Chemical Physics Letters 383 (2004) 368–375
www.elsevier.com/locate/cplett
Overlap populations, bond orders and valences for ÔfuzzyÕ atoms
I. Mayer
a
a,*
, P. Salvador
b
Institute of Chemistry, Chemical Research Center, Hungarian Academy of Sciences, P.O. Box 17, H-1525 Budapest, Hungary
b
Department of Chemistry and Institute of Computational Chemistry, University of Girona, 17071 Girona, Spain
Received 4 November 2003; in final form 14 November 2003
Published online: 5 December 2003
Abstract
Proper definitions are proposed to calculate interatomic overlap populations, bond order (multiplicity) indices and actual atomic
valences from the results of ab initio quantum chemical calculations, in terms of ÔfuzzyÕ atoms, i.e., such divisions of the threedimensional physical space into atomic regions in which the regions assigned to the individual atoms have no sharp boundaries but
exhibit a continuous transition from one to another. The results of test calculations are in agreement with the classical chemical
notions, exhibit unexpectedly small basis sensitivity and do not depend too much on the selection of the weight function defining the
actual division of the space into ÔfuzzyÕ atomic regions. The scheme is applicable on both SCF and correlated levels of theory. A free
program is available.
Ó 2003 Elsevier B.V. All rights reserved.
1. Introduction
It is often important to interpret the results of
quantum chemical calculations in genuine chemical
terms – total molecular energies carry little direct
chemical information. At the same time, one is hardly
able to comprehend the huge amount of numbers describing the wave functions, thus one needs a sort of
data compression. Already in the Huckel era, Coulson [1]
introduced the LCAO P-matrix (Ôdensity matrixÕ) for
that purpose.1
One should distinguish two approaches to the analysis of the results of ab initio calculations: one either
performs the analysis in the Hilbert space of the basis
functions assigned to the individual atoms in molecule
or in the three-dimensional (3D) physical space in which
the molecule is situated [2]. For Hilbert-space analysis
the most important steps probably were the introduction
of MullikenÕs gross and overlap population [3], WibergÕs
*
Corresponding author. Fax: +3613257554.
E-mail addresses: mayer@chemres.hu (I. Mayer),
pedro.salvador@ udg.es (P. Salvador).
1
Originally it was called Ôcharge – bond order matrixÕ, but that name
was appropriate only for simple p-electron theories with one active
orbital per atom.
0009-2614/$ - see front matter Ó 2003 Elsevier B.V. All rights reserved.
doi:10.1016/j.cplett.2003.11.048
bond index for the CNDO theory [4], its generalization
to extended Huckel [5] and ab initio theories [6], as well
as the introduction of the atomic valence index first in
the CNDO framework [7,8] and then for the ab initio
case [6]. The intimate relationship of the bond order and
valence indices to the exchange part of the second-order
density matrix has been analyzed in papers [9–11], the
latter ones also dealt with the generalization of these
indices to the wave functions accounting for electron
correlation. Although the bond order and valence indices have been applied with success to most different
chemical problems, they are obviously restricted to the
use of basis sets bearing atomic character. Thus, for
instance, they are not applicable to the calculations using plane wave basis, which became widespread in the
last time, and may meet irrecoverable difficulties if the
basis set contains diffuse functions lacking any true
atomic character. For anions, in particular, the use of
diffuse functions is mandatory, and their treatment by
using Hilbert-space based indices is virtually impossible.
(There are also other avenues of the Hilbert-space
analysis, e.g., McWeenyÕs natural hybrids [12] and their
generalization [13], the Hilbert-space based localization
criteria [14,15] or the well-known NBO scheme [16,17].
From the point of view of our subject, a larger relevance
have the proposition to compute Wiberg indices in a
I. Mayer, P. Salvador / Chemical Physics Letters 383 (2004) 368–375
L€
owdin-orthogonalized basis [18] and the so-called
ÔNatural Resonance TheoryÕ [19,20], which permits to
compute perfect ÔchemicalÕ bond order and valence indices by using a procedure which is controversial from a
puristic quantum mechanical point of view [21].)
