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Proton tunneling in calix[4]arenes: a theoretical investigation
10 August 2001
Chemical Physics Letters 343 (2001) 627±632
www.elsevier.com/locate/cplett
Proton tunneling in calix[4]arenes: a theoretical investigation Antonio Fern andez-Ramos a, Zorka Smedarchina a,*, Fabio Pichierri b a
Steacie Institute for Molecular Sciences, National Research Council of Canada, Ottawa, Ont., Canada K1A 0R6 b RIKEN Genomic Sciences Center, 1-7-22 Suehiro-cho, Tsurumi-ku, 230-0045 Yokohama, Japan Received 26 February 2001; in ®nal form 27 April 2001
Abstract The structure, energetics and vibrational force ®eld of calix[4]arene (CA) are calculated for the chiral equilibrium con®guration and the symmetric transition state for coherent quadruple proton transfer. The corresponding proton and deuteron tunneling splittings are evaluated with the instanton approach. The calculated zero-point proton tunneling splitting is compared with data reported for low-temperature NMR relaxometry experiments. Ó 2001 Elsevier Science B.V. All rights reserved.
1. Introduction Recently, Horsewill and coworkers [1,2] studied multiple proton transfer in the hydrogen bond network of calix[4]arene (CA) and p-tert-butyl calix[4]arene (tbCA). The center of the bowlshaped molecules contains four hydroxyl groups forming a chiral ring of four hydrogen bonds represented by the point group C4 . These molecules exist in two forms of opposite chirality, between which transitions are possible through exchange of the four protons along their individual hydrogen bonds (Fig. 1). In an isolated molecule of C4 symmetry, this would give rise to tunneling splitting, but if the symmetry is broken, e.g., by a medium, the exchange will be a rate process. To investigate these phenomena, Horsewill et al. used NMR relaxometry on crystalline powders at low
*
Corresponding author. Fax: +1-613-947-2838. E-mail address: [email protected] (Z. Smedarchina).
temperatures. They reported the observation of both a splitting [2] and a rate process [1,2]. The splitting corresponding to a frequency of 35 MHz in CA was derived from the observation of a corresponding peak in the magnetic ®eld dependence of the proton spin±lattice relaxation rate; this frequency showed no temperature dependence in the range 30±80 K, but the peak height increased with temperature as if subject to an activation energy of 125 K. No splitting was observed in tbCA. The observation of a peak at zero ®eld in both molecules was interpreted as evidence for an exponential rate process, the halfwidth of this peak being used to extract the rate constant. For tbCA a value of 8 107 s 1 was reported, which was found to remain constant in the range 15±21 K. No rate constant was reported for the rate process in CA, but if the published graph is interpreted in the same way as that for tbCA, the result would be similar. Both zero-®eld peaks showed an exponential temperature dependence as if subject to an activation energy of about 75 K.
0009-2614/01/$ - see front matter Ó 2001 Elsevier Science B.V. All rights reserved. PII: S 0 0 0 9 - 2 6 1 4 ( 0 1 ) 0 0 7 4 1 - 2
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A. Fernandez-Ramos et al. / Chemical Physics Letters 343 (2001) 627±632
Fig. 1. The two stereoisomers of calix[4]arene.
These observations exhibit a number of unusual features. It is not clear how the same system can show both a splitting and a corresponding rate constant, i.e., how proton exchange can be both phase-coherent and -incoherent. The observation of a tunneling splitting at temperatures much lower than required to excite hydrogen-bond vibrations suggests that it should be assigned to the zero-point splitting. Yet the signal disappears at very low temperatures. In this connection, we note that the measurements on tbCA were carried out at temperatures too low to observe tunneling splitting, which begs the question whether tbCA would also show such splitting if it were possible to perform these measurements at higher temperatures. In view of these problems, it seems useful to explore to what extent quantum-chemical and dynamics calculations can shed light on these observations. This requires an investigation of the transition state of the tunneling process, which has not been calculated before. In a recent series of papers we have investigated coherent and incoherent double and triple proton tunneling in porphine [3], 7-azaindole [4,5], guanine [5], glycine in aqueous solution [6], 3-hydroxyisoquinoline [7], a methanol±acetic acid complex [8], and a water trimer [9], respectively. We shall use the same method to investigate quadruple proton transfer in CA. First we calculate the structure, energy and force ®eld of the molecule in the stable con®guration(s) and in the transition state. Then we use
the results as input parameters for the calculation of proton tunneling splittings for ground and excited vibrational levels, and compare the results with experimental observations. 2. Dynamics of coherent proton exchange The interchange of the two stereoisomers of CA corresponds to concerted motion of four protons in a double-minimum potential. It is to be expected that the proton motion will be coupled to skeletal vibrations, so that we need to generate a multidimensional potential for the dynamics. To this end we have performed quantum-chemical calculations with GA U S S I A N 98 [10] and dynamics calculations with DO I T 1.2 [11]. We fully optimized the stationary points using density functional theory (DFT) with the B3LYP functional [12] and the cc-pVDZ basis set [13]. A Hartree±Fock (HF) calculation would be less computationally intensive, but we found that electronic correlation is very important in this case. Speci®cally, the HF method fails to produce a transition state of C4v symmetry. Selected geometric parameters obtained at the B3LYP/ cc-pVDZ level and relevant for the dynamics calculations are listed in Table 1. A more extensive report on the structural and force ®eld parameters of the two stationary points of CA will be published elsewhere.
