1. Dario Bressanini Critical stability V (Erice) 2008 http://scienze-como.uninsubria.it/bressanini Universita’ dell’Insubria, Como, Italy Boundary-condition-determined wave functions (and their nodal structure) for few-electron atomic systems

2. 2 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…

3. 3 The curse of   Currently Quantum Monte Carlo (and quantum chemistry in general) uses moderatly large to extremely large expansions for  Currently Quantum Monte Carlo (and quantum chemistry in general) uses moderatly large to extremely large expansions for  Can we ask for both accurate and compact wave functions? Can we ask for both accurate and compact wave functions?

4. 4 VMC: Variational Monte Carlo Use the Variational Principle Use the Variational Principle Use Monte Carlo to estimate the integrals Use Monte Carlo to estimate the integrals Complete freedom in the choice of the trial wave function Complete freedom in the choice of the trial wave function Can use interparticle distances into  Can use interparticle distances into  But It depends critically on our skill to invent a good  But It depends critically on our skill to invent a good 

5. 5 QMC: Quantum Monte Carlo Analogy with diffusion equation Analogy with diffusion equation Wave functions for fermions have nodes Wave functions for fermions have nodes If we knew the exact nodes of , we could exactly simulate the system by QMC If we knew the exact nodes of , we could exactly simulate the system by QMC The exact nodes are unknown. Use approximate nodes from a trial  as boundary conditions The exact nodes are unknown. Use approximate nodes from a trial  as boundary conditions+-

6. 6 Long term motivations In QMC we only need the zeros of the wave function, not what is in between! In QMC we only need the zeros of the wave function, not what is in between! A stochastic process of diffusing points is set up using the nodes as boundary conditions A stochastic process of diffusing points is set up using the nodes as boundary conditions The exact wave function (for that boundary conditions) is sampled The exact wave function (for that boundary conditions) is sampled We need ways to build good approximate nodes We need ways to build good approximate nodes We need to study their mathematical properties (poorly understood) We need to study their mathematical properties (poorly understood)

7. 7 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

8. 8 Asymptotic behavior of  Example with 2-e atoms Example with 2-e atoms is the solution of the 1 electron problem

9. 9 Asymptotic behavior of  The usual form The usual form does not satisfy the asymptotic conditions A closed shell determinant has the wrong structure

10. 10 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!

11. 11 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)

12. 12 Basis In order to build compact wave functions we used orbital functions where the cusp and the asymptotic behavior are decoupled In order to build compact wave functions we used orbital functions where the cusp and the asymptotic behavior are decoupled

13. 13 2-electron atoms Tails OK Cusps OK – 3 parameters Fragments OK – 2 parameters (coalescence wave function)

14. 14 Z dependence Best values around for compact wave functions Best values around for compact wave functions D. Bressanini and G. Morosi J. Phys. B 41, 145001 (2008) D. Bressanini and G. Morosi J. Phys. B 41, 145001 (2008) We can write a general wave function, with Z as a parameter and fixed constants k i We can write a general wave function, with Z as a parameter and fixed constants k i Tested for Z=30 Tested for Z=30 Can we use this approach to larger systems? Nodes for QMC become crucial Can we use this approach to larger systems? Nodes for QMC become crucial

15. 15 For larger atoms ?

16. 16 GVB Monte Carlo for Atoms

17. 17 Nodes does not improve 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 Was able to prove it mathematically up to N=7 (Nitrogen atom), but it seems a general feature Was able to prove it mathematically up to N=7 (Nitrogen atom), but it seems a general feature E VMC (  RHF ) > E VMC (  GVB ) E VMC (  RHF ) > E VMC (  GVB ) E DMC (  RHF ) = E DMC (  GVB ) E DMC (  RHF ) = E DMC (  GVB )

18. Is there anything “critical” about the nodes of critical wave functions?

19. 19 Critical charge Z c 2 electrons: 2 electrons: Critical Z for binding Z c =0.91103 Critical Z for binding Z c =0.91103  c is square integrable  c is square integrable <1 : infinitely many discrete bound states <1 : infinitely many discrete bound states 1≤ ≤ c : only one bound state 1≤ ≤ c : only one bound state All discrete excited state are absorbed in the continuum exactly at =1 All discrete excited state are absorbed in the continuum exactly at =1 Their  become more and more diffuse Their  become more and more diffuse

20. 20 Critical charge Z c N electrons atom N electrons atom  < 1/(N-1) infinite number of discrete eigenvalues  < 1/(N-1) infinite number of discrete eigenvalues  ≥ 1/(N-1) finite number of discrete eigenvalues  ≥ 1/(N-1) finite number of discrete eigenvalues N-2 ≤ Z c ≤ N-1 N-2 ≤ Z c ≤ N-1 N=3 “Lithium” atom Z c  2. As Z→ Z c N=3 “Lithium” atom Z c  2. As Z→ Z c N=4 “Beryllium” atom Z c  2.85 As Z→ Z c N=4 “Beryllium” atom Z c  2.85 As Z→ Z c

21. 21 Lithium atom r1r1r1r1 r2r2r2r2 r 12 r3r3r3r3 Spin  Spin  r 13 r2r2 r1r1 r3r3 Even the exaxt node seems to be r 1 = r 2, taking different cuts (using a very accurate Hylleraas expansion) Even the exaxt node seems to be r 1 = r 2, taking different cuts (using a very accurate Hylleraas expansion) Is r 1 = r 2 the exact node of Lithium ?

22. 22 Varying Z: QMC versus Hylleraas The node r1=r2 seems to be valid over a wide range of The node r1=r2 seems to be valid over a wide range of Up to c =1/2 ? preliminary results

23. 23 Be Nodal Topology r3-r4 r1-r2 r1+r2 r1-r2 r1+r2 r3-r4

24. 24 N=4 critical charge c N=3 N=4

25. 25 N=4 critical charge c  0.3502 c  0.3502 Z c  2.855 Zc (Hogreve)  2.85 N=3 N=4

26. 26 N=4 critical charge node very close to c =0.3502 preliminary results Critical Node very close to - -

27. 27 Take a look at your nodes The End