JOURNAL OF CHEMICAL PHYSICS
VOLUME 108, NUMBER 15
15 APRIL 1998
Density functional theory with an approximate kinetic energy functional
applied to study structure and stability of weak van der Waals
complexes
T. A. Wesołowski
Université de Genève, Département de Chimie Physique 30, quai Ernest-Ansermet,
CH-1211 Genève 4, Switzerland
Y. Ellinger
Laboratoire d’Etude Théorique des Milieux Extrêmes Ecole Normale Supérieure, 24,
rue Lhomond F-75231 Paris CEDEX 05, France
J. Weber
Université de Genève, Département de Chimie Physique 30, quai Ernest-Ansermet,
CH-1211 Genève 4, Switzerland
~Received 27 March 1997; accepted 13 January 1998!
In view of further application to the study of molecular and atomic sticking on dust particles, we
investigated the capability of the ‘‘freeze-and-thaw’’ cycle of the Kohn–Sham equations with
constrained electron density ~KSCED! to describe potential energy surfaces of weak van der Waals
complexes. We report the results obtained for C6H6¯X ~X5O2, N2, and CO! as test cases. In the
KSCED formalism, the exchange-correlation functional is defined as in the Kohn–Sham approach
whereas the kinetic energy of the molecular complex is expressed differently, using both the
analytic expressions for the kinetic energy of individual fragments and the explicit functional of
electron density to approximate nonadditive contributions. As the analytical form of the kinetic
energy functional is not known, the approach relies on approximations. Therefore, the applied
implementation of KSCED requires the use of an approximate kinetic energy functional in addition
to the approximate exchange-correlation functional in calculations following the Kohn–Sham
formalism. Several approximate kinetic energy functionals derived using a general form by Lee,
Lee, and Parr @Lee et al., Phys. Rev. A. 44, 768 ~1991!# were considered. The functionals of this
type are related to the approximate exchange energy functionals and it is possible to derive a kinetic
energy functional from an exchange energy functional without the use of any additional parameters.
The KSCED interaction energies obtained using the PW91 @Perdew and Wang, in Electronic
Structure of Solids ’91, edited by P. Ziesche and H. Eschrig ~Academie Verlag, Berlin, 1991!, p. 11#
exchange-correlation functional and the kinetic energy functional derived from the PW91 exchange
functional agree very well with the available experimental results. Other considered functionals lead
to worse results. Compared to the supermolecule Kohn–Sham interaction energies, the ones derived
from the KSCED calculations depend less on the choice of the approximate functionals used. The
presented KSCED results together with the previous Kohn–Sham ones @Wesołowski et al., J. Phys.
Chem. A 101, 7818 ~1997!# support the use of the PW91 functional for studies of weakly bound
systems of our interest. © 1998 American Institute of Physics. @S0021-9606~98!01215-X#
tallic complexes.2 To verify aforementioned hypothetical
mechanisms, the knowledge of the adsorption energy and
sticking coefficients of the molecules known to exist in space
would be especially valuable. The experimental data dealing
with the adsorption of small molecules on the carbonaceous
matter ~graphite or polycyclic aromatic hydrocarbons—
PAH! are sparse and do not allow to derive conclusions pertinent to astrophysical conditions. More is known about interactions between benzene and diatomic molecules such as
O2, 3,4 CO,5,6 and N2. 6–9 The nature of the intermolecular
interactions in such complexes and the mechanisms of physisorption of the aforementioned diatomics at the surface of
dust particles such as graphitic grains or PAHs are similar. In
this work, the complexes of diatomic molecules with benzene provide a series of test cases to assess the reliability of
I. INTRODUCTION
It is now well-established that considering only gasphase processes is not sufficient to understand the chemical
evolution of the interstellar medium ~ISM! or the upper atmosphere. The solid particles, present in ISM in the form of
silicates, crystalline, or amorphous carbon, ices, etc., provide
various types of surfaces acting as support and/or catalysts
for the synthesis of a number of molecules which, otherwise,
would not be formed in two-body reactions. The production
of H2 in ISM is one such typical example.1 Gas-surface interactions can also affect the composition of the interstellar
gas due to physisorption or sticking of atoms and molecules
on the surface of dust particles. Arguably, adsorption at carbonaceous surfaces leads to smaller than expected observed
abundancies of such molecular species as iron or organome0021-9606/98/108(15)/6078/6/$15.00
6078
© 1998 American Institute of Physics
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J. Chem. Phys., Vol. 108, No. 15, 15 April 1998
the computational methods to be employed to investigate
systems more relevant to ISM. The size of the surface that
should be considered—at least a pyrene molecule—
precludes the systematic use of common correlated ab initio
methods such as MP2. Alternatively, semi-empirical or molecular mechanics calculations which would be attractive
have the built-in drawback that the parametrization is derived from well-behaving systems far from the exotic species
present in space.
