J. Phys. Chem. 1996, 100, 3349-3353
3349
Calculation of UV/Vis Spectra in Solution Andreas Klamt† Bayer AG, MD-IM-FA, Q18, D-51368 LeVerkusen, Germany ReceiVed: March 2, 1995; In Final Form: July 20, 1995X
The adequate treatment of solvation effects in the calculation of UV/vis spectra is a complicated task and a general solution of this problem is still outstanding. Continuum solvation models like COSMO at least allow one to take into account a large part of the different solvent effects on electronic excitations, if the dielectric screening is properly implemented within the CI part of the underlying MO program. A comprehensive development of the respective theory is given. This ends up in a rather simple result, and it turns out that most of the former approaches to the problem have been in severe error. Geometry changes of the solute in different solvents have been taken into account for the first time and appear to be of considerable importance. A few example applications of a COSMO-MOPAC implementation are presented.
Introduction The qualitative and quantitative understanding of the different solvation effects on the electronic excitations and thus on the UV/vis spectra of organic molecules is a question of active research since 40 years ago. A good review of this area has been given by Reichart.1 The basic work on this topic has been done by Lippert2 and McRae.3 They consider the dipoledipole, dipole-induced dipole, and the dispersive interactions between solute and solvent in a classical model and come out with the finding that the energy of a given electronic transition in a solute molecule depends like
ET ) ET,0 + (f() - f(n2))a - f(n2)b
(1)
on the total dielectric constant and the squared refractive index n2 of the solvent, where the latter may be interpreted as the partial dielectric constant arising from the electronic polarizability of the solvent. The function
f() )
-1 + 1/2
(2)
here is the effective scaling factor of dielectric screening effects, i.e., polarization and screening energy, with the solvent dielectric constant . Constants a and b in eq 1 are solute-specific properties depending on the entire transition. While the second term in eq 1 usually is addressed to the change in the interaction of the wave function with the polarization during the transition, the third term is assumed due to dispersion, a fact we will contradict later in this article. The functional dependence of the solvatochromic shifts given by eq 1 has been verified by many experimental data, but due to the lack of first principal knowledge of a and b, these had to be fitted to the data. Thus it should be noted that these experiments do not really prove eq 1 but only that the solvatochromic shifts usually are composed of one part scaling with f() and another part scaling with f(n2). This will be of importance below. While the theory of the quantum mechanical calculation of electronic excitations of molecules in vacuum or gas phase is well developed, the respective theory for molecules in solvation is just growing. The explicit treatment of solvent molecules around the solute requires large numbers of molecules for a † X
E-mail: DEBAYM59@IBMMAIL. COM. Abstract published in AdVance ACS Abstracts, February 1, 1996.
0022-3654/96/20100-3349$12.00/0
realistic simulation and thus usually is restricted to force-field methods. Quantum mechanical self-consistent field (SCF) molecular orbital (MO) calculations typically are not affordable for these systems. Even more this is true for configuration interaction (CI) calculations, which would be required for a reliable estimation of electronic excitations. Therefore the use of continuum solvation models (CSM), which describe the effect of a solvent around a solute in the approximation of a dielectric continuum, is the only promising way to precede. A general overview of the basic principle and the differences of the various CSMs published during the past decade has been given by Cramer and Truhlar4 and by Tomasi and Persico.5 Even in an early stage of their work on CSMs Bonaccorsi et al. published an extension of the iterative Miertus-ScroccoTomasi (MST) model to the calculation of excitation energies.6 Only a few other applications of CSMs to the electronic excitation spectra have been published so far.7-10 All of these