What do we (not) know about Nodes

1
What do we (not) know about Nodes and where do we
go from here ?
Dario Bressanini - Georgetown University,
Washington, D.C. and Universita dellInsubria,
ITALY Peter J. Reynolds - Georgetown University,
Washington, D.C. and Office of Naval Research
PacifiChem 2000 - Honolulu, HI
2
Nodes and the Sign Problem

• So far, solutions to sign problem not proven to
  be efficient
• Fixed-node approach is efficient. If only we
  could have the exact nodes
• or at least a systematic way to improve the
  nodes ...
• we could bypass the sign problem

3
The Plan of Attack

• Study the nodes of exact and good approximate
  trial wave functions
• Understand their properties
• Find a way to parametrize the nodes using simple
  functions
• Optimize the nodes minimizing the Fixed-Node
  energy

4
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

5
The Helium Triplet

• Independent of r12
• Independent of Z He, Li, Be2,... have the same
  node
• Present in all 3S states of two-electron atoms
• The node is more symmetric than the wave function
  itself
• The wave function is not factorizablebut

r1
r12
r2
r1
r2
6
The Helium Triplet

• Implies that for 2 3S helium

• This is NOT trivial
• N is the Nodal Function
• N r1-r2 , Antisymmetric
• f unknown, totally symmetric
• The exponential is there to emphasize the
  positivity of the non-nodal factor
• The HF function has the exact node

7
Nodal Conjectures

• Which of these properties are present in other
  systems/states ?
• Some years ago J. B. Anderson found some of these
  properties in 1P He and Su H2
• Could these be general properties of the nodal
  surfaces ?
• For a generic system, what can we say about N ?

8
Helium Singlet 2 1S

• It is a 1S (1s2s) so we write
• Plot the nodes (superimposed) for different q
  using an Hylleraas expansion (125 terms)
• Plot

9
Helium Singlet 2 1S

• I.e., although , the
  node does not depend on q12 (or does very weakly)

• A very good approximation of the node is

• The second triplet has similar properties

Surface contour plot of the node
10
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 we found in the He 3S
• 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 ) -7.47803(5) a.u. Arne Anderson JCP
  1996
• Exact -7.47806032 a.u. Drake, Hylleraas
  expansion

11
The Node of the Lithium Atom

• Note that YRHF belongs to a higher symmetry group
  than the exact wave function. The node has even
  higher symmetry, since it doesnt depend on r2 or
  rij

• Â is the anti-symmetrizer, f, g and h are radial
  functions, and J is a totally symmetric function
  (like a Jastrow)
• YCI-GVB has exactly the same node, I.e., r1 r3

12
Li Atom Exact Wave Function

• The exact wave function, to be a pure 2S, must
  satisfy

• This expression is not required to vanish for r1
  r3

13
Li atom Study of Exact Node

• To study an almost exact node we take a
  Hylleraas expansion for Li with 250 terms
• Energy YHy -7.478059 a.u.
• Exact -7.4780603 a.u.

How different is its node from r1 r3 ??
14
Li atom Study of Exact Node

• The full node is a 5D object. We can take cuts
  (I.e., fix rij )
• The node seems to ber1 r3, taking different
  cuts
• Do a DMC simulation to check the attempted nodal
  crossing of the YHy node AND r1 r3

Crosses one
r3
r1
15
Li atom Study of Exact Node
Results
Out of 6106 walker moves

• 92 attempted crossing of both nodes
• 6 crossed only YHy but not r1 r3

The 6 were either in regions where the node
wasvery close to r1 r3 or an artifact of the
linear expansion
16
Li atom Study of Exact Node

• We performed a DMC simulation using a HF guiding
  function (with the r1 r3 node) and an accurate
  Hylleraas trial function (to compute the local
  energy with re-weighting)
• t 0.001 -7.478061(3) a.u. t 0.003
  -7.478062(3) a.u. Exact -7.4780603
  a.u.

Is r1 r3 the exact node of Lithium ?
17
Nodal Structure Conjecture
Strong Conjecture
Weak Conjecture

• For Lithium N(R) r1 - r3

18
Beryllium Atom
Be 1s2 2s2 1S ground state

• In 1992 Bressanini and others found that HF
  predicts 4 nodal regions JCP 97, 9200 (1992)

• Y factors into two determinantseach one
  describing a triplet Be2

• The HF node is (r1-r2)(r3-r4) and is wrong
• DMC energy -14.6576(4)
• Exact energy -14.6673

Conjecture exact Y has TWO nodal regions
19
Beryllium Atom
Be optimized 2 configuration YT

• Plot cuts of (r1-r2) vs (r3-r4)

• In 9-D space, the direct product structure
  opens up

Node is (r1-r2) x (r3-r4) ...
20
Beryllium Atom
Be optimized 2 configuration YT

• Clues to structure of additional terms? Take
  cuts...

• With alpha electrons along any ray from origin,
  node is when beta's are on any sphere (almost).
  Further investigation leads to...

Node is (r1-r2) x (r3-r4) r12 . r34 ...
21
Beryllium Atom
Be optimized 2 configuration YT

• Using symmetry constraints coupled with
  observation, full node (to linear order in rs)
  can only contain these two terms and one more

(r1-r2) x (-r13 r14 -r23 r24 )
(r3-r4) x (-r13 - r14 r23 r24 )
22
Conclusions

• Nodes are weird M. Foulkes. Seattle meeting
  1999...Maybe not Bressanini Reynolds.
  Honolulu 2000
• Exact nodes (at least for atoms) seem to
• depend on few variables
• have higher symmetry than Y itself
• resemble polynomial functions
• Possible explanation on why HF nodes are quite
  good they naturally have these properties
• It seems possible to optimize nodes directly