J. Phys. Chem. 1994,98, 5445-5459
5445
Solvent Barriers in Unimolecular Ionizations. 1. Reaction Path Analysis for Alkyl Iodides Jeffery R. Mathis and James T. Hynes' Department of Chemistry and Biochemistry, University of Colorado, Boulder, Colorado 80309-021 5 Received: January 28, 1994"
+
The s N 1 ionizations RI R+ I- of tert-butyl iodide and isopropyl iodide are investigated in several solvents theoretically. An electronically coupled twovalence bond state solute representation, combined with a (quantum) dielectric continuum solvent description, is employed to generate free energy surfaces in terms of an internuclear separation coordinate r and a collective solvent coordinate s. General free energy surface characteristics are discussed, inchding transition state energetic, structure, and location features. Decreasing transition state solvent stabilization with increasing solvent polarity is found, in contrast to the conventional Hughes-Ingold perspective, but in agreement with the Hammond postulate. In addition to exhibiting ionization barriers along the nuclear separation coordinate r, these alkyl iodides also have solvent barriers along the solvent coordinate s in the saddle point vicinities. Due to their presence, the rate constant k for the alkyl iodide ionization is quite dependent upon the approach used to compute it. Three different rate constants are examined: the harmonic and anharmonic (Le. including reaction path curvature) rate constants in a solution reaction path description and the rate constant associated with the conventional potential of mean force perspective. There are rather large reaction path curvatures, but, despite this, there is only a mild effect on the reaction rate constant. However, due to the solvent barriers, the potential of mean force perspective gives a rate inaccurate by 1.5-3 orders of magnitude. Thus, the s N 1 ionizations of t-BuI and i-PrI provide examples where conventional equilibrium solvation reaction rate methods cannot be applied; the ionization process is fundamentally a nonequilibrium one. A related feature is responsible for the breakdown of Grote-Hynes theory in one standard form due to the inapplicability of its underlying assumptions. -+
1. Introduction
This paper begins a reaction path and rate constant analysis for the solution phase sN1 ionizations
RI
+
R+ + I-
using tert-butyl iodide (t-BuI, R = (CH3)3C) and isopropyl iodide (i-PrI, R = (CH3)2HC) as model systems. Unimolecular ionizations, such as those in eq 1.1, have long been recognized'-3 as comprising one of the most fundamental chemical reaction classes, yet their theoretical understanding has been surprisingly limited. Recently, aspects of sN1 ionizations of other systemsmore specifically the reverse ion pair collapse reaction of eq 1.1-have become the subject of modern picosecond spectroscopic ~ t u d i e s further ,~ stimulating the need for their detailed understanding. The present work and its companion (the following paper in this issue)-hereafter 11-are continuations of earlier S Nionization ~ studies536 directed toward the establishment of a firm theoretical framework for these reactions. The formalism presented in ref 5 is again employed to describe the evolving solute electronic structure in solution. It involves a quantum twovalence bond (VB) state Hamiltonian for thesolute, in terms of orthonormal covalent J/c[R-I] and ionic $1 [R+-I-] basis wave functions, in the context of the dielectric continuum Kim-Hynes theory' that properly accounts for the solvent polarization and the coupling to the solute electronic structure and charge distribution. Implementation generates a twodimensional free energy surface, in terms of the reacting solute internuclear separation r and a collective solvent coordinate s, upon which the ionizations of eq 1.1 take place. It is found that both alkyl iodides have activation barriers not only along the alkyl groupiodine internuclear separation coordinate r but also along the solvent coordinate s, in all four solvents investigated. This unusual state of affairs contrasts with other alkyl halide ionizations treated in refs 5 and 6 and warrants 0
Abstract published in Advance ACS Abstracts, May 1, 1994.
