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 Efimov states in molecules and nuclei: Theoretical methods and new experiments methods and new experiments

Rome (ITALY)Rome (ITALY)19-21 Oct 200919-21 Oct 2009

Tomás González Lezana Jesús Rubayo

Soneira

http://www.3dflags.com/

Pablo VillarrealG. Delgado Barrio

S. Miret ArtésOctavio Roncero

Instituto de Física Fundamental

Franco A. Gianturco

Isabella Baccarelli

Instituto Superior y de Tecnologías y

Ciencias Aplicadas

Maykel Márquez Mijares Ricardo Pérez de

Tudela

1. Vibrational problem (J=0):DGF method

Efimov effect (He3, LiHe2)

2. Rovibrational problem (J>0):(R+V)-DGF approach

Ar3

3. Thermal properties (E(T)):PIMC, DGF methodsAr3

ContentsContents

Grebenev et al. Science 279, 2083 (1998)Toennies et al. Phys. Today 31, Feb (2001)

Motivations to study rare gas trimers:

Bulk-like properties investigated in larger (micro, nano) clusters:

Molecular superfluidity

Phase transitions

Trimers constitute extreme limits

He2X clusters exhibit to peculiar properties (Efimov effect, Borromean systems)

1. Vibrational problem (J=0)

DGF method: Hamiltonian

DGF method: Wave function

1. Alternative way to calculate of geometrical magnitudes such as average values for the area, distances ...

Pseudo-weights

Definition :

2. Analysis of the participation of different triangular arrangements on the average geometry of the system:

CollinearEquilateral

Isosceles

Scalene

The centers of the DGF satisfy the triangular condition:

|Rl – Rm| ≤ Rn ≤ Rl + Rm

Some points of the DGF might not satisfy the triangular condition:

|R1 – R2| ≤ R3 ≤ R1 + R2

R1

R2

R3

Special care for specific situations

Definition of a badness function:

Evaluation of the average value the DGF basis set:

Barrier to linearity

Ar3Ne3

c) I. Baccarelli et al. JCP 122, 84313 (2005)

c) I. Baccarelli et al. JCP 122, 144319 (2005)

d) P. N. Roy JCP 119, 5437 (2003)

For even more sophisticated methods:

(i) Talk by Sergio Orlandini after the coffee break

(ii) Orlandini et al. Mol. Phys.106, 573 (2008)

3B bound states appear through the 2B threshold as λ increases

1B + 1B + 1B

‘Usual’ system

3B

2B + 1B

En

erg

y

3B bound states disappear through the 2B threshold as λ increases

En

erg

y

3B

2B + 1B

1B + 1B + 1B

Efimov system

V3B(R1,R2,R3) = Σ λV2B(Ri)

Efimov effect

E (cm-1) <R> (A)

4He2-0.00091 51.87

4He6Li -0.00023 95.44

4He7Li -0.00195 36.80

4HeH- -0.39883 11.93

a ~ 100 Å

Candidates in Molecular Physics

Efimov state!

V=0

He3

Only He3(v=0)

He3(v=1): Efimov state

V=1

He2 more stable than He3 (v=1)

T. González-Lezana et al. PRL 82, 1648 (1999)

For λ = 1:E3B(0) = -0.1523 cm-1 E3B(1) = -0.0012 cm-1

Barletta, Kievsky (2001)Motovilov, Sofianos,Kolganova (1997, 1998)Blume, Greene, Esry (1999, 2000)Nielsen, Fedorov, Jensen (1998)

He3

He2

Spatial delocalization

No dependence on the potential

He3(0)

Ne3

Ar3 He3(1)

Pro

bab

ility

de

nsi

ty Pro

babi

lity

dens

ity

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)

4He27Li

Delfino et al., JCP 113, 7874 (2000)

λ=1λ=0.99

λ=1.05 E3B(0) = -0.0510 cm-1

E3B(1) = -0.0085 cm-1

Baccarelli et al., EPL 50, 567 (2000)

Delfino et al., JCP 113, 7874 (2000)

4He26Li

λ=0.99λ=1

λ=1.05 E3B(0) = -0.0361 cm-1

E3B(1) = -0.0055 cm-1

Baccarelli et al., EPL 50, 567 (2000)

… bad news from the experimental front !!!

