Chemical Physics Letters 407 (2005) 379–383
www.elsevier.com/locate/cplett
Treatments of non-nuclear attractors within the theory
of atoms in molecules
Diego R. Alcoba a, Luis Lain
a,*
, Alicia Torre a, Roberto C. Bochicchio
b
a
b
Dept. Quı́mica Fı́sica, Facultad de Ciencias, Universidad del Paı́s Vasco. Apdo. 644 E-48080 Bilbao, Spain
Dept. Fı́sica, Facultad de Ciencias Exactas y Naturales, Universidad de Buenos Aires, Ciudad Universitaria, 1428, Buenos Aires, Argentina
Received 18 February 2005; in final form 15 March 2005
Available online 14 April 2005
Abstract
This Letter describes simple procedures to deal with non-nuclear attractors which are found in some results arising from topological population analyses of the molecular electron density. The proposed treatments have been applied to determine atomic electron populations and bond orders in the acetylene and dilithium molecules, achieving satisfactory chemical results. A detailed
discussion of our proposals is reported.
2005 Elsevier B.V. All rights reserved.
1. Introduction
The existence of non-nuclear attractors or pseudoatoms, that is, local maxima of electron density which appear out of nuclear positions in crystals and molecules,
has been subject of controversy for some time. Several
authors attribute the occurrence of non-nuclear attractors to a basis set or method dependent effect [1–3] while
others report that these kind of attractors are a property
of the electron density, instead of artifacts of the level of
quantum chemical calculations [4,5]. The origin of this
effect has been related with interactions of homonuclear
groups at appropriate internuclear distances, although
the use of small or not properly balanced basis sets
may skip its appearance [5]. As is well-known, the theory of atoms in molecules (AIM) [2] carries out a partitioning of the physical space based on the topological
properties of the electron density function. The whole
three-dimensional space is divided into disjunct atomic
basins XA, which are defined by surfaces having zero
flux in the gradient vector field of the electron density.
*
Corresponding author. Fax: +34 946013500.
E-mail address: [email protected] (L. Lain).
0009-2614/$ - see front matter 2005 Elsevier B.V. All rights reserved.
doi:10.1016/j.cplett.2005.03.078
In this methodology each basin XA is generally associated with a determined nucleus (or nuclear attractor)
A. Hence, the appearance of non-nuclear attractors implies that the atoms and the basins cannot be put into
one-to-one correspondence with each other [6,7]. Consequently, in studies of population analysis carried out
within the framework of AIM theory the description
of bondings between ordinary atomic basins and nonnuclear attractor ones does not result obvious and special treatments are required to retrieve chemically useful
quantities. The partitionings of the molecular energy, instead of the electron density, within an AIM scheme deserve similar comments. The total electronic energy is
decomposed into one- and two-center terms; the last
ones are regarded as appropriate tools to describe interactions between two atoms. However, the presence of
non-nuclear attractors also causes the appearance of
one-center (dummy) energy components and two-center
(dummy-true) energy ones. Both types of terms need be
redistributed among other energy components to
achieve chemically meaningful bonding energies in the
molecules which present these features [8].
The aim of this Letter is to propose simple procedures, based on mappings of some overlap integral sets,
380
D.R. Alcoba et al. / Chemical Physics Letters 407 (2005) 379–383
which allow one to get rid of the non-nuclear attractors.
Our algorithms, applied to studies of population analysis, lead to results which are chemically meaningful and
provide suitable insights of chemical bondings, in agreement with the classical concepts. The Letter is organized
as follows. In the second section we describe the theoretical aspects of these algorithms. The third section has
been dedicated to report some preliminary results.
Although our formulas can be applied at any level of
theory (correlated and uncorrelated wave functions)
the reported results have been obtained at the Hartree–Fock level in systems such as C2H2 and Li2 in
which the presence of non-nuclear attractors has systematically been detected. A discussion of these results has
also been carried out in this section. Finally, the concluding remarks and the perspectives of our treatments
are pointed out in Section 4.
2. Theory
The quantities DXA and DXA XB have been termed atomic
localization index and atomic delocalization index [10],
respectively; the last one constitutes a definition of bond
order (in topological version) between the regions XA
and XB [9,11,12]. These quantities include both Coulomb and Exchange contributions when correlated wave
functions are used [13,14].
