1 / 50

Much ado about… zeroes ( of wave functions )

Universita’ dell’Insubria, Como, Italy. Much ado about… zeroes ( of wave functions ). Dario Bressanini. http://scienze-como.uninsubria.it/ bressanini. Electronic Structure beyond DFT, Leiden 2004. A little advertisement. Besides nodes, I am interested in VMC improvement Robust optimization

emery
Download Presentation

Much ado about… zeroes ( of wave functions )

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript

  1. Universita’ dell’Insubria, Como, Italy Much ado about… zeroes (of wave functions) Dario Bressanini http://scienze-como.uninsubria.it/bressanini Electronic Structure beyond DFT, Leiden 2004

  2. A little advertisement • Besides nodes, I am interested in • VMC improvement • Robust optimization • Delayed rejection VMC • Mixed 3He/4He clusters, ground and excited states • Sign problem • Other QMC topics http://scienze-como.uninsubria.it/bressanini

  3. + - Fixed Node Approximation • Restrict random walk to a positive region bounded by (approximate) nodes. • The energy is an upper bound • Fixed Node IS efficient, but approximation is uncontrolled • There is not (yet) a way to sistematically improve the nodes • How do we build a Y with good nodes?

  4. Fixed Node Approximation circa 1950 Rediscovered by Anderson and Ceperly in the ’70s

  5. Common misconception on nodes • Nodes are not fixed by antisymmetry alone, only a 3N-3 sub-dimensional subset

  6. Common misconception on nodes • They have (almost) nothing to do with Orbital Nodes. • It is (sometimes) possible to use nodeless orbitals

  7. 2 1 1 2 Common misconceptions on nodes • A common misconception is that on a node, two like-electrons are always close. This is not true

  8. Common misconceptions on nodes • Nodal theorem is NOT VALID in N-Dimensions • Higher energy states does not mean more nodes (Courant and Hilbert ) • It is only an upper bound

  9. Common misconceptions on nodes • Not even for the same symmetry species Courant counterexample

  10. 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

  11. Chaotic system Integrable system Nodes are relevant • Levinson Theorem: • the number of nodes of the zero-energy scattering wave function gives the number of bound states • Fractional quantum Hall effect • Quantum Chaos

  12. Generalized Variational Principle Upper bound to ground state Higher states can be above or below Bressanini and Reynolds, to be published

  13. 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

  14. 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 • ...building them from scratch • …improving existing nodes

  15. The Helium triplet • First 3S state of He is one of very few systems where we know the exact node • For S states we can write • For the Pauli Principle • Which means that the node is

  16. 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

  17. He: Other states • Other states have similar properties • Breit (1930) showed thatY(P e)= (x1y2 – y1x2) f(r1,r2,r12) • 2p23P e : f( ) symmetric node = (x1y2 – y1x2) • 2p3p 1P e : f( ) antisymmetricnode = (x1y2 – y1x2) (r1-r2) • 1s2p 1P o : node independent from r12(J.B.Anderson)

  18. 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

  19. Helium Nodes • Independent from r12 • Higher symmetry 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 ?

  20. 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

  21. 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 ?

  22. 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

  23. 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

  24. 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

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

  26. 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://scienze-como.uninsubria.it/bressanini/ http://archive.ncsa.uiuc.edu/Apps/CMP/

  27. 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 ?

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

  29. 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

  30. Avoided crossings Be e- gas

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

  32. 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?

  33. 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

  34. The effect of d orbitals

  35. 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

  36. 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

  37. 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

  38. 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..)

  39. He2+ molecule 3 electrons 9-1 = 8 degrees of freedom Basis: 2(1s) E=-4.9927(1) 5(1s) E=-4.9943(2) (almost exact) nodal surface of Y0 depends on r1a, r1b, r2a and r2b: higher symmetry than Y0

  40. He2+ molecule 2 Determinants EExact = -4.994598 E = -4.9932(2)

  41. He2+ molecule 3 Determinants EExact = -4.994598 E = -4.9778(3)

  42. Filippi & Umrigar JCP 1996 Li2 molecule • Adding more configuration with a small basis (double zeta STO)...

  43. %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

  44. C2 CSF 1 -75.860(1) 20 -75.900(1) Barnett et. al. 36 -75.9025(7) Barnett et. al. 1 -75.8613(8) 4 -75.8901(7) Filippi - Umrigar 1 -75.866(2) 32 -75.900(1) Lüchow - Fink Exact-75.9255 Work in progress 5(s)4(p)2(d) 1 -75.8692(5) 12 -75.9032(8) 12 -75.9038(6) Linear opt.

  45. A tentative recipe • Use a large Slater basis • But not too large • Try to reach HF nodes convergence • Use the right determinants... • ...different Angular Momentum CSFs • And not the bad ones • ...types already included

  46. Use a good basis The nodes of Hartree–Fock wavefunctions and their orbitals, Chem. Phys.Lett. 392, 55 (2004) Hachmann, Galek, Yanai, Chan and, Handy

  47. How to directly improve nodes? • Fit to a functional form and optimize the parameters (small systems) • IF the topology is correct, use a coordinate transformation (Linear? Feynman’s backflow ?)

  48. Conclusions • Nodes are worth studying! • Conjectures on nodes • have higher symmetry than Y itself • resemble simple functions • the ground state has only 2 nodal volumes • HF nodes are quite good: they “naturally” have these properties • Recipe: • Use large basis, until HF nodes are converged • Include "different kind" of CSFs with higher angular momentum

  49. Acknowledgments.. and a suggestion Silvia TarascoPeter Reynolds Gabriele Morosi Carlos Bunge Take a look at your nodes

  50. A (Nodal) song... He deals the cards to find the answers the secret geometry of chance the hidden law of a probable outcome the numbers lead a dance Sting: Shape of my heart

More Related