5244
J . Phys. Chem. 1994, 98, 5244-5248
Covalent Bond Orders and Atomic Valence Indices in the Topological Theory of Atoms in Molecules Jhnos G. Angyhn' and Michel Loos Laboratoire de Chimie ThPorique, UniversitE de Nancy I , B.P. 239, 54506 Vanoeuvre-16s-Nancy, France Istvhn Mayer Central Research Institute for Chemistry, Hungarian Academy of Sciences, H-1525 Budapest, P.O. Box 17, Hungary Received: January 5, 1994; In Final Form: March 16, 1994"
Quantum chemical counterparts are defined for chemical concepts of bond order (multiplicity) and actual valence of an atom in a molecule by performing a n atomic partitioning of the exchange part of the second-order density matrix within the framework of Bader's topological theory of atoms in molecules. Unlike previous definitions, the present results permit the formulation of the concepts of bond order and valence in a basis set independent manner and extend, therefore, their applicability to the H a r t r e e F o c k limit or even exact wave functions. The numerical results obtained indicate that, in the framework of Bader's theory, bond ionicities are much greater than usually supposed on the basis of conventional population analyses, and this is reflected by a significant reduction of covalent bond order and valence values.
the electronic density in the physical space. Not only are the atomic properties described in a coherent framework, but the One of the effects having great importance in the formation chemical bonds can also be characterized by various features of of covalent chemical bonds is the nonclassical exchange interaction the electronic charge density.17J8 In particular, an empirical between the atoms constituting the molecule. On the basis of the relationship has been established between the bond multiplicity corresponding diatomic exchange term obtained in an energy and the value of the electron density at the saddle point along the partitioning scheme,' a bond order definition has been proposed bond path." for ab initio S C F wave functions.2 Further studies revealed a Cioslowski and Mixon were the first who proposed a firstclose relationship of this parameter with the atomic partition of principles definition of bond orders in the framework of Bader's the normalization of the exchange part of the second-order density theory, using the atomic population analysis of individual occupied matrix.3-5 Also, this quantity has proved to be identical with an orbitals.1g Unfortunately their definition is not invariant in a earlier generalization of the Wiberg index6 to nonorthogonal basis strict sense with respect to unitary transformations of the sets, put forward originally in the framework of the semiempirical molecular (or natural) orbitals, being based on the use of a special EHT t h e ~ r y . ~Generalization ,~ to correlated wave functions has set of localized orbitals. The latter are selected to maximize the also been obtained by introducing an auxiliary quantity, the formal sum of squared atomic overlap integrals.20-21Accordingly, these "exchange component" of the second-order density m a t r i ~ . ~ . ~ , ~authors first perform a localized orbital transformation and then Along with bond orders, the proper definition of valence indices use a formula which has a meaning just for the localized orbitals has also been given; these quantities represent ab initio generobtained, but is not invariant with respect to unitary transformaalizations of the corresponding parameters first introduced in the tions. Of course, one gets the same localized orbitals and, framework of semiempirical CNDO theory.lOJ1 therefore, the same