Is QMC delivering its early promises?

1. Universita’ dell’Insubria, Como, Italy Some reflections on nodes and trial wave functions Is QMC delivering its early promises? Dario Bressanini http://scienze-como.uninsubria.it/bressanini QMCI Sardagna (Trento) 2006

2. 30 years of QMC in chemistry

3. The Early promises? • Solve the Schrödinger equation exactly withoutapproximation(very strong) • Solve the Schrödinger equation with controlled approximations, and converge to the exact solution (strong) • Solve the Schrödinger equation with some approximation, and do better than other methods (weak)

4. Good for Helium studies • Thousands of theoretical and experimental papers have been published on Helium, in its various forms: Small Clusters Droplets Bulk Atom

5. 4Hen 3Hem Bound L=0 Unbound Unknown L=1 S=1/2 L=1 S=1 Bound 3Hem4Hen Stability Chart 0 1 2 3 4 5 6 7 8 9 10 11 0 1 2 3 4 5 Terra Incognita 32 3He34He8 L=0 S=1/2 3He24He2 L=0 S=0 3He34He4 L=1 S=1/2 3He24He4 L=1 S=1

6. Good for vibrational problems

7. For electronic structure? Sign Problem Fixed Nodal error problem

8. The influence on the nodes of YT • QMC currently relies on YT(R) and its nodes (indirectly) • How are the nodes YT(R) of influenced by: • The single particle basis set • The generation of the orbitals (HF, CAS, MCSCF, NO, …) • The number and type of configurations in the multidet. expansion ?

9. He2+: the basis set The ROHF wave function: 1s E = -4.9905(2) hartree 1s1s’2s3s E = -4.9943(2) hartree EN.R.L = -4.9945 hartree

10. He2+: MO’s Bressanini et al. J. Chem. Phys. 123, 204109 (2005) • E(RHF) = -4.9943(2) hartree • E(CAS) = -4.9925(2) hartree • E(CAS-NO) = -4.9916(2) hartree • E(CI-NO) = -4.9917(2) hartree • EN.R.L = -4.9945 hartree

11. + E(2 csf) = -4.9946(2) hartree + E(2 csf) = -4.9925(2) hartree He2+: CSF’s 1s1s’2s3s2p2p’ • E(1 csf) = -4.9932(2) hartree 1s1s’2s3s • E(1 csf) = -4.9943(2) hartree

12. Li2 CSF E (hartree) (1sg2 1su2 omitted) -14.9923(2) -14.9914(2) -14.9933(2) -14.9933(1) -14.9939(2) -14.9952(1) E (N.R.L.) -14.9954 • Not all CSF are useful • Only 4 csf are needed to build a statistically exact nodal surface

13. A tentative recipe • Use a large Slater basis • But not too large • Try to reach HF nodes convergence • Orbitals from CAS seem better than HF, or NO • Not worth optimizing MOs, if the basis is large enough • Only few configurations seem to improve the FN energy • Use the right determinants... • ...different Angular Momentum CSFs • And not the bad ones • ...types already included

14. Dimers Bressanini et al. J. Chem. Phys. 123, 204109 (2005)

15. Is QMC competitive ?

16. 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) R12-MR-CI -37.845179 Exact (estimated) -37.8450

17. Ne Atom Drummond et al. -128.9237(2) DMC Drummond et al. -128.9290(2) DMC backflow Gdanitz et al. -128.93701 R12-MR-CI Exact (estimated)-128.9376

18. Conventional wisdom on Y • EVMC(YRHF) > EVMC(YUHF) > EVMC(YGVB) Single particle approximations Consider the N atom • YRHF = |1sR 2sR 2px 2py 2pz| |1sR 2sR| • YUHF = |1sU 2sU 2px 2py 2pz| |1s’U 2s’U| EDMC(YRHF) > ? < EDMC(YUHF)

19. Conventional wisdom on Y We can build a YRHF with the same nodes of YUHF • YUHF = |1sU 2sU 2px 2py 2pz| |1s’U 2s’U| • Y’RHF = |1sU 2sU 2px 2py 2pz| |1sU 2sU| EDMC(Y’RHF) = EDMC(YUHF) EVMC(Y’RHF) > EVMC(YRHF) > EVMC(YUHF)

20. Same Node Conventional wisdom on Y YGVB = |1s 2s 2p3| |1s’ 2s’| - |1s’ 2s 2p3| |1s 2s’| + |1s’ 2s’ 2p3| |1s 2s|- |1s 2s’ 2p3| |1s’ 2s| Node equivalent to a YUHF |f(r) g(r) 2p3| |1s 2s| EDMC(YGVB) = EDMC(Y’’RHF)

21. Nitrogen Atom • Y Param. E corr. VMC E corr. DMC • Simple RHF (1 det) 4 26.0% 91.9% • Simple RHF (1 det) 8 42.7% 92.6% • Simple UHF (1 det) 11 41.2% 92.3% • Simple GVB (4 det) 11 42.3% 92.3% • Clementi-Roetti + J 27 24.5% 93.1% Is it worth to continue to add parametersto the wave function?

22. What to do? • Should we be happy with the “cancellation of error”, and pursue it? • If so: • Is there the risk, in this case, that QMC becomes Yet Another Computational Tool, and not particularly efficient nor reliable? • VMC seems to be much more robust, easy to “advertise” • If not, and pursue orthodox QMC(no pseudopotentials, no cancellation of errors, …), can we avoid thecurse of YT ?

