J. Phys. Chem. 1993,97, 1894-7898
7894
Molecular Polarizabilities from Electronegativity Equalization Models Uri Dinur Department of Chemistry, Ben-Gurion University of the Negev, Beer-Sheva. 84105, Israel Received: January 6, 1993; In Final Form: May 17, 1993
Implications of the electronegativity equalization (EE) principle to molecular polarizabilities are discussed. An expression for molecular polarizabilities that involves simple matrix manipulations is derived based on the standard E E scheme. The polarizability within this scheme is shown to be completely determined by the atomic hardness parameters and repulsion integrals. However, it is pointed out that current EE models fail to account for polarizabilities that do not involve interatomic charge transfer, such as the perpendicular polarizability of planar molecules. A modified EE scheme is suggested in which the center of the electronic cloud is allowed to be displaced from its nucleus through a Drude model. This incorporates the notion of atomic dipoles into the theory and enables the EE scheme to be compatible with the existence of perpendicular polarizabilities in planar molecules as well as with binding in homonuclear diatomic molecules. Ab initio calculations for HF and HzO are used to test the formal results. It is found that intramolecular charge transfer between atoms is accurately reproduced by the EE model with a minimal set of parameters. The intraatomic polarization as obtained from the EE-Drude model is found to be anisotropic and oriented mainly along the perpendicular direction to the molecular plane.
1. Introduction One of the more difficult problems in the construction of accurate molecular force fields is the determination of atomic partial charges and their response to environmental changes. Neither polarizability nor conformation dependence of atomic charges can be treated in standard implementations of molecular mechanics. Yet it is clear that with the never-ending demand for more accurate force fields the response propertiesof the molecular charge distribution have to be studied and modeled. A promising approach in that regard is based on the principle of electronegativityequalization (EE). This principle, which was first suggested by Sanderson,' has been extensively studied over the years and is now well-rooted in density functional theory.*A related approach is that of Del Re.3 Both the EE principle and the Del Re approach have been implemented in molecular force field calculations of atomic partial charges by Allinger and coworkers4.s and by Scheraga and c o - w o r k e r ~with ~ ~ ~encouraging results. In thisarticle weextend theapplicationof the EE principle to the calculation of molecular polarizabilities. An intrinsic featureofthis approachthat is particularly attractive in thecontext of molecular force fields is that the molecular polarizability is obtained from atomic parameters. Several methods for breaking up the molecular polarizability into atomic components already exist in the literature, notably the model developed by Applequist,* as further modified by Birge9 and Thole,lo and the more recent model discussed by Stone." These methods do not involve the EE principle and provide alternatives to compare with. The calculation of molecular polarizability using current EE models is discussed in section 2. A general expression for molecular polarizabilities in terms of few fundamental atomic parameters is obtained. Specifically, it is shown that within current EE models the molecular polarizability is determined by the atomic hardness, by parameters that pertain to the repulsion integrals, and by the molecular geometry. However,these models cannot account for the perpendicular polarization in planar molecules. This deficiency is discussed in section 3. A modification that incorporatesthe Drude model into current EE models is proposed as a possible solution to the existing difficulty. In section 4 the formal results of sections 2 and 3 are implemented and tested on two prototypical polar molecules. Ab initio calculations of the molecular dipole moment and polarizability of HF and H20,over a range of bond lengths, are analyzed on
the basis of the two EE models. It is concluded that the EE models provide with a quantitative description of the molecular property surfaces.