approaches follow the concept of Lippert2 and McRae3 that the total polarizability of the solvent has to be split into an orientational (or) part and an inductive or electronic (el) part and that the orientational part of the ground-state polarization has to be kept fixed for the calculation of vertical electronic excitations, while the electronic part is able to adjust instantaneously to the excitation. Surprisingly most of them, i.e., including Lippert and McRae, have been making the same error in the partition of the ground-state polarization. This error has been criticized by Brady and Carr,11 and they have given a derivation of the correct partition. Unfortunately and unnecessarily, they end up with a very complicated expression for the solvatochromic shifts. This may be the reason why their treatment has not been taken up by others so far. On the basis of the work of Hynes and co-workers12 and Basilevski and Chudinov,13 Aguilar, Olivares del Valle, and Tomasi10 recently presented a rather general derivation of the theory of nonequilibrium solvation in the context of the MST model, which includes the calculation of electronic transition energies, i.e., UV/vis spectra. In contrast to the earlier work of Tomasi6 they used the correct partition, but they missed pointing out the considerable difference to the conventional treatment. In this paper an extension of the COSMO model14 to electronic excitations will be presented. A surprisingly simple and intuitive expression for the energy of excited states will be derived, which allows for a very efficient implementation of the solvation effects into the CI part of a molecular orbital program. The simplicity of the of the result also enables its © 1996 American Chemical Society
3350 J. Phys. Chem., Vol. 100, No. 9, 1996
Klamt
transfer to a nonlinear extension of the COSMO approach, i.e., the COSMO-RS approach15 introduced recently. Starting from COSMO calculations for solute and solvent, this approach allows to take into account nondielectric screening behavior of the solvent. The need for such extensions has been demonstrated by Brady and Carr.11 Only a few preliminary applications using the COSMO implementation in MOPAC16 will be given to demonstrate the principal success of the approach and some qualitative features. Obviously, due to the well-known deficiencies of the MNDO Hamiltonians available in MOPAC in the calculation of electronic excitation spectra, a really satisfying quantitative reproduction of experimental data cannot be expected. Therefore, extensive quantitative comparison with experimental data will be given as soon as a COSMO implementation in one of the more appropriate INDO methods or in an ab initio MO program becomes available. Both implementations are just in preparation. Linear Theory In this theory section we will use the notations introduced in the basic COSMO paper14 to a large extend. Thus we start with a brief summary of the basic concept of COSMO. COSMO is a molecular shaped CSM, i.e., it constructs a molecular-shaped boundary surface between the solute molecule and the surrounding dielectric medium from intersecting vdW spheres. This surface is divided in a large number m of segments, on each of which the screening charge density arising from the polarization of the medium is assumed to be constant (for details see ref 14). Now the set of the m screening charges on the segments may be represented as an m-dimensional vector q. Let A be the interaction matrix of these charges, so that Aq is the vector or the electrostatic potentials arising on the m segments due to the presence of the screening charges and 1/2qAq is the total interaction energy of the screening charges. The charge distribution of the solute molecule is represented as a set of atom-centered multipoles. Typically, monopoles, dipoles, and quadrupoles are considered. Since these are closely related to the electronic density of the molecule, we denote the vector of the n multipoles of the solute as the density vector P. Here we slightly deviate from the original COSMO notation, where only point charges had been considered which had been represented by a vector Q. Let the n × m matrix B be the interaction matrix of the n multipoles with charges on the m segments. Then BP is the potential arising on the surface due to the solute and qBP is the interaction energy of the solute molecule with the screening charges. After this introduction of the basic entities the dielectric screening charges have to be calculated for a given solute density from the dielectric theory. The basic idea of COSMO compared to other CSMs is to avoid the solution of the complicated dielectric boundary conditions and to start from the much simpler boundary condition of vanishing potential which would be valid for a conducting medium. Thus the corresponding ideal screening charges q* are given by the equations