A. Fernandez-Ramos et al. / Chemical Physics Letters 343 (2001) 627±632 Table 1 Selected structural parameters of the hydrogen bond network in calix[4]arenes calculated at the B3LYP/-ccpVDZ level
O1 H O2 H O1 O2 O1 C O1 HO2 HO1 C
Stable con®guration
Transition state con®guration
0.993 1.678 2.648 1.382 1.682 114.4
1.207 1.207 2.401 1.381 164.2 110.1
angles in degrees. Distances in A,
The main structural parameters of the stable con®guration agree satisfactorily with the available X-ray data [14]; for instance, the mean O O is calculated to be 2.65 A. A distance of 2.64 A value of 2.67 A, obtained at the BLYP/6-31G level for this distance, was reported in [15]. The two C4 minima in our calculation are separated by a transition state of C4v symmetry with barrier height 16.86 kcal/mol. Its structure, superimposed on that of the stable con®guration, is illustrated in Fig. 2. It shows that the motion of the four protons is accompanied by signi®cant skeletal displacement; for instance the O O distance is in the transition state. The reduced to 2.40 A tightening of the hydrogen bonds in the transition state, eected by displacements of the oxygens and the attached carbons in and up the bowl, obviously
Fig. 2. Superposition of the B3LYP/cc-pVDZ structures of the stable con®guration (black lines) and the transition state (gray lines) of calix[4]arene. Note the tightening of the hydrogen bond network in the transition state, which assists proton exchange.
629
facilitates tunneling. To calculate the resulting tunneling splitting, it is thus necessary to use a method that can properly account for the multidimensional character of the tunneling potential. In our approach the mode with imaginary frequency x , depicted in Fig. 3a, is chosen as the reaction coordinate and the skeletal mode contributions are included through linear coupling terms. The bowl of CA, held together by the cyclic network of four hydrogen bonds, consists of 56 atoms. The interconversion between its two stereoisomers in Fig. 1 is governed by a potentialenergy surface (PES) of 162 vibrational degrees of freedom. The DO I T code generates the complete vibrational PES in terms of the normal modes x; y of the transition state, where x denotes the reaction coordinate and y the set of transverse modes. As input it uses the standard output of electronic structure and force ®eld calculations of the stationary con®gurations along the reaction path. The PES is generated in a form suitable for instanton-dynamics calculations: it consists of a onedimensional (1D) symmetric double-well potential U x along the reaction coordinate, harmonic terms along the 161 (transverse) modes y, and mixed terms which describe the coupling between the tunneling mode and skeletal vibrations. Terms of the type Cs x2 ys correspond to coupling to symmetric (s) skeletal modes, similar to those that tighten the hydrogen bonds, and promote tunneling. Terms of the type Ca xya , which account for coupling to antisymmetric (a) modes, are
Fig. 3. Three normal modes of the transition state active in the coherent quadruple proton transfer in calix[4]arene: (a) the tunneling mode (symmetry A2 ) with frequency ix 1448 cm 1 ; (b) a totally symmetric mode (A1 ) with frequency xs 420 cm 1 , promoting tunneling; (c) a totally antisymmetric mode A2 with frequency xs 270 cm 1 , suppressing tunneling.
