6156
J. Phys. Chem. 1996, 100, 6156-6160
Electronegativity Equalization in Polyyne Carbon Chains Jerzy Cioslowski* and Martin Martinov Department of Chemistry and Supercomputer Computations Research Institute, Florida State UniVersity, Tallahassee, Florida 32306-3006 ReceiVed: September 20, 1995X
Electronegativity equalization in polyyne carbon chains with the composition Xs(sCtCs)NsY, where X, Y ) H, F, or Na, and 1 e N e 5, has been studied at the MP2/6-31G** level of theory. In contrast to charges of individual atoms, charges of the XsCtCs, sCtCs, and sCtCsY chain fragments are found to be highly additive. The observed additivity of fragment charges stems from the underlying additivity of the corresponding electronegativity vectors and softness matrices. Although the latter quantities are poorly transferable among polyyne carbon chains, they satisfy additive combination rules to a high degree of accuracy. Approximate models that involve either one-electron effective Hamiltonians or screened electrostatic interactions fail to reproduce the rigorously computed atomic softness matrices, indicating the complex nature of phenomena that control the electron flow in molecules.
Introduction Electron distributions in molecules are altered upon substitution of one atom (or functional group) by another. These substituent-induced changes can be quantitatively described with the language of density functional theory.1 Let E(Q) be the total energy of an electrically neutral system expressed as a function of the fragment charge vector Q ≡ (Q1, ..., Qn), each fragment being a single atom, a functional group, or even a molecule in a supramolecular assembly. Assuming that the function E(Q) is analytical, one can expand it in a Taylor series around Q ) 0,
E(Q) ) E(0) + Q+χ + (1/2)Q+η Q + ...
(1)
where the vector of fragment electronegativities χ and the fragment hardness matrix η (both at the electron distribution corresponding to Q ) 0) have the elements
χi )
|
∂E(Q) ∂Qi
(2)
Q)0
and
ηij )
|
∂2E(Q) ∂Qi∂Qj
(3) Q)0
respectively.2-4 At the equilibrium electron distribution Q0, the gradient of E(Q) with respect to Q vanishes,
χ + ηQ0 + ... ) 0
(4)
Provided the higher-order terms are small, the expansion (4) can be inverted, yielding
Q0 ) -η-1χ + ...
(5)
The electronegativity equalization method (EEM)3,4 is based upon eq 5 truncated after the first term. * To whom all correspondence should be addressed. X Abstract published in AdVance ACS Abstracts, March 15, 1996.
0022-3654/96/20100-6156$12.00/0
Appealing in its conceptual simplicity, the EEM approach is well suited for semiempirical calculations in which E(Q) is replaced by a model function. However, the actual dependence of the total energy on the fragment charges is not given by an analytical function, the nonanalyticity stemming from the wellknown discontinuities of the first derivative of the total energy with respect to the total number of electrons.5,6 Consequently, the assumptions behind eqs 1-5 are violated unless the chargeconservation constraint,
1+Q ) 0
(6)
where 1 ≡ (1, ..., 1) is explicitly built into the electronegativity equalization formalism.7 The imposition of this constraint results in eq 5 being replaced by
Q0 ) -σχ + ...