BP + Aq* ) 0 S q ) -A-1BP
for solvatochromic effects (see eq 2). It can be shown that this scaling approximation is highly accurate for high dielectric constants and even within an error of 10% for = 2. This approximation is the origin of the name COSMO, which stands for conductorlike screening model. Within the COSMO approach the real work is the inversion of the usually rather large matrix A, but this has to be done only once for a given solute geometry. Then we have the screening answer for any density of the molecule available by simple matrix multiplication, and we can insert the screening into the Hamiltonian and Fock matrixes and the consecutive SCF calculation is no more expensive than it is for molecules in vacuum. This makes the COSMO approach highly efficient compared to most of the other molecular shaped CSMs. Furthermore the simplicity of the model enables the calculation of analytic gradients and thus of efficient geometry optimization of the solute in the presence of the solvent, which is unique in this class of CSMs. Now it is the time to develop the theory for the calculation of electronic excitations. As the first step we have to calculate the self consistent ground state of the molecule in the solute by a COSMO-SCF calculation, at best including geometry optimization in the particular solvent. Then we end up with the ground state density P0 and the corresponding screening charges q(,P0)) f()q*(P0) ) -f() A-1BP0. Here is the total dielectric constant of the medium. For the calculation of vertical electronic excitations we then have to divide the ground-state screening charges into one part arising from the orientation of solute molecules and which thus has to be frozen during the excitation, and next the electronic part, which is assumed to follow the excitation instantaneously. According to the Clausius-Mosotti equation17 the electronic polarizability corresponds to a dielectric constant of el ) n2, where n is the refractive index of the solvent. Obviously, if there where only electronic polarizability the ground-state screening charges would be given by q(n2,P0) ) f(n2)q*(P0). But it is absolutely wrong to assume that therefore the orientational part of the ground screening charges are given by qor ) q(,P0) - q(n2,P0), since the polarization effects are by no means additive. Following the concepts of Lippert and McRae this erroneous assumption has been made in all the continuum solvation applications to excited states published so far,6-9 except the recent work of Aguilar et al.10 Instead it is the electronic susceptibility χ ) - 1, which is the additive quantity, and the total polarization or screening charges split according to the ratio of χor ) - n2 to χel ) n2 - 1 into an orientational and an electronic contribution. Hence, correctly we have
qor )
χor χor - n2 q(,P0) ) f()q*(P0) ) q*(P0) χtot χtot + 1/
(4)
2
and
qel(P0) )
n2 - 1 q*(P0) + 1/2
(5)
(3)
and the total screening energy is given by -1/2q*Aq* ) -1/2PDP, where D ) BA-1B. The screening charges and energies in a dielectric medium are now approximated by scaling the ideal, i.e., the conductor results by a dielectric scaling factor f(). The functional form of this factor is given only approximately by the dielectric theory, but accidentally the COSMO choice for this form is exactly the form usually used
Thus for water with = 80 and n2 = 1.7 we have qor = 0.97q*, while the conventional, but erroneous, treatment yields qor = 0.66q*. The next step now is quite straightforward. For an arbitrary density P ) P0 + ∆, which may be interpreted as the density of an excited state, we have to consider the potential φ′ on the surface arising from this density and from the frozen screening charges qor, i.e.
