Theoretical analysis of intramolecular double-hydrogen transfer in bridged-ring compounds

Journal of Molecular Structure, 291 (1993) 207-213 0022-2860/93/%06.00 0 1993 - Elsevier SciencePublishersB.V. All rights reserved

207

Theoretical analysis of intramolecular double-hydrogen transfer in bridged-ring compounds’ Y2 Zorka K. Smedarchina3, Willem Siebrand* Institute for Molecular Sciences, National Research Council of Canada, Ottawa, Ont., KIA 0R6, Canada Steacie

(Received 4 January 1993) Abstract Model calculationsare reported on double-hydrogenand double-deuteriumtransfer rates in two bridged-ring molecules recently investigated by Mackenzie et al. petrahedron Letters, 33 (1992) 56291.The calculations indicate that, contrary to an earlier interpretation, the two atoms are transferred by asynchronous tunnelling, the observed activation energy being representative of the energy of the biradical intermediate rather than the barrier height.

Introduction In recent papers, h4ackenzie and co-workers [1,2] reported rate constants for the double-hydrogen and double-deuterium transfer processes illustrated in Fig. 1. On the basis of the observed temperature and isotope effects, they suggested that the transfer of the two hydrogen or deuterium atoms occurred synchronously in such a way that for reaction 1 of Fig. 1 the atoms crossed the barrier classically, while for reaction 2 they crossed by quantum mechanical tunnelling. In this work, we offer an alternative interpretation of the data. As Mackenzie et al. [2] recognized, it is not easy to rationalize the proposed difference in transfer mechanism between the two reactions, since they occur in closely related molecules and are characterized by very similar kinetic parameters. These * Correspondingauthor. I Dedicated to Professor CamilleSandorfy. ’ Issued as NRCC No. 35256. 3 Visitingscientist.Permanent address:Institute for Organic Chemistry, Bulgarian Academy of Sciences, 1113 Sofia, Bulgaria.

parameters, when analyzed in terms of the Arrhenius equation, give activation energies of about 30 kcalmol-’ and frequency factors of about 10” s-l. The latter are much smaller than CH stretch frequencies. Since the barrier may be constructed starting from two CH stretch potentials associated with the CH bonds broken and formed, respectively, one would expect the frequency of transfer above the point where these two potentials merge to be of the order of the frequency of the corresponding CH frequencies, namely 10’3-1014s-‘, rather than 10” s-t as observed. Moreover, the frequency should be higher for CH than for CD, by a factor of about 2’j2, and not lower, as observed. If, however, one assumes that the transfer proceeds by tunnelling, one cannot derive the barrier height from an Arrhenius plot since these plots are curved for tunnelling reactions. The actual barrier must then be much higher than 30 kcal mol-’ and would give rise to a deuterium effect very much larger than that observed. All these di&ulties disappear if it is assumed instead that the transfer is a two-step, asynchronous

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Fig. 1. Illustration of the two moleculesand the double-hydrogentransfer reactions studied.

process. The high activation energy can then be interpreted as the energy of the intermediate, biradical structure corresponding to the transfer of one hydrogen or deuterium atom, with only minor contributions from the thermal activation of the actual tunnelling process. Measured relative to this intermediate, the barriers are low enough to account for the observed deuterium effect for both of the processes illustrated in Fig. 1. An entirely analogous situation is encountered in free-base porphyrins where the two inner hydrogens are exchanged pairwise between four pyrrole nitrogens [3-lo]. After some early controversy [3,4], there is now a consensus that this exchange occurs asynchronously down to at least lOOK, and that the observed activation energies mainly reflect the energy differences between the cis and trans isomers. For porphyrins, rate constants measured near 1OOK [7] and near 300K [5,6] lead to different activation energies, consistent with the expected contribution of the curved Arrhenius plot of the tunnelling process. The observation of a strong primary isotope effect confirms that the transfer takes place by tunnelling. In recent papers

[g-lo], we have shown that the transfer rate can be modelled quantitatively. The same model is found to account for the kinetic data obtained recently by Paquette and co-workers [ 11- 131 for double-hydrogen transfer reactions in a series of molecules structurally related to those of Mackenzie et al. [1,2]. These data are of special significance since they include rate constants for HD transfer [13] and for the reverse reaction [11,12]. The model discussed in this paper provides excellent fits to these data [ 131,which in turn provide us with useful estimates for the parameter values applicable to the data of Mackenzie et al. [2]. Model calculations In Fig. 2, we depict a schematic model potential corresponding to the reaction coordinates for the synchronous and asynchronous process. Since the reaction is reported [1,2] to be irreversible, the potential must be asymmetric such that the exothermicity 6E is large compared to kBT. For synchronous transfer, the reaction coordinate will