0022-3654/94/2098-5445%o4.50/0
a separateanalysisofthereaction pathand rateconstant. Among the consequences of the solvent barriers found is the complete breakdown of any ionization picture based on the standard potential of mean force,B-lO and an inapplicabilityof Grote-Hynes theory in one standard form." The presence of solvent barriers is also suggestive of nuclear separation coordinate dependent electron transfer, a perspective fully explored in 11. Whether or not a solvent barrier is present for any charge transfer system depends on the stabilization energy associated with the solute electronic coupling (also referred to as thequantum mechanical resonance energylz) compared to the energy necessary to "reorganize" the solvent molecules around the solute charge distribution.5-7 One can imagine the situation as follows. The energy associated with the electronic coupling 0for a given solute is the stabilization gained by mixing the covalent and ionic states. In vacuum, a molecule will be more stable when this mixing is large and the electron is delocalized, while in solution, maximum solvent stabilization is obtained for localized chargedistributions. If, at a given solute internuclear separation r, the magnitude of 0is small enough, the solvent can force the solute to localize its charge. It will be shown presently that the quantity 20/AG, < 1, where AG, is the reorganization free energy, is the necessary condition for a solvent barrier. When the ratio is less than unity, a free energy profile drawn along the solvent coordinate at fixed r will exhibit two local minima separated by a solvent barrier. The competition between the solute electronic coupling and the solvent reorganization is thus the origin of the solvent barrier for the present alkyl iodide cases. Some understanding can now be had as to why the alkyl iodides in eq 1.1-and not the other halides5-exhibit solvent barriers. From the discussion in ref 5, both iodides havevery covalent reactant state bonds, which results in a relatively small magnitude for the electronic coupling.5 In fact, the C-I bond in i-PrI is even more covalent (less ionic mixing) than that in t-BuI, so the electronic coupling for i-PrI is generally smaller, and consequently its solvent barrier is expected to be larger, as indeed is found within. 0 1994 American Chemical Society
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The Journal of Physical Chemistry, Vol. 98, No. 21, 1994
Since i-PrI ionization involves a secondary alkyl carbocation, which is not as stable as the tertiary tert-butyl cation, one can wonder whether or not this system reactsvia a unimolecular, sN1 and E l , or a bimolecular, S N and ~ E2, mechanism. Iodine, however, is a very good leaving group compared to, e.g., the other halogens,3,s and evidence supporting a unimolecular reaction mechanism has been provided byAbraham,I3as well as by others.I4 Suffice it to say that i-PrI is at least borderline, so that it should exhibit at least some sN1 character, and thus our approach is directly applicable. In addition, it is argued that considerations of this paper and its comparison are relevant to a number of other reaction classes, and in this respect i-PrI is a very useful model system. The presence of a solvent barrier for these reactions, and the added feature that its location and height are dependent on the nuclear separation coordinate, makes the determination of their rate constants a challenge. In addition to discussing the solvent barrier, how it arises, and what its energetic properties are, the present paper begins a rate analysis by examination of the free energy surface steepest descent reaction path,15termed thesolution reaction path (SRP),16 in harmonic and anharmonic reaction path curvature limits, as well as the conventional potential of mean force (PMF).*-IO In the P M F and S R P treatments, the reaction coordinate for the ionizing system is essentially the alkyl groupiodine nuclear separation coordinate r. By contrast, I1 focuses on a nuclear coordinate dependent electron transfer (ET) approach, where the reaction coordinate is instead viewed primarily as the solvent ~ o o r d i n a t e . ~ ~Two J* different ET rate constants are investigated. One corresponds to a conventional distance dependent outer sphere ET rate,17 as extended to the ionizations of eq 1.1, and the other corresponds to a new distance dependent E T rate. This latter ET rate accounts for the solute-solvent coupling present with eq 1.1, due to the inner sphere19 characteristic of both charge transfer and solute nuclear reorganization, whereas the former is shown not to incorporate this coupling. Considerable analysis is devoted to the explication of these possible ionization mechanisms, as well as the establishment of a reliable estimation of the rate. Each rate constant essentially corresponds to a different physical picture of the ionization mechanism. Through comparison of different rate expressions and their magnitudes, the likelihood (or lack thereof) of a given ionization mechanism can be accurately assessed. After consideration of all the results presented in both papers, it will be shown that the SRP gives the best estimate for the reaction rate, and thus the ionization mechanism, despite the E T nature of these ionizations. Moreover, the analysis can be applied to other inner sphere processes (see section 4 in 11), and deeper understanding is possible of, e.g., whether or not the charge transfer and solute nuclear reorganization occur in a concerted or stepwise fashion. The outline of this paper is as follows. The theoretical formulation of the problem is given in section 2. The presentation of the relevant equations is brief, as the details have been discussed e l ~ e w h e r e . ~In - ~ section 3, the calculated free energy surfaces upon which the ionizations of eq 1.1 take place are presented and discussed. Section 4 gives the results for two rate constants based on the SRP, in both harmonic and anharmonic limits. Section 5 deals with the calculation of the potential of mean force and a rate constant based on it. The potential of mean force rate constant is compared with those in section 4, and it is found to be quite inaccurate, due to an underestimate of the total activation barrier. A related feature is shown to be responsible for an inapplicability of Grote-Hynes theory11 in its usual form. Concluding remarks are offered in section 6 . 2. Theoretical Formulation The description of the solvent and its interaction with thesolute will employ the Kim-Hynes (KH) free energy formalism.7 An
Mathis and Hynes 50
--
50
40
0
E :30
s
Y, 20 L
z 10
0
0
1.5
2.5
3.5
4.5
5.5
Figure 1. Electronic coupling @(r)(-) and reorganization free energy AG,(r) (- -) behavior for t-BuI. The small magnitude of the coupling near r = 2.8-3.0 A, combined with the larger value of AG,(r), produces a solvent barrier. These quantities for i-PrI have the same qualitative behavior but differ slightly in magnitude.