Calculation of UV/Vis Spectra in Solution
J. Phys. Chem., Vol. 100, No. 9, 1996 3351
φ′ ) BP + Aqor
(6)
This potential therefore is screened by the electronic polarizability alone, i.e., by a dielectric medium of el ) n2, yielding the additional screening charges
q′ ) f(n2)A-1φ′ ) -f(n2)A-1(BP + Aqor)
(
) -f(n2)A-1B P0 + ∆ ) -f(n2)A-1B∆ -
)
- n2 P0 + 1/2
n2 - 1 -1 A BP0 ) -f(n2)A-1B∆ + qel + 1/2 (7)
Thus the total screening charge is
qtot ) q′ + qor ) -f(n2)A-1B∆ + qel + qor ) q(n2,∆) + q(,P0) (8) Apparently, we come out with the simple result that the total screening charge is composed of the total ground-state screening charge and the screening charges resulting from the electronic screening of the density difference, i.e., the system responds linearly by its electronic polarizability to the disturbance ∆ applied to the self-consistently screened ground state. Using this latter argument we easily find (a more detailed derivation is given in the Appendix) that the total screening energy, i.e., the interaction energy of the solute with the screening charges plus the energy needed to produce these screening charges is composed of three contributions:
∆E(,P0;n2,∆) ) ∆E(,P0) + ∆Bq(,P0) + ∆E(n2,∆) ) -1/2 f()P0DP0 + ∆Bq(,P0) 1
/2f(n2)∆D∆ (9)
i.e., the ground-state screening energy, the interaction energy of the density difference with the ground-state screening charge and finally the electronic screening energy of the density difference. Let us now apply eq 9 to excitation energies from the ground state to an excited state. Then to first order, i.e., assuming that the ground-state density and the excited-state density do not depend on and n2, the solvatochromic shift of a transition clearly is given by
ET ) ET,0 + f()∆exDP0 - 1/2f(n2)∆exD∆ex
(10)
Comparing this with eq 1 we find a significant difference: The f() term is not coupled to a f(n2) term in our result, while the last term is in formal accordance with eq 1. Nevertheless its origin is not due to the dispersion but due to the electronic relaxation of the system after an excitation process. As mentioned before, only the existence of contributions proportional to f() and f(n2) has been proven experimentally and thus eq 10 is in accordance with experiments as well as eq 1. Furthermore it should be noted thatsin contrast to a common interpretation in literaturesvanishing of the term proportional to f() does not imply that there is nearly no change in dipole moment during the excitation considered, but only that the interaction of this change with the ground-state screening charges is zero. In a self-consistent CSM-MO calculation, e.g., a COSMO calculation, we are not restricted to that first-order corrections for solvatochromic effects. First of all the ground state is
calculated self-consistently within a continuum of dielectric constant . Thus the induced solute dipole and higher moments depend on . Even more, in some cases, especially for zwitterions, the nature of the ground state (and consequently that of the excited states) may change drastically going from small to large values of . This implies an additional dependence of the solvatochromic shifts. The excited states have to be calculated from a CI calculation on the basis of the molecular orbitals and energies calculated from the ground-state density. The molecular orbitals and the corresponding eigenvalues automatically include the interaction with the ground state screening charges, so that in a CI the energies of the microstates only have to be corrected by the last term of eq 10, i.e., by the electronic screening energy of their density difference. This can easily be done with negligible programming and computational effort, if the screening energy is given by an explicit expression like that of COSMO or of CSMs dealing with simplified cavities. Unfortunately, the COSMO contributions to the nondiagonal elements of the CI matrix are not rigorously defined. This is due to the fact that the presence of the polarizable medium introduces an indirect interaction of charges, and thus we have additional electron-electron interactions which should be taken into account in the nondiagonal elements of the CI matrix. In general that would be feasible in close analogy to the usual Coulomb terms, but in contrast to the latter the indirect screening interactions also include an interaction of each electron with itself and such a self-interaction is incompatible with the CI concept. Only under the unrealistic assumption of an instantaneous instead of a mean-field reaction of the screening charges with respect to the electronic positions, these self-interactions could be written as one-electron integrals in the SCF calculation and thus neglected in the CI. This would require considerable work to implement the new type of integrals, which apparently is not justified. Instead we stay with the diagonal corrections in