Density functional theory ~DFT! offers an other alternative. Although calculations of that type are more and more
widely used for the study of molecules and bulk materials,
applications to weakly bound systems remain rather sparse in
the literature, except for the studies that deal with hydrogen
bonding ~see Ref. 10, for instance!. Recent DFT studies use
the formalism of Kohn and Sham ~KS!11 for which van der
Waals complexes lie at the borderline of its applicability
with several existing approximate exchange-correlation
functionals.12–17 Within the general framework of density
functional theory, the formalism of Kohn–Sham equations
with constrained electron density ~KSCED!18–20 offers yet
another alternative. The ‘‘freeze-and-thaw’’ cycle of KSCED
might be considered as an extension of the theory by Kim
and Gordon.21 The Kim–Gordon theory was originally developed for calculating potential energy surfaces of interacting closed-shell systems using electron gas approximation
applied to frozen electron densities of interacting subsystems. The KSCED formalism uses also the electron density of fragments to represent the total electron density and it
also relies on the electron gas approximation to evaluate a
fraction of the kinetic energy of the total complex. The
KSCED formalism overcomes, however, the most important
drawback of the Kim–Gordon theory since it allows the fragments’ densities to be modified18 as the KSCED formalism is
founded on the variational principle19 as discussed in the
next section. Another distinct feature of the current implementation of the KSCED formalism is the use of a gradientdependent functional to approximate both the nonadditive
component of the kinetic energy and the associated functional derivative. Recent studies of hydrogen-bonded complexes by means of the KSCED formalism showed22 that
good structural and energetic results can be obtained provided the analytical form of the approximate kinetic energy
functional is properly chosen.
In this contribution, we present the first application of
the KSCED formalism to such weakly bound systems as van
der Waals complexes.
II. METHODS
The detailed description of the current implementation
of the ‘‘freeze-and-thaw’’ cycle of the KSCED together with
the underlying formalism can be found elsewhere.22 In the
‘‘freeze-and-thaw’’ cycle the electron density of the molecular complex is obtained stepwise. In each iteration, the total
energy is minimized with respect to variations of the electron
density localized in a given fragment. The electron density of
the other fragment is frozen. In the subsequent iteration the
frozen and nonfrozen fragment invert their respective roles.
Wesołowski, Ellinger, and Weber
6079
The electron density ( r 1 ) of a fragment which interacts
with an other subsystem comprising nuclear charges and a
frozen cloud of electrons ( r 2 ) is obtained using a Kohn–
Sham-like equations with the following effective
potential:18,19
eff
V KSCED
52
(A
ZA
1
u r2RA u
E r8
1
E r8
1
d T nadd
@ r1 ,r2#
s
,
dr1
1 ~ r8 !
u r 2ru
2 ~ r8 !
dr8
u r 2ru
dr8 1V xc~ r 1 ~ r! 1 r 2 ~ r!!
~1!
N
where r 1 5 ( i 1 n (1)i u c (1)i u 2 , V xc is conventional exchangecorrelation potential (V xc5 d E xc / d r ), the T nadd
denotes the
s
nonadditive kinetic energy T nadd
@ r 1 , r 2 # 5T s @ r 1 1 r 2 #
s
2T s @ r 1 # 2T s @ r 2 # , and the index A runs through the nuclei
of subsystem 1 and subsystem 2. It is worthwhile to note that
the equation is valid regardless the fragments’ densities ~r 1
and r 2 ! do or do not overlap, and that the equation makes it
possible to minimize the total energy with respect to variations of r 1 without any knowledge of orbital representation
of r 2 . Any practical implementation of Eq. ~1! involves approximating E xc (E xc5E x 1E c ), as in the case of the standard Kohn–Sham approach, as well as approximating
T nadd
@ r 1 , r 2 # . In the present paper, a particular subset of poss
sible approximations of T nadd
is considered. The approximate
s
kinetic energy functionals assume the following analytical
form:
T appr
s @r#5
3
10
@ 3 p 2 # 2/3
Er
F ~ s ! d 3 r,
5/3
~2!
