Praha

Ostrava

Abstract

Ab initio calculations

Potential energy surface fit

Outlooks

Financial support: the Ministry of Education, Youth, and Sports of the Czech Republic (grant no. IN04125 – Centre for numerically demanding calculations of the University of Ostrava), the Grant Agency of the Faculty of Science, University of Ostrava (grant. no. 1052/2005)

Three-body contributions to the interaction energy of Ar3 have been calculated using the HF – CCSD(T) method and multiply augmented basis sets due to Dunning and co-workers [1]. The calculations have been performed using the MOLPRO 2002 suite of quantum-chemistry programs. The basis set superposition error has been treated through the counterpoise correction by Boys and Bernardi [2], the convergence to the complete basis set has been analyzed, and both the triple-excitation and core-electron contributions to the three-body energy of Ar3 have been assessed. A complete nonaditive potential has been fitted to 330 computed points and compared with other points.

Choice of points

0 10 20 30 40 50 600

10

20

30

40

50

60

70

80

90the most frequent three-body

configurations in an argon crystalAr3 configurations used in our work

[d

egre

e]

[degree]

Methods and basis setscorrelation method – CCSD(T)only valence electrons have been correlatedbasis sets – x-aug-cc-pVNZ with N = D,T,Q and x = s,d,tMOLPRO 2002 suite of ab initio programs computer – PIV 2.3 GHz, 2 GB RAM, and about 10 GB HDD requested

Analysis of convergence

2 3 4

2.5

3.0

3.5

4.0

4.5

5.0Convergence for N changing

in x-aug-cc-pVNZ

thre

e-bo

dy e

nerg

y [K

]

N in x-aug-cc-pvNz

aug-cc-pVNZ d-aug-cc-pVNZ t-aug-cc-pVNZ

1 2 3

2.5

3.0

3.5

4.0

4.5

5.0Convergence for x changing

in x-aug-cc-pVNZ

thre

e-bo

dy e

nerg

y [K

]

x in x-aug-pVNZ

x-aug-cc-pVDZ x-aug-cc-pVTZ x-aug-cc-pVQZ

9 10 11 12 13 14 15-0.04

-0.02

0.00

0.02

0.04

0.06

0.08

0.10

E3(t

avqz

) - E

3(dav

qz)

[K

]

perimeter [A]

9 10 11 12 13 14

0

2

4

6

8

10

12

14

9 10 11 12 13 14

14

16

18

20

22

24

26

Triple-excitation contributionsto the CCSD energy for d-aug-cc-pVQZ

E3(C

CS

D(T

)) -

E

3(CC

SD

)

[K]

perimeter [A]

% o

f E

3(CC

SD

(T))

perimeter [A]

9 10 11 12 13 14

0.00

0.04

0.08

0.12

0.16

9 10 11 12 13 140.1

0.2

0.3

0.4

0.5

0.6

0.7

E3(c

ore)

-

E3(v

alen

ce)

[K

]

perimeter [A]

Core-electron contributionsto the CCSD(T) energy for aug-cc-pVTZ

perimeter [A]

% o

f E

3(co

re)

Analytical formula

Calculations

6 8 10 12 14 16 18

0

20

40

60

80

6 8 10 12 14 16 18

-0.04

-0.02

0.00

0.02

0.04

= = 30o, = 120o

E3(

avtz

) - E

3(da

vqz)

[K

]

perimeter [A]

y = A1exp[-(x-x

0)/

1] + A

2exp[-(x-x

0)/

2]

perimeter [A]

resi

dues

[K]

SRLRSR VVFVV 3

kji

kj

ri

r LRRarraVRrV,,

210 cosexpexp1,, SH

Jacobi coordinates (r, R, ): r represents the “shortest” Ar-Ar separation, R denotes distance between the “remaining” Ar atom and the center-of-mass of the previous Ar-Ar fragment, and is the angle between r and R vectors.

;,,,,,,

nml

lmnlmn RrWZRrV LR

r

R

Accurate ab initio three-body potential for Accurate ab initio three-body potential for argonargon

František Karlický and René KalusFrantišek Karlický and René Kalus

Department of Physics, University of Ostrava, Ostrava, Czech RepublicDepartment of Physics, University of Ostrava, Ostrava, Czech Republic

Hypersurface

Dr

DrrDrF

for1

for21expTests

15 20 25 30 35 40 45-10

0

10

20

30

40

50

60

thre

e-bo

dy e

nerg

y [K

]

[degree]

ab initio fit u

DDD

perimeter

= 10 Å

2 3 4 5 6-20

-15

-10

-5

0

5

10

15

20

thre

e-bo

dy e

nerg

y [K

]

R [Å]

ab initio fit u

DDD

r = 3.757 Å = /2

Three-body energies for D3h geometry and perimeter = 3 x 3.757 A.