The most important approach to the 3D analysis of
molecular wave functions is the so-called Ôatoms in
moleculesÕ (AIMs) theory of Bader [22] in which the
3D space is divided into Ôatomic domainsÕ on the basis
of the topological properties of the electron density
.ð~
rÞ. It is a rich field with many important results
which we cannot survey here. We mention, however,
ngy
that A
an et al. [23] have developed a formal
mathematical mapping between the Hilbert-space and
AIM formalisms, which permitted to introduce the
bond order and and valence indices in the AIM
framework. (Later that bond order index was renamed
Ôdelocalization indexÕ and is sometimes used without
proper reference to the original work.)
The AIM method has some disadvantages, too. One
point is that it is complex and the detailed investigation
of the topology of the electron density is rather CPU
demanding, which may prevent systematic use of the
method in the routine quantum chemical applications.
The form of the atomic domains does not always agree
with our physical intuition, which does not help the
understanding of the results. A more important aspect
in our opinion is that in the AIM scheme there are sharp
boundaries between the atomic domains which contradict to the classical chemical notion of shared electrons
requiring that some electronic charge should belong simultaneously to a pair of of chemically bonded atoms.
Obviously, that may only be realized by assuming that a
part of the physical space between the atoms, and the
electron density in that part of the space, are common in
some sense for the atoms in question. (In the Hilbertspace analysis, these shared electrons may be related to
MullikenÕs overlap populations.)
Hirshfeld [24] was probably the first who dealt with
ÔfuzzyÕ atoms, i.e., such divisions of the three-dimensional physical space into atomic regions in which the
regions assigned to the individual atoms have no sharp
boundaries but exhibit a continuous transition from one
to another. For that reason he introduced for each atom
A a continuous non-negative weight function
wA ð~
rÞ P 0;
ð1Þ
satisfying in every point of the 3D space the requirement
NX
atoms
wA ð~
rÞ 1:
ð2Þ
A¼1
Actually he used the ratio of the charge density of the
free atom to that of the ÔpromoleculeÕ (assembly of noninteracting atoms placed at the positions of the nuclei
in the actual molecule), but that has no conceptual
369
importance for our considerations based on the conditions (1) and (2) above.
Inserting condition (2) into the normalization integral
of the electron density
Z
Z NX
atoms
wA ð~
rÞ.ð~
rÞ dv
N ¼ .ð~
rÞ dv A¼1
¼
Z
NX
atoms
rÞ.ð~
rÞ dv ¼
wA ð~
A¼1
NX
atoms
ð3Þ
QA ;
A¼1
one gets quite naturally the number of electrons N as a
sum of HirshfeldÕs atomic populations
Z
QA ¼ wA ð~
rÞ.ð~
rÞ dv;
ð4Þ
often also-called ÔstockholderÕsÕ atomic populations.
2. Theory
2.1. Overlap populations of ‘fuzzy atoms’
The stockholderÕs populations (4) are resulted from a
division of the space into ÔfuzzyÕ atomic regions, but they
do not reflect this fact in any immediate way, as provide
us only a single number for every atom. Thus we get no
information concerning how much charge was actually
shared between the atoms, and between what atoms it
was shared. One may, however, insert identity (2) into
the normalization integral of the electron density twice,
and write
Z
Z NX
atoms
N ¼ .ð~
rÞ dv wA ð~
rÞwB ð~
rÞ.ð~
rÞ dv
A;B¼1
Z
NX
atoms
wA ð~
rÞwB ð~
rÞ.ð~
rÞ dv ¼
A;B¼1
where
qAB ¼
Z
NX
atoms
qAB ;
ð5Þ
A;B¼1
wA ð~
rÞwB ð~
rÞ.ð~
rÞ dv:
ð6Þ
The diagonal quantities qAA may be called net atomic
populations, while the off-diagonal qAB -s ðA 6¼ BÞ represent the overlap populations and measure the extent to
which the atomic charge is shared between atoms A and
B. In fact, qAB will have an appreciable value only if
there is a part of space in which the weight factors wA
and wB significantly differ from zero simultaneously. The
quantities qAA and qAB represent perfect analogues of
MullikenÕs net atomic populations and overlap populations, respectively, except that here qAB cannot be negative owing to the inequality (1). Owing to the obvious
equality
QA ¼
NX
atoms
B¼1
qAB ;
ð7Þ
370
I. Mayer, P. Salvador / Chemical Physics Letters 383 (2004) 368–375
the ÔfuzzyÕ (stockholderÕs) atomic charges play here the
role of gross atomic populations.