2. Molecular Polarizability from Current Electronegativity Equalization Models In current models of electronegativityequalization a molecular system is viewed as consisting of atoms that have distinct properties, interact, and exchange charge. The molecular energy E is written, accordingly, as a sum of atomic energies and interatomic interactions:
In eq 1 is the energy of atom i in a virtual free and uncharged state, xi and qiare, respectively, its electronegativityand hardness in the virtual state, {qi) are the atomic charges, and U , is the interaction between sites i and j. The chemical potential pi is defined as -dE/aqi:
At equilibrium all chemical potentials are equal since there is no charge flow. If the parameters x,q are known and U, is specified the equality of the chemical potentials gives rise to Nindependent linear equations with N unknown atomic charges
where Qtot is the total molecular charge. Solving this equations determines the atomic charges (qij. This constitutes the electronegativity equalization method. At the presence of an external electric field €0 an additional term, 4,4 where , Ri is the nuclear position, is added to the right-hand side of eq 2. The atomic charges redistribute accordingly, so that the total change in the chemical potential involves both the change due to the field and the change due to adjustments in the molecular charge distribution. This change is directly related to the Fukui function (6p/6&, where u is the
0022-3654/93/2091-1894$04.~0/0 0 1993 American Chemical Society
Molecular Polarizabilities from EE Models
The Journal of Physical Chemistry, Vol. 97,No. 30, 1993 7895
local potential) which has been discussed extensively with respect to reactivity and charge transfer in acid-base interactions.2J2-16 For small fields we may write pi(eo) = pi(eo'o)
+capi/&k
6qk
+
(4)
where 6 q k is the charge change induced by the field. The electronegativity equalization is then given by
R p 0 (sa) In addition there is the condition that the external field does not change the total charge
Because of the equality of the chemical potentials at zero field (cq 3) pi(@=O) and p1(@=0)cancel out on both sides of eq 5a. Thus eq 5 lead again to N equations with N unknowns:
where
Solving eq 6 for the charge shifts 6q that are induced by the external field yields
bq = (A'A)-~A'AR-~O
(8)
Equation 8 relates the change in the ith atomic charge to all other sites through a "relay" tensor (AtA)-'AfAR.Relay tensors, albeit differently defined, were also obtained by Stonell and previously by Applequist17in their models for molecular polarizabilities. Given the charge shifts (6qk}, the induced dipole moment 6d is just
(9) Substituting eq 8 for 6q gives the polarizability in terms of the matrix A and the molecular geometry AR: (Y = (A'A)-'A'ARR (10) To explicitly evaluate CY,one has to know a p / d q (eq 7). Within the simple electrostatic model for Vi,,namely, Uij = qflj/Rij,the derivatives of the chemical potential are simply
Further modification of eq 11 is possible in the form of a semiempirical approximation for Ut,, e.g., the Ohno or Mataga repulsion integrals, in which case the polarizability also depends on the parameterization of this representation (and see section
1-10 that the ratio of the polarizability to the dipole moment depends upon the electronegativity and not the hardness.20
3. An EE-Drude Model The EE model described in the previous section accounts for charge polarization by exchanging partial charges among the various atomic sites. This picture encounters a problem in cases of planar molecules where movement of charges perpendicular to the plane is impossible and, nevertheless, the perpendicular polarizability is in general not zero. For example, mp2/6-3 1l+g(d,2p) ab initio calculations of the polarizability of water yield for an,CY,,,,, and aZz, respectively, the values 7.758.82, and 7.77 au, where the molecule lies in the yz plane with the c2 axis oriented along the z axis. The EE model discussed above necessarily predicts aXx= 0 since the charges cannot respond to a perpendicular field and move out of the molecular plane. The error according to the ab initio calculations is quite large since the perpendicular and the in-plane polarizabilities have similar magnitudes. Clearly, some modification of the EE model discussed above is desirable. One possibility is to relax the restriction of the charges to the nuclear positions by means of a simple Drude model for the motion of the electronic partial charge. The electronic charges are assumed to be attached to their cores by harmonic forces with force constants (ki). Such an approach was used, for example, by Kim and Gordon2'in their treatment of intermolecular forces. In studies of solid state it is often referred to as the shell model.22.23 It allows polarization of the atomic charge distribution (which includes the nucleus and the surrounding electrons) via the displacements (6)) of the electronic partial charges from their corresponding nuclei. Combining this model with the EE model leads to the following expression for the molecular energy:
(12) where 6i.t in the right hand side denotes the displacement coordinate of qi relative to its nucleus and where [ denotes a Cartesian component. For nonspherical atoms the harmonicforce need not be isotropic and hence k, depends upon [. It is implicitly assumed in eq 12 that ki and ai are nonzero only for negative centers (however, a more general treatment is possible). The magnitude of the displacements6 is determined by the condition
aE/as,, = o (13) Assuming that x and 7 are independent of 6, it follows from eqs 12 and 13 that
In eq 14 the derivative of the sum with respect to 61 is the force exerted on the ith electronic charge cloud by the surrounding atoms. On the basis of an electrostatic model we may write this force as
4)
Equations 7, 10, and 11 provide an analytical expression for the molecular polarizability. It suggests that the molecular polarizability depends on only few fundamental properties, namely, the atomic hardness and the repulsion integrals, along with the molecular geometry. It may also be seen from eqs 7, 10, and 11 that the polarizability and the hardness are inversely related, in similarity with previous results obtained by Politzerl* and by Vela and Gazquez.19 It may also be noted that the polarizability does not depend on the atomic electronegativities. In the case of diatomic molecules it may be verified from q s
where' :e is the field that is exerted by the molecule on site i. The displacements 6 are thus given by
Since e? depends on the coordinates of the electronic charges q 16 has to be solved iteratively, along with eq 3. By allowing the electronic charges to shift away from the nuclei one practically introduces into the EE model the element of atomic
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7896 The Journal of Physical Chemistry, Vol. 97, No. 30, 1993
dipoles. In fact, comparing the model discussed in section 2 with the models of Applequist and Stone one sees that the EE model uses only atomic charges, the Applequist model is based solely on atomic dipoles8 whereas Stone's treatment" allows for the existence of both atomic dipoles and charge. Polarization within the latter model is thus the result of interatomic charge exchange as well as intraatomic polarization. This picture which seems to represent the physics of the problem more faithfully is captured by the EE-Drude model described above. It also may be noted that the approach outlined above offers a way of applying the EE principle to homonuclear diatomic molecules. In such cases the atomiccharges are zero by symmetry and the usual approach to electronegativity equalization as reflected in eq 1 collapses. The Drude model enables binding through the displacement of the electronic partial charge in response to the penetration potentials. (This also requires that eq 1 will be rewritten explicitly in terms of the electronic partial charges with explicit distinct expressions for the repulsion and penetration integrals). An alternative approach to homonuclear diatomics, which uses electronegativities and hardnessparameters for the bonds, has been discussed by Ghosh and Parr and by Ghanty and G h ~ s h . * ~ , ~ ~ Upon application of an external homogeneous electric field eo the molecular energy (eq 13) becomes
by I and 11, respectively. For H F model I (eqs 1-3 and 10) yields the following simple expressions for the molecular properties:
In eq 23 A x is the electronegativitydifference between F and H, RHFis the bond length, e: is an external field along the bond, q~ and 9 H are the hardness parameters of F and H. ~ H isF a coupling parameter that is related to UHFin eq 1 by
= 4HqFqHF (24) and is further specified below. Note that the staticdipolemoment is obtained from eq 23 by letting ell = 0. Equations 23 m a y now be treated further in order to account for the properties surface of HF, namely, the dipole moment and polarizability as a function of the bond length. In the simplest model the atomic electronegativities and hardnesses should be independentof the bond length and only ~ H isF a function of RHF. ~ H Fis often identified with the Coulombic integral and is approximated either as a simple 1/R or by semiempirical formulas (e.g., Mataga or Ohno). Here we adopt a less restricted interpretation of ~ H and F employ a simple two term expansion in l/R:
where d is the molecular dipole moment:
viF
The chemical potential changes as in eq 4 in the previous section, and the displacements 6 change according to eq 16 except that the field that is operating on i now includes the external contributionas well, namely, eint eo. It can now be easilyverified based upon eqs 3,13 that the above model satisfies
where and qLF are parameters to be determined. Substituting in eq 23 we obtain that the dipole moment and the polarizability along the bond, as functions of the bond length, are determined by three independent parameters, Ax, 70, 41:
+
as it should. The molecular polarizability is given by
+
wherep, 8 aredetermined from theelectronegativityequalization, eq 3, and from eq 13. Taking the water molecule as an example, for a field e! that is perpendicular to the molecular plane d p / @ = 0, so that the first term on the right-hand side of (20) is zero. For the second term in eq 20 the implementation of eqs 13-16 leads to
so that koJ is thus determined uniquely by
