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Solid and gas phase properties of argon as a test
of ab initio three-body potential. František
Karlický1, Alexandr Malijevský2, Anatol
Malijevský2 and René Kalus1 1Department of
Physics, University of Ostrava, Ostrava, Czech
Republic 2Department of Physical
Chemistry, Prague Institute of Chemical
Technology, Prague, Czech Republic
Abstract
Three-body contributions to the interaction
energy of Ar3 have been calculated recently using
the HF - CCSD(T) method and multiply augmented
basis sets. Tests of new ab initio three-body
potential for argon trimer are performed in this
work. These tests consist in a comparison of
several quantities we calculated with
corresponding recent theoretical results and
available experimental values. Recently published
ab initio pair potential is used. The first part
of the tests include data on rare-gas solids - a
zero-temperature binding energy, lattice
constant, and crystal structure. The second part
of the tests is focused on the third virial
coefficient - one of thermodynamic quantities
which are influenced exclusively by two and
three-body intermolecular interactions. In
addition, the solid and gas phase properties
considered are also calculated using
semiempirical pair potential and simple
Axilrod-Teller three-body term for comparison.
Potentials
Three-body potential
Analytical formula

• Fitting method
• included 354 ab initio points
• least-square fit using Newton-Raphson method
• achieved mean deviation 0.2 cm-1 from ab initio
  points
• largest fit deviations ranging between 0.4 0.6
  cm-1 for less than 5 of points

• Ab initio methods and basis sets
• correlation method CCSD(T)
• only valence electrons have been correlated
• basis set d-aug-cc-pVQZ
• MOLPRO 2002 suite of ab initio programs

Pair potentials
Potential energy surface fit Three-body potential
has been represented by a sum of short-range
(inspired by V. Špirko et al., 1995) and
long-range terms expressed in Jacobi coordinates,
suitably damped by function F. Geometrical
factors Wlmn were derived from third-order
perturbation theory (e.g. Doran and Zucker,
1971), force constants Zlmn are taken from
previous independent ab initio calculations
(Thakkar, 1992).

• Ab initio potential of Slavícek et al. (2003)
  all-electron CCSD(T) correlation method, extended
  basis set aug-cc-pV6Z augmented with spdfg bond
  functions.
• Semi-empirical HFDID potential of Aziz (1993),
  with parameters fitted to experimental data.

Jacobi coordinates (r, R, q) 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 q is the angle between r and R vectors. ?
denotes perimeter and altß two smallest angles in
the Ar3 triangle.
Rare Gas Crystals
Theory
Crystal lattice
Comparison of this work with recent calculations (for d 3.7508 Å) Comparison of this work with recent calculations (for d 3.7508 Å) Comparison of this work with recent calculations (for d 3.7508 Å) Comparison of this work with recent calculations (for d 3.7508 Å) Comparison of this work with recent calculations (for d 3.7508 Å) Comparison of this work with recent calculations (for d 3.7508 Å) Comparison of this work with recent calculations (for d 3.7508 Å)
u2 used u3 used lattice E2 E3 EZPE EtotJ/mol
Aziz 1993 This work FCC -9083.5 612.0 767.7 -7703.8
Aziz 1993 This work HCP -9084.6 612.6 767.7 -7704.3
Aziz 1993 Lotrich et al. 1997a FCC -9082.9 569.6 760.4 -7752.9
Aziz 1993 Lotrich et al. 1997a HCP -9083.2 570.5 760.6 -7752.1
Binding energy per atom Etot EZPE E2 E3
of the crystal is a function of the nearest
neighbor separation d (or lattice constant),
where for N atoms in crystal is
Spherical symmetry of argon atoms (with closed
electronic shells) yield close-packed structures.
There are two possibilities face centered cubic
(FCC) and hexagonal closed packed (HCP) lattice.
In reality, FCC lattice is favored, theory
usually predicts HCP lattice (crystal structure
paradox).
and for zero-point energy in the quartic
oscillator approximation, using only additive
part of the potential (Horton, 1976)
FCC
HCP
Results
Number of atoms in crystal
Nearest neighbor separation and components of binding energy for optimized Etot(d) Nearest neighbor separation and components of binding energy for optimized Etot(d) Nearest neighbor separation and components of binding energy for optimized Etot(d) Nearest neighbor separation and components of binding energy for optimized Etot(d) Nearest neighbor separation and components of binding energy for optimized Etot(d) Nearest neighbor separation and components of binding energy for optimized Etot(d) Nearest neighbor separation and components of binding energy for optimized Etot(d) Nearest neighbor separation and components of binding energy for optimized Etot(d)
lattice u2 used u3 used d Å E2 E3 EZPE EtotJ/mol
FCC Aziz 1993 This work 3.7494 -9087.0 613.6 770.2 -7703.2
HCP Aziz 1993 This work 3.7501 -9085.9 613.3 768.9 -7703.7
FCC Slavicek et al. 2003 This work 3.7636 -9042.8 596.4 768.7 -7677.8
HCP Slavicek et al. 2003 This work 3.7650 -9039.5 595.1 766.1 -7678.3
FCC Aziz 1993 Doran and Zucker, 1971 3.7729 -9007.2 782.5 728.7 -7495.9
FCC Aziz 1993 Axilrod and Teller, 1943 3.7542 -9072.3 577.05 761.7 -7733.6
Comparison of this work with experiment (d 3.7555 Å, Peterson et al., 1966) Comparison of this work with experiment (d 3.7555 Å, Peterson et al., 1966) Comparison of this work with experiment (d 3.7555 Å, Peterson et al., 1966)
lattice u2 used EtotJ/mol
FCC Aziz 1993 -7703.2
HCP Aziz 1993 -7703.8
FCC Slavicek et al. 2003 -7676.7
HCP Slavicek et al. 2003 -7677.2
FCC, experiment Tessier et al. 1982 FCC, experiment Tessier et al. 1982 -7730.913
All calculations were performed in approximation
of a crystal of infinite size (N??) to eliminate
surface phenomena. In numerical calculations
finite number of atoms in sphere is used.
Neglecting remaining atoms, lying outside the
sphere, is reasonable, because the energy
correction due to these outer atoms is negligibly
small even for not too large radii (see below).
Unfortunately, theory still favors HCP over FCC.
Work in progress
Third Virial Coefficient
Theory
Results
Comparison with experiment
Comparison with other calculations
Third virial coefficient C depends on pair and
three-body potentials only this quantity is
thus well suited to test these potentials.
where
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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 Czech Republic (Grant No.
203/04/2146 )
CESTC 2006 Zakopane, Poland, 24-27 September,
2006