J . Phys. Chem. 1991, 95, 8892-8899
8892
A New and Rigorous Method for Calculatlng Intramolecular Reorganization Energies for Electron-Transfer Reactions: Applied for Self-Exchange Reactions Involvlng Alkyl and Benzyl Radtcals Kurt V. Mikkelsen,* Department of Chemistry, H . C. 0rsted Institute, Copenhagen University, Universitetsparken 5, 2100 Copenhagen 0, Denmark
Steen U. Pedersen, Henning Lund, and Peter Swanstrram Department of Chemistry, Aarhus University, Lungelandsgade 140, 8000 krhus C, Denmark (Received: December 28, 1990; In Final Form: April 15, 1991)
A new method has been developed for obtaining the reorganization energy for the changes in the intramolecular degrees of freedom during an electron-transfer reaction. This method is based on the use of analytic gradient and Hessian formalism and is an improvement over previous efforts to obtain the intramolecular reorganization energy for larger organic molecules. This approach is expected to be useful, and a preliminary application is presented. The differences in reorganization energies measured for alkyl and benzyl radicals are discussed. Experimental ittvestigations have shown the existence of significant differences in reorganization energies for self-exchange reactions of aliphatic (50 & 10 kcal mol-') and benzylic (15 f 5 kcal mol-') radicals. Most puzzling is why these large differences appear for self-exchange reactions of alkyl anions/radicals and benzyl anion/radicals. The focus is on current experimental reactions of interest: self-exchangereaction with (i) benzyl anion and benzyl radical and (ii) butyl anion and butyl radical, as donor and acceptor. The difference in reorganization energy is investigated in terms of the conventional Marcus theory dividing the reorganization energy into two separate contributions, the solvent reorganization and the intramolecular reorganization.
Introduction In the past two decades electron-transfer (ET) reactions involving organic molecules have been intensively studied.' Well-documented examples of pure outer-sphere electron transfer from bulky inorganic transition-metal complexes to organic compounds have appearede2 Different metal-organic complexes have also been applied as electron carriers. Iron porphyrin complexes have been alkylated by alkyl halides in an electroreductive reaction, and the influence of steric hindrance by the alkyl halide of the iron porphyrin has been e ~ a m i n e d . The ~ conclusion was that steric hindrance could shift the reaction mechanism from SN2to outer-sphere ET. Within the field of single-electron transfer in organic chemistry inner-sphere electron-transfer reactions are classified those where the actual electron-transfer event is strongly coupled to bond formation between the donor and the acceptor. Outer-sphere electron-transfer reactions are those that do not involve any bond formation. Another classification within this field is nonadiabatic and adiabatic electron-transfer reactions. Investigations of nonadiabatic electron-transfer reactions require information about more than one electronic state, whereas investigations of adiabatic ET reactions only require information about the electronic ground-state potential energy surfaces. Pure outer-sphere electron transfer seems to occur either when one of the reactants is sterically hindered, making additional stabilization of the transition state and bond formation between donor and acceptor impossible, or when the electron of the donor is held in an extended r-electron system. The latter is the case for most anion radicals of aromatic compounds, and this type of compound has been used in many investigations of outer-sphere ET. The self-exchange reaction of aromatic radicals has been the subject of some studies! Recently (1) Cannon, R. D. Elecfron Transfer Reactions; Butterworth: London, 1980. Eberson, L. Elecfron Transfer Reactions in Organic Chemistry; Springer-Verlag: Heidelberg, 1987. (2) Kochi, J . K.Angew. Chem., Inr. Ed. End. 1988,27, 1227-1266. and references cited therein. Kochi, J. K. Acta Chem. Scand. 1990, 44, 409. Eberson, L. J . Am. Chem. Soc. 1983, 105, 3192. (3) Lexa, D.; Mispelter, J.; SavCnt, J.-M. J . Am. Chem. Soc. 1981, 103, 6806-6812. Lexa, D.;Sevbant, J.-M.; Wang, D. L. Organometallics 1986, 5, 1428.
Eberson and Shaik reported that the E T from anion radicals of aromatic compounds might have some inner-sphere character.s This opinion was supported by comparing heterogeneous (electrochemical measurements) and homogeneous (measured in homogeneous solution) self-exchange rate constants; the homogeneous rate constants were always found to be several orders of magnitude higher than the heterogeneous rate constants. The difference in free energies of activation was accounted for by postulating some additional stabilization ( 5 kcal mol-') of the transition state in the homogeneous self-exchange reaction of the aromatic anion radicals. Very recently a paper by Grampp et al. discussed these differences in some detail, and there it was emphasized that the electric double layer in the heterogeneous measurement was bound to cause problems. They recommended that 'to test the theory homogeneous self-exchange reactions are more useful".6 In a recent paper it was found that anion radicals of aromatic compounds in fact reacted with unhindered alkyl halides with some additional stabilization of the transition state.' This conclusion was based on comparison of activation parameters for ET reactions to stereochemically hindered and unhindered alkyl halides. ET reactions between two negatively charged species have also been studied. The most simple reaction is the disproportionation reactions between two identical species (eq I), but the ET reaction
-
-+ A2A + BZ-
A'- +- A'-
A
A'- 4- Bo-
B2-
H+
BH-
(1)
(2) (3)
between different anion radicals has also been examined, e&., the ET to anion radicals of azobenzene and halosubstituted azo(4) Kojima, H.; Bard, A. J. J . Am. Chem. Soc. 1975, 97, 6817. (5) Eberson, L.; Shaik, S.S.J . Am. Chem. Soc. 1990, 112,4484. (6) Grampp, G.; Kapturkiewicz, A.; Jaenicke, W. Ber. Bunsenges. Phys.
Chem. 1990, 94,439.
(7) Daasbjerg, K.;Pedersen, S.U.;Lund, H. Acra Chem. Scand. 19!41,45, 424. (8) Amatore, C.;Gareil, M.; SavCant, J.-M. J . Electroanal. Chem. 1983, 147, I .
