Rare gas molecular trimers: From Efimov scenarios to the study of thermal properties

1. Rare gas molecular trimers: From Efimov scenarios to the study of thermal properties Tomás González Lezana Departamento de Física Atómica, Molecular y de Agregados Instituto de Física Fundamental (CSIC) MADRID (ESPAÑA) Efimov states in molecules and nuclei: Theoretical methods and new experiments Rome (ITALY) 19-21 Oct 2009

2. Instituto de Física Fundamental Franco A. Gianturco Pablo Villarreal G. Delgado Barrio Isabella Baccarelli S. Miret Artés Instituto Superior y de Tecnologías y Ciencias Aplicadas Octavio Roncero Tomás González Lezana Jesús Rubayo Soneira Maykel Márquez Mijares Ricardo Pérez de Tudela http://www.3dflags.com/

3. Contents • Vibrational problem (J=0): DGF method Efimov effect (He3, LiHe2) • Rovibrational problem (J>0): (R+V)-DGF approach Ar3 • Thermal properties (E(T)): PIMC, DGF methods Ar3

4. Motivations to study rare gas trimers: Bulk-like properties investigated in larger (micro, nano) clusters: Molecular superfluidity Phase transitions Toennies et al.Phys. Today31,Feb (2001) Grebenev et al. Science 279, 2083 (1998) Trimers constitute extreme limits He2X clusters exhibit to peculiar properties (Efimov effect, Borromean systems)

5. 1. Vibrational problem (J=0)

6. DGF method: Hamiltonian

7. DGF method: Wave function

8. 1. Alternative way to calculate of geometrical magnitudes such as average values for the area, distances ... 2. Analysis of the participation of different triangular arrangements on the average geometry of the system: Isosceles Collinear Equilateral Scalene Pseudo-weights Definition :

9. R1 The centers of the DGF satisfy the triangular condition: R2 |Rl – Rm| ≤ Rn ≤ Rl + Rm  Some points of the DGF might not satisfy the triangular condition: R3 |R1 – R2| ≤ R3 ≤ R1 + R2  Special care for specific situations

10. Definition of a badness function: Evaluation of the average value the DGF basis set:

11. Ar3 c) I. Baccarelli et al. JCP 122, 144319 (2005) d) P. N. RoyJCP 119, 5437 (2003) Ne3 c) I. Baccarelli et al. JCP 122, 84313 (2005) Barrier to linearity • For even more sophisticated methods: • Talk by Sergio Orlandini after the coffee break • (ii) Orlandini et al. Mol. Phys.106, 573 (2008)

12. 1B + 1B + 1B 2B + 1B Energy 3B ‘Usual’ system 1B + 1B + 1B 2B + 1B Energy 3B Efimov system Efimov effect V3B(R1,R2,R3) = ΣλV2B(Ri) 3Bbound statesappear through the 2B threshold as λ increases 3B bound states disappear through the 2Bthreshold as λ increases

13. Candidates in Molecular Physics a ~ 100 Å

14. Efimov state! He3 V=1 For λ = 1: E3B(0) = -0.1523 cm-1 E3B(1) = -0.0012 cm-1 V=0 Barletta, Kievsky (2001) Motovilov, Sofianos, Kolganova(1997, 1998) Blume, Greene, Esry (1999, 2000) Nielsen, Fedorov, Jensen (1998) Only He3(v=0) He2 more stable than He3 (v=1) He3(v=1): Efimov state T. González-Lezana et al. PRL 82, 1648 (1999)

15. He3 Spatial delocalization He2 No dependence on the potential

16. Ar3 Probability density He3(1) Probability density Ne3 He3(0) Probability density functions Significant differences between the geometrical features of the extremely floppy He3 system and the more localised Ar3 and Ne3 T. González-Lezana et al. JCP 110, 9000 (1999)

17. λ=1.05 λ=1 λ=0.99 4He27Li E3B(0) = -0.0510 cm-1 E3B(1) = -0.0085 cm-1 Baccarelli et al., EPL50, 567 (2000) Delfino et al., JCP113, 7874 (2000)

18. 4He26Li E3B(0) = -0.0361 cm-1 E3B(1) = -0.0055 cm-1 λ=1.05 λ=1 λ=0.99 Baccarelli et al., EPL50, 567 (2000) Delfino et al., JCP113, 7874 (2000)

19. … bad news from the experimental front !!!