One should note that the AIM theory represents a
limiting case of the present formalism: for AIM only a
single wA ð~
rÞ differs from zero in every point of space
and that is equal unity, i.e., wA ð~
rÞ ¼ 1 if ~
r is within the
atomic domain XA and it is zero otherwise. 2 Obviously,
then one has qAB ¼ 0 if A 6¼ B and QA ¼ qAA . In other
words, overlap density cannot be defined in the AIM
frame.
2.2. Bond orders and valences for ‘fuzzy’ atoms
It is known that for single determinant wave functions the second-order density matrix can be expressed
through the first-order one as
.2 ðx1 ; x2 ; x01 ; x02 Þ ¼ .1 ðx1 ; x01 Þ.1 ðx2 ; x02 Þ
.1 ðx1 ; x02 Þ.1 ðx2 ; x01 Þ:
ð8Þ
ri ; ri Þ stands for the set of the spatial and
Here, xi ¼ ð~
spin-coordinates of the ith electron. The second term on
the right-hand-side of (8) is the Ôexchange partÕ of the
second-order density matrix which originates from the
antisymmetry of the wave function and describes the socalled ÔFermi holeÕ. It is normalized as
Z Z
.1 ðx1 ; x2 Þ.1 ðx2 ; x1 Þ ds1 ds2 ¼ N ;
ð9Þ
where N is again the number of electrons and we use
notation dsi to indicate that the integration includes
summation over the spins.
If we use a set of basis orbitals fvl g, then the firstorder density matrix is expressed over the basis spinorbitals vrl by using the LCAO P-matrices (Ôdensity
matricesÕ) having the elements
r
Plm
¼
occ
X
r r
Cli
Cmi ;
r ¼ a or b;
ð10Þ
i
Cr being the matrix of orbital coefficients of spin r, as
i
Xh
a a
b b
.1 ðx1 ; x2 Þ ¼
Plm
vm ðx1 Þval ðx2 Þ þ Plm
vm ðx1 Þvbl ðx2 Þ :
l;m
N¼
X
l;m;.;s
þ
b
Plm
a
Plm
Z
Z
r1 Þv. ð~
r1 Þ dv1 P.sa
vm ð~
vm ð~
r1 Þv. ð~
r1 Þ dv1 P.sb
Z
Z
r2 Þvl ð~
r2 Þ dv2
vs ð~
vs ð~
r2 Þvl ð~
r2 Þ dv2
:
ð12Þ
Obviously, as the integrals above are nothing else than
the overlap integrals Sm. and Ssl , Eq. (12) may also be
obtained from the normalization N ¼ TrðPa þ Pb ÞS and
the idempotency relationships ðPa SÞ2 ¼ Pa S and
2
ðPb SÞ ¼ Pb S.
Now we insert identity (2) in each integral on the
right-hand-side (12) and introduce the ÔatomicÕ overlap
A
integrals Slm
calculated by using the weight function
wA ð~
rÞ of the given atom
Z
A
Slm ¼ wA ð~
rÞvl ð~
rÞvm ð~
rÞ dv:
ð13Þ
A
, Eq. (12) can
Defining matrices SA with the elements Slm
be rewritten as
NX
i
atoms X h
N¼
ðPa SA Þl. ðPa SB Þ.l þ ðPb SA Þl. ðPb SB Þ.l :
A;B¼1 l;.
ð14Þ
B
A
(S is defined analogously to S .) We may also introduce
the matrices of total density D and spin density Ps defined as
D ¼ Pa þ Pb ;
Ps ¼ Pa Pb ;
ð15Þ
and get (14) in the form
NX
NX
i
atoms
atoms X h
2
QA ¼
ðDSA Þl. ðDSB Þ.l þ ðPs SA Þl. ðPs SB Þ.l ;
A;B¼1 l;.
A¼1
PNatoms
ð16Þ
where the equality A¼1 QA ¼ N has also been utilized.
Now, we consider the contributions of a given atom A
to the two sides and define the bond order between atoms
A and B ðA 6¼ BÞ as
i
Xh
BAB ¼
ðDSA Þl. ðDSB Þ.l þ ðPs SA Þl. ðPs SB Þ.l
l;.