0022-3654/91/2095-8892%02.50/0 0 199 1 American Chemical Society
Intramolecular Reorganization Energies
The Journal of Physical Chemistry, Vol. 95, No. 22, 1991 8893
benzenes, B- (eq 2)? These electron transfers to the ambenzenes are driven/forced by the subsequent fast protonation of the azobenzene dianion (eq 3). In particular, the ET reaction from anion radicals of aromatic or heteroarcmatic compounds (A*-) to alkyl1° or aryl" halides has attracted much attention. The ET from a donor to acceptors like alkyl halides (RX) differs from traditional ET reactions in that the cleavage of the carbon-halogen bond in the alkyl halide is concerted with the transfer of the electron, This variant of the ET reaction is now known as dissociative ET; the potential energy surfaces of such reactions have been described by Morse curves. It follows that Marcus-like relations persist, but the reorganization energy for the ET includes the dissociation energy for the cleaved bond.12 The scheme of eqs 4-9 is generally considered to describe the ET reactions between aromatic anion radicals and alkyl halides. A A'-
+ RX A'-
-+
+ ek5
A'-
A
(4)
R'
+ X-
+ R' -% A + R-
R-, AR-
RH, ARH
(5)
(7)
(9)
The reduction of aromatic and heteroaromatic compounds is normally reversible (eq 4) due to stabilization of A'- through extended delocalization of the added electron in the ?r-electron system. The standard potentials of the aromatic electron donor, can be measured very easily by cyclic voltammetry (CV). The self-exchange reorganization energy, AA(0),corresponding to the self-exchange reaction (eq lo), has also been measured for many aromatic compounds. This can be done either by measuring the heterogeneous ET rate constant, kg,, for the electrode reduction of A by fast CV and use of the Marcus theory for heterogeneous ET at an inert electrodeI3 or by obtaining the homogeneous ET rate constant, kp, from ESR experiments where a line-broadening due to the self-exchange reaction is 0b~erved.l~For most aromatic and heteroaromatic compounds the measured A,(O) is close to 10 kcal mol-'. In ref 5 values of A,(O) obtained from and koh were compared and it was found that the values based on homogeneous ET data were systematically 5-6 kcal mol-' lower than values based on heterogeneous ET data.
a,
kL
A'-
+A
-+ k 10
A
A'-
The rate of the outer-sphere ET reaction between A'- and RX varies according to the Marcus theory on the difference in redox potentials between the two reactants and on the reorganization energy, A, of the ET reaction. Marcus curves (log kS vs PA) have been made for many alkyl halides.7J0Js Recently it was claimed (9) (a) Ingemann, S.; Nielsen, M. F.; Hammerich, 0.Acra Chem. Scand. 1988, B42, 583. (b) Ingemann, S.;Larsen, K. V.; Haugshoj, K. B.; Hammerich, 0. Acta Chem. Scand. 1989, 43, 981-989. (IO) (a) Simonet, J.; Michel, M. A.; Lund, H. Acfa Chem. Scand. 1975, 829,489. (b) Lund, T.; Lund, H. Acra Chem. Scad. 1987, B41, 93. (c) Andrieux, C. P.; Gallardo, 1.; SavQnt, J.-M.; Su,K. B. J. Am. Chcm. Soc. 1986, 108, 638. (d) Daasbjerg, K.;Pedersen, S.U.; Lund, H. Acra Chem. Scand. 1989, 43, 876. (1 I ) (a) Lund, H.; Michel, M. A.; Simonet, J. Acta Chem. Scand. 1974, B28,900.(b) Andrieux, C. P.; Sav€ant,J.-M.; Su, K. B. J. Phys. Chem. 1986,
90,3815 and references cited therein. (12) SavQnt, J.-M. J. Am. Chem. Soc. 1987. 109,6788. (13) Kowert, B. A.; Marcoux, L.; Bard, A. J. J. Am. Chem. Soc. 1972,94, 5538. (14) Suga, K.; Ishikawa, S.; Aoyagui, S.Bull. Chem. SOC.Jpn. 1973,46, 808 and 755. (IS) Andrieux, C. P.; Gelis, L.; SavQnt, J.-M. J. Am. Chem. Soc. 1990, 112, 786. Andrieux, C. P.; Gblis, L.; Medebielle, M.; Pinson, J.; SavCant, J.-M. J. Am. Chem. SOC.1990, 112, 3509.
that a shift in mechanism from electron transfer to sN2 was observed for the reaction between anion radicals of anthracene and butyl bromide when the temperature was changed.16 These experiments have now been repeated, and no observation of a shift in mechanism is found. The same mechanism is found in the temperature range -50 to +SO 0C.7 The competition between reduction of the alkyl radical by another ET from A (eq 7) and the coupling of the same two species (eq 6) is going to be discussed further below. Both reactions are, however, always fast compared to the initial ET reactions (eq 5). Reaction 8 is an sN2 reaction where the rate of the reaction depends on the nature of the leaving group. For all the halides except the iodides the yield of R-R and of dialkylated aromatic compounds is less than 5% of the consumed alkyl halide. The protonation of the alkyl anions (eq 9) is most likely fast due to the high pK, values of the respective alkanes. This also explains why only minor dialkylated products are isolated. Generally very little is known about the thermodynamics of alkyl radicals due to their extreme reactivity. Some attempts have been made to measure their reduction potentials to determine under which conditions alkyl radicals can be considered and under which conditions these are further reduced to the alkyl anions. Griller and Wayner have measured reduction potentials for substituted benzyl radicals, but their method, where the radicals are generated photochemically and afterward are detected voltammetrically, requires some stability of the radicals." The time from generation to detection is too long for saturated alkyl radicals to persist. Saveant and co-workers have previously measured the self-exchange reorganization energies for butyl radicals/anions by direct cyclic voltammetry of the butyl iodides.18 The method is a p plicable only if the reduction potential of the alkyl radical is more negative than the reduction potential of the alkyl halide since two distinct reduction waves have to be observed in the voltammogram. For sec-butyl iodide and tert-butyl iodide this condition was met, but for n-butyl iodide the wave of the reduction of the n-butyl radical was hidden under the reduction wave for the n-butyl iodide. Here AR.(0) was found to be 50 f 10 kcal mo1-I for sec-butyl and rert-butyl radicals. These numbers were extracted from a comparison of experimental and simulated voltammograms and by optimizing at least five reaction parameters. This procedure makes the method tedious and the accuracy of the method difficult to predict. It is possible to utilize the competition between eqs 6 and 7 to extract rate constants for the ET reaction (eq 7) from aromatic anion radicals to alkyl radicals. This has been used to estimate reduction potentials, self-exchange reorganization energies, and standard potentials for benzyl and alkyl radicals when it is assumed that eq 7 is an outer-sphere ET reaction which can be described by Marcus theory.I9 The competition between eqs 6 and 7 can be studied by linear sweep voltammetry.20 The competition parameter q = k7/(k6 k,) can be obtained by observing the catalytic increase in the reduction current of the aromatic compound. It was shown that the value of q changes from zero to one when aromatic compounds with decreasing (more negative) standard potentials are used as electron donors. Reduction potentials, E,,'/2 can be measured at q= To convert the reduction potentials to thermodynamically significant standard potentials, knowledge of k6 of the reorganization energy for the ET reaction (eq 7) is required. Rate constants for the coupling reaction between aromatic anion radicals and alkyl radicals can be measured in experiments with
+
(16) Lexa, D.; Savbant, J.-M.; Su,K. B.; Wang, D.-L. J. Am. Chem. Soc. 1988, 110, 7617-7625.