(7)
The softness matrix σ can be regarded as a fragment-discretized representation of the linear response function.8 Its properties, including a generalization to spin-polarized systems,9 have been studied recently.7 The EEM formalism attributes the substituent effects on electron distributions to differences in the fragment electronegativities in situ χ. Its practical usefulness as a predictive tool hinges upon the satisfaction of two conditions. First, the higherorder terms in eq 7 have to be sufficiently small to make their neglect possible without significantly affecting the accuracy of the computed charges. Although this assumption does not appear to be justified in the case of weakly interacting10 or dissociating systems,6 it is expected to be valid for ordinary molecules with strong bonds that are not too ionic (i.e., for species with small fragment charges). Second, both χ and η have to be either highly transferable across diverse molecules or obtainable from simple approximate models. In this paper, the potential merits and shortcomings of three such models for σ are ascertained with test calculations on polyyne carbon chains. Modeling of the Softness Matrix The elements of the softness matrix σ can be rigorously calculated from the expression7,9 © 1996 American Chemical Society
Electronegativity Equalization in Polyynes
J. Phys. Chem., Vol. 100, No. 15, 1996 6157
σij ) -∑′
boundaries to yield σ appears to be a viable approach to modeling of the softness matrix. Another possible route to approximate calculations of σ is based upon simple electrostatic arguments. The interaction energy between two fragments (usually atoms) in a molecule is presumed to be adequately described by a screened electrostatic interaction between their charges,
〈Ψ0|Q ˆ i|Ψk〉〈Ψk|Q ˆ j|Ψ0〉 + 〈Ψ0|Q ˆ j|Ψk〉〈Ψk|Q ˆ i|Ψ0〉 (8)
E0 - Ek
k
where Ψ0 and Ψk are the wave functions of the ground and the kth excited state of the molecule under study, respectively, E0 and Ek are the corresponding energies, and Q ˆ i is the fragment charge operator,6,9,10
Q ˆ i ) ∑ ∫Ω δˆ (b r l-b) r db r l
Eij ) QiQjf(Rij)
(10)
where the function f(Rij) approaches Rij-1 for large interatomic distances Rij. The resulting model function E(Q), which is fully analytical, gives rise to a hardness matrix with the elements
(9)
i
ηij ) f(Rij)
In eq 9, Ωi is the union of the atomic basins that span the ith fragment11 and the summation index l runs over all electrons. As one can readily conclude from the inspection of eq 8, the softness matrix constitutes a generalization of the atom-atom polarizability tensor π of the Hu¨ckel theory.12 Even in its simplest version, i.e., with the topological adjacency matrix employed as the effective one-electron Hamiltonian, the Hu¨ckel theory is capable of qualitatively reproducing the heteroatominduced charge alternation in conjugated hydrocarbons.13 In addition, thanks to the simplicity of the effective Hamiltonian, general rules for predicting the patterns of charge perturbation due to substituents can be derived analytically within the Hu¨ckel formalism.14 On the basis of these observations, a scheme in which the elements of the atom-atom polarizability tensor, obtained from eigenvalues and eigenvectors of a one-electron Hamiltonian constructed with semiempirical parameters assigned to atom and bond types, are combined according to the fragment
(11)
The matrix η is then converted to σ with the formula7
σ ) η-1 - (1+η-11)-1 (η-11)X(1+η-1)
(12)