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equivalent to Franck±Condon factors and have a suppressing eect on the tunneling. The coupling constants Ca;s , which are proportional to the (square of the) mode displacements between the stable con®guration and the transition state, are non-zero for 24 totally symmetric A1 modes and 15 totally antisymmetric A2 modes; the remaining 42 antisymmetric B1 ; B2 modes and 80 degenerate E modes are not linearly displaced and are therefore not included in the calculations. The mode with imaginary frequency, i.e., the reaction coordinate, is a totally antisymmetric mode of A2 symmetry. The displacement along this mode between the stable and transitionstate con®gurations represents half the tunneling distance. The dynamics of quadruple proton exchange on this 40-dimensional PES is evaluated by the instanton formalism. Since the method we use has been reviewed recently [9,16], we give here only a short outline to de®ne the required parameters; more details on the computational scheme, as well as on the original instanton literature can be found, e.g., in [11,16] and references therein. At low temperatures the tunneling dynamics in the multidimensional con®gurational space is represented by a narrow channel of trajectories centered around an extreme trajectory called instanton. In our instanton approach the zero-level tunneling splitting is given by D 0 X0 =pe
SI 0=2
;
1
where the instanton action SI 0 is calculated as the Euclidean action along the instanton trajectory at zero-temperature, and X0 is the (eective) frequency along the reaction coordinate in the well. The instanton action is built up step-by-step, starting with the evaluation of the action SI0 0 for an eective 1D motion along the reaction coordinate. It includes coupling to fast skeletal motions since the potential is taken to be vibrationally adiabatic and the mass is taken to be coordinatedependent. To this result corrections ds;a 0 accounting for coupling to slow skeletal modes are added according to SI 0
X SI0 0 P as da 0: 1 s ds 0 a
As this expression indicates, couplings to symmetric skeletal motions speed up the 1D dynamics by reducing SI0 0, whereas couplings to antisymmetric modes P contribute a Frank±Condon factor exp as a da 0=2 to the splitting. The factor as enters because the eect of antisymmetric coupling is modulated by symmetric couplings. The division of the transverse modes into the groups of `slow' or `fast' modes is based on a comparison of their frequencies with a de®ned `scaling frequency', which is generally smaller than but of the order of x . The corrections ds;a 0 are proportional to the square of the vibrational displacements between the stationary con®gurations. All parameters needed for the dynamics calculations are produced by the DO I T code from electronic structures and force ®elds for the stationary con®gurations, read directly from the output of standard quantumchemistry codes. The main dynamics parameters of the instanton calculations, based on the PES generated from the data in Table 1 at the B3LYP/cc-pVDZ level, are summarized in Table 2. The normal mode frequencies are scaled by 0.9614, which assumes that the scaling factor is the same as that for B3LYP/ 6-31G(d) [17]. Among the skeletal modes of the transition state, a group of eight symmetric modes in the range 300±900 cm 1 shows a strong enhancing eect on the tunneling; one of these modes, with frequency xs 420 cm 1 , is depicted in Fig. 3b. The cumulative eect of all symmetric Table 2 Principal dynamics parameters of coherent quadruple hydrogen/deuterium exchange in calix[4]arenes Calculated resultsa Vibrationally adiabatic barrier height kcal mol 1 Imaginary frequency, x cm 1 Eective frequency in the well, X0 cm 1 amu1=2 Tunneling distance A Parameter of enhancing coupling, ds Parameter of suppressing coupling, da Zero-level tunneling splitting cm 1 a
2
18.13/17.81 1448/1032 2835/2013 1.454/2.050 0.291/0.276 4:310=5:906 1:8 10 3 =5:4 10
7
Combined electronic structure and force ®eld calculations of the stationary points at the B3LYP/cc-pVDZ level with DO I T 1.2 dynamics calculations.
A. Fernandez-Ramos et al. / Chemical Physics Letters 343 (2001) 627±632
P
modes is given by The s ds 0 0:291. suppressing eect of the antisymmetric modes is P given by d 0 4:310. Half of this value is a a due to a single mode with frequency xa 270 cm 1 , depicted in Fig. 3c. With these parameter values we calculate a zero-point splitting D0 1:8 10 3 cm 1 , a result that is very close to the reported splitting [2] of 1:2 10 3 cm 1 in the range 30±80 K. In order to check whether excitation of lowfrequency modes can play a role in the reported splitting, we calculated mode-speci®c tunneling splittings, similar to those reported earlier for 9-hydroxyphenalenone [18], for all thermally accessible excited levels of coupled modes. Although the splitting in CA is promoted by a number of low-frequency modes, the proportionality of the correction term ds to the frequency of the coupled mode reduces the promotional eect of these modes. The same would apply, a fortiori, for coupled lattice modes. For CA, enhanced tunneling splittings of low vibrationally excited levels are only obtained for modes with