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Trial wave function construction and the nodes of trial and exact wave functions in

Trial wave function construction and the nodes of trial and exact wave functions in Quantum Monte Carlo. Dario Bressanini Universita’ dell’Insubria, Como, Italy http://www.unico.it/ ~dario. ACS National Meeting 2003 – New York. Nodes and the Sign Problem.

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Trial wave function construction and the nodes of trial and exact wave functions in

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  1. Trial wave function construction and the nodes of trial and exact wave functions in Quantum Monte Carlo Dario Bressanini Universita’ dell’Insubria, Como, Italy http://www.unico.it/~dario ACS National Meeting 2003 – New York

  2. Nodes and the Sign Problem • Fixed-node QMCis 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?

  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.

  4. Tiling Theorem (Ceperley) Impossible for ground state Nodal regions must have the same shape The Tiling Theorem does not say how many nodal regions we should expect

  5. A better Y does not mean better nodes Why? What can we do about it? 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 • 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 r2 The Helium triplet node • 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. 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

  10. 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

  11. Helium Nodes • Independent from r12 • More “symmetric” than the wave function • Some are described by polynomials in distances and/or coordinates • The HF Y, sometimes, has the correct node, or a node with the correct (higher) symmetry • Are these general properties of nodal surfaces ?

  12. 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 • Node has higher symmetry than Y • How good is the RHF node? • YRHF is not very good, however its node is surprisingly good • DMC(YRHF ) = -7.47803(5)a.u.Lüchow & Anderson JCP 1996 • Exact = -7.47806032a.u.Drake, Hylleraas expansion

  13. 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 ?

  14. 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

  15. 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

  16. 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

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

  18. Be nodal topology • Now there are only two nodal regions • It can be proved that the exact Be wave function has exactly two regions Node is (r1-r2)(r3-r4) + ... See Bressanini, Ceperley and Reynolds http://www.unico.it/~dario/ http://archive.ncsa.uiuc.edu/Apps/CMP/

  19. 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 ?

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

  21. 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

  22. r1+r2 r3-r4 r1-r2 Be model node • Second order approx. • Gives the right topology and the right shape • What's next?

  23. 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?

  24. 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

  25. The effect of d orbitals

  26. 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

  27. 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

  28. 4 Nodal Regions HF GVB 4 Nodal Regions 2 Nodal Regions CI Carbon Atom: Topology Adding determinants might not be sufficient to change the topology

  29. Carbon Atom: Energy • CSFs Det. Energy • 1 1s22s2 2p21 -37.8303(4) • 2 + 1s2 2p42 -37.8342(4) • 5 + 1s2 2s2p23d18 -37.8399(1) • 83 1s2 + 4 electrons in 2s 2p 3s 3p 3d shell 422 -37.8387(4) adding f orbitals • 7 (4f2 + 2p34f) 34 -37.8407(1) Exact -37.8450 Where is the missing energy? (g, core, optim..)

  30. %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 larger basis ... (1sg2 1su2 omitted) • GVB 8 dets -14.9907(6) 96.2(6) Estimated n.r. limit -14.9954

  31. O2 • Small basis 1 Det. -150.268(1) Filippi & Umrigar 7 Det. -150.277(1) ..................... • Large basis 1 Det. -150.2850(6) Tarasco, work in progress 2 Det. -150.2873(7) .................................. Exact -150.3268

  32. 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?

  33. Acknowledgments.. and a suggestion Silvia Tarasco Peter Reynolds Gabriele Morosi Carlos Bunge Take a look at your nodes

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