where aXb denotes a direct (dyadic) product of a and b. Equations 10-12 are widely employed in semiempirical implementations of EEM.3,4 The third approach to approximating σ relies upon the assumption of its additivity. Deviations from additivity, which determine the accuracy of softness matrices produced by this approach, can be readily assessed for a particular class of chemical systems by considering a set of four molecules, X-A-X, X-A-Y, Y-A-X, and Y-A-Y, where X and Y stand for substituents and A denotes a common fragment. Let σXX, σXY, σYX, and σYY be the corresponding softness matrices. The
TABLE 1: The Calculated MP2/6-311G** Atomic Charges of the Xs(CtC)NsY Polyyne Chains atomic charges X
Y
H H
H F
H Na
atom H C1 C2 C3 C4 C5 C6 C7 C8 C9 C10 H H C1 C2 C3 C4 C5 C6 C7 C8 C9 C10 F H C1 C2 C3 C4 C5 C6 C7 C8 C9 C10 Na
N)1
N)2
atomic charges
N)3
N)4
0.12533 0.14125 -0.12533 -0.0981 -0.12533 -0.12143 -0.12143 -0.01981
0.14608 -0.03106 -0.09071 -0.02430 -0.02430 -0.09071 -0.03106
0.14861 -0.03521 -0.08041 -0.02294 -0.01006 -0.01006 -0.02294 -0.08041 -0.03521
0.12533 0.14125 0.14515 0.13927 0.37022 0.00810 0.14905 -0.13120 0.40499 0.23264
0.14608 0.14489 -0.02964 -0.09608 -0.00894 -0.01940 0.41328 0.24673
0.14861 0.14781 -0.03451 -0.08320 -0.02217 -0.01405 0.00438 -0.01651 0.41433 0.25351
-0.66441 -0.65377 -0.65083 -0.64956 0.05448 0.10836 0.12418 0.13264 -0.74316 -0.07517 -0.05434 -0.05088 -0.18601 -0.17859 -0.12385 -0.10349 -0.57644 -0.05163 -0.03481 -0.16075 -0.08916 -0.04147 -0.52593 -0.03523 -0.16582 -0.08517 -0.85170 -0.16827
0.87470
0.88260
0.88656
0.88915
N)5
X
Y
0.15010 F F -0.03654 -0.07556 -0.02348 -0.00522 -0.00930 -0.00930 -0.00522 -0.02348 -0.07556 -0.03654 0.15010 0.14956 F Na -0.03607 -0.07740 -0.02294 -0.00741 -0.00854 -0.01312 0.00886 -0.01643 0.41483 0.25750 -0.64882 0.13794 Na Na -0.04794 -0.09287 -0.03169 -0.02680 -0.02101 -0.03752 -0.02903 -0.08422 -0.48980 -0.16796 0.89091
atom F C1 C2 C3 C4 C5 C6 C7 C8 C9 C10 F F C1 C2 C3 C4 C5 C6 C7 C8 C9 C10 Na Na C1 C2 C3 C4 C5 C6 C7 C8 C9 C10 Na
N)1
N)2
N)3
N)4
-0.65313 -0.65365 -0.65105 -0.64968 0.65313 0.23182 0.24381 0.25184 0.65313 0.42184 0.41132 0.25184 0.42184 -0.00406 -0.01580 0.23182 -0.00406 0.00040 0.41132 0.00040 0.24381 -0.01580 0.41326 0.25184 -0.65313 -0.65365 -0.65105 -0.69407 -0.67284 -0.66739 -0.10524 0.17029 0.20968 -0.08534 0.34122 0.38768 -0.57572 -0.04348 -0.14445 -0.07543 -0.53409 -0.16265
-0.64968 -0.66457 0.22752 0.39924 -0.02596 -0.02809 -0.03816 -0.08468 -0.50777 -0.16581
0.88466 0.88152 0.88569 0.81812 0.85753 0.87043 -0.81811 -0.27041 -0.22847 -0.81811 -0.27041 -0.53437 -0.27041 -0.10758 -0.27041 -0.10758 -0.53437 -0.22847
0.88831 0.87775 -0.21000 -0.51401 -0.09192 -0.06181 -0.06181 -0.09192 -0.51401 -0.21000
0.81812
0.85753
0.87043
0.87775
N)5 -0.64893 0.25612 0.41446 -0.01592 0.00667 -0.01238 -0.01238 0.00667 -0.01592 0.41446 0.25612 -0.64893 -0.66274 0.23849 0.40526 -0.02276 -0.01337 -0.02388 -0.03703 -0.03064 -0.08375 -0.49404 -0.16566 0.89014 0.88240 -0.19781 -0.50181 -0.08891 -0.04748 -0.04638 -0.04638 -0.04748 -0.08891 -0.50181 -0.19781 0.88240
6158 J. Phys. Chem., Vol. 100, No. 15, 1996
Cioslowski and Martinov
TABLE 2: Approximate Additivity of the Atomic Charge Vectors in the Xs(sCtCs)5sY (X, Y ) F, Na) Polyyne Chains atomic charge vector vector comp
0 QFF
0 QNaNa
0 QFNa
0 QNaF
1 2 3 4 5 6 7 8 9 10 11 12
-0.64893 0.25612 0.41446 -0.01592 0.00667 -0.01238 -0.01238 0.00667 -0.01592 0.41446 0.25612 -0.64893
0.88240 -0.19781 -0.50181 -0.08891 -0.04748 -0.04638 -0.04638 -0.04748 -0.08891 -0.50181 -0.19781 0.88240
-0.66274 0.23849 0.40526 -0.02276 -0.01337 -0.02388 -0.03703 -0.03064 -0.08375 -0.49404 -0.16566 0.89014