(17) (a) Wayner, D. D. M.; McPhee, D. J.; Griller, D. J. Am. Chem. Soc. 1988, 110, 132. (b) Sim, B.; Griller, D.; Wayner. D. D. M. J. Am. Chem. Soc. 1989,111,754. ( I 8) Andrieux, C. P.; Gallardo, I.; SavQnt. J.-M. J. Am. Chem. Soc. 1989. 111, 1620. (19) (a) Fuhlendorff, R.; Occhialini, D.; Pedersen, S.U.;Lund, H. Aero Chem. Scand. 1989,43,803-806. (b) Occhialini, D.; Pedersen, S.U.;Lund, H. Acta Chem. Scand. 1990,44, 715. (20) Pedersen, S. U. Acra Chem. Scand. 1987, A l l , 391. Nadjo, L.; Saveant, J.-M.; Su,K. B. J. Eledroanal. Chem. 1985, 196, 23.
8894
The Journal of Physical Chemistry, Vol. 95, No. 22, 1991
cyclizable radical "clocks". Garst2' has reported values for k6 where the aromatic anion radical is generated by sodium in dimethoxyethane, and Pedersen and Lund22have measured rate constants, kb, for electrogenerated aromatic anion radicals in N,N-dimethylformamide. The rate constants were always found to be high and quite near the diffusion-controlled limit. Differences in PAand in solvent polarity seem to play a minor role in determining the value of k6 which is in accordance with a radical/radical coupling. A knowledge of kb permits the evaluation of the total reorganization energy for the ET from the broadness of curves in the q vs PA plot, and this further allows us to convert the reduction potentials E , 1 / 2to standard potentials. The XR(0) for the alkyl radical can be calculated from X by using the Marcus cross-relation and XA(0). These measurements have been performed for different benzyl and butyl radicals. XR.(0) was found to be 20 f 5 kcal mol-' for substituted benzyl radicals and 50 f 10 kcal mol-' for butyl radicals. This means that the reduction potentials for the benzyl radicals are almost equal to their standard potentials, whereas the reduction potentials of the butyl radicals are shifted negatively by more than 500 mV relative to their respective standard potentials.19 The potentials agree reasonably well with those previously published,l'J* the difference probably being due to a slightly different estimation of the reorganization energies. It is thus known that there is a substantial difference in X(0) between benzyl and alkyl radicals. In the present investigation this difference is studied with respect to structure, to barrier of rotation, and to charge distribution for the radicals and the anions by quantum mechanical calculation for benzyl radical/anion and n-butyl radical/anion.
Theoretical Section Theoretical investigations of single ET reactions in organic chemistry have mainly utilized conventional classical electrontransfer models.' Generally the Marcus-Hush formalism for ET reactions has succeeded in obtaining reasonable agreement with experimental investigations of ET reactions. It is a classical theory since the reaction is described in terms of classical degrees of freedom. The basics of the formalism are very similar to those of activated complex t h e ~ r i e s . ~ ~ - ~ ' In this paper the classical Marcus-Hush formalism is utilized to investigate the effects of reorganization energies on the ET rates for two different systems: self-exchange reactions for (i) benzyl radical/benzyl anion and (ii) butyl radical/butyl anion. The classical Marcus-Hush formalism has been chosen, since it is commonly used in the literature of ET reactions. The use of this causes some problems, but it is a model that makes
-
(21)(a) Garst, J. F. Arc. Chem. Res. 1971,4,400-406. (b) Garst, J. F.: Barton, F. E. Tetrahedron Lett. 1969,7, 587-590. (22)Pedersen, S.U.; Lund, T. Acta Chem. Scand. 1991,45, 397. (23)Newton, M. D.; Sutin, N . Annu. Reo. Phys. Chem. 1984,35, 437. (24)Marcus, R. A.; Sutin, N. Biochim. Biophys. Acta 1985,811, 265. (25)Mikkelsen, K.V.; Ratner, M. A. Chem. Reo. 1987,87, 113. (26)Ulstrup, J. Charge Transfer Processes in Condensed Media; Springer-Verlag: Berlin, 1979. (27)DeVault, D. Quantum Mechanical Tunneling in Biological Systems; Cambridge University: New York, 1984. (28)Faraday Discuss. Chem. Soc. 1982,74. (29)Prog. Inorg. Chem. 1983, 30. (30)Kuznetsov, A. M.; Ulstrup, J.; Vorotyntsev, A. M. In The Chemical Physics ojSoloation. Parr C; Dogonadze, R. R., Kalman, E.,Kornyshev, A. A., Ulstrup, J., Eds.; Elsevier: Amsterdam, 1988;p 163. (31) Marcus, R. A. J . Chem. Phys. 1956,24, 966,976;1965, 67,853. 2889;Trans. N. Y.Acad. Sci. 1957.19.423;Annu. Reo. Phys. Chem. 1964, IS. 155; J . Chem. Phys. 1965,43. 619. (32)Hush, N.S.Trans. Faraday Soc. 1961,57,557; Prog. Inorg. Chem. 1967,8, 391;Electrochim. Acta 1968,13, 1005. (33)Levich, V. G.;Dogonadze, R. R. Dokl. Acad. Nauk. SSSR 1959,124, 123. Levich, V. G. Adu. Electrochem. Electrochem. Eng. 1966,16, 155. (34)Kestner, N.R.; Logan, J.; Jortner, J. J . Phys. Chem. 1974,78,2148. (35)Ulstrup, J.; Jortner, J . J . Chem. Phys. 1975,63,4358. 1977.26, (36)Fisher, S.F.; Van Duyne, R. P. Chem. Phys. 1974,5,183; 9.
Mikkelsen et al.
