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