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The nodes of trial and exact wave functions in Quantum Monte Carlo

The nodes of trial and exact wave functions in Quantum Monte Carlo. Dario Bressanini Universita’ dell’Insubria, Como, Italy http://www.unico.it/ ~dario Peter J. Reynolds Office of Naval Research. CECAM 2002 - Lyon. Nodes and the Sign Problem.

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The nodes of trial and exact wave functions in Quantum Monte Carlo

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  1. The nodes of trial and exact wave functions in Quantum Monte Carlo Dario Bressanini Universita’ dell’Insubria, Como, Italy http://www.unico.it/~dario Peter J. Reynolds Office of Naval Research CECAM 2002 - Lyon

  2. Nodes and the Sign Problem • So far, solutions to the sign problem have 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 • How do we build a Y with good nodes? • CI? MCSCF? Natural Orbitals (Lüchow) ?

  3. 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. • Tiling theorem

  4. Tiling Theorem (Ceperley) Impossible for ground state The Tiling Theorem does not say how many nodal regions we should expect

  5. Nodes and Configurations It is necessary to get a better understanding how CSF influence the nodes. Flad, Caffarel and Savin

  6. 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, or... • …find a way to parametrize the nodes using simple functions, and optimize the nodes directly minimizing the Fixed-Node energy

  7. 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

  8. r1 r12 r2 r1 • The wave function is not factorizablebut 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

  9. The Helium Triplet • This is NOT trivial • N = r1-r2 , Antisymmetric NodalFunction • f = unknown, totally symmetric • The HF function has the exact node • Which of these properties are present in other systems? • Are these be general properties of the nodal surfaces ? • For a generic system, what can we say about N ?

  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 Helium Singlet 1s2s 2 1S

  11. 1s3s 3S : node independent from r12 • Similar to 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 • Similar to

  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(as if Helium were separable) • The HF Y, sometimes, has the correct node, or a node with the correct (higher) symmetry

  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 we found in the He3S • Again, node has higher symmetry • 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 How different is its node from r1 = r3 ?? • We take an “almost exact” Hylleraas expansion 250 terms • Energy YHy = -7.478059 a.u. Exact = -7.4780603 a.u. • The node seems to ber1 = r3, taking different cuts, independent from r2 or rij

  15. Li atom: Study of Exact Node Numerically, we found only very small deviations from the HF node, or artifacts of the linear expansion • 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 ?

  16. Node Y(R) = 0 8-dimensional hypersurface Beryllium Atom Ground State 12 D Factor external angles 9 D After spin assignment

  17. Beryllium Atom • Y factors into two determinants each one “describing” a triplet Be+2. The node is the union of the two independent nodes. • HF predicts 4 nodal regionsBressanini et al. JCP 97, 9200 (1992) • Node: (r1-r2)(r3-r4) = 0 • The HF node is wrong • DMC energy -14.6576(4) • Exact energy -14.6673

  18. 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 ?

  19. Nodal Regions Li 2 Be 4 B 4 C 4 Ne 4 4 Li2 Clustering Algorithm

  20. Be: beyond Restricted Hartree-Fock • Hartree-Fock Y is not the most general single particle approximation • Try a GVB wave function (each electron in its own orbital) YGVB(Be) = sum of 4 Determinants

  21. Be: beyond Hartree-Fock • YGVB(Be) = sum of 4 Determinants • VMC energy improves • s2(H) improves • …but still the same node(r1-r2)(r3-r4) = 0 (r1-r2)(r3-r4) = 0 for anyf1, f2, f3, f4

  22. Plot cuts of (r1-r2) vs (r3-r4) • Of course , one would first use Be: CI expansion • What happens to the HF node in a CI expansion? • Still the same topology and the same node(same in lithium) In DMC, CI is not necessarily better than HF

  23. 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) + ...

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

  25. Be nodal topology • The clustering algorithm confirms that 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)+ ???

  26. Fitting Nodes • We would like a simple analytical approximation of the node • Ultimate goal: parametrize the node with few parameters, and directly optimize them • Useful also for diagnostic. To see if the node changes, and how, by changing basis, or expansion, or functional form. • How can we model the implicit surface Y=0 ? • The studied (simple) systems suggest polynomials of distances (plus x, y and z for L0 states)

  27. Model Node: Polynomials of ri and rij(or other simple functions) with the correct spin-space symmetry. • To find c: f(Node,c) line integral Node Fitting Nodes

  28. Fitting Nodes • Collect points on the nodal surface during a DMC or VMC walk Linear Fit: • Minimize least square deviation D2 • Discard null solution ci = 0

  29. r1+r2 r3-r4 r1-r2 Be model node • Second order approx. • Gives the right topology and the right shape

  30. Be Node: considerations • HF and GVB give the wrong topology • In CI, it seems useless to include 1s2 ns ms CSFs • 1s22s2 + 1s22p2give already the right topology • Exact Yhas only 2 nodal volumes • The nodes of the individual CSFs belong to higher symmetry groups than the exact Y • The nodes of 1s22s2 + 1s22p2belong to different symmetry groups

  31. CSFs Nodal Conjecture • If we consider the nodes of the individual CSFs as a basis for the node of the exact Y, IF we can generalize from Be, it seems necessary (maybe not sufficient) to Include configurations built with orbitals of different angular momentum and symmetry ?

  32. 4 Nodal Regions HF GVB 4 Nodal Regions 2 Nodal Regions CI Boron Atom Is it possible to change the topology of the nodal hypersurfaces by adding particular CSFs or do they merely generate deformations? Flad, Caffarel and Savin Both!

  33. 4 Nodal Regions HF 2 Nodal Regions 8 determinants GVB But energy not different than HF 2 Nodal Regions CI But energy not different than HF Li2 molecule

  34. Nodal Topology Conjecture The HF ground state of Atomic and Molecular systems has 4 Nodal Regions, while the Exact ground state has only 2 ?

  35. Node Optimization ? LiH • LiH Exact -8.0702 • Simple node –8.0673(6) optimized with derivatives • Higher terms -8.0693(6) optimized with derivatives • However • HF nodes gives practically the exact result • Big fluctuations, due to poorS(R)

  36. Conclusions • Algorithms to study topology and shape of nodes • “Nodes are weird” M. Foulkes. Seattle meeting 1999“...maybe not” Bressanini, CECAM workshop 2002 • Exact or good nodes (at least for simple systems) 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 • Hints on how to build compact MultiDet. expansions • It seems possible to optimize nodes directly • Has the ground state only 2 nodal volumes?

  37. Acknowledgments Peter Reynolds Gabriele Morosi Mose’ Casalegno Silvia Tarasco

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