0.89014 -0.16566 -0.49404 -0.08375 -0.03064 -0.03703 -0.02388 -0.01337 -0.02276 0.40526 0.23849 -0.66274
0 0 (QFF + QNaNa )
0.11674 0.02916 -00.04368 -0.05242 -0.02041 -0.02938 -0.02938 -0.02041 -0.05242 -0.04368 0.02916 0.11674
1
0 0 (QFNa + QNaF )
2
0.11370 0.03642 -0.04439 -0.05326 -0.02201 -0.03046 -0.03046 -0.02201 -0.05326 -0.04439 0.03642 0.11370
TABLE 3: The Calculated MP2/6-311G** Fragment Charges of the Xs(sCtCs)NsY Polyyne Chains
difference matrix ∆σXY,
∆σXY ≡ (1/2)(σXY + σYX) - (1/2)(σXX + σYY)
1 2
fragment charges
(13)
measures the extent to which σ is additive in these molecules. Details of Test Calculations To investigate the potential merits and shortcomings of the aforementioned approaches to the approximate evaluation of σ, a series of test calculations on polyyne carbon chains was carried out. Geometries of 30 molecules with the composition Xs(sCtCs)NsY, where X, Y ) H, F, or Na and 1 e N e 5, were optimized at the MP2/6-31G** level of theory with the GAUSSIAN 92 suite of programs.15 Atomic charges (Table 1) were computed with accuracy better than 10-5 using the recently developed variational algorithm for the determination of atomic zero-flux surfaces.16 Atomic softness matrices of the molecules under study were obtained with the previously published method based upon charge-constrained electronic structure calculations.7,9 Additivity of Fragment Softness Matrices and Fragment Electronegativities in Polyyne Carbon Chains At first glance, no simple patterns other than those due to molecular symmetry can be discerned in the computed atomic charges of polyyne carbon chains (Table 1). In particular, charge alternation along the chains is conspicuously absent. However, a closer inspection reveals approximate additivity of the atomic charge vectors, which manifests itself in the rather 0 (compare small components of the difference vector ∆QXY eq 13), 0 0 0 0 0 ≡ (1/2)(QXY + QYX ) - (1/2)(QXX + QYY ) (14) ∆QXY 0 0 0 + QYX ) and (1/2)(Q0XX + QYY ) For example, the (1/2)(QXY vectors of the most polarized polyyne chain (X ) F, Y ) Na) among the Xs(sCtCs)5sY species are quite similar, although the relative differences between some of their individual components are as large as 24% (Table 2). Atomic charges can be combined into charges of appropriately defined molecular fragments. In the case of polyyne carbon chains, the most natural choice for these fragments corresponds to partitioning of the molecules under study into XsCtCs, sCtCs, and sCtCsY units. Such a partitioning produces fragment charge vectors with relatively small components (Table 3) for which the assumptions employed in the derivation of eq 7 are well justified. The resulting improvement in the charge additivity is nicely illustrated in the case of the Xs(sCtCs)5sY
X
Y
fragment
N)2
N)3
0.00000 0.02431 HC1C2 C3C4 -0.04861 C5C6 C7C8 C2N-1C2NH 0.00000 0.024431 H F HC1C2 0.01616 0.01917 C3C4 -0.02834 C5C6 C7C8 C2N-1C2NF -0.01615 0.00918 H Na HC1C2 -0.14540 -0.05401 C3C4 -0.14079 C5C6 C7C8 C2N-1C2NNa 0.14541 0.19482 F F FC1C2 0.00001 0.00407 C3C4 -0.00813 C5C6 C7C8 C2N-1C2NF 0.00001 0.00407 F Na FC1C2 -0.16133 -0.07003 C3C4 -0.05406 C5C6 C7C8 C2N-1C2NNa 0.16134 0.18895 Na Na NaC1C2 0.00001 0.10759 C3C4 -0.21516 C5C6 C7C8 C2N-1C2NNa 0.00001 0.10759
H
H
N)4
N)5
0.03300 0.03801 -0.03299 -0.02870 -0.03299 -0.01860 -0.02870 0.03300 0.03801 0.03010 0.03608 -0.03623 -0.03035 -0.01213 -0.02166 -0.00757 0.01828 0.02352 -0.02172 -0.00285 -0.07628 -0.05849 -0.1204 -0.05854 -0.11325 0.21843 0.23315 0.01542 0.02164 -0.01541 -0.00926 -0.01541 -0.02476 -0.00926 0.01542 0.02164 -0.03781 -0.01899 -0.03613 -0.03613 -0.12284 -0.06091 -0.11439 0.21473 0.23045 0.15374 0.18279 -0.15373 -0.13639 -0.15373 -0.09276 -0.13639 0.15374 0.18279
