What do we ( not ) know about Nodes and where do we go from here ?

1. 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 2000 - Honolulu, HI

2. Nodes and the Sign Problem • So far, solutions to sign problem not proven to be efficient • Fixed-node approachis 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 NodalFunction • 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. r2 Y r1 r 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. q12 r2 r1 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 He3S • 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 = -7.47806032a.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 doesn’t 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 notrequired 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. r3 r1 r2 Crosses both 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 nodeANDr1 = r3 Crosses one r3 r1

15. Li atom: Study of Exact Node • 92 attempted crossing of both nodes • 6 crossed onlyYHy but not r1 = r3 Results 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

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-7.4780603a.u. Is r1 = r3 the exact node of Lithium ?

17. Beryllium Atom Be 1s2 2s21S ground state • In 1992 Bressanini and others found that HF predicts 4 nodal regionsJCP 97, 9200 (1992) • Y factors into two determinantseach one “describing” a triplet Be+2 • 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

18. 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) + ...

19. 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 + ...

20. Beryllium Atom Be optimized 2 configuration YT • 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 )

21. 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