where s5 u ¹ r u /2r k F ~k F being the Fermi vector!. Lee, Lee,
and Parr23 proposed to use the same analytical form of F(s)
as the one used in the conventional GGA form of the exchange functional (E x ):
E appr
x @ r # 52
3
4
@ 3 p 2 # 1/3
Er
F ~ s ! d 3 r.
4/3
~3!
Therefore for each approximate exchange energy functional
exists an associate kinetic energy functional. A particular
case of F(s)51 corresponds to the Dirac expression for the
exchange energy and Thomas–Fermi expression for the kinetic energy. Throughout this paper, the approximate kinetic
correspond to the
energy functionals used to derive T nadd
s
applied exchange energy functional. In practical applications
of the KSCED equations, the use of a kinetic energy functional which is associated with the applied exchange energy
functional offers a significant advantage, namely the KSCED
method does not involve any additional parameters compared to the standard Kohn–Sham calculations.
Since we apply the ‘‘freeze-and-thaw’’ cycle in which
the electron density of a given fragment enters alternatively
as r 1 or r 2 into the Eq. ~1!, both T s @ r 1 # and T s @ r 2 # can be
evaluated analytically. In such a case the total energy can be
expressed as:
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J. Chem. Phys., Vol. 108, No. 15, 15 April 1998
Wesołowski, Ellinger, and Weber
E @ r # 5E @ r 1 1 r 2 #
5T s @ r 1 # 1T s @ r 2 # 1T nadd
@ r1 ,r2#
s
1
2
EE
(E
1
2
~ r 1 ~ r8 ! 1 r 2 ~ r8 !!~ r 1 ~ r8 ! 1 r 2 ~ r!!
dr8 dr
u r8 2ru
A
ZA
~ r ~ r! 1 r 2 ~ r!! dr1E xc@ r 1 1 r 2 # .
u r2RA u 1
~4!
The following functionals were used:
•LDA ~Slater expression for E x and Vosko, Wilk, and
Nursair24 expression for E c !,
•B88/P86 ~E x proposed by Becke25 and E c by Perdew26!,
•PW86/P86 ~E x proposed by Perdew and Wang27 and E c by
Perdew26!,
•PW91 ~E x and E c proposed by Perdew and Wang28!.
Gaussian basis sets were used to expand one-electron
orbitals and to fit ~auxiliary functions! the electrostatic and
the exchange-correlation potentials. The orbitals were constructed using atomic basis sets with the following contraction pattern: (5211/411/1) for carbon, nitrogen and oxygen,
and (41/1) for hydrogen29 which were developed specifically
for Kohn–Sham calculations. The coefficients of the auxiliary functions were taken from Ref. 29: ~5,2;5,2! for carbon,
nitrogen, and oxygen, and ~5,1;5,1! for hydrogen atoms. This
orbital basis set together with the associated auxiliary functions will thereafter be referred to as Basis I. In order to
investigate the influence of the basis set expansion on the
DFT results, calculations were repeated using an other basis
set ~Basis II!:15 it corresponds to the (7111/411/1 * ) contraction pattern @associated with the ~4,4;4,4! auxiliary functions#
for C, N, and O, combined with the previous basis sets for
hydrogen.
In the KSCED calculations, the total electron density
was formally represented as a sum of two components ~fragments’ densities! corresponding to the two interacting molecules. The electron density of each fragment was expanded
using a limited set of atomic orbitals. Only orbitals centered
on atoms belonging to the relevant fragment were used.30
Such an expansion of the fragment’s electron density makes
it possible to polarize the fragments’ electron densities but
does not allow for any charge transfer between the fragments. Except for the O2 case ~X 3 S 2
g triplet!, the fragments
were assumed to be closed-shell singlet systems.
The grids comprised 128 radial shells. At the end of the
SCF procedure, each radial shell had either 50, 110, or 194
angular points. The ‘‘freeze-and-thaw’’ cycle was continued
until the change of subsequent KSCED energies was smaller
than 1026 Hartree, which was usually achieved in less than
five iterations.