(X, Y ) F, Na) species, where the relative differences between 0 0 individual components of the (1/2)(QFNa + QNaF ) and (1/2)‚ 0 0 (QFF + QNaNa) vectors are now less than 4% (Table 4). The observed additivity of fragment charges in polyyne carbon chains reflects the underlying additivity of the corresponding electronegativity vectors and softness matrices. Indeed, the computed fragment softness matrices are found to be poorly transferable but highly additive. For the Xs(sCt Cs)5sY (X, Y ) F, Na) species, the maximum relative difference between individual elements of the (1/2)(σFNa + σNaF) and (1/2)(σFF + σNaNa) matrices is ca. 5%, while the average difference amounts to only 0.4% (Table 5). As expected from the relative magnitudes of interactions between the terminal groups, the deviations from additivity increase with the increasing polarization and the decreasing length of the chain. However, even for the Xs(sCtCs)2sY molecules, in which the terminal XsCtCs and sCtCsY fragments are directly linked and the softness matrices have particularly simple structures,
Electronegativity Equalization in Polyynes
J. Phys. Chem., Vol. 100, No. 15, 1996 6159
TABLE 4: Approximate Additivity of the Fragment Charge Vectors in the Xs(sCtCs)5sY (X, Y ) F, Na) Polyyne Chains fragment charge vector vector comp
0 QFF
0 QNaNa
0 QFNa
0 QNaF
1 2 3 4 5
0.02164 -0.00926 -0.02476 -0.00926 0.02164
0.18279 -0.13639 -0.09276 -0.13639 0.18279
-0.01899 -0.03613 -0.06091 -0.11439 -0.23045
0.23045 -0.11439 -0.06091 -0.03613 -0.01899
1 2
1
0 0 (QFF + QNaNa )
0 0 (QFNa + QNaF )
2
0.10222 -0.07283 -0.05876 -0.07283 -0.10222
0.10573 -0.07526 -0.06091 -0.07526 -0.10573
TABLE 5: Approximate Additivity of the Fragment Softness Matrices in the Xs(sCtCs)5sY (X, Y ) F, Na) Polyyne Chains σFF 1.5077 -0.9762 -0.2263 -0.1483 -0.1570
2.2291 -0.9239 -0.1808 -0.1483
σNaNa
2.3004 -0.9239 -0.2263
2.2291 -0.9762
1.5077
1.9104 -1.1372 -0.2937 -0.1842 -0.2953
2.3932 -0.9109 -0.1610 -0.1842
2.4091 -0.9109 -0.2937
σFNa 1.5203 -0.9547 -0.2092 -0.1366 -0.2199
2.2599 -0.9149 -0.1765 -0.2139
2.3746 -0.9273 -0.3233
2.3112 -0.9174 -0.1709 -0.1662
2.3977 -1.1574
2.3548 -0.9174 -0.2600
1.9147
1.9147 -1.1574 -0.3233 -0.2139 -0.2199
2.3977 -0.9273 -0.1765 -0.1366
2.3112 -1.0567
1.7090
X
Y
σ
X
Y
σ
H F Na
H F Na
1.281 1.226 1.729
H H F
F Na Na
1.252 1.448 1.392
(
1 -1 -1 1
)
(15)
where σ is the bond softness (a reciprocal of the bond hardness κ17), the relative deviations do not exceed 6% (Table 6). Taking into account the strong influence of terminal groups on bond lengths of the carbon chains, such a high degree of additivity in σ is remarkable. Similar patterns are observed in the approximate additivity of the fragment electronegativity vectors χ, which are obtained from the expression -1
χ ) -σ Q
2.3746 -0.9149 -0.2092
2.2599 -0.9547
1.5203
2.3288 -1.0560
1.7175
(1/2)(σFNa + σNaF)
TABLE 6: The Calculated MP2/6-311G** (XsCtCs)s(sCtCsY) Bond Softnesses
σ)σ
1.9104
σNaF
(1/2)(σFF + σNaNa) 1.7090 -1.0567 -0.2600 -0.1662 -0.2261
2.3932 -1.1372
0
(16)
derived from a truncated eq 7 (note that, since σ is singular,7,9 σ-1 denotes a generalized inverse of σ and the components of χ sum to zero). The average deviations from additivity range from 2% to 6% for individual components of χ, the additivity improving with the increasing chain length.
