Transcript No Slide Title

Semiempirical modelling of helium cluster cations
I. Paidarová a), R. Polák a), F. Karlický b), I. Janeček b), D. Hrivňák b), R. Kalus b), and F. X. Gadéa c)
a) J.
b) University
Heyrovský Institute of Physical Chemistry, Praha;
of Ostrava, Ostrava; c) Université P. Sabatier, Toulouse
AIM
LEAST-SQUARES FITS FOR He3
The principal aim of the present work is to provide sufficiently accurate and still computationally
cheap tools for theoretical modelling of the intra-cluster interactions in singly charged helium
cluster cations, Hen+. This is achieved in two steps.
SIZE
p  r1,
Coordinates :
Firstly, highly accurate ab initio calculations are performed for the electronic ground state and the first two
excited states of the helium trimer cation, He3+, and the three three-body potential energy surfaces are
represented analytically using standard formulae with the adjustable parameters estimated via a leastsquares fitting procedure.
Secondly, the analytical potential energy surfaces are used in semiempirical modellings of the interactions in
larger ionized clusters of helium, Hen+. It is well known that the inclusion of the three-body interactions is
crucial for describing the intra-cluster interactions in the Hen+ complexes correctly [1]. Accordingly, a
semiempirical triatomics-in-molecules model, within which the three-body contributions are taken into
account explicitly, is proposed for the helium cluster cations. The model will be subsequently employed in
calculations of the the electronic and geometric structure of these clusters and their absorption spectra.
x1  r1 / r2, x2  (r3  r2 ) / r1
V (r1, x1, x2 ) 
Analytical formula:
(r1  r2  r3 )
kmax lmax mmax
l
m


C
exp

Ar
x
x


   klm 
1 
1 2
k
k 0 l 0 m 0
Configurations:
[1] P.J. Knowles, J.N. Murrell, E.J. Hodge, Mol. Phys. 85, 243 (1995)
+
○ - anticipated three-body configurations in Hen+ clusters (n ≤ 13)
0.9
0.8
0.7
0.6
x2
● - configurations included in fitting procedure
0.5
0.4
0.3
Restrictions imposed on the configurations
r1  1.5  10 bohr, Eground  1eV, Eexc  Eground  7eV
o
Computational
1.0
isosceles configs ( > 60 )
perpendicular <-- dissociated configs --> colinear
asymmetric <-- linear configs --> symmetric
AB INITIO CALCULATIONS ON He3
0.2
(about 1000 points for each PES)
0.1
0.0
0.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1.0
x1
o
isosceles configs ( < 60 )
method
basis set
program package
SHAPE
+
CASSCF(5,10) / icMRCI (5 active electrons in 10 active orbitals) [2]
d-aug-cc-pVTZ
MOLPRO 2000.1
[3] H.-J. Werner and P. J. Knowles, J. Chem. Phys. 89, 5803 (1988); P. J. Knowles and H.-J. Werner, Chem. Phys. Letters 145, 514 (1988)
Results
2
Potential energy surface for C2v geometries
TRIATIOMICS-IN-MOLECULES (TRIM)
+
MODEL FOR HeN
A1(2)
-7.70
-7.72
-7.74
-7.78
-7.80
-7.82
-7.84
180
160
140
-7.86
-6.8
-7.88
 = 90
-7.0
o
 = 120
-7.90
o
120
2
3
TRIM expansion of the electronic Hamiltonian
5
2
-7.4
B2(1)
-7.70
-7.8
-7.74
-7.70
2
-7.76
-6.8
 = 150
-7.0
-7.2
-7.4
 = 180
o
-7.887
-7.887
-7.890
-7.890
-7.893
-7.893
-7.896
-7.896
2.0
-7.6
2.2
2.4
2.6
o
2
-7.78
2
-7.80
2.4
-7.90
-7.8
120
2
3
-8.0
2
3
4
5
6
7
1
2
3
4
5
6
-7.80
-7.82
1 N -2 N -1 N ˆ (ABC ) N  3 N ˆ (A )