23. The curse of the YT • QMC currently relies on YT(R) • Walter Kohn in its Nobel lecture (R.M.P. 71, 1253 (1999)) “discredited” the wave function as a non legitimate concept when N (number of electrons) is large For M=109 andp=3 N=6 p = parameters per variable M = total parameters needed The Exponential Wall

24. Convergence to the exact Y We must include the correct analytical structure Cusps: QMC OK QMC OK 3-body coalescence and logarithmic terms: Often neglected Tails:

25. Asymptotic behavior of Y • Example with 2-e atoms is the solution of the 1 electron problem

26. Asymptotic behavior of Y • The usual form does not satisfy the asymptotic conditions A closed shell determinant has the wrong structure

27. Asymptotic behavior of Y Take 2N coupled electrons • In general Recursively, fixing the cusps, and setting the right symmetry… Each electron has its own orbital, Multideterminant (GVB) Structure! 2N determinants. Again an exponential wall

28. PsH – Positronium Hydride • A wave function with the correct asymptotic conditions: Bressanini and Morosi: JCP 119, 7037 (2003)

29. We need new, and different, ideas • Different representations • Different dimensions • Different equations • Different potential • Radically different algorithms • Different something Research is the process of going up alleys to see if they are blind.  Marston Bates

30. Just an example • Try a different representation • Is some QMC in the momentum representation • Possible ? And if so, is it: • Practical ? • Useful/Advantageus ? • Eventually better than plain vanilla QMC ? • Better suited for some problems/systems ? • Less plagued by the usual problems ?

31. The other half of Quantum mechanics The Schrodinger equation in the momentum representation Some QMC (GFMC) should be possible, given the iterative form Or write the imaginary time propagator in momentum space

32. Better? • For coulomb systems: • There are NO cusps in momentum space. Y convergence should be faster • Hydrogenic orbitals are simple rational functions

33. Use the Hypernode of Another (failed so far) example • Different dimensionality: Hypernodes • Given HY (R) = EY (R) build • The hope was that it could be better than Fixed Node

34. The intuitive idea was that the system could correct the wrong fixed nodes, by exploring regions where Fixed Node Fixed HyperNode Trial node Trial node Exact node Exact node Hypernodes • The energy is still an upper bound • Unfortunately, it seems to recover exactly the FN energy

35. Why is QMC not used by chemists? A little intermezzo

36. DMC Top 10 reasons • 12. We need forces, dummy! • 11. Try getting O2 to bind at the variational level. • 10. How many graduate students lives have been lost optimizing wavefunctions? • 9. It is hard to get 0.01 eV accuracy by throwing dice. • 8. Most chemical problems have more than 50 electrons. • 7. Who thought LDA or HF pseudopotentials would be any good? • 6. How many spectra have you seen computed by QMC? • 5. QMC is only exact for energies. • 4. Multiple determinants. We can't live with them, we can't live without them. • 3. After all, electrons are fermions. • 2. Electrons move. • 1. QMC isn't included in Gaussian 90. Who programs anyway? http://web.archive.org/web/20021019141714/archive.ncsa.uiuc.edu/Apps/CMP/topten/topten.html

37. Chemistry and Mathematics "We are perhaps not far removed from the time, when we shall be able to submit the bulk of chemical phenomena to calculation” Joseph Louis Gay-Lussac - 1808 “The underlying physical laws necessary for the mathematical theory of a large part of physics and the whole of chemistry are thus completely known, and the difficulty is only that the exact application of these equations leads to equations much too complicated to be soluble” P.A.M. Dirac - 1929

38. Nature and Mathematics “il Grande libro della Natura e’ scritto nel linguaggio della matematica, e non possiamo capirla se prima non ne capiamo i simboli“ Galileo Galilei Every attempt to employ mathematical methods in the study of chemical questions must be considered profoundly irrational and contrary to the spirit of chemistry… If mathematical analysis should ever hold a prominent place in chemistry – an aberration which is happily almost impossible – it would occasion a rapid and widespread degeneration of that science. Auguste Compte

39. Orthodox QMC A Quantum Chemistry Chart J.Pople The more accurate the calculations became, the more the concepts tended to vanish into thin air (Robert Mulliken)

40. NOT DIRECTLY OBSERVABLES ILL-DEFINED CONCEPTS d- d+ d- d- d+ Chemical concepts • Molecular structure and geometry • Chemical bond • Ionic-Covalent • Singe, Double, Triple • Electronegativity • Oxidation number • Atomic charge • Lone pairs • Aromaticity

41. Nodes • 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 Should we concentrate on nodes? Checked on small systems: L, Be, He2+. See also Mitas

42. Avoided crossings Be e- gas Stadium

43. Nodal topology • The conjecture (which I believe is true) claims that there are only two nodal volumes in the fermion ground state • See, among others: • Ceperley J.Stat.Phys63, 1237 (1991) • Bressanini and coworkers. JCP97, 9200 (1992) • Bressanini, Ceperley, Reynolds, “What do we know about wave function nodes?”, in Recent Advances in Quantum Monte Carlo Methods II, ed. S. Rothstein, World Scientfic (2001) • Mitas and coworkers PRB72, 075131 (2005) • Mitas PRL 96, 240402 (2006)

44. If has 4 nodes has 2 nodes, with a proper Avoided nodal crossing • At a nodal crossing, Y and Y are zero • Avoided nodal crossing is the rule, not the exception • Not (yet) a proof...

45. He atom with noninteracting electrons

47. Casual similarity ? First unstable antisymmetric stretch orbit of semiclassical linear heliumalong with the symmetric Wannier orbit r1 = r2 and various equipotential lines

48. Casual similarity ? Superimposed Hylleraas node

49. How to directly improve nodes? • Fit to a functional form and optimize the parameters (maybe for small systems) • IF the topology is correct, use a coordinate transformation

50. Coordinate transformation • Take a wave function with the correct nodal topology • Change the nodes with a coordinate transformation (Linear? Feynman’s backflow ?) preserving the topology Miller-Good transformations