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Ground and excited states for exotic three-body atomic systems

Ground and excited states for exotic three-body atomic systems. Lorenzo Ugo ANCARANI Laboratoire de Physique Moléculaire et des Collisions Université Paul Verlaine – Metz Metz, France FB19 - Bonn, 1 September 2009 Collaborators: Gustavo GASANEO and Karina RODRIGUEZ

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Ground and excited states for exotic three-body atomic systems

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  1. Ground and excited states for exotic three-body atomic systems Lorenzo Ugo ANCARANI Laboratoire de Physique Moléculaire et des Collisions Université Paul Verlaine – Metz Metz, France FB19 - Bonn, 1 September 2009 Collaborators: Gustavo GASANEO and Karina RODRIGUEZ Universidad Nacional del Sur, Bahia Blanca, Argentine Dario MITNIK Universidad de Buenos Aires, Buenos Aires, Argentine

  2. OUTLINE • Generalities • Angularly correlated basis • Results for three-body exotic systems - ground state - excited states • Simple function – predictive tool for stability • Concluding remarks

  3. m3,z3: heaviest and positively charged m2,z2: light and negatively charged m1,z1: lightest and negatively charged m1,z1 r13 r12 m3,z3 r23 m2,z2 THREE-BODY PROBLEM OF ATOMIC SYSTEMSBOUND STATES REDUCED MASSES: Schrödinger Equation No analytical solution !

  4. z3=2 z3=1

  5. NO ANALYTICAL SOLUTION CONSTRUCTION OF A TRIAL WAVAFUNCTION • NUMERICAL • VERY LARGE number of parameters • Functional form ? • ENERGY: very good (ground state) ANALYTICAL - SIMPLE (few parameters) - GOOD FUNCTIONAL FORM - ENERGY : not so good (ground state) • INTERMEDIATE • (compromise) • Limited number of parameters • Functional form ? • ENERGY: good (also excited states?) • Practical for applications • (e.g. collisions)

  6. e- ki k1 e- k0 e- e- k2 DOUBLE IONISATION : (e,3e) Final Channel Initial channel (e,3e) A e- + He He++ + e- + e- + e- e- (E1, k1) He++ 4-body problem (6 interactions) e- (Ei,ki) Detection in coincidence: FDCS He e- (E0, k0) e- (E2, k2)

  7. e- r1 r12 He2+ (Z=2) r2 e- First Born Approximation (FBA) Interaction Momentum transfer : 3-body CONTINUUM problem 3-body BOUND problem Ground state of He

  8. FUNCTIONAL FORM OF WF • Asymptotic behaviour - one particle far away from the other two - all particles far away from each other • Close to the two-body singularities (r13=0, r23=0, r12=0) (Kato cusp conditions) Important for calculations of - double photoionization(Suric et al , PRA, 2003) - expectation values of singular operators (annihilation,Bianconi, Phys lett B,2000) • Triple point (all rij=0)

  9. ANGULARLY CORRELATED BASIS C3 MODEL FOR DOUBLE CONTINUUM (Garibotti and Miraglia, PRA (1980); Brauner, Briggs and Klar, JPB (1989)) Sommerfeld parameters: - Correct global asymptotic behaviour - OK with Kato cusp conditions

  10. Non-relativistic Schrödinger Equation S states - Hylleraas Equation : 3 interparticle coordinates No analytical solution !

  11. DOUBLE BOUND FUNCTIONS ANALOG TO THE C3 DOUBLE CONTINUUM m1,z1 r13 r12 m3,z3 r23 m2,z2 (Ancarani and Gasaneo, PRA, 2007) For two light particles 1,2 (z1<0, z2<0) and a third 3 heavy particle (z3>0)

  12. (Gasaneo and Ancarani, PRA, 2008) Basis functions: By construction: - Angularly correlated (r12) - Parameter-free: three quantum numbers (n1, n2, n3) - OK with Kato cusp conditions ANGULAR CORRELATED CONFIGURATION INTERACTION (ACCI)

  13. CALCULATIONS of - energies of ground and excited states - mean values of <(rij)p> with p>0 or <0 ALL RESULTS are in Hartree atomic units (ENERGY:1 a.u.=27.2 eV) SELECTION: compared to « numerically exact » values when available (obtained with hundreds/thousands of variational parameters)

  14. RESULTS (infinite m3): GROUND STATE Configurations included: 1s1s+(1s2s+2s1s)+2s2s Angular correlation: n3 up to 5 M = number of linear coefficients