* Be C
w
Q'
RC
Figure 1. Potential energy surfaces for a self-exchange electron-transfer reaction. RC is a generalized reaction coordinate, and Q* corresponds to the location of the transition state. The reorganization energy is shown as h and the free energy of activation by AG#,
an immediate connection to experimental data. The electron-transfer system is represented by potential energy surfaces as shown in Figure 1. The transition state corresponds to the configuration at Q*,and in the activated complex theories the electron transfer occurs at the intersection point between the potential energy curves for the initial and final states. The electronic coupling between the donor and acceptor molecules is assumed to be so large that there is no interference with abovelying potential energy surfaces. This type of ET reaction is termed adiabatic, whereas ET reactions that require knowledge of other ptential energy surfaces are termed nonadiabatic. The expression for the ET rate constant is kET= Z exp(-AG*/ksT)
(1 1)
where Z is either the collision frequency in a bimolecular reaction or the frequency of the active vibrational mode in a monomolecular reaction. The exponential factor contains the free activation energy AG* and the thermal energy kBTas expected for activated complex rate constants (kBis Boltzmann's constant). The free energy of activation AG* is that associated with the ET reaction between the solvated donor and acceptor. This rate constant does not include quantum effects and a ~ s u m e s ~ ~that - ~ O(i) the transition conserves energy and that activated states of the reactants and products have the same energy; (ii) the Franck-Condon principle is valid, so that the change in electronic structure occurs at a fixed nuclear momentum and position; (iii) the electronic coupling between donor and acceptor is assumed to be large enough so that the reactants are converted into products with unit probability in the vicinity of Q*; and (iv) solvent dynamics are negligible and the solvent is assumed to be in thermodynamic equilibrium during (37)Schmidt, P.P.J . Chem. Soc.. Faraday Trans. 2 1973,69,1104: J . Electroanal. Chem. 1977,82,29. (38)Ratner. M. A.: Madhukar. A. Chem. Phvs. 1978.30. 201. (39j Dogonadze, R.' R.;Kuznetsov, A. M. S&f. Sci. 1975,6, 1; Phys. Status Solidi B 1972,54,439. (40)Larsson, S.J . Am. Chem. Soc. 1981, 103,4034;Chem. Phys. Lett. 1982,90, 136;J . Chem. Soc., Faraday Trans. 2 1983,79, 1375;J . Phys. Chem. 1984,88,1321. (41)Newton, M. D. Inr. J . Quantum Chem. Symp. 1980,14, 363;Ado. Chem. Ser. 1982,No. 198,255;J . Phys. Chem. 1986, 90,3134. (42)Mikkelsen, K. V.; Dalgaard, E.; Swanstrom, P. J . Phys. Chem. 1987, 91,3081. Mikkelsen, K.V.; Ratner, M. A. Inr. J . Quantum Chem. Symp. 1987,21,341:1988,22,707; J . Phys. Chem. 1989,93,1759;J . Chem. Phys. 1989,90,4237. (43)Mikkelsen, K. V.; Agren, H.; Jensen, H.,J. Aa.; Helgaker, T. J . Chem. Phys. 1988,89,3086. Medina-Llanos, C.; Agren, H.:Mikkelsen, K. V.: Jensen. H. J. Aa. J . Chem. Phys. 1989,90, 6422. Mikkelsen, K. V.; Agren, H. .J. Phys. Chem. 1990, 94,6220.
The Journal of Physical Chemistry, Vol. 95, No. 22, 1991 88%
Intramolecular Reorganization Energies the reaction. It is also assumed that the coupling between the donor and acceptor is not too large and can be neglected in the calculation of the free energy of activation. If the coupling were very large, the free energy profile would change in the vicinity of the transition state and the activation energy would be decreased by an amount similar to the coupling energy. Conventionally, the free energy of activation is determined without such considerations. Marcus calculated the free energy of activation using statistical mechanical arguments and concentrated on the nuclear rearrangement necessary for the description of the activated state. The nuclear configuration included the hydration shells of the donor and acceptor and the polarization effects in the solvent. The free energy of activation is given bys0
AG* = (A
+ AG0)2/4X
(12)
where AGO and X are the free energy change and the reorganization energy for the ET reaction, respectively. The reorganization energy is the energy required to change the equilibrium nuclear configuration before electron transfer to the equilibrium configuration after electron transfer, while keeping the transferring electron on the donor. The reorganization energy is separated into two contributions, the inner-sphere Xi (intramolecular) and the outer-sphere X, (solvent) reorganization free energy.30 X = Xi
+ A,
= 72zjkkHjkqjqk
(14)
The modes in eq 14 are general modes, whereas for normal modes, Xi can be written Xi = Y Z j k ~ j Q t
(15)
In both cases kH is Hooke's law force constant which is approximated by a reduced or averaged force constant for the mode. This force constant is given as an average between the force constant for the reactants (f)and the force constant for the products Ctp). For a given mode kH is given by22-24 kH =
2yp(f+ p)-l
(1 / 8 d [
J71 /€,p)(AD(€,p))2dr - $0 /%t)(AD(fJSdr]
(17) The dielectric medium is characterized by two dielectric constants, the optical dielectric constant (%) and the static dielectric constant (est). The optical dielectric constant refers to the optical polarization, whereas the static dielectric constant refers to the total polarization. The total polarization is the vector sum of the optical and inertial polarization vectors. The difference in the electric displacement vectors for the reactant and the product states is given by
W e ) = Wp,e) - D(p,,c)
A, = ( 1 / 8 4 [ ( 1 / c o p-) (1/€st)1J(AD)2dr
The vibrational displacement coordinates are represented by q,. The outer-sphere reorganization energy, A,, is the energy for reorganizing the solvent for an electron-transfer reaction. The outer solvent is the solvent oustide that part of the ET system considered the inner-sphere. A, depends on the properties of the outer solvent and can be calculated by using various models. Usually the outer solvent is approximated by a dielectric medium, and depending on the representation of the dielectric function for the medium, one can have either advanced or very simple descriptions of the outer solvent. These expressions are more or less reminiscent of the work by Born.45 The charge distributions associated with the reactants and products induce polarization charges in the outer solvent. Usually this polarization is made up of two parts, the optical polarization and the inertial polarization. The optical polarization represents the response from the solvent due to the electronic degrees of freedom. This polarization responds rapidly to changes in the charge distributions for the molecular ET system. The inertial polarization stems from the vibrational-orientational motion of the solvent molecules and responds much more slowly. Therefore, it is the inertial polarization that contributes to the free energy of activation, and in terms of the dielectric Green function formalism by Newton and Friedman the solvent reorganization energy is given as4* (44) Mikkelsen, K. V.; Newton, M . p. Submitted for publication. (45) Born, M. 2.Phys. 1920, I , 45. Agren, H.; Mikkelsen, K. V. Theor. Chem., in press.