2
i
Xh
ðPa SA Þl. ðPa SB Þ.l þ ðPb SA Þl. ðPb SB Þ.l ;
l;.
ð17Þ
ð11Þ
Substituting (11) into (9), carrying out the summations
over the spins and performing trivial rearrangements, we
obtain in terms of spatial orbitals
the total valence of atom A as
X
ðDSA Þl. ðDSA Þ.l ;
VA ¼ 2QA ð18Þ
l;.
2
The existence of so-called Ônon-nuclear attractorsÕ, i.e., maxima of
electron density which cannot be put into correspondence with any
nucleus in the molecule, raises conceptual questions and makes
inapplicable the AIM theory for systems like Li2 or acetylene in
several basis sets.
and the free valence of atom A as the difference
X
BAB :
FA ¼ VA B
B6¼A
ð19Þ
I. Mayer, P. Salvador / Chemical Physics Letters 383 (2004) 368–375
It can be shown with a somewhat involved analysis that
in the single determinant (UHF) case the free valence
can be expressed via the spin density
X
FA ¼
ðPs SA Þl. ðPs SA Þ.l ðUHFÞ;
ð20Þ
l;.
and vanishes for the closed shell RHF case for which the
spin density is zero. In the RHF case – similarly to the
Hilbert-space and AIM counterparts [6,23] – the total
valence of an atom equals the sum of all its bond orders
X
VA ¼
BAB ðRHFÞ:
ð21Þ
B
B6¼A
The AIM indices [23] again represent special cases of the
above definitions. They can be written exactly in the
form of the present equations, if the integrals (13) with
the weight functions wA ð~
rÞ are replaced by integrals over
the atomic domains XA .
As one of us proposed nearly two decades ago [10] for
the case of Hilbert-space analysis, the same Eqs. (17)–(19)
should be used in order to define bond order, total and
free valence indices also if electron correlation is taken
into account. This procedure is equivalent to defining an
auxiliary ÔHartree–Fock likeÕ second-order exchange
0
0
0
0
density matrix .exch
2 ðx1 ; x2 ; x1 ; x2 Þ ¼ .1 ðx1 ; x2 Þ.1 ðx2 ; x1 Þ
expressed in terms of the first-order density matrix in
analogy to Eq. (8), and to use it for calculating the indices. As the first-order density matrix computed from a
correlated wave function does not have the idempotency
property, in that case the free valence cannot be expressed
any more via the spin density and it does not exactly
vanish even for closed shell molecules. The same scheme
has been rediscovered most recently [25] on the basis of a
detailed analysis of the second-order density matrix.
We wish not enter here into a detailed discussion of
the possible definitions of the bond order and valence
indices in the correlated case, mention only briefly that
the alternative approach were to use, instead of the
above exchange density matrix, the whole difference
between the actual second-order density matrix
.2 ðx1 ; x2 ; x01 ; x02 Þ and the first (ÔdirectÕ) term .1 ðx1 ; x01 Þ.1
ðx2 ; x02 Þ on the right-hand-side of (8). The index calculated in this manner reflects not only the ÔFermi holeÕ
connected with the antisymmetry but also the ÔCoulomb
holeÕ which is due to the electron correlation. Equivalent
to this is the use the correlation of the fluctuations of
atomic populations [26]. 3 (The two types of definition
coincide for single determinant wave functions.)
No doubt, such a parameter may give useful information about the physical behaviour of the correlated
system. It may not, however, put into correspondence
with the chemical notion of bond order (multiplicity). It
3
Apparently the authors of [25] have misunderstood the message of
our paper [10] when attributing it to this group. That paper was
devoted to demonstrate the inadequacy of this approach.
