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Dario Bressanini

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

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Dario Bressanini

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

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

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

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

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

  6. Long term motivations • 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 • The exact wave function (for that boundary conditions) is sampled • We need ways to build good approximate nodes • We need to study their mathematical properties (poorly understood)

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

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

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

  10. Asymptotic behavior of Y • In general Recursively, fixing the cusps, and setting the right symmetry… Each electron has its own orbital, Multideterminant (GVB) Structure!

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

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

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

  14. Z dependence • Best values around for compact wave functions • 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 ki • Tested for Z=30 • Can we use this approach to larger systems? Nodes for QMC become crucial

  15. For larger atoms ?

  16. GVB Monte Carlo for Atoms

  17. Nodes does not improve • 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 • Was able to prove it mathematically up to N=7 (Nitrogen atom), but it seems a general feature • EVMC(YRHF) > EVMC(YGVB) • EDMC(YRHF) = EDMC(YGVB)

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

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

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

  21. Spin  r13 Spin  r1 r12 r3 r2 Spin  r2 r1 r3 Lithium atom Is r1 = r2 the exact node of Lithium ? • Even the exaxt node seems to be r1 = r2, taking different cuts (using a very accurate Hylleraas expansion)

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

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

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

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

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

  27. The End Take a look at your nodes

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