Benchmark calculations

Computational

method CASSCF(5,10) / icMRCI (5 active electronsin 10 active orbitals) [3]

basis set d-aug-cc-pVTZprogram package 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

Potential energy curves for the electronic ground state and the first two excited states calculated for C2v geometries

○ – our calculations

+ – M. F. Satterwhite and G. I. Gellene, J. Phys. Chem. 99, 1339 (1995)subplots – comparison with literatura data

Potential energy surfaces for C2v geometries

A detailed plot of the C2v PES for the electronic ground state

Ab initioAb initio calculation on He calculation on He33++ of interest for semiempirical of interest for semiempirical

modelling of Hemodelling of Henn++

Ivana PaidarovIvana Paidarováá a)a), Rudolf Pol, Rudolf Poláákk a)a), Franti, Františšek Karlickek Karlický ý b)b), Daniel Hriv, Daniel Hrivňákňák bb)), and René Kalus , and René Kalus bb))

a) a) J. HeyrovskJ. Heyrovskýý Institute of Physical Chemistry, Praha, Institute of Physical Chemistry, Praha, b) b) University of Ostrava, OstravaUniversity of Ostrava, Ostrava

Aim

The principal aim of the present calculations is to provide highly The principal aim of the present calculations is to provide highly accurate potential energy surfaces (PES) for the electronic ground accurate potential energy surfaces (PES) for the electronic ground state and the first two excited states of the Hestate and the first two excited states of the He33

++ ion to be employed ion to be employed

in subsequent semiempirical modellings of larger helium cluster in subsequent semiempirical modellings of larger helium cluster cations, Hecations, Henn

++. It is well known that the . It is well known that the diatomics-in-moleculediatomics-in-molecule (DIM) (DIM)

approach, which performs well for the heavier rare gases, fails approach, which performs well for the heavier rare gases, fails remarkably even for the smallest Heremarkably even for the smallest Henn

++. It is argued that this is mainly . It is argued that this is mainly

due to the neglect of three-body interactions due to the neglect of three-body interactions [1][1] within the DIM within the DIM framework and, consequently, the three-body contributions to the framework and, consequently, the three-body contributions to the HeHenn

++ interaction energy have to be extracted from interaction energy have to be extracted from ab initioab initio

calculations and included in semiemirical models for them to become calculations and included in semiemirical models for them to become acceptably accurate. This can be done, acceptably accurate. This can be done, e. g.e. g., within the , within the triatomics-in-triatomics-in-moleculesmolecules (TRIM) approach (TRIM) approach [2][2], which represents a natural , which represents a natural generalization of the DIM method.generalization of the DIM method.

[1] [1] P.J. Knowles, J.N. Murrell, E.J. Hodge, Mol. Phys. 85, 243 (1995)P.J. Knowles, J.N. Murrell, E.J. Hodge, Mol. Phys. 85, 243 (1995)[2] this poster session, D. Hriv[2] this poster session, D. Hrivňák ňák et al.et al., , Semiempirical modelling of HeSemiempirical modelling of Henn

++ clusters. clusters.

Grant No. 203/04/2146 of the Grant Agency of the Czech Republic

Potential energy surface

Computational

Coordinates

where r1 ≤ r2 ≤ r3 are inter-atomic distances.

Analytical formula

Computationally cheap Morse potential (Easymp → 0)

with (X = D, A, R)

Configurations

○ - anticipated three-body configurations in Hen+ clusters (n ≤ 13)

● - configurations included in fitting procedure

Least-square fits

rough optimization: genetic algorithm (with binary encodedstrings)

fine-tuning: Levenberg-MarquardtNewton-Raphson (program by V. Špirko)

Results

Examples of 1D fits for a representative set of geometries and for the electronic ground state

● - ab initio points from economy calculations—— - least-square fitssubplots - deviations of the least-square fits from the ab initio data

Distribution of the least-square fits residues for electronic ground state points

Dependence of the Morse potential parameters on He3+

shape (electronic ground state)

Economy calculations

Computational

method Equation-Of-Motion Coupled Clusters [4]

basis set d-aug-cc-pVTZ [5]

program package ACES II

[4] J.F. Stanton and R.J. Bartlett, J. Chem. Phys. 98, 7029 (1993)[5] Basis set converged results were obtained with daug-cc-pVTZ basis set, in the series of aug-cc-pVXZ calculations, X=D,T,Q.