371
is sufficient to recall that for WeinbaumÕs classical wave
function [27] of the H2 molecule (full CI in minimal
basis set of Slater orbitals with optimized exponents) the
Hilbert-space index computed with inclusion of Coulomb hole was only 0.39, while the analogue of (17) gave
0.95 [10]. WeinbaumÕs wave function gives a very fair
description of the H2 molecule, accounting for some
84% of its total binding energy. In our opinion, one
must not call a parameter Ôbond orderÕ, if it gives such a
small value for such a pretty good wave function. This is
the case, even if other correlated calculations of H2 [28]
give results in which the deviation from unity is not so
dramatic. The same conclusion can be drawn from the
results presented in [25] in the AIM framework: bond
indices exceeding 5 have been obtained for N2 and F2 by
using the definition including the Coulomb hole, while
the Ôexchange onlyÕ definition produced chemically reasonable numbers. We may conclude, therefore, that the
use of definitions of type (17)–(19) are much more
ÔchemicalÕ. They are also much cheaper, as require only
the first-order density matrix and not the second-order
one. Thus the calculation of bond orders and valences in
the correlated case does not require more computational
effort than in the SCF one.
3. Test calculations
We have written a small program, using BeckeÕs
method of multicenter numerical integration [29] which
combines ChebyshevÕs integration of the radial function
with LebedevÕs quadrature [30] of the angular part and
introduces a weight factor of every center in every point
of space. The routine for Lebedev quadrature has been
downloaded from [31]. The program uses as sole input
the Ôformatted checkpoint fileÕ generated in a Gaussian
run; for interfacing and performing Hilbert-space analysis parts of the program [32] have been adapted. We
have made available our program for downloading [33].
It would exceed the scope of the present study to
perform any systematic search for the most adequate
weight function wA ð~
rÞ. The results of the test calculations presented here have been obtained in the simplest
possible way, by using BeckeÕs weight function originally
proposed for the purposes of doing effective numerical
integration. Following his recipe, we used the Slater–
Bragg effective atomic radii [34] and accepted his suggestion to increase the radius of hydrogen to the value
. However, for fluorine we use the value 0.9 A
,
0.35 A
representing the average of the covalent and ionic radii.
For sake of completeness, we give the algorithm of
computing BeckeÕs wA ð~
rÞ in Appendix A. Except a few
cases, we applied the stiffness parameter k ¼ 3, as suggested by Becke. The program itself can be used with
any weight functions; we have also performed a few
372
I. Mayer, P. Salvador / Chemical Physics Letters 383 (2004) 368–375
Table 1
Atomic and overlap populations, valences and bond orders calculated by the Ôfuzzy atomsÕ formalism and BeckeÕs weight function
H2
N2
HF
CO
H2 O
NH3
B2 H6
SO
SO2
SO3
CH4
C2 H 6
C2 H 4
C2 H 2
C6 H 6
C60
a
H
N
H
F
C
O
O
H
N
H
B
Hbr
Ht
S
O
S
O
S
O
C
H
C
H
C
H
C
H
C
H
C
Gross atomic
populations
Net atomic
populations
Valences
A
A
B
A
B
0.875
6.513
0.456
9.298
5.603
7.463
7.789
0.722
6.682
0.763
4.852
0.622
0.786
15.683
7.410
15.230
7.444
14.969
7.419
5.529
0.811
5.412
0.808
5.437
0.808
5.497
0.723
5.311
0.778
0.873
6.514
0.459
9.294
5.609
7.456
7.781
0.723
6.672
0.762
4.844
0.621
0.787
15.672
7.435
15.219
7.463
14.951
7.437
5.525
0.812
5.409
0.809
5.432
0.809
5.492
0.725
5.302
0.778
1.000