ko,x = 402/ff, (22) where a, is the molecular perpendicular polarizability and qo is the partial charge of the oxygen.21 4. Applications to HzO and HF In this section we implement the EE models discussed above with respect to two prototypical polar molecules, H20 and HF. The two models discussed in sections 2 and 3 are denoted hereafter
I where qo QF q H - 2&, and q1 = qHF. These expressions (eq 26) were tested against ab initio calculations of the dipole moment and polarizability of H F at various bond lengths in the vicinity of the equilibriumvalue. The ab initio calculations were carried out with the Gaussian 90 program26at the MP2/6-311+G(d,2p) level of theory. The molecular dipole moments and polarizabilities were fitted with the formulas of eq 24 and the three parameters were determined via a least-squares procedure. The results are shown in Table I. Clearly, model I accounts for the property surface of HF with very small deviations. Eighteen quantum mechanical observables are nicely fitted with only three parameters, of which two are common to the dipole moment and the polarizability. Alternatively stated, the dipole moment and polarizability are seen to be related to each other in the form predicted by model I. We note also that the resulting value of Ax, 0.23 1, is in the range found in other works (e.&, refs 27 and 28) and that, likewise, the value of 70 is comparable to expectations based on standard values of hardness parameters*and on the use of the Coulomb law for qHF. The results shown above suggest that polarization of the H F bond occursby means of charge transfer from one atom to another. However, as discussed in section 3, if charge transfer is the only mechanism for polarization, then the perpendicular polarization
Molecular Polarizabilities from EE Models
The Journal of Physical Chemistry, Vol. 97, No. 30, 1993 7897
TABLE I: Dipole Moments and Polarizabilities of HF and H2W HF I I1
0.231 0.255
-0.567 -0.62
0.015 0.01
ab initio d
RHF
‘YL
0.056
0.248
I d
I1
‘Yl
all
d
‘YL
‘Yl
-0.639 -0.677 -0.716 -0.754 -0.796 -0.834 -0.873 -0.91 1 -0.95
0 0 0 0 0 0 0 0 0
4.189 4.719 5.279 5.872 6.553 7.212 7.901 8.623 9.375
-0.636 -0.675 -0.714 -0.753 -0.796 -0.835 -0.874 -0.913 -0.952
3.149 3.145 3.142 3.138 3.135 3.131 3.128 3.125 3.122
4.446 4.928 5.438 5.976 6.596 7.195 7.823 8.479 9.165
1.82
100.0
3.1 1
2.06
6.15
1.11
‘YII
HF 0.80 0.85 0.90 0.95 1.00 1.05 1.10 1.15 1.20
-0.661 -0.694 -0.728 -0.762 -0.798 -0.832 -0.865 -0.897 -0.928
2.853 2.922 2.992 3.062 3.138 3.208 3.278 3.346 3.413
4.531 4.938 5.393 5.899 6.51 7.127 7.798 8.523 9.299
rms dev (%)
HzO I I1
0.1072 0.1146
0.4741 0.5057
0.0762 0.0822
ab initio ROH
d
axx
0.90 0.92 0.94 0.96 0.98 1.00 1.02 1.04 1.06
-0.749 -0.755 -0.761 -0,767 -0.773 -0.778 -0.783 -0.788 -0.792
7.543 7.612 7.680 7.746 7.810 7.872 7.932 7.991 8.048
0.0564
0.6
I ‘YYY
7.752 8.089 8.444 8.817 9.208 9.618 10.047 10.495 10.961
I1
azz
d
axx
‘YYY
‘Yzz
d
‘Yxx
‘YYY
4x2
7.118 7.330 7.548 7.771 7.999 8.233 8.472 8.716 8.965
-0.743 -0.750 -0.756 -0.763 -0.771 -0.778 -0.786 -0.794 -0.802
0.OOO 0.000 O.OO0 O.OO0 0.000 0.000 0.000 0.000 0.000
7.796 8.146 8.504 8.870 9.244 9.625 10.014 10.410 10.814
7.087 7.309 7.536 7.768 8.006 8.248 8.495 8.748 9.005
-0.741 -0.748 -0.755 -0.763 -0.770 -0.778 -0.786 -0.794 -0.802
8.249 8.127 8.009 7.895 7.784 7.677 7.575 7.476 7.381
8,011 8.323 8.645 8.974 9.312 9.658 10.012 10.374 10.744
7.061 7.274 7.491 7.712 7.938 8.168 8.402 8.640 8.882
rms dev (%) 0.66 100.0 0.70 0.29 0.78 5.70 1.98 0.81 a R in angstroms, d and a in atomic units. The water molecule is in the yz plane, with the bisector along the z axis. The HOH angle is 106.1O. of H F should be zero. For this purpose we now consider model I1 (eqs 3, 12, and 13). Equation 24 still holds for the dipole moment except that R H Fis now replaced by RHF+ 6 ~ .The displacement 6~ has to be determined along with q F in a selfconsistent manner. For a given displacement the charges are found from q F = A x / ( ~ $ / R H-FTO), then a new displacement is found from 6 = qFeF/k (eq 16), where t~ is the total field operating on the fluorinecharge distribution. This process repeats itself until the difference between successive displacements is smaller than a preset number (10-6 in the present calculations). The field t that determines the displacement is composed from the external contribution as well as the internal field generated by the hydrogens. The latter field is given in the present model by ‘int -qHV?HF (27) The implementation of model I1 was first attempted with an isotropicharmonic force. It was found that a single forceconstant that correctly reproduces the perpendicular polarizability yields an in-plane polarizability that is too large. The fact that model I successfullyaccounts for the polarization along the bond implies that practically no intraatomic polarization occurs. This further means that the fluorine atomic charge distribution is polarizable (in an intraatomic sense, via a Drude model) in the direction perpendicular to the bond but not along the bond. Thus when both charge transfer and intraatomic polarization are allowed the net results is a parallel polarizability that is too large. Several possible solutions that pertain to the form of the internal field ( ~ H F )and of the electronegativity x as a function of R and 6 were considered. It was found simplest however to assume an anisotropic model and allow for different force constants for the in-plane and perpendicular directions. (In that respect we note