  15. RESULTS (infinite m3): EXCITED STATES (Gasaneo and Ancarani, PRA, 2008) Even get the doubly excited state: 2s21S ( E(M=20)=-0.7659 ) • All states obtained - form an orthogonal set • - satisfy two-body Kato cusp conditions • Good energy convergence • Can be systematically be improved by increasing M

  16. RESULTS (finite m3): GROUND AND EXCITED STATES Configurations included: 1s1s+(1s2s+2s1s)+(1s3s+3s1s)+2s2sand n3=1,2,3,4,5  M=30

  17. RESULTS (finite m3): GROUND AND EXCITED STATES Configurations included: 1s1s+(1s2s+2s1s)+(1s3s+3s1s)and n3=1,2  M=10

  18. ACCI WITH EXTRA CORRELATION (method suggested by Rodriguez et al., JPB 2005+2007) • Same methodology (only linear parameters, analytical, …) • Set of orthogonal functions, satisfying Kato cusp conditions • Even better energy convergence (Drake, 2005)

  19. ACCI WITH EXTRA CORRELATION (Rodriguez, Ancarani, Gasaneo and Mitnik, IJQC, 2009) GROUND STATE:only 1s1s included (n1=n2=1) and n3=1,2 (Frolov, PRA, 1998)

  20. ACCI WITH EXTRA CORRELATION (Rodriguez, Ancarani, Gasaneo and Mitnik, IJQC, 2009) GROUND STATE:only 1s1s included (n1=n2=1) and n3=1,2 (Drake, 2005) (Frolov, PRA, 2000)

  21. ACCI WITH EXTRA CORRELATION (Rodriguez, Ancarani, Gasaneo and Mitnik, IJQC, 2009) GROUND STATE:only 1s1s included (n1=n2=1) and n3=1,2 (Drake, 2005) (Frolov, Phys.Lett. A, 2006)

  22. ACCI WITH EXTRA CORRELATION (Rodriguez, Ancarani, Gasaneo and Mitnik, IJQC, 2009) GROUND STATE:only 1s1s included (n1=n2=1) and n3=1,2 D. Exotic systems : n=1: positronium Ps- n∞ : negative Hydrogen ion H-

  23. Ps- H- (Frolov and Yeremin, JPB, 1989) (Rodriguez et al, Hyperfine Interactions, 2009)

  24. SIMPLE FUNCTION WITHOUT PARAMETERS e- r1 r12 Z r2 e- No ground state for H-!! (Ancarani, Rodriguez and Gasaneo ,JPB, 2007) • Pedagogical • Without nodes • With both radial (r1,r2) and angular (r12) correlation • Satisfies all two-body cusp conditions • Sufficiently simple: analytical calculations of copt(Z) and mean energy E(Z) • Without parameters (only Z); • Rather good energies, and predicts a ground state forH-!!

  25. GENERALISATION TO THREE-BODY SYSTEMS r1 r12 r2 SIMPLE FUNCTION WITHOUT PARAMETERS m1,z1 3 masses mi and 3 charges zi : Reduced masses: m3,z3 m2,z2 (Ancarani and Gasaneo ,JPB , 2008) Same properties (in particular: analytical!) Same form for any system

  26. z3=2 z3=1 (Ancarani and Gasaneo ,JPB , 2008)

  27. PREDICTIVE CHARACTERSTABILITY OF EXOTIC SYSTEMS (Ancarani and Gasaneo ,JPB , 2008) 3 masses mi and 3 charges zi : with m1 the lightest Stability condition: Example:m1=m2 et z1=z2 z2/z3= -1 Critical charge for a given r : Nucleus of virtual infinite mass:

  28. Summary • Angularlycorrelated basis - satisfytwo-body Kato conditions - orthogonal set of wavefunctions - onlylinearparameters (relativelysmall M) –> fast and efficient - cansystematicallyimproveenergies ( + possibility of including more correlationthrough r13ir23jr12k ) - canbeused for otherpotentials! • Ground and excited S states for normal and exoticthree-body systems • Simple function– predictivetool for stability Future … L>0 states … atomic systems with N > 3 bodies … molecular systems

  29. Thanks for listening

  30. Optimisation d’une fonction d’essai Valeurs moyennes: Energie moyenne: Variance: Autres valeurs moyennes: Théorème du Viriel: Energie locale: Fluctuations moyennées

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