1. Dario Bressanini Qmc in the Apuan Alps III (Vallico sotto) 2007 http://scienze-como.uninsubria.it/bressanini Universita’ dell’Insubria, Como, Italy The quest for compact and accurate 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 ?

8. 8 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? If so: If so:  Is there the risk, in this case, that QMC becomes Yet Another Computational Tool, and not particularly efficient nor reliable? 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 ?

9. 9 The curse of   QMC currently heavily relies on  T (R) QMC currently heavily relies on  T (R) Walter Kohn in its Nobel lecture (R.M.P. 71, 1253 (1999)) tried to “discredit” 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)) tried to “discredit” 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

10. 10 The curse of   Current research focusses on Current research focusses on  optimizing the energy for moderately large expansions (good results)  Exploring new trial wave function forms, with a moderately large number of parameters (good results) Is it hopeless to ask for both accurate and compact wave functions? Is it hopeless to ask for both accurate and compact wave functions?

11. 11 A Quantum Chemistry Chart J.Pople The more accurate the calculations became, the more the concepts tended to vanish into thin air (Robert Mulliken) Orthodox QMC

12. 12 Li 2 J. Chem. Phys. 123, 204109 (2005) -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 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 (1  g 2 1  u 2 omitted)

13. 13 A tentative recipe Use a large Slater basis Use a large Slater basis  But not too large  Try to reach HF nodes convergence Orbitals from CAS seem better than HF, or NO Orbitals from CAS seem better than HF, or NO 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

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

15. 15 Carbon Atom: Energy CSFsDet.Energy CSFsDet.Energy 1 1s 2 2s 2 2p 2 1-37.8303(4) 1 1s 2 2s 2 2p 2 1-37.8303(4) 2 + 1s 2 2p 4 2-37.8342(4) 2 + 1s 2 2p 4 2-37.8342(4) 5 + 1s 2 2s 2p 2 3d 18 -37.8399(1) 5 + 1s 2 2s 2p 2 3d 18 -37.8399(1) 83 1s 2 + 4 electrons in 2s 2p 3s 3p 3d shell 422-37.8387(4) 83 1s 2 + 4 electrons in 2s 2p 3s 3p 3d shell 422-37.8387(4) adding f orbitals 7(4f 2 + 2p 3 4f) 34-37.8407(1) 7(4f 2 + 2p 3 4f) 34-37.8407(1) R12-MR-CI-37.845179 Exact (estimated)-37.8450

16. 16 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 Is “orthodox” QMC competitive ?

17. 17 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 and fragments: Usually neglected

18. 18 Asymptotic behavior of  Example with 2-e atoms Example with 2-e atoms is the solution of the 1 electron problem

19. 19 Asymptotic behavior of  The usual form The usual form does not satisfy the asymptotic conditions A closed shell determinant has the wrong structure

20. 20 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. Again an exponential wall

21. 21 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)

22. 22 Basis In order to build compact wave functions we used basis functions where the cusp and the asymptotic behavior are decoupled In order to build compact wave functions we used basis functions where the cusp and the asymptotic behavior are decoupled Use one function per electron plus a simple Jastrow Use one function per electron plus a simple Jastrow Can fix the cusps of the orbitals. Very few parameters Can fix the cusps of the orbitals. Very few parameters

23. 23 GVB for atoms

24. 24 GVB for atoms

25. 25 GVB for atoms

26. 26 GVB for atoms

27. 27 GVB for atoms

28. 28 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 )

29. 29 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 )

30. 30 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

31. 31 Nitrogen Atom  Param. E corr. VMC E corr. DMC  Param. E corr. VMC E corr. DMC Simple RHF (1 det)842.7%92.6% Simple RHF (1 det)842.7%92.6% Simple UHF (1 det)11 41.2%92.3% Simple UHF (1 det)11 41.2%92.3% Simple GVB (4 det) 1142.3%92.3% Simple GVB (4 det) 1142.3%92.3% Clementi-Roetti + J2724.5%93.1% Clementi-Roetti + J2724.5%93.1% Is it worth to continue to add parameters to the wave function?

32. 32 GVB for molecules Correct asymptotic structure Correct asymptotic structure Nodal error component in HF wave function coming from incorrect dissociation? Nodal error component in HF wave function coming from incorrect dissociation?

