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  m 3,z 3 : heaviest and positively charged  m 2,z 2 : light and negatively charged  m 1,z 1 : lightest and negatively charged m 1,z 1 m 2,z 2 r 13 r 23 m 3,z 3 m 3,z 3 r 12 THREE-BODY PROBLEM OF ATOMIC SYSTEMS BOUND STATES REDUCED MASSES: Schrödinger Equation No analytical solution !

4 z 3 =1 z 3 =2

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

6 (e,3e) e-e- kiki k0k0 e-e- k1k1 e-e- k2k2 e-e- A Initial channel Final Channel DOUBLE IONISATION : (e,3e) e - + He He ++ + e - + e - + e - HeHe e - (E i,k i ) e - (E 0, k 0 ) e - (E 2, k 2 ) e - (E 1, k 1 ) He ++ 4-body problem (6 interactions) Detection in coincidence: FDCS

7 Momentum transfer : Interaction 3-body BOUND problem Ground state of He 3-body CONTINUUM problem First Born Approximation (FBA) r1r1r1r1 e-e-e-e- r2r2r2r2 e-e-e-e- He 2+ (Z=2) r 12

8 Asymptotic behaviour - one particle far away from the other two - all particles far away from each other Close to the two-body singularities (r 13 =0, r 23 =0, r 12 =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 r ij =0) FUNCTIONAL FORM OF WF

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

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

11 DOUBLE BOUND FUNCTIONS ANALOG TO THE C3 DOUBLE CONTINUUM (Ancarani and Gasaneo, PRA, 2007) For two light particles 1,2 (z 1 0) r 13 r 23 m 3,z 3 m 3,z 3 r 12 m 1,z 1 m 2,z 2

12 ANGULAR CORRELATED CONFIGURATION INTERACTION (ACCI) (Gasaneo and Ancarani, PRA, 2008) By construction: - Angularly correlated (r 12 ) - Parameter-free: three quantum numbers (n 1, n 2, n 3 ) - OK with Kato cusp conditions Basis functions:

13 CALCULATIONS of - energies of ground and excited states - mean values of 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 m 3 ): GROUND STATE Configurations included: 1s1s+(1s2s+2s1s)+2s2s Angular correlation: n 3 up to 5 M = number of linear coefficients

15 RESULTS (infinite m 3 ): EXCITED STATES (Gasaneo and Ancarani, PRA, 2008)  All states obtained - form an orthogonal set - satisfy two-body Kato cusp conditions  Good energy convergence  Can be systematically be improved by increasing M Even get the doubly excited state: 2s 2 1 S ( E (M=20) = )

16 RESULTS (finite m 3 ): GROUND AND EXCITED STATES Configurations included: 1s1s+(1s2s+2s1s)+(1s3s+3s1s)+2s2s and n 3 =1,2,3,4,5  M=30

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

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

19 ACCI WITH EXTRA CORRELATION (Rodriguez, Ancarani, Gasaneo and Mitnik, IJQC, 2009) (Frolov, PRA, 1998) GROUND STATE: only 1s1s included (n 1 =n 2 =1) and n 3 =1,2

20 ACCI WITH EXTRA CORRELATION (Rodriguez, Ancarani, Gasaneo and Mitnik, IJQC, 2009) (Frolov, PRA, 2000) (Drake, 2005) GROUND STATE: only 1s1s included (n 1 =n 2 =1) and n 3 =1,2

21 ACCI WITH EXTRA CORRELATION (Rodriguez, Ancarani, Gasaneo and Mitnik, IJQC, 2009) (Frolov, Phys.Lett. A, 2006) (Drake, 2005) GROUND STATE: only 1s1s included (n 1 =n 2 =1) and n 3 =1,2

22 ACCI WITH EXTRA CORRELATION (Rodriguez, Ancarani, Gasaneo and Mitnik, IJQC, 2009) GROUND STATE: only 1s1s included (n 1 =n 2 =1) and n 3 =1,2 D. Exotic systems : n=1: positronium Ps- n  ∞ : negative Hydrogen ion H-

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

24 SIMPLE FUNCTION WITHOUT PARAMETERS Pedagogical Without nodes With both radial (r 1,r 2 ) and angular (r 12 ) correlation Satisfies all two-body cusp conditions Sufficiently simple: analytical calculations of c opt (Z) and mean energy E(Z) Without parameters (only Z); Rather good energies, and predicts a ground state for H - !! No ground state for H - !! r1r1r1r1 r2r2r2r2 Z r 12 e- e- e- (Ancarani, Rodriguez and Gasaneo,JPB, 2007)

25 r1r1r1r1 r2r2r2r2 r 12 GENERALISATION TO THREE-BODY SYSTEMS 3 masses m i and 3 charges z i : Reduced masses: Same properties (in particular: analytical!) Same form for any system m1,z1 m1,z1 SIMPLE FUNCTION WITHOUT PARAMETERS m2,z2 m2,z2 m3,z3 m3,z3 (Ancarani and Gasaneo,JPB, 2008)

26 z 3 =1 z 3 =2 (Ancarani and Gasaneo,JPB, 2008)

27 PREDICTIVE CHARACTER STABILITY OF EXOTIC SYSTEMS 3 masses m i and 3 charges z i : with m 1 the lightest Stability condition: Example: m 1 =m 2 et z 1 =z 2 z 2 /z 3 = -1 Critical charge for a given r : Nucleus of virtual infinite mass: (Ancarani and Gasaneo,JPB, 2008)

28 … L>0 states … atomic systems with N > 3 bodies … molecular systems Summary Future

29 Thanks for listening

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32 Optimisation d ’ une fonction d ’ essai Energie moyenne: Variance: Autres valeurs moyennes: Th é or è me du Viriel: Valeurs moyennes: Energie locale: Fluctuations moyenn é es

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