Two geometrical arrangements of the complexes were
considered: the first one (A) has a C 2 v ~or C s ! symmetry,
whereas the other one (B) corresponds to a C 6 v structure. In
the CO case, two C 6 v structures are possible: B1 with the
carbon of CO pointing toward the benzene ring and B2 with
the oxygen atom pointing to the benzene ring. Structures A
and B will be also referred to as a ‘‘parallel’’ and a ‘‘per-
TABLE I. The intermolecular distance (R 0 ) and the binding energy (E int)
of the C6H6¯O2 complex at the parallel orientation derived from the
‘‘freeze-and-thaw’’ cycle of KSCED calculations using different approximate functionals. See the text for the description of functional parametrizations.
Method
KSCED/LDA
KSCED/B88/P86
KSCED/PW86/P86
KSCED/PW91
R 0 @Å#
E int @kcal/mol#
3.55
no minimum
3.63
3.30
20.52
no minimum
20.15
21.13
pendicular’’ arrangement, respectively. All interaction energy curves corresponded to rigid geometries of the interacting molecules: R CC51.42 Å, and R CH51.10 Å, for
benzene; R OO51.21 Å, R NN51.098 Å, and R CO51.128 Å,
for the diatomics under investigation.
Finally, the basis set superposition error ~BSSE! on
binding energies derived from supermolecule Kohn–Sham
calculations was estimated according to the counterpoise
method31,32 applied to each point of the energy curves.
The ‘‘freeze-and-thaw’’ cycle was performed using our
implementation of the KSCED equations into the deMon
program developed by Salahub and collaborators.33
III. RESULTS AND DISCUSSION
Our previous studies of the complexes of our interest by
means of the supermolecule Kohn–Sham calculations17
showed that the interaction energies and the equilibrium geometries depend strongly on the choice of the exchange
functional used. In particular, the B88 functional led to the
potential energy curves without minimum whereas the LDA
one led to too strong attraction between interacting molecules in all three cases. The PW86/P86 and PW91 functionals led to similar results, although the PW91 ones agreed
slightly better with available experimental measurements.
The similarity between the interaction energy curves derived
from the KS/PW86/P86 and KS/PW91 calculations indicate
that the regions where F PW91(s) and the F PW86(s) differ significantly (s.5) contribute negligibly to the energetics of
the complex. In the KSCED formalism, the analytical form
of the gradient-dependent contributions F(s) affects both the
exchange as well as the nonadditive kinetic energy. It is
worthwhile, therefore, to study the effect of F(s) on the
KSCED energies in more detail. The first part of the Results
sections aims on the rationalization of the differences between the KSCED energies obtained using different analytical forms of F(s). In the second part of Sec. III the structure
TABLE II. The intermolecular distance (R 0 ) and the binding energy (E int)
of the C6H6¯N2 complex at the parallel orientation derived from the
‘‘freeze-and-thaw’’ cycle of KSCED calculations using different approximate functionals. See the text for the description of functional parametrizations.
Method
KSCED/LDA
KSCED/B88/P86
KSCED/PW86/P86
KSCED/PW91
R 0 @Å#
E int @kcal/mol#
3.48
no minimum
3.41
3.15
20.69
no minimum
20.50
21.47
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TABLE III. The intermolecular distance (R 0 ) and the binding energy (E int)
of the C6H6¯CO complex at the parallel orientation derived from the
‘‘freeze-and-thaw’’ cycle of KSCED calculations using different approximate functionals. See the text for the description of functional parametrizations.
TABLE V. Numerical components of the interaction energy ~in @kcal/mol#!
for the C6H6¯N2 complex at 3.2 Å separation and at the parallel orientation
calculated using different approximate functionals. See the text for the description of functional parametrizations.
Method
Method
KSCED/LDA
KSCED/B88/P86
KSCED/PW86/P86
KSCED/PW91
R 0 @Å#
E int @kcal/mol#
3.46
no minimum
3.34
3.12
20.96
no minimum
20.67
21.8
6081
KSCED/LDA
KSCED/B88/P86
KSCED/PW86/P86
KSCED/PW91
E int(E x )
E int(Tnad)
E int(E x )1E int(Tnad)
26.48
2.49
23.74
22.20
7.13
4.70
5.66
3.37
0.65
7.20
1.92
1.56
and the energy of studied van der Waals complexes are studied by means of the KSCED calculations applying F PW91(s)
and Eqs. ~2!–~3!.
