What do we know about nodes of wave functions ?

1. What do we know about nodes of wave functions ? Dario Bressanini Universita’ dell’Insubria, Como, Italy http://www.unico.it/~dario Trento 2003 – Few Body Critical Stability

2. Time evolution Diffusion Birth-Death Quantum Monte Carlo • 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 ofY.

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

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. r1 r12 r2 r1 r2 The Helium Triplet • 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

10. 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 q12 r2 r1 Surface contour plot of the node Other He states: 1s2s 2 1S and 2 3S

11. He: Other states • 1s2s 3S : (r1-r2) f(r1,r2,r12) • 1s2p 1P o : node independent from r12(J.B.Anderson) • 2p23P e : Y = (x1y2 – y1x2) f(r1,r2,r12) • 2p3p1P e : Y = (x1y2 – y1x2) (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 He3S • 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 ) = -7.47803(5)a.u.Lüchow & Anderson JCP 1996 • Exact = -7.47806032a.u.Drake, Hylleraas expansion

14. r3 r1 r2 Li atom: Study of Exact Node • The node seems to ber1 = r3, taking different cuts, independent from r2 or rij • We take an “almost exact” Hylleraas expansion 250 term • a DMC simulation with r1 = r3 node and good Y to reduce the variancegives • DMC-7.478061(3)a.u.Exact-7.4780603a.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 Ymight 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. Plot cuts of (r1-r2) vs (r3-r4) Beryllium Atom • HF predicts 4 nodal regionsBressanini 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. • The HF node is wrong • DMC energy -14.6576(4) • Exact energy -14.6673

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

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

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 http://www.unico.it/~dario/ http://archive.ncsa.uiuc.edu/Apps/CMP/ Node is (r1-r2) (r3-r4)+ ???

21. Hartree-Fock Nodes • YHF has always, at least, 4 nodal regions for 4 or more electrons • It might have Na! Nb! Regions • Ne atom: 5! 5! = 14400 possible regions • Li2 molecule: 3! 3! = 36 regions How Many ?

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

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. r1+r2 r3-r4 r1-r2 Be model node • 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 -14.6672(2)+1s2 2p2 • Umrigar et al. -14.66718(3)+1s2 2p2 • Huang et al. -14.66726(1)+1s2 2p2opt • Casula & Sorella -14.66728(2)+1s2 2p2 opt • Exact -14.6673555 • 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 anddifferent symmetry groups than the exact Y

27. Be numbers • HF -14.6565(2)1s2 2s2 • GVB node same 1s1s' 2s2s' • Luechow & Anderson -14.6672(2)+1s2 2p2 • Umrigar et al. -14.66718(3)+1s2 2p2 • Huang et al. -14.66726(1)+1s2 2p2 opt • Casula & Sorella -14.66728(2)+1s2 2p2 opt • Bressanini et al. -14.66733(7)+1s2 3d2 • Exact -14.6673555

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. %CE • HF -14.9919(1) 97.2(1) +8 -14.9914(1) 96.7(1) • + -14.9933(1) 98.3(1) +4 -14.9933(1) 98.3(1) • + -14.9952(1) 99.8(1) Li2 molecule, large basis Adding CFS with a large basis ... (1sg2 1su2 omitted) • GVB 8 dets -14.9907(6) 96.2(6) Estimated n.r. limit -14.9954

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