Is QMC delivering its early promises?

1. Universita’ dell’Insubria, Como, Italy The quest for compact and accurate trial wave functions Is QMC delivering its early promises? Dario Bressanini http://scienze-como.uninsubria.it/bressanini TTI III (Vallico sotto) 2007

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. 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 ?

10. 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

11. 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

12. + 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

13. 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

14. 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

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

16. Is QMC competitive ?

17. 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

18. 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

19. 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

20. 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:

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

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

23. 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

24. Basis 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

25. GVB for atoms

26. GVB for atoms

27. GVB for atoms

28. GVB for atoms

29. GVB for atoms

30. 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)

31. 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)

32. 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)

33. 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?

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

35. GVB for molecules Localized orbitals

36. GVB Li2 Wave functions VMC DMC HF 1 det compact -14.9523(2) -14.9916(1) GVB 8 det compact -14.9688(1) -14.9915(1) CI 3 det compact -14.9632(1) -14.9931(1) GVB CI 24 det compact -14.9782(1) -14.9936(1) CI 3 det large basis -14.9933(2) CI 5 det large basis -14.9952(1) E (N.R.L.) -14.9954 Improvement in the wave function but irrelevant on the nodes,

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

38. Different coordinates • Li2 ground state • Compact MOs built using elliptic coordinates

39. Li2 Wave functions VMC DMC HF 1 det compact -14.9523(2) -14.9916(1) HF 1 det elliptic -14.9543(1) -14.9916(1) CI 3 det compact -14.9632(1) -14.9931(1) CI 3 det elliptic -14.9670(1) -14.9937(1) E (N.R.L.) -14.9954 Some improvement in the wave function but negligible on the nodes,

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

41. HF LCAO H Li Different coordinates • This is an explicit example of a phenomenon already encountered in bigger 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. • Using elliptical coordinates is the right way to proceed • Future work will explore if this effect might be important in the construction of many body nodes

42. Playing directly with nodes? • 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 • The nodes seem to be smooth and “simple” • Can we “expand” the nodes on a basis? He2+

43. He2+: “expanding” the node • It is a one parameter Y !! Exact

44. “expanding” nodes • This was only a kind of “proof of concept” • 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???????

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

46. 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

47. 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 ?

48. 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

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

50. 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