frequencies above 250 cm 1 , which are not signi®cantly populated under the reported experimental conditions [2]. As a result the splitting of excited levels with nonnegligible Boltzmann populations is found to be not measurably dierent from the zero-point splitting. Hence our calculations indicate that proton tunneling is not the direct cause of the temperature dependence of the NMR signals observed in [2]. As a possible test for the interpretation of the experimental data, we extended the calculations to a CA isotopomer with the four hydrogen-bonded protons replaced by deuterons. The results are summarized in Table 2. Parameters related to the synchronous motion of the tunneling atoms change in accord with the doubled mass of the tunneling particle. Thus the imaginary frequency and the eective in the well are reduced pfrequency by a factor of 2, while the (mass-weighted) tunneling distance is increased by the same factor. Since the vibrations coupled to the tunneling mode are skeletal, their eect remains almost unchanged, as re¯ected in the values of the coupling parameters ds;a . The calculated zero-point tunneling splitting equals D0 5:4 10 7 cm 1 , indicating a
631
low-temperature deuterium isotope eect of about 3:3 103 . The predicted frequency of the resonance peak of this isotopomer is therefore about 16 KHz. 3. Conclusion The present calculations address proton transfer in an isolated molecule, whereas the experiments are performed on crystalline powders. However, there are good reasons to assume that the in¯uence of the lattice will be small. The transfer occurs inside a large molecule and the hydroxyl groups are far removed from hydroxyl groups in neighboring molecules. Proton transfer rates in free base porphyrins, where the protons are also located inside a large molecule were found to be the same in solution and in the solid [19±21]. We therefore expect our isolated molecule results to be applicable, within the margin of error of the calculations, to the solid-state data. In other words, the calculations support the interpretation of the NMR data of [2] in terms of a quadruple proton tunneling splitting, provided the splitting is assigned to the zero-point level. As to the question of whether proton exchange in the same system can be both phase-coherent and -incoherent, our interpretation of the experiments on tbCA diers from that of the authors of [2]. In the systems at hand the levels are sharp, tunneling is a rare event and the tunneling motion cannot be the mechanism that randomizes the dipole±dipole interactions and thus be responsible for the width of the peaks in the spin±lattice relaxation rate. The central peaks of comparable width observed in CA and tbCA are due to incoherent processes but these are random dipole±dipole interactions with external spins. Therefore, their widths cannot be related to the rate of incoherent tunneling in the cyclic hydrogen-bond network. In this case, tunneling can produce relaxation only by a coherent proton motion which results in the peak of the spin±lattice relaxation rate at the resonant frequency x D0 . A complete account of our interpretation of the experiments on CA and tbCA and of the tunneling dynamics will be given elsewhere.
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References [1] D.F. Brougham, R. Caciuo, A.J. Horsewill, Nature 397 (1999) 241. [2] A.J. Horsewill, N.H. Jones, R. Caciuo, Science 291 (2001) 100. [3] Z. Smedarchina, M.Z. Zgierski, W. Siebrand, P.M. Kozlowski, J. Chem. Phys. 109 (1998) 1014. [4] A. Fern andez-Ramos, Z. Smedarchina, W. Siebrand, M.Z. Zgierski, J. Chem. Phys. 114 (2001) 7518. [5] Z. Smedarchina, W. Siebrand, A. Fernandez-Ramos, L. Gorb, J. Leszczynski, J. Chem. Phys. 112 (2000) 566. [6] A. Fern andez-Ramos, Z. Smedarchina, W. Siebrand, M.Z. Zgierski, J. Chem. Phys. 113 (2000) 9714. [7] A. Fern andez-Ramos, Z. Smedarchina, M.Z. Zgierski, J. Chem. Phys. 113 (2000) 2662. [8] A. Fern andez-Ramos, Z. Smedarchina, J. RodriguezOtero, J. Chem. Phys. 114 (2001) 1567. [9] Z. Smedarchina, A. Fernandez-Ramos, W. Siebrand, J. Comp. Chem. 22 (2001) 787. [10] M.J. Frisch et al., GA U S S I A N 98, Revision A.3. [11] Z. Smedarchina, A. Fernandez-Ramos, W. Siebrand, M.Z. Zgierski, DO I T 1.2, a computer program to calculate
[12] [13] [14] [15] [16] [17] [18] [19] [20] [21]
hydrogen tunneling rate constants and splittings, National Research Council of Canada, http://gold.sao.nrc.ca/sims/ software/doit1.2/index.html. A.D. Becke, J. Chem. Phys. 98 (1993) 5648. D.E. Woon, T.H. Dunning Jr., J. Chem. Phys. 98 (1993) 1358. R. Ungaro, A. Pochini, G.D. Andreetti, V. Sangermano, J. Chem. Soc. Perkin Trans. 2 (1984) 1979. R.J. Bernardino, B.J. Costa Cabral, J. Phys. Chem. A 103 (1999) 9080. W. Siebrand, Z. Smedarchina, M.Z. Zgierski, A. Fernandez-Ramos, Int. Rev. Phys. Chem. 18 (1999) 5. A.P. Scott, L. Radom, J. Phys. Chem. 100 (1996) 16502. A. Fernandez-Ramos, Z. Smedarchina, M.Z. Zgierski, W. Siebrand, J. Chem. Phys. 109 (1998) 1004. T.J. Butenho, C.B. Moore, J. Am. Chem. Soc. 110 (1988) 8336. J. Braun, M. Schlabach, B. Wehrle, M. Kocher, E. Vogel, H.-H. Limbach, J. Am. Chem. Soc. 116 (1994) 6593. J. Braun, H.-H. Limbach, P.G. Williams, H. Morimoto, D.E. Wemmer, J. Am. Chem. Soc. 118 (1996) 7231.