The following part of Sec. III describes the numerical
studies aiming on analyzing the effect of approximate functional parametrizations on the KSCED results. The potential
energy curves corresponding to the parallel orientation of
interaction molecules were derived from the KSCED calculations applying several approximate functionals. The characteristics of the minima at the potential energy curves ~interaction energy and the equilibrium intermolecular distance!
are collected in Tables I–III. In line with rapported previously supermolecule KS results,17 the B88 exchange functional leads to the potential energy curves without minima.
In the case of all three complexes, the PW91 functional leads
to the deepest potential energy minima, the depths and geometries of which agree very well with available experimental results ~see also Tables VII–IX!. The PW86/P86, PW91
functionals lead to the KSCED interaction energies which
agree within the range of 1 kcal/mol with the ones derived
from KSCED/LDA calculations. In the case of the supermolecular KS calculations, the differences between the LDA
energies and the ones obtained using gradient-dependent
functionals ~PW86/P86! or ~PW91! are larger and amount to
about 2 kcal/mol.17 The smaller effect of gradient-dependent
terms on the KSCED interaction energies than on the KS
ones results from the cancellation of errors in approximate
terms. In the KS case, the behavior of the gradient-dependent
terms at small electron densities ~large s! affects significantly
the final energies resulting in reducing the intermolecular
attraction derived at the LDA level.17 In the KSCED case,
owning to the gradient-dependent terms the nonadditive kinetic energy becomes less repulsive. The opposite signs of
gradient-dependent contributions results therefore in the
smaller total effects of gradient-dependent terms on the interaction energies ~see Tables IV–VI!.
The results presented above provide a practical confir! are the
mation that the PW91 functionals ~E x , E c , and T nadd
s
functionals of choice for KSCED studies complexes of our
interests. These results are in line with the conclusions of our
previous studies aimed on selecting the most accurate
exchange-correlation functional for studies of van der Waals
complexes17 and aimed on selecting the most accurate kinetic energy functional to be used to derive
d T nadd
@ r 1 , r 2 # / d r 1 in a number of test complexes.22
s
In the following section, the KSCED results obtained by
means of the PW91 functionals ~exchange, correlation, and
nonadditive kinetic energies! are presented. Parallel and perpendicular arrangements of interacting molecules were considered to find the structure of the complex with the lowest
energy. The basis set effect on the KSCED results was analyzed and compared to the effect on the KS results.
The KSCED binding energies of the C6H6¯O2 complex,
obtained using both atomic basis sets agree very well with
the experimental results ~Table VII!. Upon changing the basis set, the KSCED binding energies are negligibly affected.
The KSCED energies agree within the error bars with the
results derived from spectrophotometric measurements by
Goodling et al.3 The agreement of the KSCED binding energy with the one derived from photoionization experiments
by Grover et al.4 is also reasonable. The experimental structure of this complex is not available. The KSCED calculations predict that the parallel structure is more stable than the
perpendicular one and that the equilibrium distance is about
3.3 Å. These results are also in a very good agreement with
the ones derived from the MP2 calculations by Granucci and
Persico34 reporting a binding energy equal to 1.24 kcal/mol
at parallel arrangement and for an intermolecular separation
equal to 3.36 Å. At the perpendicular arrangement the binding energy and equilibrium distance derived from the MP2
calculations amount to 0.87 kcal/mol and 3.9 Å, respectively.
The Kohn–Sham supermolecular calculations predict also
that the parallel arrangement is more stable than the perpen-
TABLE IV. Numerical components of the interaction energy ~in @kcal/mol#!
for the C6H6¯O2 complex at 3.2 Å separation and at the parallel orientation
calculated using different approximate functionals. See the text for the description of functional parametrizations.
TABLE VI. Numerical components of the interaction energy ~in @kcal/mol#!
for the C6H6¯CO complex at 3.2 Å separation and at the parallel orientation calculated using different approximate functionals. See the text for the
description of functional parametrizations.