(19)
One can restate this in the form46 XO = [ ( I
/cop)
- (1 /%t)l&?(rD*rA,d)
(20)
where rD and rA describe the size of the donor and acceptor, respectively, and the distance between the donor and acceptor is d. The function g(rD,rA,d)can be expressed by using various models. One model locates the two compounds in two separate spherical cavities embedded in a dielectric medium with the centers of the spherical cavities separated by the distance d. Other models enclose the donoracceptor encounter in one cavity, also embedded in a dielectric medium. Expressions for the solvent reorganization energy have been worked out for spherical and ellipsoidal cavities. For two spherical reactants embedded in a dielectric medium separated by the distance d, one obtains the following expression for the solvent reorganization energy A, = - ( I / € s ~ ) I ( A ~ ) ~+ [ (( 1~/ 2/ r~A ~) D -)
(1/8r)[(1/cop)
(16)
(18)
where D(p , c) is the electric displacement vector due to the presence of a charge distribution ppr and a dielectric medium represented by a dielectric constant t. The charge distribution p and p r are for the reactant and product states, respectively. denerally the relation between the displacement vector and the charge distribution depends on the electric function for the medium. In most cases image effects are neglected, and in such cases the displacement vector is independent of the dielectric function so that eq 17 can be expressed as
(13)
The inner-sphere reorganization energy is thus the energy required to reorganize the intramolecular degrees of freedom and is calculated from the parameters of the inner-sphere vibrational modes, and changes in bond lengths and bond angles. By assuming the harmonic approximation for the displacements of the internal coordinates, Xi can be written Xi
A, =
(21) where At- is the number of electrons transferred during the process. The expression (eq 21) for the solvent reorganization energy is valid only for situations where the intermolecular distance d is larger than the sum of the two radii rDand rA. This expression has been refined, and for any distance between the donor and acceptor one has46 A',
= (1/8r)[(1/Eop) - (l/t,t)l(Ae)Z[(1/2rD)+ ( I / z r A ) - (1 /d) - t r D / ( 8 - rD2) + rA/(d2 - rA2) 1/ d In ( d rD)(d r A ) / ( 8- r D 2 ) ' / ' ( 8 - rA2)'/2]]( 2 2 )
+
+
For a bimolecular ET reaction in solution the donor and acceptor compounds form an encounter complex. The two spherical cavities are then in contact and d = rD + rA;assuming rD = rA = r, we obtain the following expression for the solvent reorganization energy using eq 22. A,' = ( 1 / 8 ~ ) [ ( l / e ~ J - ( 1 / d l ( A e ) 2 [ 1 / 6 r ( 3In 3
- 1)1 (23)
The continuum approach for calculating the solvent reorganiz(46) German, E. D.; Kuznetsov, A. M. Electrochim. Acta 1981,26, 1595. Kharkats, Y. I. J . Chem. Sac., Faraday Trans. 2 1974, 70, 1345. (47) Grampp, G.; Jaenicke, W. Ber. Bunsenges. Phys. Chem. 1984.88, 325. Grampp, G.; Jaenicke, W . J . Chem. SOC.,Faraday Trans. 2 1985,81. 1035. Grampp, G.; Harrer, W.; Jaenicke, W. J . Chem. Soc.,Faraday Trans. I 1987,83, 161. (48) Newton, M. D.; Friedman, H J . Chem. Phys. 1988,88,4460; 1988, 89, 3400 (erratum).
8896 The Journal of Physical Chemistry, Vol. 95, No. 22, 1991
vation energy has been extended to nonspherical compounds. The donor and acceptor compounds are then assumed to have an ellipsoidal shape. Inner-sphere reorganization energies have mainly been calculated for donor/aoceptor systems containing metal ions and ligands. Very few attempts have been reported to calculate the inner-sphere reorganization energies for ET systems consisting of organic compounds. Grampp and Jaenicke have employed a method based on a semiempirical relation between charge densities and force constants.4' This approach employs a number of approximations, but it certainly illustrates the difficulties in calculating the inner-sphere reorganization energy. In this paper a new approach is presented for calculating the inner-sphere reorganization energy for ET reactions. This method is based on modern electronic structure formalism, namely analytical methods for calculating the molecular gradients and Hessian.49 The total energy for a molecular compound can be expressed as a Taylor function of the nuclear degrees of freedom, Le., the vibrational degrees of freedom q = (qi)around a given reference configuration E(q) = E(qo) + GTAq + AqTHAq
(24)
where q is a vector containing the internal coordinates for the molecule. The superscript T indicates transposition of the vector. The vector qo contains the internal coordinates for the molecule at the reference configuration and Aq is the difference vector given by (25) Aq = q - 40 The first and second derivatives are the gradient and Hessian matrices:
For the optimized structure the gradient vector vanishes and the Hessian is termed the force constant matrix. The force constant matrix is given by the Hessian matrix for the optimized molecular structure.
F = H(q = qq)
(28)
The fully optimized molecular equilibrium structure is determined for the reactants and the products in the ET reaction. At the same time the force constant matrix is determined and used in conjunction with the change in the internal coordinates to obtain a projected force constant for the reactants and products undergoing an ET reaction. The vectors qp and q:.contain the values of the internal coordinates for the two optimized molecular structures for the reactant and the products, respectively. The projected force constant is given by
/" = AqTFAq/lAq12
(29)
where 1Aq1 is the norm of the vector Aq. Hereby we have also included the off-diagonal terms in the Hessian matrix, and this goes beyond the standard treatment where one performs a summation over each bond length and angle that changes during the reaction. In the standard scheme one sums for all bonds the product of the force constant and the square of the changes for each bond. Thereby one does not include the coupling between the different vibrational modes. The calculations presented in this paper have shown that off-diagonal terms in the Hessian are not only nonzero but also important. The force constants obtained for the reactants and products are projected along the reaction coordinate as seen in eq 29. The inner-sphere reorganization energy is given by The expression for the intramolecular reorganization energy does (49) Geometrical Derivatives of Energy Surfaces and Molecular Properties: Jsrgensen, P.; Simons, J., Eds.; Reidel: Dordrecht, 1986.