bond orders, starting from any sets of orbitals These quantities are closely related to the Mulliken-type which leave invariant the many-electron wave function, so no partitioning of the overlap density;2 for that reason the term actual arbitrariness takes place. In spite of such practical "Mulliken-Mayer analysis" (MMA) is sometimes adopted for invariance of the algorithm, we find this situation unsatisfactory them.12 The concept of Mulliken populations, and thus MMA, from a conceptual point of view. In our opinion, the definition corresponds to dividing the electronic charge between different of a quantity of physical or chemical significance should possess basis orbitals, Le. a partitioning in the Hilbert-space spanned by an explicit unitary invariance, being expressed, e.g., through the the LCAO basis functions. Although such a scheme works (unitary-invariant) density matrix elements. This holds true, even reasonably well for truly atomic basis functions like minimal if the definition in question leads to reasonable numerical results.22 basis sets or the "well-balanced" 6-31G** basis, it often fails for The purpose of the present paper is to show that, with the larger basis sets.I3J4 For example, it is not applicable to nonsecond-order density matrix as a starting point, a general definition atom-centered bases, and nonphysical results can be obtained if of bond orders and atomic valence indices can be worked out the basis set contains diffuse functions which are spread over the within the framework of Bader's theory. The resulting indices whole molecule and, therefore, cannot really be attributed to any are invariant with respect to the choice of orbitals, making thus particular atom in the molecule. All these shortcomings could unnecessary the preliminary calculation of any localized orbitals. easily be avoided by performing a partitioning of the electronic Of course, the unique and invariant answer furnished by our charge in the physical space, which can be done in a basis set formula can be evaluated also on a particular set of localized independent manner. orbitals, and it may be very instructive as an interpretative tool. The theory of atoms in molecules (AIM), elaborated by Bader It is important to note that unlike other indices, based on Hilbertand his co-workers,lsJ6 is based on such a direct partitioning of space partitioning, the indices defined here converge smoothly to their appropriate limiting value with the extension of the basis set. e Abstract published in Advance ACS Abstracts, April 15, 1994.
Introduction
0022-3654/94/2098-5244%04.50/0 0 1994 American Chemical Society
The Journal of Physical Chemistry, Vol. 98, No. 20, 1994 5245
Topological Theory of Atoms in Molecules
The diatomic contributions give rise to the bond order, BAB:
Theory The bond order can be related to the normalization of the exchange part of the second-order density m a t r i ~ : ~ . ~
JV =
s s dx,
dx, pl(x2;x1) P I ( X I ; X ~ )
(1)
where pI(xI;x2)is the first-order density matrix. For singledeterminant wave functions the normalization is equal to N , the number ofelectrons: N = N . In thecorrelated case, the function pI(x2;xI)p ~ ( x ~ ; x used z ) in (1) is simply the auxiliary quantity mentioned in the introduction; in these circumstances no direct physical meaning is attributed to N . The first-order density matrix can be written in terms of molecular (or natural) orbitals as
while the spin-free part of the monoatomic contribution is equal to the atomic valence index, VA: vA
= 2qA - c f l i n j ( $ i b j ) A
(4jbi)A
(10)
ij