Method
KSCED/LDA
KSCED/B88/P86
KSCED/PW86/P86
KSCED/PW91
E int(E x )
E int(Tnad)
E int(E x )1E int(Tnad)
25.58
3.05
23.73
22.69
6.30
4.21
5.07
3.37
0.72
7.27
1.34
0.69
Method
KSCED/LDA
KSCED/B88/P86
KSCED/PW86/P86
KSCED/PW91
E int(E x )
E int(Tnad)
E int(E x )1E int(Tnad)
26.81
2.90
23.93
22.31
7.55
4.99
5.90
3.98
0.74
7.89
1.97
1.67
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J. Chem. Phys., Vol. 108, No. 15, 15 April 1998
Wesołowski, Ellinger, and Weber
TABLE VII. Geometry ~relative orientation and the intermolecular distance
R 0 @Å#! and the binding energy @kcal/mol# of the C6H6¯O2 complex derived from supermolecule Kohn–Sham ~KS! and ‘‘freeze-and-thaw’’ cycle
of KSCED calculations.
Structure A
Structure B
R0
E int
R0
E int
KS/PW91
KS/PW91~BSSE!
KS/PW91a
KS/PW91~BSSE!a
3.36
3.44
3.39
3.69
21.65
20.53
21.31
20.67
3.71
4.47
3.91
4.12
21.20
20.47
20.77
20.51
KSCED/PW91/PW91
KSCED/PW91/PW91a
3.33
3.30
21.10
21.13
3.66
3.74
20.89
20.75
Method
TABLE VIII. Geometry ~relative orientation and the intermolecular distance
R 0 @Å#! and the binding energy @kcal/mol# of the C6H6¯N2 complex derived from supermolecule Kohn–Sham ~KS! and ‘‘freeze-and-thaw’’ cycle
of KSCED calculations. R 0 is the distance between the centers of interacting
molecules.
Structure A (C 2 v )
R0
E int
R0
E int
KS
KS~BSSE!
KSa
KS~BSSE!a
3.51
3.59
3.56
3.83
21.43
20.40
21.40
20.74
4.15
4.66
4.10
4.34
21.17
20.32
20.51
20.28
KSCED
KSCEDa
3.25
3.15
21.37
21.47
3.72
3.51
20.73
20.84
Exp.b
Exp.c
Exp.d
3.5
3.3
Method
E int521.260.3
E int521.6560.32
Exp.b
Exp.c
a
Basis II.
Reference 3: the structure is not experimentally determined.
c
Reference 4: the structure is not experimentally determined.
b
Structure B (C 6 v )
21.43
20.9260.07
a
Basis II.
References 7, 9.
c
Reference 6.
d
Reference 8.
b
dicular one. However, the binding energies are smaller than
the corresponding energies obtained from either the experiments, the best available the MP2 calculations, or the
KSCED calculations. In a contrary to the KSCED results, the
ones derived from the supermolecule Kohn–Sham calculations are more basis set sensitive ~the intermolecular separations, in particular!.
The KSCED results obtained for C6H6¯N2 exhibit the
similar trends as the ones obtained for the C6H6¯O2 complex ~Table VIII!. The parallel structure of the complex is
more stable than the perpendicular one. The KSCED results
are in excellent agreement with the ones derived from supersonic molecular jet spectroscopy experiments by Nowak
et al.,6 reporting a binding energy equal to 1.43 kcal/mol for
an intermolecular separation between parallelly arranged
molecules equal to 3.3 Å. Other experimental groups reported slightly different values for the binding energy ~0.92
kcal/mol8! or the equilibrium intermolecular distance ~3.5
Å7!. The characteristics of the KSCED potential energy
curves are very similar to ones derived from the MP2 calcu-
lations by Hobza et al.35 The MP2 binding energies amount
to 1.69 kcal and 0.85 kcal/mol for the parallel and the perpendicular complexes, respectively. The corresponding intermolecular separations amount to 3.46 Å and 3.8 Å. In a
contrary to the supermolecule Kohn–Sham results, the
KSCED potential energy curves are not affected significantly
by the choice of the basis set.
