1. Dario Bressanini QMCI Sardagna (Trento) 2008 http://scienze-como.uninsubria.it/bressanini Universita’ dell’Insubria, Como, Italy Some considerations on nodes and trial wave functions

2. 2 30+ years of QMC in chemistry

3. 3 The Early promises? Solve the Schrödinger equation exactly without approximation (very strong) Solve the Schrödinger equation exactly without approximation (very strong) Solve the Schrödinger equation with controlled approximations, and converge to the exact solution (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) Solve the Schrödinger equation with some approximation, and do better than other methods (weak)

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

5. 5 Good for vibrational problems

6. 6 For electronic structure? Sign Problem Fixed Nodal error problem

7. 7 The influence on the nodes of   QMC currently relies on  T (R) and its nodes (indirectly) QMC currently relies on  T (R) and its nodes (indirectly) How are the nodes  T (R) of influenced by: How are the nodes  T (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  The functional form of  T (R) ?

8. 8 Improving   Current Quantum Monte Carlo research focuses on Current Quantum Monte Carlo research focuses on  Optimizing the energy  Adding more determinants (large number of parameters)  Exploring new trial wave function forms (moderately large number of parameters) »Pfaffians, Geminals, Backflow... Node are improved (but not always) only indirectly Node are improved (but not always) only indirectly

9. 9 Adding more determinants Use a large Slater basis Use a large Slater basis  Try to reach HF nodes convergence Orbitals from MCSCF are good Orbitals from MCSCF are good Not worth optimizing MOs, if the basis is large enough Not worth optimizing MOs, if the basis is large enough Only few configurations seem to improve the FN energy Only few configurations seem to improve the FN energy Use the right determinants... Use the right determinants... ...different Angular Momentum CSFs And not the bad ones And not the bad ones ...types already included

10. 10 Li 2 -14.9954 E (N.R.L.) E (N.R.L.) -14.9952(1) -14.9939(2) -14.9933(1) -14.9933(2) -14.9914(2) -14.9923(2) E (hartree) CSF CSF Not all CSF are useful Not all CSF are useful Only 4 csf are needed to build a statistically exact nodal surface Only 4 csf are needed to build a statistically exact nodal surface Bressanini et al. J. Chem. Phys. 123, 204109 (2005) (1  g 2 1  u 2 omitted)

11. 11 Dimers Bressanini et al. J. Chem. Phys. 123, 204109 (2005)

12. 12 Convergence to the exact  We must include the correct analytical structure We must include the correct analytical structure Cusps: 3-body coalescence and logarithmic terms: QMC OK Tails: Often neglected

13. 13 Asymptotic behavior of  Example with 2-e atoms Example with 2-e atoms is the solution of the 1 electron problem

14. 14 Asymptotic behavior of  The usual form The usual form does not satisfy the asymptotic conditions A closed shell determinant has the wrong structure

15. 15 Asymptotic behavior of  In general In general Recursively, fixing the cusps, and setting the right symmetry… Each electron has its own orbital, Multideterminant (GVB) Structure! Take 2N coupled electrons 2 N determinants. An exponential wall

16. 16 Basis In order to build compact wave functions we used basis functions where the cusp and the asymptotic behavior is decoupled In order to build compact wave functions we used basis functions where the cusp and the asymptotic behavior is decoupled Use one function per electron plus a simple Jastrow Use one function per electron plus a simple Jastrow

17. 17 PsH – Positronium Hydride A wave function with the correct asymptotic conditions: A wave function with the correct asymptotic conditions: Bressanini and Morosi: JCP 119, 7037 (2003)

18. 18 GVB for atoms

19. 19 GVB for atoms

20. 20 GVB for atoms

21. 21 GVB for atoms

22. 22 GVB for atoms

23. 23 GVB for molecules Correct asymptotic structure Correct asymptotic structure Is there a nodal error component in HF wave function coming from incorrect dissociation? Is there a nodal error component in HF wave function coming from incorrect dissociation?