The sum of bond orders connecting atom A with its partners is equal to the (covalent) valence index, VA, in a closed-shell (RHF) S C F case: V A=
x
BAB
(closed shells)
B#A
nf P ( s ~ P) * ( s ~ ) I4i(r1) 4;(r,) ( 2 ) where np is the occupation number of the spin-orbital J.i(x) = qJiW
4s).
After integratingover spinvariables and introducing a partition of spatial integrals over atomic basins, QA,
s
dx =
Jn, A
dr
s
ds
In the case of open-shell (UHF) S C F wave functions, the equality (11) does not hold, and the remaining term can be interpreted as the free valence index, FA: FA = Z n ; n ; ( 4 j b j ) A ( $ # [ ) A
(UHF)
(12)
ij
For more general, correlated wave fundtions, the free valence index FAcan be defined as the difference of the valence index VA and the sum of the bond orders:
(3)
one obtains
In Bader’s theory the boundaries of atomic basins, QA, are determined by the zero-flux condition, Vp(r).n(r) = 0, applied to the electronic density p(r).15316 The elements of the atomic overlap matrix (AOM) of the orbitals, qJi(r), appearing in (4), are defined by integrating over such an atomic domain:23
Let us stress here that the atomic overlap matrix (AOM) is defined by the integrals of the molecular orbitals in the atomic domain of a given atom and should be distinguished from the ordinary overlap matrix of the atomic basis orbitals. The electronic charge, qA, of atom A is
and the number of electrons, N , is equal to the sum of atomic charges, N = C A q A . Introducing the auxiliary quantities ni = np + nf and n: = n; - nf, one can write the term depending on the occupation numbers as
In the case of a one-determinant wave function, the normalization in (1) is N = Z A q A , so one can separate mono- and diatomic contributions, leading to the following relationship for a fixed atom A:
F~ = V, - C
B A B B#A
(general case)
(13)
We note that integrals of the exchange part of the secondorder density matrix over atomic domains have already been used to characterize the Fermi-correlation and the degree of localization (number of electron pairs) contained by a spatial domain.24Similar integrals over pairs of domains measure the extent of delocalization between pairs of atoms, a quantity which can be regarded as related to the bond order, although it was not so stated in the original arti~le.2~ Comparison with Cioslowski’s Definition. For the definition of bond orders, Cioslowski and Mixon19 have proposed to determine first a set of population-localized orbitals, which maximize thesum of the squareof atomic overlapmatrix elements and calculate then the bond orders by the formula
This expression is related to ours (9) in the following way:
In effect, Cioslowski’s formula (14) is not invariant with respect to unitary transformations of the orbitals, just because the contributions of the nondiagonal atomic overlap matrix elements are neglected. On the basis of population-localized orbitals, these nondiagonal atomic overlap matrix elements become small, and for strictly localizable systems they are even negligible. Although the interpretative value of Cioslowski’s bond order definition has proved to be quite good:5-29 its determination is complicated by the calculation of the specific set of localized spin-orbitals. This represents particular inconveniences in the case of inherently delocalized systems, like benzenoid hydrocarbons, where the bond order can be obtained only after the different Kekult structures have been averaged.19 This fact stresses again that the definition in (14) cannot be correct from a rigorous theoretical point of view. Furthermore, in Cioslowski’s scheme the definition of atomic valence indices is not straightforward either: for example, due to the neglected off-diagonal AOM contributions, the relationship in (1 1) between the bond orders and atomic valence indices does not hold.