Mikkelsen et al. TABLE I: Bond Lengths (angstroms) for the Carbodarbon Bonds in Benzyl Anion and Benzyl Radical bond benzyl anion benzyl radical CI-C2 C2C3 c3-c4 c4-c5
1.3979 1.3617 1.4580 1.3531
1.3872 1.3839 1.3976 1.4738
not include the nuclear tunneling corrections. It is important to note at this point that the method derived in this section makes use of projected force constants. These force constants are related to generalized reaction coordinates that describe the overall changes in the intramolecular degrees of freedom. These are not force constants for separate vibrations but generalized reaction coordinates that correspond to motion of a much larger mass than the ones involved in standard schemes. Due to the larger mass corresponding to the generalized reaction coordinates, tunneling corrections are less important in the present treatment than in the standard schemes. Our method employs an ab initio electron description for obtaining the optimized molecular structures and force constants. At this level reorganization energies for the inner-sphere have been evaluated for very few systems and always only for small metal complexes. Such systems are very simple to investigate in this respect since it is a question of optimizing the bond lengths between the metal ion and the ligands. During this optimization, which consists of pointwise total energy calculations for different bond lengths between the metal ion and the ligand, one also obtains the second derivative of the total energy with respect to the metal-ligand distance. This is then related to the force constant through appropriate mass weighting, and in this case both the force constant and the changes in the molecular structure are obtained relatively easily. This process is much more difficult for larger organic compounds, since it usually involves changes in a large number of intramolecular degrees of freedom so the force constant cannot be obtained as simply as for the metal ion-ligand systems. Structural and Other Related Aspects for the Self-Exchange Reactions. A full optimization is performed of the molecular structure for four different compounds participating in two self-exchange reactions:
-
CH,CH,CH,CH,- + CH3CH2CH2CH2' CH,CH2CH,CH; + CH3CH2CH2CH2- (32) These reactions are symmetric and termed self-exchange reactions. Our aim is to discuss the difference in the self-exchange reorganization energies for the two types of radicals in terms of structure, charge distribution, rotational barriers, and force constants. Standard methods are applied for optimizing the molecular structures providing the equilibrium bond lengths and angles, force constant matrix, and usual properties such as total energy and charge distribution. At first, the aromatic reactants in eq 3 1 are considered. For both species planar structures are obtained, with the CHI methylene group located in the same plane as the phenyl group. This enables bonding interaction between the phenyl T system and the 2p orbital of the CH2 group. This effect is most significant for the anion, and for this compound one notices a substantial charge delocalization from the CH2 group into the aromatic T system. This substantially stabilizes the anion, and the rotation of the CH2 group around the carbonarbon bond is expected to be hindered. The spin-restricted Hartree-Fock (HF) method is applied,51 (50) Holstein, T. In Tunneling in Biological Systems; Chance, B., DeVault, D. C., Frauenfelder, H., Marcus, R. A,, Schrieffler, J. B., Sutin, N., Eds.; Academic Press: New York, 1980. (51) Binkley, J. S.; Frisch, M. J.: DeFrees, D. J.; Rahgavachari, K.; Whiteside, R. A.; Schlegel, H. B.; Fluder, E. M.; Pople, J. A. Gaussian 82; Carnigie-Mellon University: Pittsburgh, PA, 1982. Hehre, W. J.; Radom, L.; Schleyer, P. v. R.; Pople, J. A. Ab-initio Molecular Orbital Theory: Wiley: New York, 1986.
The Journal of Physical Chemistry, Vol. 95, No. 22, 1991 8897
Intramolecular Reorganization Energies
C6H5CH2 anion
-0,s ,
TABLE 11: Bond Lengths (angstroms) and Angles (dcgnes) for the Carbon-Carbon Bonds in Butyl Anion and Butyl Radical bond butyl anion butyl radical Cl-C2 1S638 1.5199 1 S756 C2-C3 1.5502 c3-c4 1.5475 1.5412 c 1-c2-c3 1 18.33 1 12.48 111.78 c2-c3-c4 1 13.58 102.70 alfl 118.91 105.38 alf2 109.22 108.70 alf3 109.70 110.74 1 1 1.28 alf4
-om{ -0,70f 0
n-Butyl anion
- .
I
.
20
.
I
40
- .
I
60
- -
I
-
.
80
I
-0,586 7
100
Twlrt rnglr (drgroor) -0,588
I
El
1
C6H5CH2 radical -0,828
I
m I
-0,828
-0,594
7 30
0
60
90
120
150
180
Twlrt rnglr
n-Butyl
-0,838
radical
-0,820
- 0 , 8 3 8 , . . I . . I . . I . . I . . 0 20 40 80 80
100
T w l d onglr (dogrror)
Figure 2. Total energy for the benzyl anion and the benzyl radical as a function of the rotation angle around the Ca-CI bond.
since the spin-unrestricted H F calculations implied higher states of different multiplicity giving rise to average values for the S2 operator very different from 0 (singlet) and 0.75 (doublet). For some of the spin-unrestricted HF calculations average values of about 1.22 are obtained for the S2 operator, indicating improper contributions to the electronic wave function from states with higher multiplicity. The proper multiplicity is ensured by applying the spin-restricted HF for both compounds. The STO-3G basis set has been chosen for these preliminary calculations because this is sufficient for a reasonable geometry optimization of the compounds of eqs 31 and 32. In Table I the bond lengths between the carbon atoms in both the benzyl anion and the benzyl radical are presented. The changes are most noticeable in the vicinity of the CH2group. The structure of the radical can be divided into an aromatic phenyl system and a CH2group with some bonding ?r interaction between the two systems. The benzyl anion has a much larger aromatic system which includes the CH2 group, and the bond length between the carbon atoms numbered a and l is substantially shorter than the corresponding one for the benzyl radical. Figure 2 presents the total energy for the two compounds as a function of the rotation angle around the Ca-Cl bond. At the angle Oo the molecular structures are the fully optimized ones and are of planar conformation. The CH2 group is rotated in steps of 15' to 90°, where in the latter case the plane of the CH2group is perpendicular to the aromatic plane. The calculations involve full optimization of the molecular structures while the rotation angle is kept fixed. The energy rotation barrier is substantially higher for the anion than for the radical. The benzyl anion is expected to have a rigid planar conformation, whereas the benzyl radical has a more freely rotating CH2 group.
-0,823 0
30
80 90 120 150 Twlrt rnglr (drgroor)
180
Figure 3. Total energy for the n-butyl anion and the butyl radical as a function of the rotation angle around the CI-C2 bond.