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-I

I

REXXION COORDINATE Fig. 2. Sketch of the potentials governing the reactions depicted in Fig. 1 for asynchronous transfer (-) transfer (- - -).

roughly to the symmetric CH stretching coordinate of the hydrogens to be transferred; for asynchronous transfer, it corresponds to a local CH stretching coordinate. In both cases, the potential will be constructed from two Morse potentials whose avoided crossing defines the height of the barrier. The frequency and anharmonicity of the CH potentials will be taken from spectroscopic data [14] and their separation is taken to be equal to Mackenzie’s estimate [2]. An essential feature of our model is the recognition that this separation, and thus the tunnelling distance, is not stationary but oscillates as a result of the molecular vibrations. The presence of low-frequency, high-amplitude bending and twisting modes of the molecular framework will permit the hydrogen atoms to tunnel at the moment when the tunnelling distance goes through a minimum. Thermal excitation of these modes should have a much larger effect on the tunnelling rate than population of the highfrequency, low-amplitude CH modes. Many conventional treatments fail to take this into account which leads to a serious underestimation of the tunnelling contribution to the transfer. The rate constant for asynchronous doublehydrogen transfer from the initial state to the intermediate state at energy AE, and then from this

correspond

and synchronous

state to the final state with energy -SE written in the form e(T)

=

2k;)kE) kg) + kg)

can be

(1)

exp(-AE/kBT)

where ki) and k E’ denote the rate constants for tunnelling through the first and second barrier, respectively. In view of the asymmetry of the potential implied by the irreversibility of the transfer, we simplify Eq. (1) to kH(T) FZ2kg)exp(-AE/k*T) as

(24

where the temperature-dependent tunnelling rate constant kg) is governed by the reduced mass m w 1 of the hydrogen atom and the (unknown) barrier height. The rate for asynchronous tunnelling of two deuterium atoms is analogously . es(T)

= 2k$) exp(-AE/kBT)

(2b)

For synchronous tunnelling of two hydrogen atoms, the rate constant k,(T) equals the tunnelling rate kzH. In view of the larger mass (m = 2) and higher barrier, we have kzH << kg), but this advantage of single hydrogen transfer is offset by the Boltzmann factor exp(-AE/kBT). Thus, in

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Z.K. Smeakrchina,

general, synchronous tunnelling is expected to prevail at low temperatures and asynchronous tunnelling at high temperatures. To compare the two mechanisms at intermediate temperatures, we need a model for the calculation of (energy-conserving) tunnelling rate constants. The model we shall use is a simplified version of the golden-rule model applied earlier to innerhydrogen exchange in porphyrins [9,10]. Simplification is possible since the temperature in the range for which data are available is high relative to the frequency of skeletal modes affecting the tunnelling distance. We therefore treat these modes classically and write for the tunnelling rate constants &H7 kD&ZH

Or k2D)

[15-171

where p(R) is the distribution of distances R between the donor and acceptor carbon atoms. For harmonic oscillators, we have [18] p(R) =

[27r/,42(T)]-1’2 exp[-(R

- @2/A2(T)]

(4) where i? is the equilibrium value and A’(T) the mean square amplitude of R. Elsewhere [19] we have shown that, in simple cases, the system of skeletal modes can be represented in a limited temperature interval by a single effective mode, so that A2(T) = (h/M0)coth(AS2/2kBT)

(5)

where M and fl are the reduced mass and frequency of the effective mode, respectively. The tunnelling rate for fixed R is governed by the overlap between the CH vibrations of the tunnelling atom in the initial and final state of the transfer. Since these vibrations have frequencies that are high relative to kBT, their vibrational quantum numbers will be given by v, = 0 and vi = AE/hw, and by vi = 0 and vf = GE/liw for asymmetric and symmetric tunnelling, respectively, if w is the relevant CH frequency and the subscripts i, r and f denote the initial, intermediate and final state, respectively. In the present case, where both CH

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Mol. Struct., 297 (1993) 207-213

stretching and CH bending will be important, use an effective frequency [ 141

we

w = w,tr&h cos2 p + %$nd sin2 p

(6)

where p is the angle between the CH bond and the line connecting the donor and acceptor carbon atoms. The tunnelling rate constant for fixed R can now be written as the following golden rule [10,14]: k,(R) = (2~ltz)A(J)IS,,(R)12/tiR