The KSCED results obtained for the C6H6¯CO exhibit
trends similar to the ones obtained for the C6H6¯O2 and
C6H6¯N2 complexes ~Table IX!. As in the previous cases,
the change of the basis set does not affect significantly the
KSCED results. The supermolecule Kohn–Sham results are
more dependent on the atomic basis set. The parallel structure is more stable than the perpendicular one and the char-
TABLE IX. Geometry ~relative orientation and the intermolecular distance R 0 @Å#! and the binding energy
@kcal/mol# of the C6H6¯CO complex ~Structures A, B1, and B2! derived from supermolecule Kohn–Sham
~KS! and ‘‘freeze-and-thaw’’ cycle of KSCED calculations. R 0 is the distance between the centers of interacting
molecules.
Structure A (C 2 v )
Structure B1 (C 6 v )
Method
R0
E int
R0
E int
R0
E int
KS
KS~BSSE!
KSa
KS~BSSE!a
3.64
3.71
3.70
3.9
21.38
20.61
21.1
20.7
4.24
4.39
4.38
4.51
20.91
20.22
20.16
20.07
3.97
4.47
4.18
4.27
20.97
20.29
20.27
20.26
KSCED
KSCEDa
3.17
3.12
21.79
21.8
3.78
3.51
20.44
20.75
3.71
3.64
20.63
20.60
Exp.b
Exp.c
3.24
3.44
21.73
a
Structure B2 (C 6 v )
Basis II.
Reference 6. ~The experimental structure is almost parallel.!
c
Reference 5. ~The experimental structure is almost parallel.!
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J. Chem. Phys., Vol. 108, No. 15, 15 April 1998
acteristics of the potential energy curve are in excellent
agreement with experimental results derived from supersonic
molecular jet spectroscopy experiments by Nowak et al.6
The structure derived from rotational spectroscopy by Brupbacher and Bauder5 is characterized by a larger intermolecular separation amounting to 3.44 Å, which is not supported
by the present calculations.
In all three studied complexes, the nonadditive kinetic
energy contributes significantly to the binding energy. This
contribution calculated by means of the gradient-dependent
kinetic energy functional is repulsive and amounts to 2.4,
4.4, and 5.4 kcal/mol at the most stable conformations of the
C6H6¯O2, C6H6¯N2, and C6H6¯CO complexes, respectively. In line with the previous results for hydrogen-bonded
complexes,22 the gradient-less contribution to the nonadditive kinetic energy calculated using the Thomas–Fermi functional is even more repulsive. At equilibrium geometries, the
value of the nonadditive kinetic energy calculated using the
gradient-less kinetic energy functional is larger by about
10% than the value calculated using the gradient-dependent
kinetic energy functional.
IV. CONCLUSIONS
The KSCED/PW91 equilibrium geometries and the energies are in a good agreement with available experimental
data and with available results obtained from high-level postHartree–Fock ab initio calculations. Other considered functionals led to worse results. In particular, the B88 exchange
functional, which is known to be not applicable for weakly
interacting systems, leads to qualitatively wrong KSCED results. The presented results provide a link between our earlier
studies which showed that for van der Waals complexes of
our interests the PW91 exchange-correlation functional applied within the Kohn–Sham framework leads to good structures and energies17 and other studies showing that the
F PW91(s) provides the best approximation to be used to construct the kinetic energy functional needed to derive
d T nadd
@ r 1 , r 2 # / d r 1 . 22
s
Given the fact that the exact analytical form of functionals used is not known, it is worthwhile to stress that accuracy
requirements for approximate functionals are different in the
KS and in the KSCED formalisms. Opposite to the KS case,
in which the quality of the results depends only on the accuracy of E xc , the KSCED results depend on the accuracy of
the sum E xc1T nadd
@ r 1 , r 2 # . Therefore, cancellations of ers
rors might be possible in the KSCED case as it is illustrated
by smaller effects of gradient corrections on the KSCED
energies than on the KS ones.
For all studied complexes, both the supermolecule
Kohn–Sham and the ‘‘freeze-and-thaw’’ KSCED/PW91 calculations predict that the parallel arrangement of interacting
Wesołowski, Ellinger, and Weber
6083
molecules is more stable than the perpendicular one. Finally,
the KSCED results depend less on the basis set than the
BSSE corrected KS ones.
ACKNOWLEDGMENTS
The authors are grateful to Professor D. R. Salahub for
providing the copy of the deMon program. Financial support
by the Federal Office for Education and Science, acting as
Swiss COST office, is greatly acknowledged. This work is
also a part of the Project No. 20-49037.96 of the Swiss National Science Foundation. The support of CNRS-IDRIS is
acknowledged.