H

H




N  2 A1 B  A1C B 1
2 A1
180
160
140
-7.88
180
160
140
R [boh
r]
1
-7.78
-7.86
-7.88
2.6
A1(1)
-7.84
-7.86
2.2
-7.76
B2(1)
-7.82
-7.84
2.0
ˆ
H
TRIM
-7.74
-7.90
[d
eg.
]
-8.0
-7.72
A1(2)
120
2
2
3
R [boh
r]
[d
eg.
]
-7.72
E [hartree]
-7.6
E [hartree]
E [hartree]
100
4
R [boh
r]
-7.2
[d
eg.
]
E [hartree]
-7.76
100
4
Localized (diabatic) basis set
5
A1(1)
100
4
5
-7.70
7
-7.72
R [bohr]
 (kN )  1s1 1s2
-7.74
○ – our calculations
+ – M. F. Satterwhite and G. I. Gellene, J. Phys. Chem. 99, 1339 (1995)
-7.78
-7.80
1sN ,
k  1, , N
-7.82
-7.84
180
160
140
-7.86
subplots – comparison with literatura data
1sk
-7.88
-7.90
120
2
3
100
4
R [boh
r]
 [
deg
.]
E [hartree]
-7.76
5
TRIM Hamiltonian matrices for HeN+
( ABC )
( ABC )
H AA
 U11
A detailed plot of the C2v / C∞v PESes for the electronic ground state
H
( ABC )
BB
U
( ABC )
22
( ABC )
( ABC )
HCC
 U33
global minimum
R1 = R2 = 2.339 bohr
-7.860
-7.877
100
-7.884
-7.890
-7.887
-7.871-7.876-7.879
2.2
2.3
2.4
2.5
2.6
2.7
2.1
2.2
2.3
100
-7.876
-7.869
2.1
Emin = -7.8961 hartree
-7.895
-7.881
R [bohr]
 [d
2.5
140
eg.
]
2.6
2.7
160
180
R
[bo
]
hr
-7.885
k 1
0
-7.890
2.2
3.0
2.8
2.6
-7.895
2.4
120
2.8
E (j ABC )   xkj  (kABC )
Example – TRIM Hamiltonian matrices for He4+
2.0
2.0
2.8
2.0
2.2
2.2
2.4
2.6
2.8
3.0
2.4
2.4
2.6
R [b
1
ohr
R1 [bohr]
2.2
]
2.8
3.0
(123)
(123)
(123)
 U11
U12
U13
0 
 (123)

(123)
(123)
U12
U22
U23
0 
(123)


H (He4 )  (123)
(123)
(123)
 U13
U23
U33
0 

(123) 
0
0
Eneut 
 0
]
-7.891
120
global minimum
R1 = R2 = 2.341 bohr
2.4
-7.880
hr
-7.892
-7.8955
[b
o
-7.885
2.6
2
-7.880
H
( ABC )
EF
3
-7.875
R2 [bohr]
-7.889
-7.894
( ABC )
( ABC )
( ABC )
HBC
 HCB
 U23
j 1
-7.8949
E [hartree]
 [deg.]
-7.875
U
( ABC )
13
-7.870
-7.870
-7.886
H
( ABC )
CA
2.8
R
-7.896
E
( ABC )
neut
H
( ABC )
AC
Ukl( ABC )   E (j ABC )x(kjABC )x(ljABC ),
-7.8944
-7.865
-7.882
160
140
H
3.0
Emin = -7.8970 hartree
E [hartree]
180
( ABC )
DD
( ABC )
( ABC )
( ABC )
H AB
 HBA
 U12
3
2.0
C∞v
C2v
etc.
Equilibrium structure of He3+ (comparison with literature)
Independent inputs for the TRIM method
method
Emin
[hartree]
Re
[bohr]
De
[eV]
QICSD(T), aug-cc-pVTZ [3]
QICSD(T), aug-cc-pVQZ [3]
-7.896672
-7.902103
2.340
2.336
2.598
2.640
MRD-CI, cc-pVTZ [4]
-7.8954
2.34
2.59
this work
-7.897021
2.339
[3] M. F. Satterwhite and G. I. Gellene, J. Phys. Chem. 99, 1339 (1995)
[4] E. Buonomo et al., Chem. Phys. Letters 259, 641 (1996)
Fourth International Conference on Photodynamics, Havana, 6 – 10 February, 2006
2.639
Eneut (ABC) … ground-state energies of neutral triatomic fragments (calculated using semiempirical
two- and three-body potentials for helium)
Ej(ABC)
… energies of ionic triatomic fragments for their electronic ground state (j = 1) and the
lowest two excited states (j = 2,3) (calculated using the analytical formula given above
and fitted to the ab initio data presented in the left panel)
xkj
… expansion coefficients of the eigenstates of ionic triatomic fragments with respect to
the localized basis set (approximate estimates are obtained using the diatomic-inmolecules method: since the triatomic contributions to the total interaction energy of
He3+ represent only a small perturbation to the triatomic electronic Hamiltonian, this
approximation should give energies accurate up to the first order of the perturbation
theory)
Grant No. 203/04/2146 of the Grant Agency of the Czech Republic