The appearance of non-nuclear attractors, in the following denoted as X, provides the existence of
DXX and DXX XA terms, depending on the matrix elements
S ij ðXX Þ, which have not a clear chemical meaning. In
these situations, the AIM theory is not straightforwardly applicable [7]. In order to get rid of this problem
we propose to perform a redistribution of the elements
S ij ðXX Þ of each basin XX according to some criteria that
relate them with the chemically useful regions. To get
this purpose we define effective overlap matrices
i
ðS eff Þj ðXA Þ which allow one to rewrite Eq. (3) as
i
X X Xh
i
1 i1 k
N¼
Dj Dl 22 Dikjl ðS eff Þj ðXA Þ
XA
As is well-known, the number of electrons N of a
determined molecule can be expressed by means of
traces of reduced density matrices
i
Xh
1 i1 k
N¼
Dj Dl 22 Dikjl dij dkl
ð1Þ
i;j;k;l
in which {i, j, k, l, . . .} is a set of orthonormal molecular
orbitals and 1 Dij and 2 Dikjl are the matrix elements of
the spin-free first- and second-order reduced density
matrices,
respectively (note that the trace of the matrix
2
D is N2 ). We will consider the partitioning of the whole
real space, X, according to the BaderÕs atomic regions
XA [2]. Taking into account that this partitioning holds
X = [AXA and XA \ XB = ;(" A, B; A 6¼ B), the Kronecker deltas can be written in the form
X
X
dij ¼
hijjiXA ¼
S ij ðXA Þ;
ð2Þ
XA
XA
S ij ðXA Þ
are the elements of the overlap
where hijjiXA ¼
matrices calculated over regions XA. The substitution
of these deltas in Eq. (1) lead to [9]
i
X X Xh
1 i1 k
N¼
Dj Dl 22 Dikjl S ij ðXA ÞS kl ðXB Þ:
ð3Þ
XA
XB
i;j;k;l
This equation suggests the partitioning
X
X
N¼
DXA þ
DXA XB ;
XA
ð4Þ
i;j;k;l
k
ðS eff Þl ðXB Þ
ðXA ; XB 6¼ XX Þ
ð7Þ
and following this scheme, Eqs. (5) and (6) are therefore
reformulated substituting the S ij ðXA Þ elements for their
i
counterpart ones ðS eff Þj ðXA Þ.
We propose several different procedures to evaluate
i
the ðS eff Þj ðXA Þ matrix elements. In a first method these
quantities are formulated as
ðS eff Þlm ðXA Þ ¼ S lm ðXA Þ þ S lm ðXX Þ
l
ðS eff Þm ðXA Þ ¼ S lm ðXA Þ
ðl 2 AÞ;
ð8Þ
ðl 62 AÞ;
ð9Þ
in which the greek letters l,m, . . . are the usual atomic
functions. The symbol l 2 A means that the atomic
function l is centered on the nucleus A which is a feature
used in the well-known techniques of Mulliken population analysis [15]. Once this step has been performed the
matrix elements ðS eff Þlm ðXA Þ are transformed back into
the molecular orbital basis set, giving rise to matrix elei
ments ðS eff Þj ðXA Þ which are used in Eqs. (5) and (6).
The evaluation of the effective overlap matrix elements
described in Eqs. (8) and (9) is very simple for computational purposes. However it has the drawback that the
resulting overlap matrices are not symmetric. Thus, alternatively to this treatment we propose to express
X 1 l
1
ðS eff Þlm ðXA Þ ¼ S lm ðXA Þ þ
ðS 2 Þk ðXX ÞðS 2 Þkm ðXX Þ ðk 2 AÞ;
k
XA <XB
ð10Þ
where
DXA ¼
XB
1
2
X
ð1 Dij 1 Dkl 22 Dikjl ÞS ij ðXA ÞS kl ðXA Þ;
ð5Þ
i;j;k;l
DXA XB ¼ 2
X
ð1 Dij 1 Dkl 22 Dikjl ÞS ij ðXA ÞS kl ðXB Þ
ðXA < XB Þ:
i;j;k;l
ð6Þ
l
Þk ðXX Þ
where the calculation of the matrix elements ðS
is
performed following the same procedure that is used in
the well-known Löwdin orthogonalization method [16]
and where the index k has been restricted to the atomic
functions centered on the nucleus A. The effective overlap matrix elements expressed in the molecular orbital
basis set, ðS eff Þij ðXA Þ, are again calculated performing
D.R. Alcoba et al. / Chemical Physics Letters 407 (2005) 379–383
the corresponding transformation and used in Eqs. (5)
and (6). As can easily be checked formula (10) turns
out to be symmetric under the permutation of indices.