(7)

where J is the electronic coupling integral driving the transfer and l/tin is the density of final states. The factor A(J) replaces the standard weak coupling factor J2 in the case of strong and intermediate coupling (J 2 hw) [9,10]. This completes the description of the model. Since no information on the vibrational force field is available, the choice of parameter values will necessarily be somewhat arbitrary. For the CH and CD modes, the choice is relatively straightforward; using standard spectroscopic data together with the geometry reported in ref. 11, we obtain from Eq. (6) WH= 3000~0s~ 38 + 1700 sin2 38 = 2500 cm-’ and w. = 2500 cm-’ /2; M 185Ocm-‘. To calculate the overlap integrals, we assume that the CH(D) vibrations behave as Morse oscillators with anharmonicity constants derived from the value [20] X = -55 cm-’ typical for CH stretching. Thus we set XH = -55 cm-‘, x,, = x2, = -28cm-’ and &o = -14cm-‘. From the average non-bonding C..C distance given by Mackenzie et al., [2], we obtain a tunnelling distance I = 1.7OA if we assume tetrahedral angles and a standard CH bond length < = 1.08 A. To estimate the reduced mass and frequency of the “effective” low-frequency skeletal mode that brings the reacting carbon atoms together, we note that two types of motions will be involved, namely bending and twisting of the rings that are drawn as vertical planes in Fig. 1. For synchronous tunnelling only the former will be effective whereas both may contribute to asynchronous tunnelling. Assuming an

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-li -2 -3

-4I\\ -5 -_

--__

--__

-6 -7

--__

I

I

I

I

I

2.5

2.6

2.7

2.8

2.9

1000/T Fig. 3. Temperature dependence of the rate of HH (upper curves) and DD (lower curves) tunnelling for asynchronous (-) and synchronous (- - -) transfer in compound 1. Parameter values used: qr/n = 2500/1850, X@ = 2XR = -5Scm-t, Xg = 2X; = -28cm-‘, r = 1.70& t = l.OSA, J= O.O8eV, R, = @cm-‘, M= SOmu, R, = 70cm-‘, AE=17.9kcalmol-‘, 6E = 5 kcalmol-‘.

effective mass M = 50mn in both cases, we use the respective effective frequencies as adjustable parameters. The two remaining parameters, AE and J are taken from our analysis of the data of Paquette

et al. [ 131.The energy of the intermediate biradical is estimated from the observed activation energy E, by means of the empirical relation AE M E,5.5 kcalmol-‘. This leads to AE = 19.2 kcal mol-’ and 25.8 kcal mol-’ for compounds 1 and

-3.5-

-5.5

I 1.95

I

I

2.05

2.15

2. 5

1000/T Fig. 4. Temperature dependence of the rate of HH (upper curve) and DD (lower curve) tunnelling for asynchronous DD transfer in compound 2. Parameters as for Fig. 3, except AE = 20.6 kcalmol-‘, J x 0.06eV.

HH and

212

2, respectivley. The electronic coupling J is found by extrapolation of the value J = 0.22 eV, found to be appropriate for a tunnelling distance T = 1.48& to the value J = O.O8eV, for I = 1.70A by means of a formula reported earlier [14]. We now try to fit the resulting equation for the temperature-dependent rate constant to kr.n.t and kDD for the two molecules by varying a single parameter, 0, or &, the effective framework frequency in the two models. It turns out that the fitting is straightforward if we assume asynchronous tunnelling. The effective frequency R, = 60cm-i is low but not unreasonably so, considering that twisting of the heavily chlorinated rings will be the most effective distortion for promotion of the transfer. With slight adjustments of the AE values to 17.9 kcalmol-’ and 20.6 kcal mol-’ for compounds 1 and 2, respectively, and of J to 0.06eV for compound 2, we obtain the fits depicted by the solid lines in Figs. 3 and 4. The situation is quite different if we assume synchronous tunnelling. This assumption leads to a temperature dependence that is much too weak and a deuterium isotope effect that is much too strong. A typical result is depicted by the broken lines in Fig. 3, obtained by a choice of parameters that fits the observed rate constant kzH at a temperature of 353 K. In general, the same parameters have been used as for the asynchronous mechanism; the exothermicity SE has been arbitrarily set equal to 5 kcalmol-’ and R, has been adjusted to 70cm-‘. For synchronous tunnelling, symmetric bending rather than asymmetric twisting would be expected to be the most effective vibration in promoting transfer. Therefore we expect the actual value of R, to be considerably higher than 7Ocm-’ which would lead to a reduction in the calculated value of kZH and kzo. The model calculations also allow us to obtain the rate of transfer in the classical limit where all transferring atoms cross the top of the barrier. It is found that this contribution is entirely negligible for both hydrogen and deuterium in the molecules considered. A reasonable conclusion is thus that

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the transfer process proceeds by two-step, asynchronous tunnelling, the same mechanism as that operative in the related molecules studied by Paquette et al. [14] and in the unrelated free base porphyrins [IO]. In all these cases the high activation energies combined with a relatively weak deuterium effect suggests strongly that the tunnelling must be asynchronous. Acknowledgements This paper is dedicated to Camille Sandorfy whose friendship and support we deeply appreciate. We wish to thank Leo Paquette for bringing these data to our attention.

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