5246 The Journal of Physical Chemistry, Vol. 98, No. 20, 1994
AngyIn et al.
TABLE 1: Bond Orders from Topological (AIM) and Hilbert-Space (MMA) Partitioning Calculated from SCF Wave Functions Using 6-316” and 6-31++C** Basis Sets at 6-31C* Optimized Geometries. Cioslowski’s Results19 Are in Parentheses BAB(AIM)
molecule
bond A-B HF H-F LiF Li-F LiH Li-H N2 N-N
co
C-O HCN H-C N-C H-N HNC H-N N-C H-C C2H6 C-H
c-c
C2H4 C-H C-C C2H2 C-H
c-c
C6H6 C-C (ortho) C-C (meta) C-C (para) C-H
R A ~
6-31G*//6-31G*
BAB(MMA)
6-31++Gt*//6-31G*
6-31G*//6-31G*
6-3 1++G**//6-3 1G*
0.9110
0.480 (0.506)
0.422 (0.433)
0.715
0.824
1.5550
0.170 (0.167)
0.164 (0.160)
0.642
0.563
1.6360
0.201 (0.207)
0.204 (0.210)
0.969
0.710
1.0784
3.037 (3.038)
3.041 (3.045)
2.796
3.706
1.1138
1.509 (1.509)
1.524 (1.524)
2.307
2.325
1.0587 1.1325
0.900 (0.922) 2.242 (2.241) 0.074 (0.073)
0.921 (0.922) 2.233 (2.232) 0.087 (0.089)
0.863 2.934 0.012
0.898 2.862 0.041
0.9852 1.1543
0.656 (0.653) 1.679 (1.692) 0.020 (0.021)
0.638 (0.641) 1.700 (1.719) 0.020 (0.023)
0.769 2.398 0.029
0.794 2.61 1 0.073
1.0856 1.5275
0.966 (0.984) 1.013 (1.047)
0.966 (0.982) 0.987 (1.018)
0.962 0.975
0.980 0.994
1.0760 1.3169
0.972 (1.005) 1.984 (1.958)
0.984 (1.01 3) 1.881 (1.918)
0.956 1.983
0.955 2.180
1.0570 1.1855
0.953 (0.990) 2.885 (2.920)
0.977 (1.015) 2.860 (2.897)
0.872 3.186
0.853 3.365
1.3862 2.4009 2.1724 1.0756
1.399 (1.435) 0.073 (0.195) 0.101 (0.117) 0.963 (0.987)
1.386 (1.430) 0.072 (0.191) 0.098 (0.1 18) 0.977 (0.999)
1.454 -0.009 0.094 0.942
2.084 -0.219 0.189 0.647
Quiterecently Fulton has proposeda “sharing i n d e ~ ” , 3which ~?~~ can be written in terms of spin-orbitals $ i ( x ) = $v(r)u(s) as
or pv
ij
It is easy to see that the sharing index is exactly half of our bond order in the case of SCF wave functions, while for noninteger occupation numbers (natural orbitals) the scaling between the two indices is more complicated. The motivation for Fulton’s work was quite similar to ours: a chemically meaningful index should be an invariant property of the many-electron wave function, which can be interpreted, if one wishes, in terms of u or T , localized or delocalized orbitals, and so on, but has its own significance without reference to any particular set of orbitals. Comparison with the MMA Definition. The previous formulas are valid for arbitrary wave functions, including the case when, e.g., numerical HartreeFockorbitals areused. In order toclarify the relationship between the MMA definitions and the present ones, the bond order and valence indices defined above should also be expressed by using an atomic orbital basis. The basis set expansion of the molecular (or natural) spinorbitals is
Au
+
where the total density matrix is P = Pa Pb and the spin density matrix is Ps = P a - Po. Similarly, for the atomic valence index,
In these formulas, ( x , , J x Vis) ~an atomic overlap integral of the basis orbitals over the atomic domain defined in full analogy to (5). The definitions of MMA can be recovered by replacing the atomic overlap matrices of the atomic orbitals by the corresponding “Mulliken-partition” according to the recipe A
(X,hv)A
* Tu s,v
(22)
where the role of the 7; function is to introduce a summation restriction: 7,” is 1 if xr is located on atom A, and 0 otherwise. In this manner one obtains the replacement scheme
The definition of the “density matrix” for electrons of spin u (u = a or 8) is