The changes in the molecular structure are also observed as a function of the rotation angle. For both compounds we observe significant changes in the bond length between the carbon atoms Ca and C1 and in the charge on the carbon atom (Ca) in the CH2 group. For the benzyl anion there are large changes in the bond length and the charge on the carbon atom Ca. The bond length goes from being of highly double bonding character to a single carbon-carbon bond length when the CH2group is perpendicular to the aromatic plane. At the same time the charge on the carbon atom Ca increases by a factor of 1.6. The benzyl anion proceeds from being a highly delocalized system spanning all seven carbon atoms to a system with a higher degree of charge localization, consisting of an aromatic system plus a CHI group. The benzyl radical does not exhibit a change of similar magnitude when the CH2 group is rotated, but a similar trend is observed as for the benzyl anion. The opportunity of bonding interactions between the aromatic system and the CH2 group is decreased when the CH2 group is rotated. The aliphatic species in the self-exchange reaction (eq 32) are of a different character compared with the ones discussed above. They are much more floppy and do not contain an aromatic system
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The Journal of Physical Chemistry, Vol. 95, No. 22, 19'91
TABLE 111: Solvent and Inner Reorganization Energies for Benzyl Radical/Anion and Butvl RadicrVAnion
radical benzyl butyl
?/A 3.48 3.37
AJkcal mol-I &'/kcal 22.1 22.8
mol-' 16.9 17.5
hi/kcal mol-' 13.4 53.8
"Size calculated from density data for toluene and butane and by assuming a spherical structure. enabling charge delocalization. Again, a full optimization of the molecular structure is performed, and Table I1 shows some of the quantities for the butyl anion and the butyl radical. It is mainly in the area of the terminal CH2group that one observes differences between the two structures. The bond length CI-C2 and angle alfl differ significantly, giving rise to very different structures for the butyl anion and butyl radical. We do not observe any significant delocalization of the extra charge on the butyl anion. This extra charge is mainly located on the terminal CH2 group. Larger structural differences are observed between the two aliphatic compounds than between the two benzylic systems. Also for the aliphatic compounds the CH2 group is rotated around the terminal carbon-arbon bond C1-C2. As before, we perform a full optimization for the molecular structures while keeping the rotation angle fixed. In Figure 3 the total energy as a function of the rotation angle is presented. For both anion and radical a function is obtained with a periodicity of about 120' and a lower energy barrier for the rotation than that for the aromatic systems. Again, the energy barrier is larger for the anion than for the radical. This may be attributed to the increased electron-electron repulsion that occurs when the CH2 group is rotated away from its equilibrium conformation and toward neighboring groups. The charge distribution is virtually independent of the rotation angle. Also, the molecular structures change very little while the CH, group is rotated. Calculation of the Solvent Reorganization Energy and the Inner-Sphere Reorganization Energy. This section focuses on the origin of the difference in the reorganization energies for the two reactions 3 1 and 32. The reorganization energy contains contributions from the outer solvent and the internal degrees of freedom as seen from eq 13. The solvent reorganization energy is calculated by using expressions 21 and 23. All of these expressions utilize a model that describes the outer solvent as a dielectric medium. The expressions contain quantities representing the sizes of the donor and acceptor participating in the ET reaction. The sizes of the four compounds in reactions 31 and 32 are estimated, and these quantities are presented in Table 111. The two reactions take place in N,N-dimethylformamide, and the static and optical dielectric constants for this solvent are 36.7and 2.04, respectively. In Table 111 we present the calculated solvent reorganization energies for the two reactions using the expressions 21 and 23. The difference between the solvent reorganization energies for the two reactions is very small. As mentioned in the Introduction, the difference in reorganization energy for the two reactions is large and therefore this difference is not explained only by the difference in the solvent reorganization energy. This approach does not take account of specific solvent effects. Differences in specific solvent effects could arise due to the different types of charge distribution: a localized vs a delocalized charge distribution. This aspect is currently being investigated, and the preliminary conclusion is that a given molecule with a localized charge distribution is slightly better solvated than the same molecule with a delocalized charge d i ~ t r i b u t i o n .The ~ ~ difference is about 10 kcal mol-'. So again this does not explain the large difference in reorganization energy for the two reactions. The solvent reorganization energies are only calculated by using the continuum model with the donor and acceptor placed in separate spherical cavities. The four compounds are of similar molecular size. Therefore, the difference in solvent reorganization energy for the two reactions does not change significantly when a continuum model, in which the compounds are located in elliptic cavities, is used.
Mikkelsen et al. Next the inner-sphere reorganization energy is considered. This reorganization energy expresses the energy involved in changing the bond length and angles for the donor and acceptor molecules. As shown under Theoretical Section, the evaluation of the inner-sphere reorganization energy requires determination of the force constants and the changes in the internal degrees of freedom. The calculations determining the fully optimized molecular structures also determine the force constant matrix for this configuration. The structures of the reactants and products are also known since the ET reactions are self-exchange reactions. The structures of the four compounds were determined by performing a full optimization of the molecular structures, and from these the changes in the internal degrees of freedom are obtained. The force constant matrix is projected according to eq 29 by using the vector describing the differences in the internal degrees of freedom going from the reactants to the products. The projected force constant matrix then gives the force constant for changing the internal degrees of freedom from the reactant state to the product state. The average force constant is calculated by using eq 16 to be 0.9892 and 0.1812au for reactions 31 and 32, respectively. The norm of the vector Aq is 0.208 and 0.9735au for reactions 31 and 32,respectively. These values enable the calculation of the inner-shell reorganization energy by using expression 30. The obtained values for Ai are 13.4 and 53.8 kcal mol-' for reactions 31 and 32,respectively.
Discussion Very significant differences have been found in structure and charge distribution for the two types of radicals/anions investigated here. In the benzyl radical/anion a highly delocalized charge distribution is observed and the bond to the benzylic carbon adapts some double-bond character. The charge delocalization is also manifested in the rather large energy barrier for rotating the CH2 group and the flatness of the structure found for the benzyl anion as well as for the benzyl radical. On the other hand, the butyl radical/anion behaves quite differently. There is no significant charge delocalization in the butyl anion, and the expected pyramidal structure is found here. Considerable reorganization of the structure is therefore needed to form the flat structure of the butyl radical. It is obvious that no double-bond character is found for the butyl anion. According to the calculated values for A,, and &' for the systems, the difference in solvent reorganization energy for the two selfexchange reactions is insignificant. The A,' is considered to be more correct than A,, as discussed previously. The A,' can, however, be refined by taking into account the delocalization of charge. A calculation for the benzene anion shows,52however, that the contribution from this refinement is quite limited since for the benzene anion it is found to be less than 3 kcal mol-'. For even further refinement the calculation of A,' should take into account the specific solvation of the system, but the increase in computational time would be ~ubstantial.~, The method developed for calculating Ai makes use of accurate analytical gradient and Hessian formalism. In contrast to the conventional methods, this method takes into account the couplings between the intramolecular modes by utilizing the full Hessian matrix. The average force constant is obtained by projecting the force constant matrix along the reaction path. The use of this method has been demonstrated by calculating intramolecular reorganization energies for the two self-exchange reactions in eqs 31 and 32. The calculations clearly show that the difference in reorganization energy stems from the reorganization of the intramolecular degrees of freedom. The origin of this is due to structural differences previously discussed. The absolute values of Ai are of a preliminary status since they were obtained with a basis set which is designed for geometry optimizations but not for reliable energy calculations. Inclusion of correlation energy and flexible basis sets will clearly improve (52) Linse, P.; Mikkelsen, K. V. J . Phys. Chem. 1991, 95, 4843; work in progress.