3.103
0.899
0.899
2.766
2.766
2.337
1.236
3.218
1.186
3.719
0.996
1.022
2.622
2.622
4.979
2.694
7.086
2.633
3.939
1.115
4.151
1.103
4.056
1.094
3.934
1.072
4.252
1.078
1.000
3.109
0.914
0.914
2.779
2.779
2.368
1.254
3.256
1.200
3.718
0.996
1.023
2.614
2.614
4.959
2.690
7.045
2.634
3.937
1.116
4.151
1.105
4.054
1.101
3.935
1.082
4.255
1.086
1.000
7.000
0.579
9.421
6.070
7.930
8.169
0.915
7.190
0.937
5.405
0.764
0.916
16.139
7.863
16.167
7.915
16.353
7.882
6.133
0.967
6.131
0.956
6.124
0.938
6.125
0.875
6.082
0.918
B
1.000
7.000
0.583
9.418
6.077
7.924
8.163
0.918
7.186
0.938
5.400
0.764
0.917
16.119
7.882
16.143
7.927
16.316
7.894
6.130
0.967
6.129
0.957
6.120
0.940
6.122
0.878
6.080
0.920
6.00a
5.24a
4.35a
Overlap
populations
Bond orders
A
B
A
B
H–H
N–N
H–F
0.125
0.487
0.124
0.127
0.486
0.124
1.000
3.103
0.899
1.000
3.109
0.914
C–O
0.467
0.468
2.766
2.779
O–H
0.190
0.191
1.169
1.184
N–H
0.162
0.171
1.073
1.085
B–Hbr
B–Ht
B–B
S–O
0.071
0.129
0.154
0.454
0.071
0.129
0.155
0.447
0.460
0.943
0.846
2.622
0.460
0.943
0.845
2.614
S–O
0.470
0.462
2.490
2.480
S–O
0.461
0.455
2.362
2.348
C–H
0.151
0.152
0.985
0.984
C–C
C–H
C–C
C–H
C–C
C–H
C–C
C–H
C–C(6,6)
C–C(5,6)
0.286
0.140
0.389
0.147
0.477
0.151
0.312
0.136
0.285
0.141
0.388
0.148
0.478
0.152
0.312
0.137
1.130
0.951
1.976
0.963
2.865
0.986
1.440
0.937
1.128
0.951
1.960
0.967
2.856
0.991
1.436
0.940
0.28a
0.24a
1.42a
1.13a
Basis sets: 6-31G** (A) and 6-311++G** (B).
Single point, 6-31G basis, smaller (20 by 50) integration grid. (No symmetry is utilized.)
calculations by using two other ones, as will be briefly
discussed below.
BeckeÕs Chebyshev–Lebedev integration scheme is
simple and indeed very effective. A rather modest
number of 3300 grid points (30 radial by 110 angular)
per atom provides the necessary accuracy. Thus the
CPU-time requirement of the scheme is also modest: a
complete Hilbert space and Ôfuzzy atomsÕ analysis of
benzene without using any symmetry (i.e., performing
the calculations independently for all the 12 atoms and
66 atomic pairs) required less than 1 min on a laptop
when 6-31G** basis (120 basis orbitals) has been used
and less than 1.5 min for the 6-311++G** basis (174
basis functions). 4
4
In this calculation, the SCF program reduced the number of
independent functions to 173 owing to the near linear dependence of
the diffuse functions. As a consequence, the Mulliken populations
became badly non-symmetric, despite the fact that the full D6h
symmetry was used. The ÔfuzzyÕ calculations, however, gave symmetric
results within the limits of the numerical ÔnoiseÕ.
Table 1 present the results of SCF calculations for a
number of molecules by using two basis sets. Inspection
of the results indicates that they are in agreement with
the classical chemical notions, and are practically independent of the basis set. The unexpectedly small basis
sensitivity of the method is further illustrated in Table 2
presenting data for nine different basis sets. In all calculations the geometries were fully optimized for the
given basis set.
In the Hilbert space analysis using balanced basis sets
of pronounced atomic character, e.g., 6-31G**, one
usually obtains bond orders and valences somewhat
lower than the classical integer values. In the case of
Ôfuzzy atomÕ analysis, the valences are usually slightly
higher than the nominal values. This deviation is reduced, if one uses a left soft cutoff function – e.g., turns
to the value k ¼ 4 in BeckeÕs function. The effect is,
however, minor and does not influence the chemical
picture one obtains of a molecule.