Chemical Physics Letters 343 (2001) 627±632
www.elsevier.com/locate/cplett
Proton tunneling in calix[4]arenes: a theoretical investigation Antonio Fern andez-Ramos a, Zorka Smedarchina a,*, Fabio Pichierri b a
Steacie Institute for Molecular Sciences, National Research Council of Canada, Ottawa, Ont., Canada K1A 0R6 b RIKEN Genomic Sciences Center, 1-7-22 Suehiro-cho, Tsurumi-ku, 230-0045 Yokohama, Japan Received 26 February 2001; in ®nal form 27 April 2001
Abstract The structure, energetics and vibrational force ®eld of calix[4]arene (CA) are calculated for the chiral equilibrium con®guration and the symmetric transition state for coherent quadruple proton transfer. The corresponding proton and deuteron tunneling splittings are evaluated with the instanton approach. The calculated zero-point proton tunneling splitting is compared with data reported for low-temperature NMR relaxometry experiments. Ó 2001 Elsevier Science B.V. All rights reserved.
1. Introduction Recently, Horsewill and coworkers [1,2] studied multiple proton transfer in the hydrogen bond network of calix[4]arene (CA) and p-tert-butyl calix[4]arene (tbCA). The center of the bowlshaped molecules contains four hydroxyl groups forming a chiral ring of four hydrogen bonds represented by the point group C4 . These molecules exist in two forms of opposite chirality, between which transitions are possible through exchange of the four protons along their individual hydrogen bonds (Fig. 1). In an isolated molecule of C4 symmetry, this would give rise to tunneling splitting, but if the symmetry is broken, e.g., by a medium, the exchange will be a rate process. To investigate these phenomena, Horsewill et al. used NMR relaxometry on crystalline powders at low
*
Corresponding author. Fax: +1-613-947-2838. E-mail address: [email protected] (Z. Smedarchina).
temperatures. They reported the observation of both a splitting [2] and a rate process [1,2]. The splitting corresponding to a frequency of 35 MHz in CA was derived from the observation of a corresponding peak in the magnetic ®eld dependence of the proton spin±lattice relaxation rate; this frequency showed no temperature dependence in the range 30±80 K, but the peak height increased with temperature as if subject to an activation energy of 125 K. No splitting was observed in tbCA. The observation of a peak at zero ®eld in both molecules was interpreted as evidence for an exponential rate process, the halfwidth of this peak being used to extract the rate constant. For tbCA a value of 8 107 s 1 was reported, which was found to remain constant in the range 15±21 K. No rate constant was reported for the rate process in CA, but if the published graph is interpreted in the same way as that for tbCA, the result would be similar. Both zero-®eld peaks showed an exponential temperature dependence as if subject to an activation energy of about 75 K.
0009-2614/01/$ - see front matter Ó 2001 Elsevier Science B.V. All rights reserved. PII: S 0 0 0 9 - 2 6 1 4 ( 0 1 ) 0 0 7 4 1 - 2
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A. Fernandez-Ramos et al. / Chemical Physics Letters 343 (2001) 627±632
Fig. 1. The two stereoisomers of calix[4]arene.
These observations exhibit a number of unusual features. It is not clear how the same system can show both a splitting and a corresponding rate constant, i.e., how proton exchange can be both phase-coherent and -incoherent. The observation of a tunneling splitting at temperatures much lower than required to excite hydrogen-bond vibrations suggests that it should be assigned to the zero-point splitting. Yet the signal disappears at very low temperatures. In this connection, we note that the measurements on tbCA were carried out at temperatures too low to observe tunneling splitting, which begs the question whether tbCA would also show such splitting if it were possible to perform these measurements at higher temperatures. In view of these problems, it seems useful to explore to what extent quantum-chemical and dynamics calculations can shed light on these observations. This requires an investigation of the transition state of the tunneling process, which has not been calculated before. In a recent series of papers we have investigated coherent and incoherent double and triple proton tunneling in porphine [3], 7-azaindole [4,5], guanine [5], glycine in aqueous solution [6], 3-hydroxyisoquinoline [7], a methanol±acetic acid complex [8], and a water trimer [9], respectively. We shall use the same method to investigate quadruple proton transfer in CA. First we calculate the structure, energy and force ®eld of the molecule in the stable con®guration(s) and in the transition state. Then we use
the results as input parameters for the calculation of proton tunneling splittings for ground and excited vibrational levels, and compare the results with experimental observations. 2. Dynamics of coherent proton exchange The interchange of the two stereoisomers of CA corresponds to concerted motion of four protons in a double-minimum potential. It is to be expected that the proton motion will be coupled to skeletal vibrations, so that we need to generate a multidimensional potential for the dynamics. To this end we have performed quantum-chemical calculations with GA U S S I A N 98 [10] and dynamics calculations with DO I T 1.2 [11]. We fully optimized the stationary points using density functional theory (DFT) with the B3LYP functional [12] and the cc-pVDZ basis set [13]. A Hartree±Fock (HF) calculation would be less computationally intensive, but we found that electronic correlation is very important in this case. Speci®cally, the HF method fails to produce a transition state of C4v symmetry. Selected geometric parameters obtained at the B3LYP/ cc-pVDZ level and relevant for the dynamics calculations are listed in Table 1. A more extensive report on the structural and force ®eld parameters of the two stationary points of CA will be published elsewhere.