24. 24 GVB for molecules Localized orbitals

25. 25 GVB Li 2 E (N.R.L.) E (N.R.L.) -14.9936(1) GVB CI 24 det compact -14.9632(1) CI 3 det compact -14.9688(1) GVB 8 det compact -14.9523(2) HF 1 det compact VMC Wave functions -14.9916(1) -14.9915(1) -14.9931(1) -14.9782(1) -14.9933(2) -14.9952(1) -14.9954 DMC CI 3 det large basis CI 5 det large basis Improvement in the wave function but irrelevant on the nodes, but irrelevant on the nodes,

26. 26 GVB in QMC Conclusions Conclusions  The quality of the wave function improves, giving better VMC energies …  … but the nodes are not changed, giving the same QMC energies  Bressanini and Morosi J. Chem. Phys. 129, 054103 (2008)

27. 27 Conventional wisdom on  E VMC (  RHF ) > E VMC (  UHF ) > E VMC (  GVB ) E VMC (  RHF ) > E VMC (  UHF ) > E VMC (  GVB ) Single particle approximations  RHF = |1s R 2s R 2p x 2p y 2p z | |1s R 2s R |  RHF = |1s R 2s R 2p x 2p y 2p z | |1s R 2s R |  UHF = |1s U 2s U 2p x 2p y 2p z | |1s’ U 2s’ U |  UHF = |1s U 2s U 2p x 2p y 2p z | |1s’ U 2s’ U | Consider the N atom E DMC (  RHF ) > ? ? < E DMC (  UHF )

28. 28 Conventional wisdom on  We can build a  RHF with the same nodes of  UHF  UHF = |1s U 2s U 2p x 2p y 2p z | |1s’ U 2s’ U |  UHF = |1s U 2s U 2p x 2p y 2p z | |1s’ U 2s’ U |  ’ RHF = |1s U 2s U 2p x 2p y 2p z | |1s U 2s U |  ’ RHF = |1s U 2s U 2p x 2p y 2p z | |1s U 2s U | E DMC (  ’ RHF ) = E DMC (  UHF ) E VMC (  ’ RHF ) > E VMC (  RHF ) > E VMC (  UHF )

29. 29 Conventional wisdom on  Node equivalent to a  UHF |f(r) g(r) 2p 3 | |1s 2s| E DMC (  GVB ) = E DMC (  ’’ RHF )  GVB = |1s 2s 2p 3 | |1s’ 2s’| - |1s’ 2s 2p 3 | |1s 2s’| + |1s’ 2s’ 2p 3 | |1s 2s|- |1s 2s’ 2p 3 | |1s’ 2s| Same Node

30. 30 What to do? Should we be happy with the “cancellation of error”, and pursue it? Should we be happy with the “cancellation of error”, and pursue it? After all, the whole quantum chemistry is built on it! After all, the whole quantum chemistry is built on it! If not, and pursue orthodox QMC (no pseudopotentials, no cancellation of errors, …), can we avoid the curse of  T ? If not, and pursue orthodox QMC (no pseudopotentials, no cancellation of errors, …), can we avoid the curse of  T ?

31. 31 The curse of the   QMC currently relies on  T (R) QMC currently relies on  T (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 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 p = parameters per variable M = total parameters needed For M=10 9 and p=3  N=6 The Exponential Wall

32. 32 Numbers and insight “The more accurate the calculations became, the more the concepts tended to vanish into thin air “ (Robert Mulliken) There is no shortage of accurate calculations for few-electron systems There is no shortage of accurate calculations for few-electron systems  −2.90372437703411959831115924519440444669690537 a.u. Helium atom (Nakashima and Nakatsuji JCP 2007) However… However…

33. 33 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

34. We need new, and different, ideas A little intermezzo (for the students)

35. 35 We need new, and different, ideas Research is the process of going up alleys to see if they are blind. Marston Bates Different representations Different representations Different dimensions Different dimensions Different equations Different equations Different potential Different potential Radically different algorithms Radically different algorithms Different something Different something

36. 36 Just an example Try a different representation Try a different representation Is some QMC in the momentum 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 ?