In the above procedures the matrix elements
i
ðS eff Þj ðXA Þ have been formulated by means of partitionings of the S ij ðXX Þ matrix in the Hilbert space spanned
by the basis function set. However, an alternative partitioning of those matrix elements can also be performed
in the ordinary three-dimensional space. In this case the
decomposition of S ij ðXX Þ matrix elements is carried out
following a fuzzy atom approach [17–19]. This method
has successfully been used in recent studies of population
analysis [7] as well as in partitionings of molecular energy
[20,21]. According to the mathematical framework of this
technique a non-negative continuous weight function
wA(r) is introduced for each atom A. These weight functions, which are mutually overlapping, measure the degree in which a determined point of space r is
considered to belong to atom A and fulfill the condition
X
wA ðrÞ ¼ 1 ðA 6¼ X Þ;
ð11Þ
A
which let us to write
X
S ij ðXX Þ ¼
hijwA ðrÞjjiXX
ðA 6¼ X Þ
ð12Þ
A
Consequently, the ðS eff Þij ðXA Þ matrix elements are formulated as
i
ðS eff Þj ðXA Þ ¼ S ij ðXA Þ þ hijwA ðrÞjjiXX
ðA 6¼ X Þ:
ð13Þ
The resulting effective overlap matrices are symmetric
and no transformations into the atomic basis set are
needed so that expression (13) can directly be used in
formulas (5) and (6).
The above reported relationships can also be generalized considering that several non-nuclear attractors may
appear in the topological partitionings. In this general
l
case Eq.P(8) must be rewritten as ðS eff Þm ðXA Þ ¼
l
l
S m ðXA Þ þ X S m ðXX Þðl 2 AÞ, and the counterpart versions of Eqs. (10) and (13) can straightforwardly be
formulated. In the next section we report results arising
from all these treatments.
3. Results and discussion
As mentioned in the introduction, our population
analyses can be applied at any level of theory provided
the second-order reduced density matrix elements are
known. However, in order to simplify the calculations,
we have limited our numerical tests to the closed-shell
Hartree–Fock [self-consistent field (SCF)] case in which
the second-order reduced density matrix is formulated
according to the well-known relationship
2
Dikjl ¼
1 1 i1 k 1 1 i1 k
D D D D:
2 j l 4 l j
381
The C2H2 and Li2 are molecules in which the occurrence of non-nuclear attractors in several basis sets is
well-known [5,7,22]. Hence, we have chosen these systems as appropriate test examples to apply the above
described treatments. The SCF molecular orbitals and
the overlap integrals calculated over the Bader regions
S ij ðXA Þ have been obtained from a modified version of
the GAUSSIAN 94 code [23]. The numerical integrations
of the expressions hijwA jjiXX have been carried out with
a modified PROAIM program [24] which uses the weight
functions reported in [25]. These weight functions, which
satisfy Eq. (11), depend on both the empirical Slater–
Bragg atomic radii of the atoms composing the molecule
under study and an iteration order k [25] which defines
the cutoff profiles of the functions. According to [20]
we have chosen the value k = 3 and increased the radius
of hydrogen to the value 0.35 Å. The reported results
have been obtained with the basis sets 6-31G(d,p) and
6-31G. For both studied systems (C2H2 and Li2 molecules), the geometries were optimized for the corresponding basis set within SCF wave functions. In the
case of the Li2 molecule a stretched geometry has also
been studied using identical procedure.
Table 1 reports the results for the C2H2 molecule
obtained from the classical AIM approach (columns 2
and 3) and the above three proposed treatments (last
six columns) denoted in the Table as DiXA and DiXA XB
(i = b, c, d). All these results have been calculated with
the 6-31G(d,p) basis set, except those written in parenthesis in the Table which have been obtained using the
6-31G one. As can be observed, in the 6-31G(d,p) basis
set the standard AIM description of this molecule exhibits a non-nuclear attractor in the midpoint of the internuclear distance of the C–C bond. Therefore, the
resulting atomic populations of the C atoms and the
magnitude of the C–C bond order are cumbersome from
a chemical point of view. However, any of the manipulations proposed in the preceding section overcome
these shortcomings, leading to atomic populations of
the C atoms and C–C bond orders which are in better
agreement with the classical properties of this molecule,
i.e., the multiplicity of the bond C–C. No significant
changes have been found within our treatments for the
atomic populations of the H atoms as well as for the
C–H bond order, which allows one to interpret that
the performed redistributions of the S ij ðXX Þ matrix elements mainly produce changes on the XC regions. On
the other hand, the results obtained with the basis set
6-31G for this hydrocarbon within the classical AIM
treatment show that no non-nuclear attractor appears
in the calculation and the numerical values found for
the ðDXAa Þ and ðDXA XBa Þ quantities turn out to be similar
to those obtained with the 6-31G(d,p) basis set within
any of our treatments. This indicates a weak basis
dependence of the AIM approach provided effective
overlaps are used.