A. Fernandez-Ramos et al. / Chemical Physics Letters 343 (2001) 627±632 Table 1 Selected structural parameters of the hydrogen bond network in calix[4]arenes calculated at the B3LYP/-ccpVDZ level
O1 H O2 H O1 O2 O1 C O1 HO2 HO1 C
Stable con®guration
Transition state con®guration
0.993 1.678 2.648 1.382 1.682 114.4
1.207 1.207 2.401 1.381 164.2 110.1
angles in degrees. Distances in A,
The main structural parameters of the stable con®guration agree satisfactorily with the available X-ray data [14]; for instance, the mean O O is calculated to be 2.65 A. A distance of 2.64 A value of 2.67 A, obtained at the BLYP/6-31G level for this distance, was reported in [15]. The two C4 minima in our calculation are separated by a transition state of C4v symmetry with barrier height 16.86 kcal/mol. Its structure, superimposed on that of the stable con®guration, is illustrated in Fig. 2. It shows that the motion of the four protons is accompanied by signi®cant skeletal displacement; for instance the O O distance is in the transition state. The reduced to 2.40 A tightening of the hydrogen bonds in the transition state, eected by displacements of the oxygens and the attached carbons in and up the bowl, obviously
Fig. 2. Superposition of the B3LYP/cc-pVDZ structures of the stable con®guration (black lines) and the transition state (gray lines) of calix[4]arene. Note the tightening of the hydrogen bond network in the transition state, which assists proton exchange.
629
facilitates tunneling. To calculate the resulting tunneling splitting, it is thus necessary to use a method that can properly account for the multidimensional character of the tunneling potential. In our approach the mode with imaginary frequency x , depicted in Fig. 3a, is chosen as the reaction coordinate and the skeletal mode contributions are included through linear coupling terms. The bowl of CA, held together by the cyclic network of four hydrogen bonds, consists of 56 atoms. The interconversion between its two stereoisomers in Fig. 1 is governed by a potentialenergy surface (PES) of 162 vibrational degrees of freedom. The DO I T code generates the complete vibrational PES in terms of the normal modes x; y of the transition state, where x denotes the reaction coordinate and y the set of transverse modes. As input it uses the standard output of electronic structure and force ®eld calculations of the stationary con®gurations along the reaction path. The PES is generated in a form suitable for instanton-dynamics calculations: it consists of a onedimensional (1D) symmetric double-well potential U x along the reaction coordinate, harmonic terms along the 161 (transverse) modes y, and mixed terms which describe the coupling between the tunneling mode and skeletal vibrations. Terms of the type Cs x2 ys correspond to coupling to symmetric (s) skeletal modes, similar to those that tighten the hydrogen bonds, and promote tunneling. Terms of the type Ca xya , which account for coupling to antisymmetric (a) modes, are
Fig. 3. Three normal modes of the transition state active in the coherent quadruple proton transfer in calix[4]arene: (a) the tunneling mode (symmetry A2 ) with frequency ix 1448 cm 1 ; (b) a totally symmetric mode (A1 ) with frequency xs 420 cm 1 , promoting tunneling; (c) a totally antisymmetric mode A2 with frequency xs 270 cm 1 , suppressing tunneling.