8899
J . Phys. Chem. 1991, 9 5 , 8 8 9 9 - 8 9 0 9 the energy calculations and we are currently investigating these issues. The method represents clearly an improvement compared to previous methods. In the near future the method will be applied to self-exchange reactions, where the geometries for the donor and acceptor are known from experimental investigations. The method can also
be applied to larger organic molecules since some semiempirical electronic structure models give methods for evaluating and utilizing the molecular gradient and Hessian. Registry No. Benzyl anion, 18860-1 5-6; benzyl radical, 2 154-56-5; butyl anion, 79431-01-9; butyl radical, 2492-36-6.
Molecular Slmulatlons of Anion-Exchange Systems: Quaternary Methylammonium Ion Functional Group with Various Anions D. Y. Paithankar and N.-H. L. Wang* School of Chemical Engineering, Purdue University, West Lafayette, Indiana (Received: November 9, 1990)
47907
Molecular simulationshave been used to understand the binding in ion-exchange systems. Monte Carlo simulationsin conjunction with statistical perturbation theory have been performed to evaluate the relative free energy of association of various ions toward quaternary methylammonium ion (QMA). The systems simulated consist of QMA with an anion in aqueous solution. The differences in association of QMA with fluoride and chloride, chloride and bromide, acetate and acetamide, and two forms of aspartic acid have been studied. Information on both the energetics and structure is obtained. The ratio of association constants of fluoride and chloride ions cannot be predicted closely. But the predicted ratio of association constants of chloride and bromide agrees with the experimental observations. Also, close agreement is obtained between the predicted ratio of association constants of acetate and acetamide with the experimental data. The simulations of two forms of aspartic acid confirm the previous experimental evidence that oppositely charged groups in the neighborhood of binding sites on an amino acid molecule during ion-exchange can inhibit binding. The structures of the solute-QMA pairs in water in the cases of halides, acetate, and the form of aspartate in which the oppositely charged groups are apart are solvent separated, whereas they are contact pair for acetamide and the form of aspartate in which the charged groups are neighbors. Charged groups with higher affinities to QMA can have larger equilibrium distances from QMA than similar neutral groups with lower affinities. Overall, the technique of molecular simulations with statistical perturbation theory is shown to be useful in understanding the fundamentals of ion exchange.
Introduction
Adsorption and chromatography processes are widely used in chemical and biochemical separations.’ A key issue in the successful applications of adsorption and chromatographic processes is the sorbent selectivity.* The selectivity depends upon intermolecular interactions between sorbent, solute, and solvent molecules. The interactions come about as a complex interplay of various intermolecular interactions, viz., electrostatic interactions, hydrogen bonding, hydrophobic interactions, complexation, and chelation. They are strongly influenced by the prevalent conditions such as pH, salt, and temperature. Also, the presence of solvent has a significant effect on the interactions. These intermolecular interactions are strongly dependent on the geometries of the solutes and the functional group. For instance, protein molecules have a complex three-dimensional structure, and the location of the amino acid residues including the charged residues in the three-dimensional structure is important in determining the strength of various interactions. Because of the numerous varieites and the complex coupling of the various intermolecular interactions in a three-dimensional space, it has not been possible to establish general rules for the prediction of selectivities from macroscopic measurements of selectivities and other observations. A variety of theoretical approaches has been suggested in interpreting and predicting ion-exchange equilibria. Helfferich has discussed these. approaches in detail in his treatise on ion exchange.’ To use most of these approaches, one must perform at least some ( I ) Asenjo, J. A. Separation Process in Eiofechnology; Marcel Dekker: New York, 1990. (2) King, C. J. Separation and Purification: Critical Needs and Opporfunities; National Research Council; National Academy Press: Washington, D.C., 1987. (3) Helfferich, F. Ion Exchange; McGraw-Hill: New York, 1962,
experiments and fit the parameters since the parameters cannot be obtained a priori. Paulef suggested the use of Bjerrum theory5 to interpret and predict selectivities in ion-exchange equilibria. In the model suggested by Pauley, the essential feature is the electrostatic interaction between the counterion and the fixed ionic functional group. Thus, to estimate the selectivity, one needs to determine the association constant between the functional group and the ion to be sorbed. Bjerrum theory has been used to predict the association constants for ion pairs for infinitely dilute solutions. This theory has some drawbacks. In this theory, the association constant depends only on the charge on the ions and the dielectric constant of the solvent, the solvent being treated as a continuum. According to this theory, the association constants for tetraalkylammonium salts are predicted to decrease with increasing size.6 But the reverse trend occurs in water. The reversal has been attributed to water-structure enforced ion pairing’ in which larger, less hydrophilic ions are forced into the same cage by water in order to decrease the disruption of the hydrogen-bonded network of water. The recent studies of aqueous electrolyte solutions on a molecular level have predicted many properties not previously predicted by theories in which the solvent is treated as a continuum. Molecular simulations is a promising tool for understanding intermolecular interaction^.**^ These calculations, however, need (4) Pauley, J. L. J. Am. Chem. Soc. 1954, 76, 1422. ( 5 ) Davies, C. W. Ion Association; Butterworths Scientific Publications: London, 1962. (6) Accascina, F.; Goffredi, M.; Triolo, R. Z . Phys. Chem. (Munich) 1972, 81, 148. (7) Diamond, R. W. J . Phys. Chem. 1963, 67, 2513. (8) van Gunsteren, W.F.; Weiner, P. K.Computer Simulation ofEiomolecular Sysrems; ESCOM: Leiden, The Netherlands, 1989. (9) Jorgensen, W. L. Acc. Chem. Res. 1989, 22, 184.
0022-3654191 12095-8899SO2.5010 -~ ., . 0 1991 I
I
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American Chemical Society