If we take into account that the atomic overlap integrals are non-negligible for those contributions which involvebasis functions on the given center, the analogy is quite clear. By inserting (23) into (19)-(21), one recovers easily the bond order and valence definitions of MMA:
which yields directly
or
The Journal of Physical Chemistry, Vol. 98, No. 20, 1994 5241
Topological Theory of Atoms in Molecules
TABLE 2 Atomic Charges (QA = ZA- 9A) and Valence Indices from Topological (AIM) and Hilbert-Spce (MMA) Partitioning Calculated from SCF Wave Functions Using 6-316" and 6-31++6** Basis Sets at 6-31G*-Optimized Geometries molecule atom A HF H F LiF F Li LiH
6-31G*//6-31GZ
6-31++G**//6-31G*
QA (AIM)
QA (MMA)
VA (AIM)
VA (MMA)
QA (AIM)
QA ( M M N
VA (AIM)
VA ( M M N
+0.724 -0.725
+0.517 -0.517
0.474 0.479
0.715 0.715
+0.764 -0.764
+0.417 -0.4 17
0.421 0.421
0.824 0.824
-0.946 +0.946
-0.660 +0.660
0.171 0.170
0.642 0.642
-0.951 +0.949
-0.735 +0.735
0.161 0.163
0.563 0.563
-0.909 +0.908
-0.177 +0.177
0.199 0.201
0.969 0.969
-0.908 +0.907
-0.536 +0.536
0.202 0.204
0.711 0.710
N
0.000
0.000
3.037
2.796
0.000
0.000
3.041
3.706
C
+1.403
H Li N2
co
0
-1.404
+0.268 -0.268
1.508 1.508
2.307 2.307
+1.399 -1.399
+0.258 -0.258
1.523 1.524
2.325 2.325
HCN H C N HNC
+0.243 +1.239 -1.481
+0.313 +0.066 -0.379
0.974 3.141 2.316
0.875 3.797 2.946
+0.197 +1.281 -1.478
+0.239 +0.130 -0.369
1.008 3.154 2.320
0.939 3.761 2.903
+0.579 +1.224 -1.803
+0.425 +O. 152 -0.588
0.677 1.699 2.334
0.867 3.405 2.683
f0.597 +1.281 -1.813
+0.358 +0.107 -0.465
0.659 1.721 2.338
0.798 3.168 2.427
+0.075 -0.025
-0.476 +0.159
3.899 1.016
3.821 0.930
+0.232 -0.078
-0.31 1 +0.104
4.017 1.123
3.904 0.949
-0.035 +0.018
-0.353 +0.176
3.985 1.084
3.864 0.937
+0.070 -0.035
-0.224 +0.112
3.986 1.113
4.123 0.955
-0.177 +0.177
-0.276 +0.276
3.899 1.016
4.082 0.901
-0.132 +0.132
-0.194 +O. 194
3.910 1.053
4.304 0.956
-0.014 +0.014
-0.200 +0.200
4.1 15 1.084
3.981 0.927
+0.038 -0.038
-0.141 +0.141
4.116 1.120
4.872 0.980
H C N C2H6 C H C2H6 C H C2H2 C H C6H6 C H and
The presence of basis functions of strongly non-atomic character destroys obviously the close correspondence between the two sets of definitions.
Examples and Discussion A series of simple molecules has been used to illustrate the utility of the new definitions of bond order and valence indices. All the geometries have been optimized at the S C F level of theory with the 6-31G* basis set, using the Gaussian '92 series of programs,32 and in order to investigate the basis set sensibility, the bond order analysis has been carried out with both 6-3 1G* and 6-31++G** bases. Atomic overlap matrices have been calculated by the EXTREME and PROAIM programs of Bader and co-workers.23 The MMA indices have been programmed in our Gaussian '92 version. The bond orders, valences, and atomic charges calculated for some simple molecules are shown in Tables 1 and 2. Let us note that evaluation of atomic charges, bond orders, and valences in the AIM framework requires numerical integrations. Owing to the complexity of the surfaces separating atomic basins, these integrations inevitably lead to small numerical uncertainties (inaccuracies). This is reflected by the results pertinent to the diatomic molecules in Table 2. In fact, for a diatomic the charges of the two atoms obviously should have equal absolute values. Similarly, in the framework of the closed-shell single-determinant (RHF) scheme, their valences should also be equal, according to (1 1). However, the deviations obtained usually are of the order of 10-3 or less (the largest deviation being 0.0046 for the valences of H and F in HF) and so essentially can be neglected.