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A. Fernandez-Ramos et al. / Chemical Physics Letters 343 (2001) 627±632
equivalent to Franck±Condon factors and have a suppressing eect on the tunneling. The coupling constants Ca;s , which are proportional to the (square of the) mode displacements between the stable con®guration and the transition state, are non-zero for 24 totally symmetric A1 modes and 15 totally antisymmetric A2 modes; the remaining 42 antisymmetric B1 ; B2 modes and 80 degenerate E modes are not linearly displaced and are therefore not included in the calculations. The mode with imaginary frequency, i.e., the reaction coordinate, is a totally antisymmetric mode of A2 symmetry. The displacement along this mode between the stable and transitionstate con®gurations represents half the tunneling distance. The dynamics of quadruple proton exchange on this 40-dimensional PES is evaluated by the instanton formalism. Since the method we use has been reviewed recently [9,16], we give here only a short outline to de®ne the required parameters; more details on the computational scheme, as well as on the original instanton literature can be found, e.g., in [11,16] and references therein. At low temperatures the tunneling dynamics in the multidimensional con®gurational space is represented by a narrow channel of trajectories centered around an extreme trajectory called instanton. In our instanton approach the zero-level tunneling splitting is given by D 0 X0 =pe
SI 0=2
;
1
where the instanton action SI 0 is calculated as the Euclidean action along the instanton trajectory at zero-temperature, and X0 is the (eective) frequency along the reaction coordinate in the well. The instanton action is built up step-by-step, starting with the evaluation of the action SI0 0 for an eective 1D motion along the reaction coordinate. It includes coupling to fast skeletal motions since the potential is taken to be vibrationally adiabatic and the mass is taken to be coordinatedependent. To this result corrections ds;a 0 accounting for coupling to slow skeletal modes are added according to SI 0
X SI0 0 P as da 0: 1 s ds 0 a
As this expression indicates, couplings to symmetric skeletal motions speed up the 1D dynamics by reducing SI0 0, whereas couplings to antisymmetric modes P contribute a Frank±Condon factor exp as a da 0=2 to the splitting. The factor as enters because the eect of antisymmetric coupling is modulated by symmetric couplings. The division of the transverse modes into the groups of `slow' or `fast' modes is based on a comparison of their frequencies with a de®ned `scaling frequency', which is generally smaller than but of the order of x . The corrections ds;a 0 are proportional to the square of the vibrational displacements between the stationary con®gurations. All parameters needed for the dynamics calculations are produced by the DO I T code from electronic structures and force ®elds for the stationary con®gurations, read directly from the output of standard quantumchemistry codes. The main dynamics parameters of the instanton calculations, based on the PES generated from the data in Table 1 at the B3LYP/cc-pVDZ level, are summarized in Table 2. The normal mode frequencies are scaled by 0.9614, which assumes that the scaling factor is the same as that for B3LYP/ 6-31G(d) [17]. Among the skeletal modes of the transition state, a group of eight symmetric modes in the range 300±900 cm 1 shows a strong enhancing eect on the tunneling; one of these modes, with frequency xs 420 cm 1 , is depicted in Fig. 3b. The cumulative eect of all symmetric Table 2 Principal dynamics parameters of coherent quadruple hydrogen/deuterium exchange in calix[4]arenes Calculated resultsa Vibrationally adiabatic barrier height kcal mol 1 Imaginary frequency, x cm 1 Eective frequency in the well, X0 cm 1 amu1=2 Tunneling distance A Parameter of enhancing coupling, ds Parameter of suppressing coupling, da Zero-level tunneling splitting cm 1 a
2
18.13/17.81 1448/1032 2835/2013 1.454/2.050 0.291/0.276 4:310=5:906 1:8 10 3 =5:4 10
7
Combined electronic structure and force ®eld calculations of the stationary points at the B3LYP/cc-pVDZ level with DO I T 1.2 dynamics calculations.
A. Fernandez-Ramos et al. / Chemical Physics Letters 343 (2001) 627±632
P
modes is given by The s ds 0 0:291. suppressing eect of the antisymmetric modes is P given by d 0 4:310. Half of this value is a a due to a single mode with frequency xa 270 cm 1 , depicted in Fig. 3c. With these parameter values we calculate a zero-point splitting D0 1:8 10 3 cm 1 , a result that is very close to the reported splitting [2] of 1:2 10 3 cm 1 in the range 30±80 K. In order to check whether excitation of lowfrequency modes can play a role in the reported splitting, we calculated mode-speci®c tunneling splittings, similar to those reported earlier for 9-hydroxyphenalenone [18], for all thermally accessible excited levels of coupled modes. Although the splitting in CA is promoted by a number of low-frequency modes, the proportionality of the correction term ds to the frequency of the coupled mode reduces the promotional eect of these modes. The same would apply, a fortiori, for coupled lattice modes. For CA, enhanced tunneling splittings of low vibrationally excited levels are only obtained for