Results

Comparison of potential energy curves for the first three electronic states and for selected C2v geometries with the benchmark results

● – economy calculations

○ – benchmark calculations

subplots – a detailed view of local minimum

Differences between the economy and benchmark calculations for two selected geometries (C2v and D∞h)

A detailed plot of the C∞v PES for the electronic ground state

The C∞v PES is extremely flat for the electronic ground state

Grant No. 203/06/XXXX (submitted) of the Grant Agency of the Czech Republic

Equilibrium structure of He3+

(comparison with literature)

method Emin Re De[hartree] [bohr] [eV]

QICSD(T), aug-cc-pVTZ [6]-7.896672 2.340 2.598 QICSD(T), aug-cc-pVQZ [6] -7.902103 2.336 2.640

MRD-CI, cc-pVTZ [7] -7.8954 2.34 2.59

this work, benchmark -7.897021 2.339 2.639 this work, economy -7.896084 2.341 2.639 (?)

[6] M. F. Satterwhite and G. I. Gellene, J. Phys. Chem. 99, 1339 (1995) [7] E. Buonomo et al., Chem. Phys. Letters 259, 641 (1996)

23

45

-7.90

-7.88

-7.86

-7.84

-7.82

-7.80

-7.78

-7.76

-7.74

-7.72

-7.70

100

120

140160

180

2B2(1)

2A1(2)

2A1(1)

23

45

-7.90

-7.88

-7.86

-7.84

-7.82

-7.80

-7.78

-7.76

-7.74

-7.72

-7.70

100

120

140160

180

2B2(1)

E [hart

ree]

23

45

-7.90

-7.88

-7.86

-7.84

-7.82

-7.80

-7.78

-7.76

-7.74

-7.72

-7.70

100

120

140160

180

2A1(1)

E [hart

ree]

23

45

-7.90

-7.88

-7.86

-7.84

-7.82

-7.80

-7.78

-7.76

-7.74

-7.72

-7.70

100

120

140160

180

2A1(2)

E [hart

ree]

2.12.2

2.32.4

2.52.6

2.72.8

100120

140

160

180

-7.895

-7.890

-7.885

-7.880

-7.875

-7.870

-7.865

-7.860

R [bohr] [deg.]

E [h

artr

ee]

-7.896

-7.894

-7.892

-7.891

-7.889

-7.887

-7.886

-7.884

-7.882

-7.881

-7.879

-7.877

-7.876

-7.876

-7.871

-7.869

2.1 2.2 2.3 2.4 2.5 2.6 2.7 2.8

100

120

140

160

180

global minimumR

1 = R

2 = 2.339 bohr

Emin

= -7.8970 hartree

[d

eg.]

R [bohr]

-7.8949

-7.8955

2.0 2.2 2.4 2.6 2.8 3.02.0

2.2

2.4

2.6

2.8

3.0

R2 [

bo

hr]

R1 [bohr]

-7.8944

global minimumR

1 = R

2 = 2.341 bohr

Emin

= -7.8961 hartree

2.02.2

2.42.6

2.83.0

-7.895

-7.890

-7.885

-7.880

-7.875

-7.870

2.0

2.2

2.4

2.6

2.83.0

E [h

artr

ee]

R 2 [b

ohr]

R1 [bohr]

2.02.2

2.42.6

2.8

3.0

-8.0

-7.8

-7.6

-7.4

2.0

2.2

2.4

2.62.8

3.0

E [h

artr

ee]

