What do we (What do we (notnot) know ) know about Nodesabout Nodesand where do we go from and where do we go from here ?here ?

Dario BressaniniDario Bressanini - Georgetown - Georgetown University, Washington, D.C. and University, Washington, D.C. and Universita’ dell’Insubria, ITALYUniversita’ dell’Insubria, ITALY

Peter J. ReynoldsPeter J. Reynolds - Georgetown - Georgetown University, Washington, D.C. and University, Washington, D.C. and Office of Naval ResearchOffice of Naval Research

PacifiChem 2000 - PacifiChem 2000 - Honolulu, HIHonolulu, HI

Nodes and the Sign Nodes and the Sign ProblemProblem

•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

The Plan of AttackThe 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

The Helium Triplet 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

),,( 1221 rrr

),,(),,( 12121221 rrrrrr

•Which means that the node is

02121 rrorrr

The Helium TripletThe 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 factorizable

but

r1

r2

r1

2

021 rr

r1

r2

021 rr

)(),(),,( 12211221 rrrrrr

),,(211221

1221),(),,( rrrferrNrrr

The Helium TripletThe Helium Triplet

),,(211221

1221),(),,( rrrferrNrrr ),,(211221

1221),(),,( rrrferrNrrr • Implies that for 2 3S helium

•This is NOTNOT 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

Nodal ConjecturesNodal 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 u H2

•Could these be general properties of the nodal surfaces ?

•For a generic system, what can we say about N ? )()( RR f

Exact eN )()( RR fExact eN

Helium Singlet 2 Helium Singlet 2 11SS

• It is a 1S (1s2s) so we write

• Plot the nodes (superimposed) for different using an Hylleraas expansion (125 terms)

• Plot

),,( 1221 rr

r1

r2

),,( rr

r

Helium Singlet 2 Helium Singlet 2 11SS

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

r1

r2

),,( 1221 rr

• A very good approximation of the node is constrr 4

241

Surface contour plot of the node

• The second triplet has similar properties

Lithium Atom Lithium Atom Ground StateGround State

)(1)(2)(1)(2)(1)(2)(1)(1 21331321 rsrsrsrsrsrsrsrsRHF

•The RHF node is r1 = r3

if two like-spin electrons are at the same distance from the nucleus then =0

•This is the same node we found in the He 3S

•How good is the RHF node?

RHF is not very good, however its node is surprisingly good (might it be the exact one?)

DMC(RHF ) = -7.47803(5) a.u. Arne & Anderson JCP 1996

Exact = -7.47806032 a.u. Drake, Hylleraas expansion

The Node of the Lithium The Node of the Lithium Atom Atom

310 rrRHF

•Note that RHF belongs to a higher symmetry group than the exact wave function. The node has even higher symmetry, since it doesn’t depend on r2 or rij

iiiiiGVBCI rrrJrhrgrfA ),,()()()(ˆ 231312321

•Â is the anti-symmetrizer, f, g and h are radial functions, and J is a totally symmetric function (like a Jastrow)

• CI-GVB has exactly the same node, I.e., r1 = r3

LiLi Atom: Exact Wave Atom: Exact Wave Function Function

),,,,,( 231312321 rrrrrrLi

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

),,,,,(),,,,,(

),,,,,(),,,,,(

131223132121323123

132312312231312321

rrrrrrfrrrrrrf

rrrrrrfrrrrrrf

• This expression is not required to vanish for r1 = r3

• To study an “almost exact” node we take a Hylleraas expansion for Li with 250 terms

• Energy Hy = -7.478059 a.u.Exact = -7.4780603 a.u.

321231312321

ˆ rrrkjilmnHy errrrrrA

How different is its node from r1 = r3 ??

LiLi atom: Study of atom: Study of Exact NodeExact Node

LiLi atom: Study of atom: Study of Exact NodeExact 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 Hy node AND r1 = r3

r3r1

r2

r1

r3

Crosses both

Crosses one

LiLi atom: Study of atom: Study of Exact NodeExact Node

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

ResultsResults

0.7 0.75 0.8 0.85 0.9 0.95 10.7

0.75

0.8

0.85

0.9

0.95

1

Out of 6*106 walker moves:Out of 6*106 walker moves:

The 6 were either in regions where the node wasvery close to r1 = r3 or an artifact of the linear expansion

• 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)

• = 0.001 -7.478061(3) a.u. = 0.003 -7.478062(3) a.u. Exact -7.4780603 a.u.

Is r1 = r3 the exact node of Lithium ?

LiLi atom: Study of atom: Study of Exact NodeExact Node

Beryllium AtomBeryllium Atom

Be 1s2 2s2 1S ground state In 1992 Bressanini and others

found that HF predicts 4 nodal regions JCP 97, 9200 (1992)

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

factors into two determinantseach one “describing” a triplet Be+2

Conjecture: exact has TWO nodal regions

Beryllium AtomBeryllium Atom

Be optimized 2 configuration T

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) + ...

Beryllium AtomBeryllium Atom

Be optimized 2 configuration T

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 + ...

Beryllium AtomBeryllium Atom

Be optimized 2 configuration T

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

(r1-r2) x (-r13 + r14 -r23 + r24 ) +

(r3-r4) x (-r13 - r14 +r23 + r24 )

ConclusionsConclusions

•“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 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