that the polarizability of the free fluorine atom is slightly anisotropic, the 6-3 11+g(d,2p) values being 2.188,2.188, 1.955 au). The results are shown in Table I. As seen, 27 observables, which now also sample the perpendicular polarizability, are well reproduced by model I1 with five parameters. The deviation with respect to cyI, which is higher than the deviations in d and all, can be easily reduced by making kl a function of RHF.In fact the degree of anisotropyof k that is obtained (Table I) implies that the model now incorporates the bonding into the parameter k so that the latter is not entirely an atomic property. The bond length variation of kl is then expected since upon dissociation the molecular polarizability changes and becomes the sum of the atomic polarizabilities of F and H, which is 4.15 au (vs -3 in the molecule). Another possibility is to allow the hydrogen to have a bond length dependent polarizability. This is because at the end of the dissociation process the hydrogen atom regains its electron and becomes polarizable with a nonzero parameter k of its own. Finally, kll is less sensitive to the bond length than kl since, based upon the results of model I, the parallel polarization is governed by charge transfer, with relatively little intraatomic polarization. The former mechanism is well reproduced by the EE model. The above analysis was repeated for HzO. The equations for the dipole moment and polarizability are analogous to eq 26 so that the property surfaces are again predicted to be reproduced by three or five parameters, depending on the model. We first discuss model I. As Table I shows model I with three parameters accounts very well for the in-plane properties. This again means that the bond polarization is brought about by charge migration between sites rather than intraatomic polarization. It should be noted that the polarization along the bisector is due to charge migration from the oxygen to the hydrogen whereas ayyinvolves
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7898 The Journal of Physical Chemistry, Vol. 97, No. 30, 1993 charge migration along the hydrogens. Model I is still capable of accounting for the polarization in both directions, along with the dipole moment surface. As in the case of H F it was found that in order to account for the perpendicular polarization it necessary to assume different force constants for the perpendicular and in-plane directions. Here k~ was less sharply determined, and the value of 0.6 was imposed and the other parameters optimized. As in H F the perpendicular polarization is less well reproduced than the planar components and its bond length dependence is incorrect, strongly suggesting that kl of either 0 or H should be made bond-length dependent. However, even within the current minimal representation the deviations are not worse than 10% for crl and are -3% or less for the other observables (total of 36). On the basis of these results, it may be concluded that EE models are capable of a quantitative reproduction of molecular property surfaces.
perpendicular polarizabilities in H F and H20. It is concluded that EE models may provide a quantitative representation of property surfaces.
5. Conclusions In this work we have studied the implications of the electronegativity equalization principle to molecular polarizability. It was found that current models enable one tocalculate analytical molecular wlarizabilities from atomic hardness Darameters and repulsion htegrals. It was also pointed out tha't whereas such models account for molecular polarizabilities by means of charge transfer, part of the polarization of the molecular charge distribution must be ascribed to intraatomic polarization of the electronic partial charge. It was shown that such a mechanism may be incorporated into current models by augmenting the EE principle with a Drude model. In this extended model the electronic partial charges are attached to the nuclei by harmonic forces and are thus allowed to shift away from the nuclei in response to external fields. This enables planar molecules to have perpendicular polarizabilities. Comparisons with ab initio calculations for H F and water have shown that the expressions derived for the molecular dipole moment and polarizability as functions of the bond length are accurate to within 3% over the range considered. The results were interpreted to imply that (within this model) bond polarization occurs mainly through charge transfer. The results of the EE-D model may be similarly interpreted. It was also found that anisotropic harmonic forces have to be assumed in order to account for both planar and
A I,7 7
Acknowledgment. This work was supported in part by the Basic Research Foundation administered by the Israel Academy of Sciences and Humanities. References and Notes (1) Sanderson, R. T. Science 1955, 121, 207. (2) Parr, R. G.; Yang, W. Density functional theory of atoms and
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