R 2 [b

ohr]R1 [bohr]

-8.0

-7.8

-7.6

-7.4

-7.2

-7.0

-6.8

1 2 3 4 5 6 7-8.0

-7.8

-7.6

-7.4

-7.2

-7.0

-6.8

1 2 3 4 5 6 7

2.0 2.2 2.4 2.6

-7.896

-7.893

-7.890

-7.887

2.0 2.2 2.4 2.6

-7.896

-7.893

-7.890

-7.887

2.0 2.2 2.4 2.6-7.893

-7.890

-7.887

-7.884

2.2 2.4 2.6 2.8-7.878

-7.875

-7.872

-7.869

E [h

artr

ee]

= 90o = 120o

= 150o

R [bohr]

= 180o

-8.0

-7.8

-7.6

-7.4

-7.2

-7.0

-6.8

1 2 3 4 5 6 7-8.0

-7.8

-7.6

-7.4

-7.2

-7.0

-6.8

1 2 3 4 5 6 7

2.0 2.2 2.4 2.6

-7.896

-7.893

-7.890

-7.887

2.0 2.2 2.4 2.6

-7.896

-7.893

-7.890

-7.887

E [h

artr

ee]

= 90o = 120o

= 150o

R [bohr]

= 180o

2 3 4 5 6 71 2 3 4 5 6 7 8-20

0

20

40

60

80

100 = 180o

R [bohr]

E2

E1

E3

Eec

onom

y -

Ebe

nchm

ark

[meV

] = 90o

1 1 1 2 2 3 2 1, / , ( ) /r r r r r r SIZE SHAPE

1 1 2 e 1 1( , , ) exp 2 ( ) 2exp ( )e e e eV r D A r R A r R

( )e 1 2 .X k m

kmk m n

X

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.00.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

1

isosce

les config

s ( >

60

o)

2

asymmetric <-- linear configs --> symmetric

isosceles configs ( < 60o)

pe

rpe

ndic

ula

r <

-- d

isso

cia

ted

co

nfig

s -

-> c

olin

ea

r

0.00

0.05

0.10

0.15

0.20

0.25

0.00

0.02

0.04

0.06

0.08

0.10

1 2 3 4 5 6 7

0.1

0.2

0.3

0.4

0.5

0.6

1 2 3 4 5 6 7

0.00

0.02

0.04

0.06

0.08

0.10

1 2 3 4 5 6 7-60

-30

0

30

60

1 2 3 4 5 6 7-60

-30

0

30

60

1 2 3 4 5 6 7-60

-30

0

30

60

1 2 3 4 5 6 7-60

-30

0

30

60

E [h

artr

ee]

1 = 1.0,

2 = 1.0 (D

inf,h)

1 = 0.7,

2 = 1.0 (C

inf,v)

1 = 1.0,

2 = 0.0 (D

3,h)

R1 [bohr]

1 = 0.7,

2 = 0.5 (C

s)

resi

du

es

[me

V]

re

sid

ue

s [m

eV

]

re

sid

ue

s [m

eV

]

re

sid

ue

s [m

eV

]

-100 -80 -60 -40 -20 0 20 40 600

2

4

6

8

10

ab

un

da

nce

[%

]

residues [meV]

0.00.2

0.40.6

0.81.0

0.04

0.06

0.08

0.10

0.5

0.6

0.7

0.8

0.91.0

De

[har

tree

]

1

2

0.00.2

0.40.6

0.81.0

2.0

2.2

2.4

2.6

2.8

3.0

0.5

0.6

0.7

0.8

0.91.0

Re [b

oh

r]

1

2

0.00.2

0.40.6

0.81.0

1.05

1.10

1.15

1.20

1.25

0.5

0.6

0.7

0.8

0.91.0

Ae

[1/b

ohr]

1

2