Inspecting the results for SO, SO2 and SO3 one
may see that they correspond to the classical notion of
I. Mayer, P. Salvador / Chemical Physics Letters 383 (2004) 368–375
373
Table 2
Basis set dependence of net atomic populations, overlap populations and bond orders calculated by the Ôfuzzy atomsÕ formalism and BeckeÕs weigh
function
Net atomic populations
Overlap populations
Bond orders
C
H
C–C
C–H
C–C
C–H
Ethane
STO-3G
6-31G
6-31G(d,p)
6-311G(d,p)
6-311++G(d,p)
cc-pVDZ
cc-pVTZa
aug-cc-pVDZ
6-311++G(3df,pd)a
5.439
5.433
5.412
5.409
5.409
5.409
5.405
5.410
5.405
0.809
0.810
0.808
0.809
0.809
0.809
0.808
0.808
0.808
0.285
0.282
0.286
0.285
0.285
0.284
0.287
0.285
0.286
0.136
0.136
0.140
0.141
0.141
0.141
0.141
0.141
0.141
1.138
1.128
1.130
1.128
1.128
1.126
1.132
1.130
1.131
0.955
0.951
0.951
0.951
0.951
0.951
0.951
0.951
0.951
Ethylene
STO-3G
6-31G
6-31G(d,p)
6-311G(d,p)
6-311++G(d,p)
cc-pVDZ
cc-pVTZa
aug-cc-pVDZ
6-311++G(3df,pd)a
5.460
5.456
5.437
5.432
5.432
5.434
5.425
5.434
5.427
0.797
0.790
0.789
0.790
0.789
0.789
0.789
0.788
0.789
0.376
0.380
0.389
0.389
0.388
0.388
0.393
0.390
0.393
0.140
0.143
0.147
0.147
0.148
0.147
0.148
0.148
0.148
2.024
1.979
1.976
1.966
1.960
1.966
1.964
1.960
1.965
0.955
0.961
0.963
0.965
0.967
0.965
0.966
0.966
0.966
Acetylene
STO-3G
6-31G
6-31G(d,p)
6-311G(d,p)
6-311++G(d,p)
cc-pVDZ
cc-pVTZa
aug-cc-pVDZ
6-311++G(3df,pd)a
5.515
5.525
5.497
5.492
5.492
5.496
5.487
5.495
5.489
0.749
0.723
0.723
0.726
0.725
0.723
0.724
0.723
0.723
0.448
0.456
0.477
0.478
0.478
0.478
0.482
0.478
0.482
0.144
0.148
0.151
0.152
0.152
0.151
0.152
0.152
0.153
2.894
2.863
2.865
2.859
2.856
2.862
2.857
2.857
2.859
0.979
0.985
0.986
0.990
0.991
0.987
0.990
0.989
0.989
O
H
O–H
O–H
7.816
7.852
7.789
7.783
7.781
7.779
7.768
7.767
7.770
0.761
0.713
0.722
0.725
0.723
0.727
0.725
0.726
0.723
0.163
0.179
0.190
0.189
0.191
0.190
0.194
0.193
0.194
1.091
1.137
1.169
1.175
1.184
1.170
1.188
1.193
1.191
Water
STO-3G
6-31G
6-31G(d,p)
6-311G(d,p)
6-311++G(d,p)
cc-pVDZ
cc-pVTZa
aug-cc-pVDZ
6-311++G(3df,pd)a
a
Using 10 f-orbitals.
divalent, tetravalent and hexavalent sulfur, the latter
two being typical hypervalent atoms. (The 6-31G**
Hilbert-space valences of sulfur are 1.81, 3.48 and
5.18, respectively, in these three molecules.) The
atomic populations, however, indicate that the ratio of
the atomic radii of the sulfur and oxygen – and perhaps of some other atoms – may need some adjustment, similar to that performed for fluorine. That
work is underway.
Fig. 1 displays some results for the ethylene molecule
dissociating into two triplet methylenes, as calculated
with a (4,6) CAS wave function by using 6-31G** basis
set. One may see that the C–C bond order which is
nearly two at the equilibrium distance is gradually decreases and tends to zero at the large distances – as it
should. Simultaneously with this, there appears a free
valence on the carbon, tending to a limit close to two at
the large distances, in agreement with the fact that there
are two unpaired electrons in the triplet methylene. The
sum of the C–C bond order and of the carbon
free valence is almost constant, thus the carbon atom
remains practically four-valent during the whole
374
I. Mayer, P. Salvador / Chemical Physics Letters 383 (2004) 368–375
determined from the 6-31G** reference density to a
molecular calculation performed by using diffuse functions, then completely wrong results could be obtained.
Obviously, one has always use the ÔpromoleculeÕ calculated with the same basis set as used for the molecule.
We are not sure that this would be worth of the necessary big effort.
