1. Trento 2003 – Few Body Critical Stability
What do we know about nodes of wave functions ?
Dario Bressanini
Universita’ dell’Insubria, Como, Italy
Trento 2003 – Few Body Critical Stability

2. Quantum Monte Carlo
Time evolution
Diffusion
Birth-Death
The time dependent Schrödinger equation is similar to a diffusion equation
We simulate exactly the imaginary-time Schrödinger equation
Y is interpreted as a concentration of fictitious particles, called walkers. Computes the exact energy

3. The Fermion Problem + - Wave functions for fermions have nodes.
Diffusion equation analogy is lost. Need to introduce positive and negative walkers.
The (In)famous Sign Problem
Restrict random walk to a positive region bounded by nodes. Unfortunately, the exact nodes are unknown. + -
Use approximate nodes from a trial Y. Kill the walkers if they cross a node.
The energy depends ONLY from the nodes of Y.

4. Nodes What do we know about wave function nodes?
Very little ....
NOT fixed by (anti)symmetry alone. Only a 3N-3 subset
Very very few analytic examples
Nodal theorem is NOT VALID
Higher energy states does not mean more nodes (Courant and Hilbert )
They have (almost) nothing to do with Orbital Nodes. It is possible to use nodeless orbitals.

5. Tiling Theorem (Ceperley)
Impossible for ground state
The Tiling Theorem does not say how many nodal regions we should expect

6. Nodes and Configurations
A better Y does not mean better nodes
Why? What can we do about it?

7. 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
Find a way to parametrize the nodes using simple functions, and optimize the nodes directly minimizing the Fixed-Node energy

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

9. The Helium Triplet r1 r12 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

10. 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

11. He: Other states 1s2s 3S : (r1-r2) f(r1,r2,r12)
1s2p 1P o : node independent from r12 (J.B.Anderson)
2p2 3P e : Y = (x1 y2 – y1 x2) f(r1,r2,r12)
2p3p 1P e : Y = (x1 y2 – y1 x2) (r1-r2) f(r1,r2,r12)
1s2s 1S : node independent from r12
1s3s 3S : node independent from r12

12. Helium Nodes Independent from r12
More “symmetric” than the wave function
Some are described by polynomials in distances and/or coordinates
The same node is present in different states
The HF Y, sometimes, has the correct node, or a node with the correct (higher) symmetry
Are these general properties of nodal surfaces ?

13. 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
This is the same node present in the He 3S
Again, node has higher symmetry than Y
How good is the RHF node?
YRHF is not very good, however its node is surprisingly good (might it be the exact one?)
DMC(YRHF ) = (5) a.u. Lüchow & Anderson JCP 1996
Exact = a.u. Drake, Hylleraas expansion

14. 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 ?

15. 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

16. 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

17. 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

18. Be: CI expansion Plot cuts of (r1-r2) vs (r3-r4)
What happens to the HF node in a good CI expansion?
Plot cuts of (r1-r2) vs (r3-r4)
In 9-D space, the direct product structure “opens up”
Node is (r1-r2)(r3-r4) + ...

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

20. Be nodal topology Now there are only two nodal regions
It can be proved that the exact Be wave function has exactly two regions
See Bressanini, Ceperley and Reynolds
Node is (r1-r2) (r3-r4) + ???

21. 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 ?

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

23. 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

24. 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?

25. 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?

26. 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

27. 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

28. 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

29. Li2 molecule, large basis
Adding CFS with a large 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

30. Conclusions Exact or good nodes (at least for simple systems) seem to
depend on few variables
have higher symmetry than Y itself
resemble simple functions
Possible explanation on why HF nodes are quite good: they “naturally” have these properties
Use large basis, until HF nodes are converged
Include "different" CSFs
Has the ground state only 2 nodal volumes?

31. Acknowledgments Silvia Tarasco Peter Reynolds Gabriele Morosi
Carlos Bunge

32. Take a look at your nodes
... a suggestion
Take a look at your nodes