the CI matrix. From this matrix we calculate the multiconfiguration states of our system. This leads to a first approximation to the excitation energies. For a second and probably more exact estimate of the screening energy of the resulting states and hence their excitation energies, we can easily calculate the resulting multiconfiguration densities of these states and use these for the evaluation of the electronic screening correction, i.e., the third term in eq 10. If we now subtract the diagonal COSMO contributions from the eigenvalues and then add this new estimate of the screening correction, we end up with a second estimate of the excitation energies. Finally, we could also neglect the diagonal corrections in the CI matrix and solely apply the multiconfiguration correction to the resulting states. Thus we have three different estimates for the excitation energies, of which the second is probably the most accurate one. From the comparison of the three results we may derive an estimate for the magnitude of error arising from these conceptual problems. Nonlinear Extension Using COSMO-RS A critical microscopic consideration of the situation of a solute surrounded by solvent molecules reveals clearly that the dielectric theory should not be applicable to the orientational polarization of the solvent in this situation.15 Instead a theory has been developed (COSMO-RS15) starting from the ideal screening charge densities of solute and solvent in a conductor. The screening in a solvent is described by the pairing of surface segments having these ideal screening charge densities. By this theory it is demonstrated why the screening energies derived from the dielectric continuum models give reasonable results
3352 J. Phys. Chem., Vol. 100, No. 9, 1996
Klamt
TABLE 1: Experimental and Calculated Excitation Energies of Dyes in Solutiona calculated exp. dye
)2
) 4.8
vacuum geometry ) 80
CI size
S1
3.75
3.24
(3,1)
S2
5.38
5.12
(3,1)
S3
2.58
2.34
(5,2)
2.43
2.74
(4,2)
2.00
2.81
(5,2)
vacuum
)2
) 80
)2
4.02
3.84 3.90 3.90
3.69 3.77 3.77
3.66 3.74 3.74
3.42 3.45 3.45
4.94
4.83 4.81 4.81
4.68 4.66 4.66
4.82 4.80 4.80
4.63 4.61 4.61
3.17
3.03 3.03 3.03
2.89 2.89 2.89
3.02 3.02 3.02
2.78 2.77 2.77
2.40 S4
3.08 S5
2.10
2.89 2.89 2.89
) 4.8
optimized geometry
2.27 2.56 2.54
2.48 2.75 2.72
2.76 2.76 2.76
2.61 2.61 2.61
2.38 2.68 2.70
) 4.8
) 80
2.18 2.52 2.47
2.65 2.89 2.86
2.26 2.39 2.38
2.47 2.88 2.85
a Experimental data are taken from ref 1. The three calculated data in each box correspond to the three levels of approximation described in the text. All energies are in electronvolts. The CI size is given in the MOPAC notation.
for several important solvents like water and nonpolar solvents, despite the principal inadequacy of the dielectric theory. For other solvents, e.g., acetone and benzene, considerable deviations from a dielectric behavior should appear, and these are described quantitatively by the COSMO-RS formalism. By a systematic data analysis Brady and Carr11 have demonstrated that the parameters and n2 alone are not capable of describing the solvatochromic effects. This is a clear indication that nonlinear corrections beyond the dielectric theory are necessary for a general description of solvatochromism. The use of COSMO-RS may be a promising approach for such a generalized description of solvatochromic effects. COSMO-RS gives a statistical thermodynamic treatment of the screening problem. From this no unique screening charge density arises, but a mean screening charge distribution 〈q(P0)〉 can be derived for the ground state of the molecule. If we take advantage of the finding expressed by eqs 9 and 10 that the total solvatochromic shift of an arbitrary excitation is given by the interaction energy of the density change with the ground state screening charges plus the dielectric electronic screening energy of this density change, we only need to replace the dielectric screening charges q(,P0) by 〈q(P0)〉 in eq 9 and thus end up with the expression
ET ) ET,0 + ∆exB〈q(P0)〉 - 1/2f(n2)∆exD∆ex
(11)
for the mean value of the solvatochromic shift of an electronic excitation. The calculation of the last term in eq 11 stays unchanged, since the electronic polarization is well described by the dielectric theory. For the numerical implementation as a first step an SCF calculation has to be done, using the mean screening charges 〈q(P0)〉 as the source of an external potential. Now the situation is comparable to that at the end of the groundstate COSMO calculation: We have only to correct the diagonal elements of the CI matrix for the electronic screening energy of the density difference and thus find the mean excitation