1.7175 -1.0560 -0.2663 -0.1753 -0.2199
2.3288 -0.9211 -0.1765 -0.1753
2.3746 -0.9211 -0.2663
As discussed in the previous section of this paper, other approaches to the approximate evaluation of σ are possible in principle. To ascertain the potential usefulness of these alternative schemes, several calculations involving either effective oneelectron Hamiltonians or screening functions f(Rij), eqs 10 and 11, were carried out. Tight-binding Hamiltonians parametrized by six adjustable parameters, namely, four resonance integrals (for the XsC, CsC, CtC, and CsY bonds) and two Coulomb integrals (for the X and Y heteroatoms; the Coulomb integrals for carbons can be set to zero without any loss of generality), were found incapable of furnishing reasonably accurate atomic softness matrices. Equally disappointingly, atomic softness matrices were satisfactorily reproduced only when unphysical screening functions f(Rij) that do not decay monotonically with the interatomic distance Rij were allowed. Conclusions Charge-constrained calculations on polyyne carbon chains terminated with various substituents uncover interesting trends in electronegativity equalization. In contrast to charges of individual atoms, charges of the XsCtCs, sCtCs, and sCtCsY chain fragments are found to be highly additive. This difference in additivity appears to be related to the excessive magnitudes of atomic charges, which are on average much larger than those of the fragment charges, that invalidate the assump-
TABLE 7: The Calculated MP2/6-311G** Atomic Softness Matrix of the Hs(sCtCs)5sH Polyyne Chain H
C1
C2
C3
C4
C5
C6
C7
C8
C9
C10
H
0.745 00.461 -0.137 -0.003 -0.052 -0.001 -0.033 -0.002 -0.024 -0.003 -0.023 -0.006
2.317 -1.172 -0.063 -0.236 -0.025 -0.128 -0.019 -0.090 -0.015 -0.085 -0.023
2.032 -0.544 -0.101 -0.006 -0.040 0.006 -0.023 0.004 -0.015 -0.003
2.141 -0.995 -0.083 -0.177 -0.036 -0.101 -0.023 -0.090 -0.024
2.105 -0.571 -0.093 -0.006 -0.036 0.006 -0.019 -0.002
2.132 -0.975 -0.093 -0.177 -0.040 -0.128 -0.033
2.132 -0.571 -0.083 -0.006 -0.025 -0.001
2.105 -0.995 -0.101 -0.236 0.052
2.141 -0.544 -0.063 -0.003
2.032 -1.172 -0.137
2.317 -0.461
0.745
6160 J. Phys. Chem., Vol. 100, No. 15, 1996 tions behind the EEM approach, eq 7. The observed additivity of fragment charges has its roots in the additivity of the corresponding electronegativity vectors and softness matrices. The errors in χ, and σ calculated from approximate combination rules do not exceed 6% despite a very poor transferability of both quantities among polyyne chains. Approximate models that involve either one-electron effective Hamiltonians or screened electrostatic interactions fail to reproduce the rigorously computed atomic softness matrices, indicating the complex nature of phenomena that control electron flow in molecules. This failure underscores the need for further research aimed at the development of more sophisticated implementations of the electronegativity equalization method. Acknowledgment. This work was partially supported by the National Science Foundation under Grant CHE-9224806, the Florida State University through time granted on its Cray Y-MP digital computer, and the US DOE through its Supercomputer Computations Research Institute. References and Notes (1) For recent reviews on DFT, see: Parr, R. G.; Yang, W. Density Functional Theory of Atoms and Molecules; Oxford University Press: New York, 1989. Kryashko, E. S.; Luden˜a, E. V. Energy Density Functional Theory of Many-Electron Systems; Kluwer: Dordrecht, 1990. Density Functional Methods in Chemistry; Labanowski, J. K., Andzelm, J. W., Eds.; Springer: New York, 1991. (2) Parr, R. G. Int. J. Quantum Chem. 1984, 26, 687. Sanderson, R. G. Science 1951, 114, 670. Parr, R. G.; Pearson, R. G. J. Am. Chem. Soc. 1983, 105, 7512.
