Ahead...
The HF ground state of Atomic and Molecular systems has 4 Nodal Regions, while the Exact ground state has only 2

30. Avoided crossings Be
e- gas

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

32. Be numbers HF node -14.6565(2) 1s2 2s2 GVB node same 1s1s' 2s2s'
Luechow & Anderson (2) s2 2p2
Umrigar et al (3) +1s2 2p2
Huang et al (1) +1s2 2p2 opt
Casula & Sorella (2) +1s2 2p2 opt
Exact
Including 1s2 ns ms or 1s2 np mp configurations does not improve the Fixed Node energy...
...Why?

33. Be Node: considerations
... (I believe) they give the same contribution to the node expansion
ex: 1s22s2 and 1s23s2 have the same node
ex: 2px2, 2px3px and 3px2 have the same structure
The nodes of "useful" CSFs belong to higher and different symmetry groups than the exact Y

34. The effect of d orbitals

35. Be numbers HF -14.6565(2) 1s2 2s2 GVB node same 1s1s' 2s2s'
Luechow & Anderson (2) s2 2p2
Umrigar et al (3) +1s2 2p2
Huang et al (1) +1s2 2p2 opt
Casula & Sorella (2) +1s2 2p2 opt
Bressanini et al (7) s2 3d2
Exact

36. CSF nodal conjecture WARNING: Conjecture Ahead...
If the basis is sufficiently large, only configurations built with orbitals of different angular momentum and symmetry contribute to the shape of the nodes
This explains why single excitations are not useful

37. Carbon Atom: Topology
4 Nodal Regions HF
GVB
4 Nodal Regions
Adding determinants might not be sufficient to change the topology
2 Nodal Regions CI

38. Carbon Atom: Energy CSFs Det. Energy 1 1s22s2 2p2 1 -37.8303(4)
5 + 1s2 2s 2p23d (1)
83 1s2 + 4 electrons in 2s 2p 3s 3p 3d shell (4)
adding f orbitals
7 (4f2 + 2p34f) (1)
Exact
Where is the missing energy? (g, core, optim..)

39. He2+ molecule Basis: 2(1s) E=-4.9927(1) 5(1s) E=-4.9943(2)
3 electrons
9-1 = 8 degrees of freedom
Basis: 2(1s) E= (1)
5(1s) E= (2)
(almost exact) nodal surface of Y0 depends on
r1a, r1b, r2a and r2b: higher symmetry than Y0

40. He2+ molecule
2 Determinants
EExact =
E = (2)

41. He2+ molecule
3 Determinants
EExact =
E = (3)

42. Li2 molecule
Adding more configuration with a small basis (double zeta STO)...
Filippi & Umrigar JCP 1996

43. Li2 molecule, large basis
Adding CFS with a larger basis ... (1sg2 1su2 omitted)
HF (1) 97.2(1)
%CE
(1) 96.7(1)
GVB 8 dets (6) 96.2(6)
(1) 98.3(1)
(1) 98.3(1)
(1) 99.8(1)
Estimated n.r. limit

44. C2 CSF 1 -75.860(1) 20 -75.900(1) Barnett et. al.
(8) (7) Filippi - Umrigar
(2) (1) Lüchow - Fink
Exact
Work in progress 5(s)4(p)2(d)
(5) (8)
(6) Linear opt.

45. A tentative recipe Use a large Slater basis
But not too large
Try to reach HF nodes convergence
Use the right determinants...
...different Angular Momentum CSFs
And not the bad ones
...types already included

46. Use a good basis
The nodes of Hartree–Fock wavefunctions and their orbitals, Chem. Phys.Lett. 392, 55 (2004)
Hachmann, Galek, Yanai, Chan and, Handy

47. How to directly improve nodes?
Fit to a functional form and optimize the parameters (small systems)
IF the topology is correct, use a coordinate transformation (Linear? Feynman’s backflow ?)

48. Conclusions Nodes are worth studying! Conjectures on nodes Recipe:
have higher symmetry than Y itself
resemble simple functions
the ground state has only 2 nodal volumes
HF nodes are quite good: they “naturally” have these properties
Recipe:
Use large basis, until HF nodes are converged
Include "different kind" of CSFs with higher angular momentum

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

50. 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