extensive discussion of this formalism has been presented el~ewhere,j-~ but in the interest of internal coherence the details will be summarized. In theKH picture, thesolvent interacts with the soluteelectronic chargedistributionviaiwo types of polarizations: the (quantized) electronic polarization Pd associated with solue!t electronic motion and the (classical) orientational polarization Po, associated w$h rotations and translations of individual solvent molecuies. Pel always maintains an equilibrium with the s o l ~ t e but , ~ Po, is in general out of equilibrium, i.e. the solute ele_ctron_icmotion time scale is usually much faster' than that for Po,. Po, fluctuations can be accounted for by a collective solvent coordinate5-6J68 according to
where em and €0 are the solvent optical and static dielectric constants, k is a location in the solvent, and $c(k;r) and &I(k;r) are the vacuum electric fields arising from the solute covalent and iznic valence bond states, $c[R-I] and $q[R+I-], respectively. Po, is completely equilibrated when u = 0 to the covalent state charge distribution and when u = 1 to the ionic state charge distribution.20 The reorganization free energies, A G t ( r ) and AG,(r) given by
corresponding to Fel and ?or, respectively, depend on the solute charge distribution and thus will be r-dependent. Displayed in Figure 1 is the AG,(r) result for t-BuI in CH3CN as an example. AG,(r) is largest at large separations, corresponding to a fully ionic solute electronic character, and smoothly goes to zero in the limit of zero ~eparation.~.6 AG,(r) for i-PrI has the same qualitative behavior but is slightly larger due to the smaller isopropyl cavity size compared to that of tert-butyl. It is shown in the Appendix that use of eq 2.1 for ?or g'ives rise to a kinetic coupling term ku, where the dot signifies a time derivative, and that transformation of the solvent coordinate u to a new solvent coordinate s according to s = (AG,(r)/AG:)'/2u
(2.3)
where AG: is a reference value for AG,(r), diagonalizes the kinetic energy for the system and thus allows for straightforward
The Journal of Physical Chemistry, Vol. 98, No. 21, 1994 5447
Solvent Barriers in Unimolecular Ionizations. 1 interpretation of the dynamic~.~2.23This (r,s)coordinate system will therefore be adopted throughout the remainder of this paper. This is a different coordinate system than the (r,u) coordinate system employed p r e v i o ~ s l y (although ~*~ u in refs 5 and 6 was actually called s). There, it made no difference because the focus was on the free energy surface saddle point characteristics and not on the dynamics; the free energy surface saddle point characteristics (whichinclude the reactant, product, and transition states), e.g., ionic character and free energy, are exactly the same in either the (r,u) or (r,s) coordinate system.23 The solute electronic structure in solution is described by the two valence bond state wave function
in terms of orthonormal covalent state +&) and ionic state +I(r) basis wave functions,5-6 the vacuum energies for which are given by
e(')=
+
obtained from the nonlinear SchrMinger equation in ref 7a. The solution of eq 2.1 1 gives the solute charge distribution, i.e., the solute electronic structure, for any point (r,s) and thus the charge distribution and free energy G(r,s) (eq 2.6) for nonequilibrium solvation For the special case of equilibrium solvation
dG(r,s)/ds= 0
(+c(r)l@(r)l+c(r) );
@ ( r ) = (+I(r)Ip(r>I+I(r)) (2.5) where W(r)is the solute vacuum Hamiltonian. The two st$e coefficients (CC,CI] depend not only on r but also on s, since Po, fluctuations can alter the solute charge di~tribution.~ By invoking the variational principle' that the actual solute wave function is the one which minimizes the free energy, a nonlinear Schrainger equation for Q is obtained, the exact form of which is not needed here, and is given in ref 7a. Using Q obtained from that Schrainger equation, the general free energy function is given by5-7,23
G(r,s) = v(r,s)- ' / 2 A V ( r , s ) x- /3(r)y + 1/4AGf(r)yZf+
fl-e Y+P
where
V(r,s)=
where the (r,s)notation in eq 2.4 has been suppressed. Both x and y have values ranging from -1 to 1 . The case x = 1 and y = 0 corresponds to the completely covalent RI reactant state, while x = -1, y = 0 is the completely ionic R+I- product state. The case x = 0, y = 1 corresponds to a completely delocalized intermediate state between the two reactant and product extremes 01< 0 describes the higher energy antisymmetric state nor relevant here7). x and y are found by solving
'/,(e",