Ar3

Geometries

3B effects found by analysis of different triangular arrangements

I. Baccarelli et al. JCP 122, 144319 (2005)

Geometries

M. Casalegno et al. JCP 112, 69 (2000)

He2H-

Configurations in our basis set with the larger values of P(k)

j

H--CG distance

r(H--CG) = 1.9-2.5 bohr

r(H--CG) = 13.2 bohr

r(H--CG) = 10.8 bohr

r(H--CG) = 10.1 bohrr(H--CG) = 10.7 bohr

r(H--CG) = 9.2 bohr

2. Rotational problem (J>0)

General procedure

For an asymmetric rotor:

we construct the Hamiltonian matrix:

With the symmetry-adapted rovibrational basis:

?

We assume:

General procedure

depends on R1, R2 and R3

Márquez-Mijares et al., CPL 460, 471 (2008)

Baccarelli et al., Phys. Rep. 452, 1 (2007)

HC [1] HC (this work) DGF

(v1,v2ℓ )

(L,K )E [cm-1] (v1,v2

ℓ )() E [cm-1] (k, , ) E [cm-1]

(0,00)(0,0) -252.24 (0,00)(0)A1’ -252.23 (1,0,III ) -252.23

(0,11)(0)E’ -229.78 (2-3,0,II-I ) -229.79

(1,00)(0,0) -221.80 (1,00)(0)A1’ -221.79 (4,0, III) -221.79

(0,20)(0,0) -209.48 (0,20)(0)A1’ -209.48 (5,0, III) -209.48

(0,22)(0)E’ -209.32 (6-7,0,II-I ) -209.33

(1,11)(0)E’ -202.58 (8-9,0,II-I ) -202.58

(2,00)(0,0) -195.99 (2,00)(0)A1’ -195.97 (10,0, III) -195.98

(1,11)(0)E’ -193.21 (11-12,0,II-I ) -193.22

-191.29 (2,10)(0)A1’ -191.29 (13,0, III) -191.28

(0,33)(0) A2’ -187.76 (14,0, I ) -187.76[1] F. Karlický et al. JCP 126, 74305 (2007)

Ar3J=0

Assignment ℓ ↔ k,Ω for the J=0 case

~

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., CPL 460, 471 (2008)

J=20

J=0

Ar3

Structural features of the bound states for J=0 seems to be also present at larger values of J

Márquez-Mijares et al., JCP 130, 154301 (2009)

DGF HC[1]

k A [MHz] B [MHz] B’ [MHz]

C [MHz] (v1,v2ℓ ) B [MHz] C [MHz]

1 1739.26 1739.17 1739.21 861.32 (0,00) 1738.35 863.32

2-3 1713.43 1712.97 1713.20 837.77 (0,11) 1697.59 785.88

4 1692.88 1692.07 1692.48 831.62 (1,00) 1691.92 834.58

5 1688.91 1688.22 1688.57 812.15 (0,20) 1596.78 809.92

6-7 1688.16 1686.76 1687.46 809.40 (0,22) 1694.31 627.73

8-9 1672.23 1666.65 1669.44 806.15 (1,11) 1630.11 644.37

10 1672.66 1660.20 1666.43 782.32 (2,00) 1653.20 782.53

11-12 1688.71 1660.45 1674.46 771.56 (0,31) 1601.10 541.76

[1] F. Karlický et al. JCP 126, 74305 (2007)

Rotational constants

Adjustable parameters

2x10-4

3x10-2

1x10-4

1x10-2

1x10-1

2x10-3

1x10-1

7x10-2

rms

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

DGF result requires large values of J to converge

Good agreement up to T=20-25 K

Ar3

Sudden raise of the PIMC result with a larger Rc

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

Morse A

Morse B

1. the participation of more diffuse structures

2. Continuum (dissociation?)

Lack of all relevant states in the DGF result

The increase in E(T) is then related to:

Morse A

Morse A + Morse B

Ar+Ar+Ar

Ar2+Ar

Energy

The trimer seems to explore different configurations during the MC propagation

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

Some estimates:PIMC

Equilateral Linear

Ar2 + Ar

Ar3 T = 22 K

Equilateral region:

θ1 = θ2 = θ3 = 60 deg.

R1 = R2 = R3 = 3.7 Å

Linear region:

θ1 = θ2 = 0 deg.; θ3 = 180 deg.

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

Tsai & Jordan JCP 99, 6957 (1993)

Ar3

Ar13

Specific Heat

Peak of phase transition (?) with more spatially extended geometries

No proper peakDistinct behaviour with Rc

• 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

ConclusionsConclusions

FIS2007-62006