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D.R. Alcoba et al. / Chemical Physics Letters 407 (2005) 379–383
Table 1
Calculated electron populations for the C2H2 molecule using the 6-31G(d,p) and 6-31G basis sets at SCF level
DXA a
C
H
X
C–C
DXA XB a
3.492
(4.200)
0.353
(0.330)
0.484
DXA b
DXA XB b
C–X
DXA XB c
DXA d
4.157
4.156
4.162
0.351
0.353
0.353
1.549
(2.873)
0.962
(0.965)
1.086
C–H
DXA c
DXA XB d
2.906
2.863
2.875
0.979
0.983
0.980
In parenthesis, values calculated in the 6-31G basis set.
a
Standard AIM theory.
b
Calculated using Eqs. (8) and (9).
c
Calculated using Eq. (10).
d
Calculated using Eq. (13).
Table 2
Calculated electron populations for the Li2 molecule using the 6-31G(d,p) basis set at SCF level
DXA a
Li
X
Li–Li
Li–X
DXA XB a
2.083
0.685
DXA b
DXA XB b
2.500
0.172
0.488
DXA c
DXA XB c
2.500
1.000
DXA d
DXA XB d
2.506
1.000
1.012
Distance Li–Li 2.81 Å (equilibrium).
a
Standard AIM theory.
b
Calculated using Eqs. (8) and (9).
c
Calculated using Eq. (10).
d
Calculated using Eq. (13).
Table 3
Calculated electron populations for the Li2 molecule using the 6-31G(d,p) basis set at SCF level
DXA a
Li
X
Li–Li
Li–X
0
Li–X
0
X–X
DXA XB a
2.153
0.099
DXA b
DXA XB b
2.500
0.308
0.248
0.247
0.198
DXA c
DXA XB c
2.500
1.000
DXA d
DXA XB d
2.499
1.000
0.998
Distance Li–Li 3.81 Å (stretched).
a
Standard AIM theory.
b
Calculated using Eqs. (8) and (9).
c
Calculated using Eq. (10).
d
Calculated using Eq. (13).
Table 2 shows the results for the Li2 molecule calculated at the equilibrium distance (2.81 Å). The existence
of a non-nuclear attractor in the middle of the Li–Li
bond which disappears at internuclear distances less
than 2.38 Å has previously been reported [22]. The
above proposed treatments remove the quantities related with this non-nuclear attractor and again lead to
satisfactory results, particularly in the description of
the Li–Li bond order which turns out to be 1.0. The re-
sults arising from a stretched configuration of this molecule (at a Li–Li distance 3.81 Å) are collected in Table
3. This geometry presents two non-nuclear attractors,
0
denoted as X and X , each one situated at 1.67 Å of a
Li atom. The numerical determinations obtained in this
last system show that our methodology can also be applied to systems possessing multiple non-nuclear attractors. As can be observed in the Table, the behavior of
our treatments is also successful in describing atomic
D.R. Alcoba et al. / Chemical Physics Letters 407 (2005) 379–383
populations and bond orders in this stretched molecule
and the results arising from each of them are again
similar.
4. Concluding remarks and perspectives
In conclusion, in this Letter we have proposed three
different simple treatments to get rid of non-nuclear
attractors which appear in some results arising from
the AIM theory. All these proposals are based on the
formulation of effective overlap matrices which constitute the appropriate tools to redistribute the electron
density assigned to the pseudoatoms. The results
obtained within the study of population analysis of the
test molecules C2H2 and Li2 show a good agreement
with the genuine chemical knowledge. No significant differences have been found in the numerical determinations from the three proposed methods. However, we
are currently studying in our laboratories the application of this methodology to the topological partitioning
of the molecular energy. The preliminary results show
that the partitioning of this quantity allows one to carry
out a deeper discussion on the behavior of the different
treatments proposed in this Letter. Another possibility
for dealing with non-nuclear attractors could be the
use of eccentric orbitals [26] or the corresponding natural orbitals [27]. This approach deserves interest and will
be explored in future developments.
Acknowledgments
This work has been partly supported by the spanish
Ministry of Education (BQU2003-00359), the Universidad del Pais Vasco (00039.310-15261/2003), the Universidad de Buenos Aires (X-024) and the CONICET
(Argentina, PIP No. 02151/01). D.R.A. thanks the Gobierno Vasco for the financial support of a stay in Bilbao.
383
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