Cioslowski and Martinov (3) Baekelandt, B. G.; Mortier, W. J.; Lievens, J. L.; Schoonheydt, R. J. Am. Chem. Soc. 1991, 113, 6730. vanGenechten, K. A.; Mortier, W. J.; Geerlings, P. J. Chem. Phys. 1987, 86, 5063. Mortier, W. J.; Ghosh, S. K.; Shankar, S. J. Am. Chem. Soc. 1986, 108, 4315. Ohwada, K. J. Phys. Chem. 1993, 97, 1832. (4) Nalewajski, R. F. J. Phys. Chem. 1985, 89, 2831. Korchowiec, J.; Nalewajski, R. F. Int. J. Quantum Chem. 1992, 44, 1027. Nalewajski, R. F. J. Am. Chem. Soc. 1984, 106, 944. (5) Parr, R. G.; Bartolotti, L. J. J. Phys. Chem. 1983, 87, 2810. Perdew, J. P.; Parr, R. G.; Levy, M.; Balduz, J. L., Jr. Phys. ReV. Lett. 1982, 49, 1691. Gyftopoulos, E. P.; Hatsopoulos, G. N. Proc. Natl. Acad. Sci. U.S.A. 1968, 60, 786. (6) Cioslowski, J.; Stefanov, B. B. J. Chem. Phys. 1993, 99, 5151. (7) Cioslowski, J.; Martinov, M. J. Chem. Phys. 1994, 101, 366. (8) Berkowitz, M.; Parr, R. G. J. Chem. Phys. 1988, 88, 2554. Handler, G. S.; March, N. J. Chem. Phys. 1975, 63, 438. (9) Cioslowski, J.; Martinov, M. J. Chem. Phys. 1995, 102, 7499. (10) Cioslowski, J. Int. J. Quantum Chem. 1994, 49, 463. (11) Bader, R. F. W. Atoms in Molecules: A Quantum Theory; Oxford University Press: Oxford, 1990. (12) Coulson, C. A.; Longuet-Higgins, H. C. Proc. R. Soc. London 1947, A191, 39. (13) Coulson, C. A.; Longuet-Higgins, H. C. Proc. R. Soc. London 1947, A192, 16. (14) Gutman, I. Theor. Chim. Acta 1979, 50, 287. (15) Frisch, M. J.; Trucks, G. W.; Head-Gordon, M.; Gill, P. M. W.; Wong, M. W.; Gomperts, R.; Andres, J. L.; Raghavachari, K.; Binkley, J. S.; Gonzalez, C.; Martin, R. L.; Fox, D. J.; Defrees, D. J.; Baker, J.; Stewart, J. J. P.; Pople, J. A. GAUSSIAN 92, Revision B; Gaussian Inc.: Pittsburgh, PA, 1992. (16) Cioslowski, J.; Stefanov, B. B. Mol. Phys. 1995, 84, 707. Stefanov, B. B.; Cioslowski, J. J. Comput. Chem. 1995, 16, 1394. (17) Cioslowski, J.; Mixon, S. T. J. Am. Chem. Soc. 1993, 115, 1084.
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