Differences between t-aug-cc-pVQZ and d-aug-cc-pVQZ.

References[1] see, e.g., T. H. Dunning, J. Phys. Chem. A 104 (2000) 9062.[2] S. F. Boys and F. Bernardi, Mol. Phys. 19 (1970) 553.[3] inspired by V. Špirko and W. P. Kraemer, J. Mol. Spec. 172 (1995) 265.[4] see, e. g., M. B. Doran and I. J. Zucker, Sol. St. Phys. 4 (1971) 307.[5] A. J. Thakkar et al., J. Chem. Phys. 97 (1992) 3252.[6] P. Slavicek et al., J. Chem. Phys. 119 (2003) 2102.

The geometry of Ar3 has been described by the perimeter of the

Ar3 triangle, p, and by the two smaller angles in this triangle, and .Angles and used in our ab initio calculations have been chosen so that they spread over all the relevant three-body geometries in both an fcc and an hcp argon crystal.

Contributions due to triple-excitations and core correlations.

For each choice of Ar3 geometry, three-body energies have been calculated at the CCSD(T) + d-aug-cc-pVQZ level for a sufficiently broad range of perimeters. The calculations seem to be converged within about several tenths of K at this level.

To save computer time, the following procedure has been adopted: A small set of energies has been calculated for a small number of perimeters, at both

the aug-cc-pVTZ and the d-aug-cc-pVQZ level. This has usually taken 7 – 10 days of computer time.

Differences between the aug-cc-pVTZ and the d-aug-cc-pVQZ energies have been fitted to an analytical formula.

A much larger set of energies has been calculated (20 – 40) at the aug-cc-pVTZ level (about one hour of computation).

Finally, the aug-cc-pVTZ energies have been corrected.

Differences between the aug-cc-pVTZ and d-aug-cc-pVQZ energies for a selected C2v geometry, and the corresponding least-square fit.

Examples of results

Our three-body potential (2-D cut) for argon compared with Axilrod-Teller-Muto uDDD term

The least-square fit compared with additionally calculated points - 1D cut along a line in the -plane (see section “choice of points”), perimeter of Ar3 triangle is fixed. The Axilrod-Teller (uDDD) term is also added.

1D cut for fixed r,Jacobi coordinates

r

R

Three-body potential has been represented by a sum of short-range [3] and long-range terms expressed in Jacobi coordinates, suitably damped by function F. Geometrical factors Wlmn were derived from third-order perturbation theory [4], force constants Zlmn are taken from previous independent ab initio calculations [5].

Methodleast-square fit using Newton-Raphson method

achieved standard deviation 0,4 K

maximum fit deviation 1 K from ab initio points

This work is the first step in a series of calculations of the three-body non-additivities we plan for heavier rare gases, argon – xenon. The aim of these calculations consists in obtaining reliable three-body potentials for heavier rare gases to be employed in subsequent molecular simulations.Firstly, a thorough analysis of methods and computations is proposed for the argon trimer. In addition to the all-electron calculations presented here, which use Dunning’s multiply augmented basis sets of atomic orbitals, further calculations will be performed for argon including mid-bond functions, effective-core potentials, and core-polarization potentials.Secondly, properly chosen mid-bond functions, optimized for dimers, will be combined with effective-core potentials and with core-polarization potentials to calculate three-body energies for krypton and xenon. Our previous work on krypton and xenon two-body energies [6] indicates that this way may be rather promising.Our potential will be tested by computing experimentally measurable results: vibrational spectra of trimers, third virial coefficient, crystal structures and their binding energies.

8 10 12 14 16-300

-200

-100

0

corrected aug-cc-pVTZ

60 - 60 - 60

perimeter [Å]10 12 14 16 18

0

100

200

300

aug-cc-pVTZ, d-aug-cc-pVQZ

0 - 0 - 180

thre

e-bo

dy e

nerg

y [K

]

8 10 12 14 16-300

-200

-100

0

DDD+DDQ+DQQ+DD)+QQQ

45 - 45 - 90

V3(r = 3.757 Å,R,) sections of three-body potential for argon

l, m, n = D-dipole, Q-quadrupole, O-octupole

V3(r,R, = /2) sections of three-body potential for argon comparison

our potential Axilrod-Teller-Muto uDDD term