energies from the CI. Hence there should not be any technical problems combined with the calculation of absorption spectra within the COSMO-RS formalism. Even more, the fluctuations in the screening charge distribution should give rise to a line broadening in the spectra. In principle, it should also be possible to calculate this part of the line broadening, but this might become rather expensive, since a separate SCF and CI calculation has to be performed for each screening charge vector taken into account. On the other hand, in each of these calculations the Green function D, the calculation of which is most expensive, can be reused, as long as the geometry is kept fixed. Some Applications To demonstrate the principal performance of the theory presented above, we now give a few example applications. These are done with an extension of the public domain COSMO implementation in MOPAC,16 and we generally have used the AM1 Hamiltonian.18 It should again be noted that AM1 is not very well suited for the calculation of electronic spectra, but we used it since no implementation of COSMO in one of the probably more appropriate INDO Hamiltonians is available yet. Thus we should not expect a perfect agreement of calculated and observed spectra, but we may consider whether the solvatochromic trends are well reproduced and have an estimate of the importance of the different contributions to the solvatochromic shift and of the accuracy of the approximations. We have chosen five examples (S1 to S5 in Table 1) from Reichardt’s book.1 For these, spectral data in different solvents are available, and considerable solvent effects have been observed. All structures have been optimized in the gas phase first. Then consecutive geometry optimizations in media of dielectric constants ) 2, 4.8, and 80 have been performed using COSMO, in order to simulate alkane, chloroform, and water as solvent. Then we started a CI calculation in vacuum and for the three values of , once at frozen gas-phase geometry
Calculation of UV/Vis Spectra in Solution and once for the appropriately optimized structures. All three levels of the CI-COSMO approximation have been evaluated. The results are presented in Table 1. In all of the test cases considered, we were able to reproduce the solvatochromic shifts qualitatively. In example S5 we could reproduce the change in the sign of the solvatochromic shift from alkane to chloroform compared to chloroform to water. Especially for those examples having a significant change in the mesomeric structure between a non-polar and a polar solvent (S1, S3, S5) the importance of the geometry optimization of the solute in the particular solvent was clearly demonstrated. The geometry changes, i.e., mainly the changes of the bond lengths which go along with a mesomeric change, could not be taken into account if the geometry is fixed at its gas-phase minimum. It turns out that these changes imply an effect which is comparable to the screening effect at fixed geometry. Thus it is in general of questionable use to calculate solvatochromic shifts at fixed gas-phase geometries, as has been done almost exclusively so far. COSMO is the only generally available continuum solvation model which allows for efficient and realistic geometry optimization in solvents and thus is uniquely suited for the calculation of solvatochromic effects. The comparison of the three different levels of approximation gives a rather clear picture: In all cases considered the results of methods 2 and 3, which calculate the shift on basis of the final CI densities of the states, are very close (within 0.03 eV), while method 1, which only takes into account the energy changes of the various configurations involved in the CI, often leads to considerable deviations of up to 0.4 eV. Since we can be quite sure that the methods 2 and 3 are closer to the physical truth, we have to disregard method 1. It should be noted that Rauhut et al.9 stay on this insufficient level. Even more, the coincidence of the results of the other two methods demonstrates that the diagonal changes in the CI matrix are of minor importance for the nature of the excited states. It should be noted that nearly all MO calculations of UV spectra so far have been calculated in vacuum but usually are compared with experimental data from nonpolar solvents. Our results demonstrate that there is a considerable systematic error in such a treatment, since the shift from gas phase to nonpolar solvents often is about half of the magnitude of the solvatochromic shifts between nonpolar and polar solvents. Thus, realistic calculations of UV spectra for dyes in solution necessarily require the consideration of the solvent, even if only nonpolar solvents are studied. Appendix For those doubting the short argumentation regarding the excitation energies, a comprehensive derivation of the resulting formula is given below: Let us start from the solute in its ground state, having the energy E1 ) E0 - 1/2f()P0DP0, where E0 is the electronic energy of the ground state and the latter term is the screening energy.