4.5
4
V(C)
3.5
3
2.5
2
B(CC)
F(C)
1.5
4. Summary
B(CH)
1
0.5
0
1
1.5
2
2.5
3
3.5
4
4.5
5
C–C distance [Å]
Fig. 1. C–C and C–C bond orders, total and free carbon valences for
the dissociation of the ethylene molecule into two triplet methylenes,
treated at the (4,6) CAS level of theory by using 6-31G** basis set.
dissociation. 5 (The C–H bond order stays nearly constant at a value close to one.)
In a few cases we modified BeckesÕs function on the
basis of investigating the electron density along the
straight line connecting the atoms which are chemically bonded. We have used the position of the extremum of the density (usually a minimum, but
maximum in the cases for which there is a non-nuclear attractor) to determine the ratio of atomic radii
entering BeckeÕs formula – instead of using the fixed
radii. With this scheme we have got perfect results for
hydrocarbons and several other systems. However,
investigation of boron compounds BH, BH3 , B2 H6 ,
which are pretty well described by using BeckeÕs wA ð~
rÞ,
led to very bad results for all parameters: independently of the basis set used, practically all the valence
electrons are attributed to the hydrogen(s). It appears
that a similar effect is observed at the AIM level [35],
so it is to be attributed to some peculiarities of the
electron density of the boron compounds. Nonetheless, this observation prevent us to recommend such a
scheme for general use.
We have also made a few calculations by using
HirshfeldÕs original recipe. The reference densities were
obtained from 6-31G** ROHF calculations of the free
atoms, by performing an angular averaging. We could
conclude that if the same basis is used for molecules,
then the results are quite similar to those which can be
obtained by a softened version of BeckeÕs function
(k ¼ 2). However, if one tried to use the weight factors
5
Note, that the ground state of methylene is the triplet; contrary to
the CAS scheme, the RHF method is only able to describe dissociation
of ethylene into two singlet methylenes and results in divalent carbons
with no free valences.
We have adapted the calculation of ab initio quantum chemical bond order (multiplicity) indices, total and
free atomic valences to the case of Ôfuzzy atomsÕ. A
proper definition of the overlap density applicable in
that framework is also proposed. The scheme is applicable on both SCF and correlated levels of theory. The
results of test calculations are in agreement with the
classical chemical notions, exhibit very little basis sensitivity and depend not too much on the selection of the
weight function defining the actual division of the space
into ÔfuzzyÕ atomic regions.
Acknowledgements
This work has been partly supported by the Hungarian Scientific Research Fund, Grant OTKA T34812,
and by the Improving the Human Potential Program,
Access to Research Infrastructures, under Contract No.
HPRI-1999-CT-00071 Access to CESCA and CEPBA
Large Scale Facilities established between The European
Community and CESCA/CEPBA, making possible a
short stay of I.M. in Girona. Furthermore, this work
has been partially funded through the Spanish DGES
Project No. BQU2002-04112-C02-02.
Appendix A
BeckeÕs wA ð~
rÞ is an algebraic function which strictly
satisfies requirement (2). In addition, it gives exactly
wA ¼ 1 on the ÔownÕ nucleus A – all the other wB -s are
zero there. It can be formulated in terms of the following
definitions:
PA ð~
rÞ
;
wA ð~
rÞ ¼ P
P
rÞ
B B ð~
Y PA ð~
rÞ ¼
0:5 1 mðkÞ ðrA ; rB Þ ;
B
B6¼A
2 mðlÞ ðrA ; rB Þ ¼ mðl1Þ ðrA ; rB Þ 1:5 0:5 mðl1Þ ðrA ; rB Þ ;
mð0Þ ðrA ; rB Þ ¼ lðrA ; rB Þ þ aAB 1 ½lðrA ; rB Þ2 ;
I. Mayer, P. Salvador / Chemical Physics Letters 383 (2004) 368–375
lðrA ; rB Þ ¼ ðrA rB Þ=RAB ;
rA ¼ j~
r ~
RA j;
rB ¼ j~
r: ~
RB j;
aAB ¼ 0:25ð1 v2AB Þ=vAB ;
RAB ¼ j~
RA : ~
RB j;
but jaAB j 6 0:5;
vAB ¼ rA0 =rB0 :
Here, k is a fixed parameter of the procedure (number of
iterations) determining the stiffness of the cutoff (usually
k ¼ 3), and the rA0 -s are the fixed atomic radii. (~
RA are the
radius-vectors of the nuclei.)
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