33. 33 GVB for molecules Localized orbitals

34. 34 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) -14.9952(1) -14.9954 DMC CI 5 det large basis Improvement in the wave function but irrelevant on the nodes, but irrelevant on the nodes,

35. 35 Different coordinates The usual coordinates might not be the best to describe orbitals and wave functions The usual coordinates might not be the best to describe orbitals and wave functions In LCAO need to use large basis In LCAO need to use large basis For dimers, elliptical confocal coordinates are more “natural” For dimers, elliptical confocal coordinates are more “natural”

36. 36 Different coordinates Li 2 ground state Li 2 ground state Compact MOs built using elliptic coordinates Compact MOs built using elliptic coordinates

37. 37 Li 2 E (N.R.L.) E (N.R.L.) -14.9632(1) CI 3 det compact -14.9523(2) HF 1 det compact -14.9916(1) -14.9931(1) -14.9954 CI 3 det elliptic -14.9937(1) -14.9670(1) HF 1 det elliptic -14.9543(1) -14.9916(1) VMC Wave functions DMC Some improvement in the wave function but negligible on the nodes, but negligible on the nodes,

38. 38 Different coordinates It might make a difference even on nodes for etheronuclei It might make a difference even on nodes for etheronuclei Consider LiH +3 the 2s  state: Consider LiH +3 the 2s  state: HF LCAO H Li The wave function is dominated by the 2s on Li The wave function is dominated by the 2s on Li The node (in red) is asymmetrical The node (in red) is asymmetrical However the exact node must be symmetric However the exact node must be symmetric

39. 39 Different coordinates This is an explicit example of a phenomenon already encountered in other systems, the symmetry of the node is higher than the symmetry of the wave function This is an explicit example of a phenomenon already encountered in other systems, the symmetry of the node is higher than the symmetry of the wave function The convergence to the exact node, in LCAO, is very slow. The convergence to the exact node, in LCAO, is very slow. Using elliptical coordinates is the right way to proceed Using elliptical coordinates is the right way to proceed HF LCAO H Li Future work will explore if this effect might be important in the construction of many body nodes Future work will explore if this effect might be important in the construction of many body nodes

40. 40 Playing directly with nodes? It would be useful to be able to optimize only those parameters that alter the nodal structure It would be useful to be able to optimize only those parameters that alter the nodal structure A first “exploration” using a simple test system: A first “exploration” using a simple test system: He 2 + The nodes seem to be smooth and “simple” The nodes seem to be smooth and “simple” Can we “expand” the nodes on a basis? Can we “expand” the nodes on a basis?

41. 41 He 2 + : “expanding” the node Exact It is a one parameter  It is a one parameter 

42. 42 “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 ….  Waierstrass theorem  The goal is to have only few linear parameters to optimize  Will it work???????

43. 43 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

44. 44 Conclusions The wave function can be improved by incorporating the known analytical structure… with a small number of parameters The wave function can be improved by incorporating the known analytical structure… with a small number of parameters … but the nodes do not seem to improve  … but the nodes do not seem to improve  It seems more promising to directly “manipulate” the nodes. It seems more promising to directly “manipulate” the nodes.

45. 45 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

46. 46 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 ?

47. 47 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

48. 48 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

49. 49 Nodes Should we concentrate on nodes? Checked on small systems: L, Be, He 2 +. See also Mitas 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 quite good: they “naturally” have these properties

50. 50 Be Nodal Topology r3-r4 r1-r2 r1+r2 r1-r2 r1+r2 r3-r4

51. 51 Avoided crossings Be e - gas Stadium

52. 52 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)

53. 53 Nodal Regions NeLiBe B C Li 2 24 4 4 4 422 2 2 2 2

54. 54 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... Not (yet) a proof...If has 4 nodes has 2 nodes, with a proper In the generic case there is no solution to these equations

55. 55 He atom with noninteracting electrons

56. 56

57. 57 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

58. 58 Superimposed Hylleraas node Casual similarity ?

59. 59 How to directly improve nodes? “expand” the nodes and optimize the parameters “expand” the nodes and optimize the parameters IF the topology is correct, use a coordinate transformation IF the topology is correct, use a coordinate transformation

60. 60 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? Backflow ?) preserving the topology Change the nodes with a coordinate transformation (Linear? Backflow ?) preserving the topology Miller-Good transformations

61. 61 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

62. 62 Think Different!