M. J. Barlow and J. Silk, Astrophys. J. 207, 131 ~1976!.
P. Marty, G. Serra, B. Chaudret, and I. Ristorcelli, Astron. Astrophys. 282,
916 ~1994!.
3
W. A. Goodling, K. R. Serak, and P. R. Ogilby, J. Phys. Chem. 95, 7868
~1991!.
4
J. R. Grover, G. Hagenow, and E. A. Walters, J. Chem. Phys. 97, 628
~1992!.
5
Th. Brupbacher and A. Bauder, J. Chem. Phys. 99, 9394 ~1993!.
6
R. Nowak, J. A. Menapace, and E. R. Bernstein, J. Chem. Phys. 89, 1309
~1988!.
7
Th. Weber, A. M. Smith, E. Riedle, and E. W. Schlag, Chem. Phys. Lett.
175, 79 ~1990!.
8
B. Ernstberger, H. Krause, H. J. Neusser, Z. Phys. D 20, 189 ~1991!.
9
Y. Ohshima, H. Kohguchi, and Y. Endo, Chem. Phys. Lett. 184, 21
~1991!.
10
P. Sule and A. Nagy, J. Chem. Phys. 104, 8524 ~1996!.
11
W. Kohn and L. J. Sham, Phys. Rev. A 140, 1133 ~1965!.
12
S. Kristyan and P. Pulay, Chem. Phys. Lett. 229, 175 ~1994!.
13
J. M. Perez-Jaroda and A. D. Becke, Chem. Phys. Lett. 233, 134 ~1995!.
14
B. I. Lundquist, Y. Andersson, H. Stiao, S. Chan, and D. C. Langreth, Int.
J. Quantum Chem. 56, 247 ~1995!.
15
E. Ruiz, D. R. Salahub, and A. Vela, J. Phys. Chem. 100, 12265 ~1996!.
16
P. Hobza, J. Šponer, and T. Reschel, J. Comput. Chem. 16, 1315 ~1996!.
17
T. A. Wesołowski, O. Parisel, Y. Ellinger, and J. Weber, J. Phys. Chem. A
101, 7818 ~1997!.
18
P. Cortona, Phys. Rev. B 44, 8454 ~1991!.
19
T. A. Wesołowski and A. Warshel, J. Phys. Chem. 98, 5183 ~1993!.
20
T. A. Wesołowski and J. Weber, Chem. Phys. Lett. 248, 71 ~1996!.
21
R. G. Gordon and Y. S. Kim, J. Chem. Phys. 56, 3122 ~1972!.
22
T. A. Wesołowski, J. Chem. Phys. 106, 8516 ~1997!.
23
H. Lee, C. Lee, and R. G. Parr, Phys. Rev. A 44, 768 ~1991!.
24
S. H. Vosko, L. Wilk, and M. Nursair, Can. J. Phys. 58, 1200 ~1980!.
25
A. D. Becke, Phys. Rev. A 38, 3098 ~1988!.
26
J. P. Perdew, Phys. Rev. B 33, 8822 ~1986!.
27
J. P. Perdew and Y. Wang, Phys. Rev. B 33, 8800 ~1986!.
28
J. P. Perdew and Y. Wang, in Electronic Structure of Solids’ 91, edited by
P. Ziesche and H. Eschrig ~Academie Verlag, Berlin, 1991!, p. 11.
29
F. Sim, D. R. Salahub, S. Chin, and M. J. Dupuis, J. Chem. Phys. 95, 4317
~1990!.
30
See Ref. 22 for the detailed discussion of the methods to expand the
fragments’ electron densities. In this paper, the KSCED~m! variant is
applied.
31
S. F. Boys and F. Bernardi, Mol. Phys. 19, 553 ~1970!.
32
P. Hobza and R. Zahradnik, Chem. Rev. 88, 871 ~1988!.
33
A. St-Amant, Ph. D. Thesis, Université de Montréal ~1992!.
34
G. Granucci and M. Persico, Chem. Phys. Lett. 205, 331 ~1993!.
35
P. Hobza, O. Bludský, H. L. Selze, and E. W. Schlag, J. Chem. Phys. 98,
6223 ~1993!.
1
2
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