37. 37 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

38. 38 Better? For coulomb systems: For coulomb systems: There are NO cusps in momentum space.  convergence should be faster There are NO cusps in momentum space.  convergence should be faster Hydrogenic orbitals are simple rational functions Hydrogenic orbitals are simple rational functions

39. 39 Another (failed so far) example Different dimensionality: Hypernodes Different dimensionality: Hypernodes Given H  (R) = E  (R) build Given H  (R) = E  (R) build Use the Hypernode of Use the Hypernode of The hope was that it could be better than Fixed Node The hope was that it could be better than Fixed Node

40. 40 Hypernodes Exact node Trial node Fixed Node Exact node Trial node Fixed HyperNode The energy is still an upper bound The energy is still an upper bound Unfortunately, it seems to recover exactly the FN energy Unfortunately, it seems to recover exactly the FN energy The intuitive idea was that the system could correct the wrong fixed nodes, by exploring regions where

41. 41 Feynman on simulating nature Nature isn’t classical, dammit, and if you want to make a simulation of Nature, you’d better make it quantum mechanical, and by golly it’s a wonderful problem, because it doesn’t look so easy” Nature isn’t classical, dammit, and if you want to make a simulation of Nature, you’d better make it quantum mechanical, and by golly it’s a wonderful problem, because it doesn’t look so easy” Richard Feynman 1981 Richard Feynman 1981

42. 42 Nodes Should we concentrate on nodes? Conjectures on nodes Conjectures on nodes  have higher symmetry than  itself  resemble simple functions  the ground state has only 2 nodal volumes  HF nodes are often a god starting point

43. 43 How to directly improve nodes? Fit to a functional form and optimize the parameters ( maybe for small systems ) Fit to a functional form and optimize the parameters ( maybe for small systems ) IF the topology is correct, use a coordinate transformation IF the topology is correct, use a coordinate transformation

44. 44 He 2 + : “expanding” the node Exact It is a one parameter  It is a one parameter 

45. 45 “expanding” nodes This was only a kind of “proof of concept” This was only a kind of “proof of concept” It remains to be seen if it can be applied to larger systems It remains to be seen if it can be applied to larger systems  Writing “simple” (algebraic?) trial nodes is not difficult ….  The goal is to have only few linear parameters to optimize  Will it work???????

46. 46 Coordinate transformation Take a wave function with the correct nodal topology Take a wave function with the correct nodal topology Change the nodes with a coordinate transformation (Linear? Feynman’s backflow ?) preserving the topology Change the nodes with a coordinate transformation (Linear? Feynman’s backflow ?) preserving the topology Miller-Good transformations

47. 47 The need for the correct topology Using Backflow alone, on a single determinant  is not sufficient, since the topology is still wrong Using Backflow alone, on a single determinant  is not sufficient, since the topology is still wrong More determinants are necessary (only two?) More determinants are necessary (only two?)

48. 48 Be Nodal Topology r3-r4 r1-r2 r1+r2 r1-r2 r1+r2 r3-r4

49. 49 Avoided crossings Be e - gas Stadium

50. 50 Nodal topology The conjecture (which I believe is true) claims that there are only two nodal volumes in the fermion ground state The conjecture (which I believe is true) claims that there are only two nodal volumes in the fermion ground state See, among others: See, among others:  Ceperley J.Stat.Phys 63, 1237 (1991)  Bressanini and coworkers. JCP 97, 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 PRB 72, 075131 (2005)  Mitas PRL 96, 240402 (2006)

51. 51 Nodal Regions NeLiBe B C Li 2 24 4 4 4 422 2 2 2 2

52. 52 Avoided nodal crossing At a nodal crossing,  and  are zero At a nodal crossing,  and  are zero Avoided nodal crossing is the rule, not the exception Avoided nodal crossing is the rule, not the exception Not (yet) a proof... (any help is appreciated) Not (yet) a proof... (any help is appreciated)If has 4 nodes has 2 nodes, with a proper

53. 53 He atom with noninteracting electrons

54. 54

55. 55 Casual similarity ? First unstable antisymmetric stretch orbit of semiclassical linear helium along with the symmetric Wannier orbit r 1 = r 2 and various equipotential lines

56. 56 Superimposed Hylleraas node Casual similarity ?

57. 57 A QMC song... He deals the cards to find the answers the sacred geometry of chance the hidden law of a probable outcome the numbers lead a dance Sting: Shape of my heart

58. 58 Take a look at your nodes! Think Different