20. Geometries 3B effects found by analysis of different triangular arrangements Ar3 I. Baccarelli et al. JCP 122, 144319 (2005)

21. r(H--CG) = 13.2 bohr r(H--CG) = 10.1 bohr r(H--CG) = 1.9-2.5 bohr r(H--CG) = 10.7 bohr r(H--CG) = 9.2 bohr r(H--CG) = 10.8 bohr Geometries He2H- M. Casalegno et al. JCP 112, 69 (2000) H--CG distance Configurations in our basis set with the larger values of P(k)j

22. 2. Rotational problem (J>0)

23. ? General procedure We assume: For an asymmetric rotor: With the symmetry-adapted rovibrational basis: we construct the Hamiltonian matrix:

24. depends on R1, R2and R3 General procedure Baccarelli et al., Phys. Rep. 452, 1 (2007) Márquez-Mijares et al., CPL460, 471 (2008)

25. J=0 Ar3 [1] F. Karlický et al. JCP 126, 74305 (2007) Assignment ℓ↔ k,Ωfor theJ=0 case

26. ~ Procedure to assign symmetry: For each value of k and Ω (and therefore ℓ) : The symmetry for the vibrational part : The symmetry for the rotational part : Márquez-Mijares et al., CPL460, 471 (2008)

27. Ar3 J=20 Structural features of the bound states for J=0 seems to be also present at larger values of J J=0 Márquez-Mijares et al., JCP130, 154301 (2009)

28. rms 2x10-4 3x10-2 1x10-4 1x10-2 7x10-2 1x10-1 2x10-3 1x10-1 Adjustable parameters Rotational constants [1] F. Karlický et al. JCP 126, 74305 (2007)

29. 3. Thermal properties: E(T) and Cv(T)

30. DGF result requires large values of J to converge Good agreement up to T=20-25 K More pronounced increase for the free cluster Previously reported: T=20 K Liquid-gas transition (Etters & Kaelberer 1975) T=28 K beginning of diffusive motion (Leitner, Berry & Whitnell 1989) Energy Ar3 Sudden raise of the PIMC result with a larger Rc

31. Ar+Ar+Ar Ar2+Ar Morse A +Morse B Morse A Energy Morse B Morse A The increase in E(T) is then related to: • the participation of more diffuse structures • Continuum (dissociation?) Lack of all relevant states in the DGF result

32. Some estimates: PIMC D(Ar-Ar) = 99 cm-1 E(Ar2) ~ -84 cm-1 E(Ar2) + E(Ar2) E(Ar3) for the equilateral ground state ~ -252 cm-1 The trimer seems to explore different configurations during the MC propagation

33. Equilateral Linear Ar2 + Ar

34. Equilateral region: Linear region: θ1 = θ2 = θ3 = 60deg. θ1 = θ2 = 0 deg.; θ3 = 180 deg. R1 = R2 = R3 = 3.7 Å R1 = R2 = 3.7 Å; R3 = 7.5 Å Atom+Diatom region: θ1 = 60 deg.; θ2 = 0 deg.; θ3 = 120 deg. R1 = 3.7 Å;R2 = 8 Å;R3 = 12 Å Ar3 T = 22 K

35. Ar3 Ar13 Specific Heat Tsai & JordanJCP 99, 6957 (1993) Peak of phase transition (?) with more spatially extended geometries No proper peak Distinct behaviour with Rc

36. Conclusions FIS2007-62006 • Study of energy spectrum by means of a DGF approach • Efimov behaviour for He3 and LiHe2 • Rovibrational analysis performed with a V+R scheme and a DGF-based method. Comparison with an exact hyperspherical coordinates method reveals good performance even for large values of J • DGF and PIMC thermal analysis: • Appearance of liquid-like/diffusive behaviour with T • DGF description requires more rovibrational states