J. Phys. Chem., Vol. 100, No. 9, 1996 3353 Let us then freeze the reorientational part of the screening charges, i.e., qor ) ( - n2)/( + 1/2)q*. As long as we only freeze qor this does not change the energy of the system. As a next step we switch off the electronic polarizability, i.e., we remove qel. Obviously in this state 2 the system has an energy E2 which is different from E1. This energy can be calculated from the reverse step, i.e., by starting from the solute with frozen qor and screening it by the electronic polarizability. Apparently this step brings us back to state 1. Since the potential of the solute together with qor is φ′ ) BP0 + Aqor the electronic screening energy of state 2 is -1/2f(n2)φ′A-1φ′ and we have E2 ) E1 + 1/2f(n2)φ′A-1φ′. Starting from state 2, let us now consider an excitation which changes the density to P ) P0 + ∆. Let ET,0 be the change in the electronic energy corresponding to the excitation. Due to the presence of the frozen orientational screening charges, an additional energy ∆Bqor is required for the excitation, so that the energy of this excited state now is E3 ) E2 + Et,0 + ∆Bqor. Finally we may switch on the electronic polarizability again. Thus we gain the screening energy -1/2f(n2)φ′′A-1φ′′ with φ′′ ) BP + Aq0 ) φ′ + B∆. Hence the energy of our final state is E4 ) E3 - 1/2f(n2)φ′′A-1φ′′. Summarizing the energy changes of the different steps and making a few rearrangements yields our final result for the excitation energy ET in a solvent: ET ) ET,0 + ∆Bq0 1/ f(n2)∆D∆. 2 References and Notes (1) Reichardt, C. SolVents and SolVation Effects in Organic Chemistry; VCH: Weinheim, 1990. (2) McRae, E. G. J. Phys. Chem. 1957, 61, 562. (3) Lippert, E. Z. Elektrochem. 1957, 61, 952. (4) Cramer, C. J.; Truhlar, D. G. In ReViews in Computational Chemistry; Lipkowitz, K. B., Boyd, D. B., Eds.; VCH Publishers: New York, Vol. 6, in press. (5) Tomasi, J.; Persico, M. Chem. ReV. 1994, 94, 2027. (6) Bonaccorsi, R.; Cimiraglia, R., Tomasi, J. J. Comput. Chem. 1983, 4, 567. (7) Karelson, M. M.; Zerner, M. C. J. Phys. Chem. 1992, 96, 6949. (8) Fox, T.; Ro¨sch, N.; Zauhar, R. J. J. Comput. Chem. 1993, 14, 253. (9) Rauhut, G.; Clark, T.; Steinke, T. J. Am. Chem. Soc. 1993, 115, 9174. (10) Aguilar, M. A.; Olivares del Valle, F. J.; Tomasi, J. J. Chem. Phys. 1993, 98, 7375. (11) Brady, J. E.; Carr, P. W. J. Phys. Chem. 1985, 89, 5759. (12) Kim, H. J., Hynes, J. T. J. Chem. Phys. 1992, 96, 5088. Gehlen, J. N.; Chandler, D.; Kim, H. J.; Hynes, J. T. J. Phys. Chem. 1992, 96, 1748. (13) Basilevski, M. V.; Chudinov, G. E. Chem. Phys. 1991, 157, 327. (14) Klamt, A.; Schu¨u¨rmann, G. J. Chem. Soc., Perkin Trans. 2 1993, 799. (15) Klamt, A. J. Phys. Chem. 1995, 99, 2224. (16) Stewart, J. J. P. MOPAC program package (MOPAC7/MOPAC93), QCPE-N0. 455, 1993. Stewart, J. J. P. MOPAC93.00 Manual, Fujitsu Ltd., Tokyo, Japan, 1993. (17) Boettcher, C. J. F. Theory of Electric Polarization; Elsevier: Amsterdam, 1973. (18) Dewar, M. J. S.; Zoebisch, E. G.; Healy, E. F.; Stewart, J. J. P. J. Am. Chem. Soc. 1985, 107, 3902.
JP950607F