As it can be seen from Tables 1 and 2, the multiplicity of apolar covalent chemical bonds is usually reflected quite well by both AIM and MMA schemes (see e.g., ethane, ethylene, acetylene, and nitrogen molecules), while the difference between AIM and MMA results increases with increasing ionic character of the bonds involved. Both the smaller atomic charges and the greater bond orders indicate that the MMA scheme overemphasizes thecovalent with respect to the ionic character of chemical bonds. This feature can immediately be related to the fact that the overlap population is halved in the Mulliken's population analysis inherent in the MMA scheme: half of the overlap electron population is attributed to the "cation" even if its distribution in the physical space is considerably shifted to the "anion". It is, however, worth pointing out in this context that this "halving" of the overlap population is the only mathematically consistent choice as far as partioning in the Hilbert-space is concerned.2 The effects of bond ionicity is illustrated by the highly ionic LiF and LiH molecules, which have an AIM bond order of about 0.2, while the corresponding MMA indices are 0.6 and 0.7, respectively. This difference in bond orders is in line with a big difference in the net atomic charges given by the two schemes. Bader's theory predicts an electron to be almost completely transferred from Li to F o r H , the charges being above 0.9 in both molecules. (This conclusion remains valid also if bond functions are added to the basis.) At the same time, significantly smaller Mulliken charges are obtained; they are scattered in a wide range and are strongly basis dependent. One can see in other cases as well that the basis set dependence of AIM bond orders is quite small, while the use of Mulliken partition sometimes leads to rather large variations. Considering the MMA result, however, one has to keep in mind that the data shown in Tables 1 and 2 may somewhat exaggerate the shortcomings of the MMA scheme, which has
5248 The Journal of Physical Chemistry, Vol. 98, No. 20, 1994 TABLE 3: Covalent AIM Bond Order and Atomic Valence in the Nz Molecule Calculated in the 6-316* Basis Set Using Different Wave Functions method
SCF
MP2
CID
CAS(4.4)
BNN VN
3.037 3.037
2.779 3.178
2.840 3.161
2.824 3.058
TABLE 4 Evolution of the AIM Bond Order and Valence during the Dissociation of the Hz Molecule, As Calculated from a CAS(2,2) Wave Function, Using the 6-31G** Basis Set R (4 0.7534 1.oooo 2.0000 3.0000 6.0000
BHH 0.958 0.910 0.326 0.029 0.000
VH 1.006 1.010 1.032 1.010 1.ooo
E (a4 -1.149 6368 -1.126 4584 -1.014 1616 -0.997 4538 -0.996 4658
proved to be really useful in many practical applications, see e.g. refs 33-40. In fact, even the 6-31G* basis is somewhat "unbalanced" in the sense that it contains polarization functions on the heavy atoms but not on the hydrogens. In this sense the basis 6-31 ++G**, containing diffuse functions, which are lacking truly atomic character, is even worse, and this can lead to serious consequences. For instance, the AIM bond order in benzene is 1.4, which coincides with the 6-31G* MMA index, but the latter increases to the unphysical value of 2.08 when the 6-31++G** basis set is used. A similar effect can be observed for the N2 molecule. The small deviations with respect to Cioslowski's bond orders can be explained by the neglect of nondiagonal contributions in his scheme. Obviously, the importance of these terms is small in well-localizable systems, and it increases with the delocalized character of the electronic structure. It should be emphasized once again that for systems like benzene Cioslowski's bond orders can becalculatedonly by artificial averaging of thevaluesobtained for different Kekult structures. Finally, it should be emphasized that the definitions discussed above are applicable also to correlated wave functions. The preliminary results obtained for the N2 molecule using different correlated wave functions (Table 3) and for the dissociation of the HZmolecule at the CAS level (Table 4) completely correspond to the expectations. At the equilibrium interatomic distance, the difference between the total valence and the bond order gives a measure of how much the system being studied deviates from the idealized closed-shell R H F case with doubly filled orbitals. When the internuclear distance increases, one can follow the process of bond dissociation. In the H2 molecule, the total valence of hydrogen atoms is practically equal to 1 at every distance. This valence is used almost completely for bond formation at the equilibrium interatomic distance, but a greater and greater part of it becomes free as the internuclear separation increases and the bond order decreases. At very large distances we have two independent monovalent hydrogen atoms, which do not form any bonds, and their valence is therefore completely free (BAB= 0, VA = FA = 1).
Conclusions We consider extremely important the fact that one can give quantum mechanical counterparts for such genuine chemical concepts as bond order (multiplicity) and actual valence of an atom in a molecule. The results described above permit the
Angydn et al. formulation of these concepts in a basis set independent manner and extend, therefore, their applicability to the Hartree-Fock limit or even exact wave functions. The numerical results obtained indicate that in the framework of Bader's AIM theory the bond ionicities can be much greater than the values usually assumed, which is accompanied by a significant reduction of covalent bond orders and atomic valences.
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