modes with frequencies above 250 cm 1 , which are not signi®cantly populated under the reported experimental conditions [2]. As a result the splitting of excited levels with nonnegligible Boltzmann populations is found to be not measurably dierent from the zero-point splitting. Hence our calculations indicate that proton tunneling is not the direct cause of the temperature dependence of the NMR signals observed in [2]. As a possible test for the interpretation of the experimental data, we extended the calculations to a CA isotopomer with the four hydrogen-bonded protons replaced by deuterons. The results are summarized in Table 2. Parameters related to the synchronous motion of the tunneling atoms change in accord with the doubled mass of the tunneling particle. Thus the imaginary frequency and the eective in the well are reduced pfrequency by a factor of 2, while the (mass-weighted) tunneling distance is increased by the same factor. Since the vibrations coupled to the tunneling mode are skeletal, their eect remains almost unchanged, as re¯ected in the values of the coupling parameters ds;a . The calculated zero-point tunneling splitting equals D0 5:4 10 7 cm 1 , indicating a
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low-temperature deuterium isotope eect of about 3:3 103 . The predicted frequency of the resonance peak of this isotopomer is therefore about 16 KHz. 3. Conclusion The present calculations address proton transfer in an isolated molecule, whereas the experiments are performed on crystalline powders. However, there are good reasons to assume that the in¯uence of the lattice will be small. The transfer occurs inside a large molecule and the hydroxyl groups are far removed from hydroxyl groups in neighboring molecules. Proton transfer rates in free base porphyrins, where the protons are also located inside a large molecule were found to be the same in solution and in the solid [19±21]. We therefore expect our isolated molecule results to be applicable, within the margin of error of the calculations, to the solid-state data. In other words, the calculations support the interpretation of the NMR data of [2] in terms of a quadruple proton tunneling splitting, provided the splitting is assigned to the zero-point level. As to the question of whether proton exchange in the same system can be both phase-coherent and -incoherent, our interpretation of the experiments on tbCA diers from that of the authors of [2]. In the systems at hand the levels are sharp, tunneling is a rare event and the tunneling motion cannot be the mechanism that randomizes the dipole±dipole interactions and thus be responsible for the width of the peaks in the spin±lattice relaxation rate. The central peaks of comparable width observed in CA and tbCA are due to incoherent processes but these are random dipole±dipole interactions with external spins. Therefore, their widths cannot be related to the rate of incoherent tunneling in the cyclic hydrogen-bond network. In this case, tunneling can produce relaxation only by a coherent proton motion which results in the peak of the spin±lattice relaxation rate at the resonant frequency x D0 . A complete account of our interpretation of the experiments on CA and tbCA and of the tunneling dynamics will be given elsewhere.
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References [1] D.F. Brougham, R. Caciuo, A.J. Horsewill, Nature 397 (1999) 241. [2] A.J. Horsewill, N.H. Jones, R. Caciuo, Science 291 (2001) 100. [3] Z. Smedarchina, M.Z. Zgierski, W. Siebrand, P.M. Kozlowski, J. Chem. Phys. 109 (1998) 1014. [4] A. Fern andez-Ramos, Z. Smedarchina, W. Siebrand, M.Z. Zgierski, J. Chem. Phys. 114 (2001) 7518. [5] Z. Smedarchina, W. Siebrand, A. Fernandez-Ramos, L. Gorb, J. Leszczynski, J. Chem. Phys. 112 (2000) 566. [6] A. Fern andez-Ramos, Z. Smedarchina, W. Siebrand, M.Z. Zgierski, J. Chem. Phys. 113 (2000) 9714. [7] A. Fern andez-Ramos, Z. Smedarchina, M.Z. Zgierski, J. Chem. Phys. 113 (2000) 2662. [8] A. Fern andez-Ramos, Z. Smedarchina, J. RodriguezOtero, J. Chem. Phys. 114 (2001) 1567. [9] Z. Smedarchina, A. Fernandez-Ramos, W. Siebrand, J. Comp. Chem. 22 (2001) 787. [10] M.J. Frisch et al., GA U S S I A N 98, Revision A.3. [11] Z. Smedarchina, A. Fernandez-Ramos, W. Siebrand, M.Z. Zgierski, DO I T 1.2, a computer program to calculate
[12] [13] [14] [15] [16] [17] [18] [19] [20] [21]
hydrogen tunneling rate constants and splittings, National Research Council of Canada, http://gold.sao.nrc.ca/sims/ software/doit1.2/index.html. A.D. Becke, J. Chem. Phys. 98 (1993) 5648. D.E. Woon, T.H. Dunning Jr., J. Chem. Phys. 98 (1993) 1358. R. Ungaro, A. Pochini, G.D. Andreetti, V. Sangermano, J. Chem. Soc. Perkin Trans. 2 (1984) 1979. R.J. Bernardino, B.J. Costa Cabral, J. Phys. Chem. A 103 (1999) 9080. W. Siebrand, Z. Smedarchina, M.Z. Zgierski, A. Fernandez-Ramos, Int. Rev. Phys. Chem. 18 (1999) 5. A.P. Scott, L. Radom, J. Phys. Chem. 100 (1996) 16502. A. Fernandez-Ramos, Z. Smedarchina, M.Z. Zgierski, W. Siebrand, J. Chem. Phys. 109 (1998) 1004. T.J. Butenho, C.B. Moore, J. Am. Chem. Soc. 110 (1988) 8336. J. Braun, M. Schlabach, B. Wehrle, M. Kocher, E. Vogel, H.-H. Limbach, J. Am. Chem. Soc. 116 (1994) 6593. J. Braun, H.-H. Limbach, P.G. Williams, H. Morimoto, D.E. Wemmer, J. Am. Chem. Soc. 118 (1996) 7231.