What do we (not) know about Nodes and where do we go from here ? Dario Bressanini - Georgetown University, Washington, D.C. and Universita’ dell’Insubria, ITALY Peter J. Reynolds - Georgetown University, Washington, D.C. and Office of Naval Research PacifiChem Honolulu, HI

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

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

The Helium Triplet First 3 S 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

The Helium Triplet Independent of r 12 Independent of Z: He, Li +, Be 2+,... have the same node Present in all 3 S states of two-electron atoms The node is more symmetric than the wave function itself The wave function is not factorizable but r1r1 r2r2 r 12 r1r1 r2r2

The Helium Triplet Implies that for 2 3 S helium NOT This is NOT trivial N is the Nodal Function N = r 1 -r 2, 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 Conjectures Which of these properties are present in other systems/states ? Some years ago J. B. Anderson found some of these properties in 1 P He and  u H 2 Could these be general properties of the nodal surfaces ? For a generic system, what can we say about N ?

Helium Singlet 2 1 S It is a 1 S ( 1 s 2 s ) so we write Plot the nodes (superimposed) for different  using an Hylleraas expansion (125 terms) Plot r1r1 r2r2 r 

Helium Singlet 2 1 S I.e., although, the node does not depend on   (or does very weakly) r1r1   r2r2 A very good approximation of the node is Surface contour plot of the node The second triplet has similar properties

Lithium Atom Ground State The RHF node is r 1 = r 3 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 3 S 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 ) = (5) a.u. Arne & Anderson JCP 1996 Exact = a.u. Drake, Hylleraas expansion

The Node of the Lithium Atom 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 r 2 or r ij  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., r 1 = r 3

Li Atom: Exact Wave Function The exact wave function, to be a pure 2 S, must satisfy This expression is not required to vanish for r 1 = r 3

To study an “almost exact” node we take a Hylleraas expansion for Li with 250 terms Energy  Hy = a.u. Exact = a.u. How different is its node from r 1 = r 3 ?? Li atom: Study of Exact Node

The full node is a 5D object. We can take cuts (I.e., fix r ij ) The node seems to be r 1 = r 3, taking different cuts Do a DMC simulation to check the attempted nodal crossing of the  Hy node AND r 1 = r 3 r3r3 r1r1 r2r2 r1r1 r3r3 Crosses both Crosses one

Li atom: Study of Exact Node 92 attempted crossing of both nodes 6 crossed only  Hy but not r 1 = r 3 Results Out of 6*10 6 walker moves: The 6 were either in regions where the node was very close to r 1 = r 3 or an artifact of the linear expansion

We performed a DMC simulation using a HF guiding function (with the r 1 = r 3 node) and an accurate Hylleraas trial function (to compute the local energy with re-weighting)  = (3) a.u.  = (3) a.u.  Exact a.u. Is r 1 = r 3 the exact node of Lithium ? Li atom: Study of Exact Node

 For Lithium N ( R ) = r 1 - r 3 Strong Conjecture Nodal Structure Conjecture Weak Conjecture

Beryllium Atom Be 1 s 2 2 s 2 1 S ground state  In 1992 Bressanini and others found that HF predicts 4 nodal regions JCP 97, 9200 (1992)  The HF node is ( r 1 - r 2 )*( r 3 - r 4 ) and is wrong DMC energy (4) Exact energy   factors into two determinants each one “describing” a triplet Be +2 Conjecture: exact  has TWO nodal regions

Beryllium Atom Be optimized 2 configuration  T  Plot cuts of (r 1 -r 2 ) vs (r 3 -r 4 )  In 9-D space, the direct product structure “opens up” Node is (r 1 -r 2 ) x (r 3 -r 4 ) +...

Beryllium 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 (r 1 -r 2 ) x (r 3 -r 4 ) + r 12. r

Beryllium 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: (r 1 -r 2 ) x (- r 13 + r 14 - r 23 + r 24 ) + (r 3 -r 4 ) x (- r 13 - r 14 + r 23 + r 24 )

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  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