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Ground and excited states for exotic three-body atomic systems

Lorenzo Ugo ANCARANI

Laboratoire de Physique Moléculaire et des CollisionsUniversité 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

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OUTLINE

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

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m3,z3: heaviest and positively charged m2,z2: light and negatively charged m1,z1: lightest and negatively charged

mm11,z,z11

mm22,z,z22

rr1313

rr2323

mm33,z,z33

rr1212

THREE-BODY PROBLEM OF ATOMIC SYSTEMS

BOUND STATES

REDUCED MASSES:

][ 321321zzz mmm

Schrödinger Equation

No analytical

solution !

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z3=1 z3=2

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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)

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(e,3e)(e,3e)

e- kik0

e-

k1

e-

k2

e-

A

Initial channelInitial channel Final ChannelFinal Channel

DOUBLE IONISATION : (e,3e)

e- + He He++ + e- + e- + e-

He

e- (Ei,ki)

e- (E0, k0)

e- (E2, k2)

e- (E1, k1)

He++

4-body problem (6 interactions)

21021

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dEdEddd

d

Detection in coincidence:

FDCS

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Momentum transfer :

Interaction

3-body BOUND problem Ground state of He

D6

D9

3-body CONTINUUM problem

First Born Approximation (FBA)

rr11

ee--

rr22

ee--

HeHe2+2+

(Z=2)(Z=2)

rr1212

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• 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)

FUNCTIONAL FORM OF WF

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(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

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Non-relativistic Schrödinger Equation

No analytical

solution !

S states - Hylleraas Equation :

3 interparticle coordinates

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DOUBLE BOUND FUNCTIONS ANALOG TO THE C3 DOUBLE CONTINUUM

(Ancarani and Gasaneo, PRA, 2007)

For two light particles 1,2 (z1<0, z2<0) and a third 3 heavy particle (z3>0)

rr1313

rr2323

mm33,z,z33

rr1212

mm11,z,z11

mm22,z,z22

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ANGULAR CORRELATED CONFIGURATION INTERACTION (ACCI)

(Gasaneo and Ancarani, PRA, 2008)

By construction: - Angularly correlated (r12)

- Parameter-free: three quantum numbers (n1, n2, n3)

- OK with Kato cusp conditions

Basis functions:

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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)

)1( eme

H

HE

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RESULTS (infinite m3): GROUND STATE

Configurations included: 1s1s+(1s2s+2s1s)+2s2s

Angular correlation: n3 up to 5

M = number of linear coefficients

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RESULTS (infinite m3): 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: 2s2 1S ( E(M=20)=-0.7659 )

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RESULTS (finite m3): GROUND AND EXCITED STATES

Configurations included: 1s1s+(1s2s+2s1s)+(1s3s+3s1s)+2s2sand n3=1,2,3,4,5 M=30

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RESULTS (finite m3): GROUND AND EXCITED STATES

Configurations included: 1s1s+(1s2s+2s1s)+(1s3s+3s1s)and n3=1,2 M=10

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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 2005+2007)

(Drake, 2005)

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ACCI WITH EXTRA CORRELATION(Rodriguez, Ancarani, Gasaneo and Mitnik, IJQC, 2009)

(Frolov, PRA, 1998)

GROUND STATE: only 1s1s included (n1=n2=1) and n3=1,2

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ACCI WITH EXTRA CORRELATION(Rodriguez, Ancarani, Gasaneo and Mitnik, IJQC, 2009)

(Frolov, PRA, 2000)

(Drake, 2005)

GROUND STATE: only 1s1s included (n1=n2=1) and n3=1,2

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ACCI WITH EXTRA CORRELATION (Rodriguez, Ancarani, Gasaneo and Mitnik, IJQC, 2009)

(Frolov, Phys.Lett. A, 2006)

(Drake, 2005)

GROUND STATE: only 1s1s included (n1=n2=1) and n3=1,2

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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-

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(Frolov and Yeremin, JPB, 1989)

Ps-

H-

(Rodriguez et al, Hyperfine Interactions, 2009)

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SIMPLE FUNCTION WITHOUT PARAMETERS

• 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 for H- !!

No ground state for H- !!

rr11

rr22

ZZrr1212

e-e-

e-e-

(Ancarani, Rodriguez and Gasaneo ,JPB, 2007)

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rr11

rr22

rr1212

GENERALISATION TO THREE-BODY SYSTEMS

3 masses mi and 3 charges zi :

Reduced masses:

Same properties (in particular: analytical!)

Same form for any system

m1,z1m1,z1

SIMPLE FUNCTION WITHOUT PARAMETERS

m2,z2m2,z2

m3,z3m3,z3][ 321321zzz mmm

][ 321321zzz mmm

(Ancarani and Gasaneo ,JPB , 2008)

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z3=1 z3=2

(Ancarani and Gasaneo ,JPB , 2008)

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PREDICTIVE CHARACTER

STABILITY OF EXOTIC SYSTEMS

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: 0,3 rm

0,0 0,2 0,4 0,6 0,8 1,00,97

0,98

0,99

1,00

1,01

1,02

1,03

1,04

E/E

[m2m

3]

r=m2/m

3

(Ancarani and Gasaneo ,JPB , 2008)

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… L>0 states

… atomic systems with N > 3 bodies

… molecular systems

SummarySummary

FutureFuture

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Thanks for listening

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Optimisation d’une fonction d’essai

Energie moyenne:

Variance:

Autres valeurs moyennes:

Théorème du Viriel:

Valeurs moyennes:

Energie locale:

Fluctuations moyennées

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