Conformational Analysis of 2,2′-Bithiophene Revisited: The

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Conformational Analysis of 2,2′-Bithiophene Revisited: The Maximum Entropy Method Applied to Large Sets of H-H and 13C-H Partially Averaged Dipolar Couplings Giorgio Cinacchi* School of Chemistry, UniVersity of Bristol, Cantock’s Close, Bristol BS8 1TS, England ReceiVed: December 7, 2009; ReVised Manuscript ReceiVed: June 20, 2010

Two recently determined sets of partially averaged H-H and 13C-H NMR dipolar couplings of 2,2′-bithiophene as a solute in nematic liquid-crystalline solvents have been analyzed by means of the maximum entropy method. This investigation originated to ascertain the real information content of the new 13C-H data. To this end, a procedure has been employed which consists in adding the experimental data one after the other and monitoring, during each of these steps, the behavior of the minimum of the associated work function as well as that of the root-mean-square deviation. It turns out that nearly all 13C-H data do not provide new pieces of information with respect to a subset of seven H-H dipolar couplings. Inclusion of the very few which do, however, alters significantly the conformational distribution function calculated taking into account the sole H-H dipolar couplings. The final form of this function is characterized by a prominent peak corresponding to the trans conformation and two smaller peaks corresponding to the cis conformation and of the conformation in which the two rings are mutually orthogonal. The form of the conformational distribution function is the result of a considerable orientational-conformational coupling. The implications of the present results on the form of the torsional potential-energy function of the isolated molecule are discussed. I. Introduction Determination of molecular conformational distributions has always been a subject of great relevance in chemistry. Of general and long-standing importance is the influence that a fluid medium may have on the form of the confomational distribution function of a molecule dissolved therein. Unfortunately, the experimental determination of the latter function is very complicated: only indirect measurements are possible, and the quantities directly measurable are linked to the conformational distribution function via a functional relationship. Nuclear magnetic resonance (NMR) spectroscopy in liquidcrystalline solvents (LX-NMR) is a valuable technique to study the structure and orientational order that a molecule adopts in condensed fluid media.1-3 In the case of a flexible molecule, it offers also the possibility to investigate the coupling between orientational order and the conformations adopted in the anisotropic fluid solvent.2,3 In the specific case of LX-NMR spectroscopy, the abovementioned functional relationship is

Dijexp )

∫ dΩ ∫ dΦP(Ω, Φ)Dij(Ω, Φ)

(1)

where Dexp ij is the experimentally determined dipolar coupling between nuclei i and j, P(Ω, Φ) is the relevant probability density distribution, which is a function of orientational variables, collected under the symbol Ω, and structural variables, collected under the symbol Φ, and Dij(Ω, Φ) is the value of the ij dipolar coupling in the orientational-structural state defined by (Ω, Φ). Its explicit expression, in Hertz, is * To whom correspondence should be addressed. E-mail: giorgio. [email protected].

Dij(Ω, Φ) ) -

µ0γiγjp 2

8π

P2(Bˆ · nˆ)

P2(cos θij(Ω, Φ)) rij3 (Φ)

(2)

where µ0 is the vacuum magnetic permittivity, γi and γj are the respective magnetogyric ratios, p ) h/(2π), with h being the Planck constant, P2() is the second Legendre polynomial, Bˆ is b, direction, and nˆ is the unit vector along the magnetic field, B the nematic director; θij is the angle that the vector joining the b. The nematic two nuclei, whose length is rij, forms with B director n is aligned (transverse to) Bˆ if the liquid crystal is of positive (negative) diamagnetic susceptibility. In order to extract pieces of information on P(Ω, Φ), there are two possibilities: either to develop a model for the probability density distribution that contains a few parameters whose value is to be determined by fitting to the experimental data (as it is done, e.g., within the additive potential (AP) method; representative examples of its application are provided in ref 4) or to resort to the maximum entropy (ME) principle.5 In the second case, the Lagrange multiplier minimization of the information theory entropy functional with the constraints provided by the normalization condition and the experimental data leads to an exponentiated form of the probability density distribution. If the experimental data are dipolar couplings, this is explicitly given by6-9

exp[ P(Ω, Φ) )

∑ λijDij(Ω, Φ)] ij

Z

(3)

with Z being the normalization constant. One way to determine the set of {λij} consists in minimizing a suitably defined functional

10.1021/jp911603t  2010 American Chemical Society Published on Web 07/15/2010

Conformational Analysis of 2,2′-Bithiophene Revisited

F ) ln Z -

∑ λijDijexp

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(4)

ij

It has been proven that if the experimental data are linearly independent, then F is a concave function of λij.10,11 In this case, the minimum is stable and unique. Otherwise, the minimization process is not stable. Therefore, the minimization of F provides a straightforward means to determine the linear independence, and thus the real information content, of the experimental data. The ME method is completely a posteriori, depending exclusively on the quality and quantity of the experimental data. In the case of LX-NMR, the conformational information content of the set of dipolar couplings certainly increases with the orientational order. However, highly ordered molecules are hard to find in real experiments. The natural way to obtain a more complete data set, i.e., adding more dipolar couplings, is experimentally more feasible thanks to advances in NMR spectrum analysis. The situation could not be so easy, however, because the added experimental data could be, within the ME context, linearly dependent on those already determined; thus, they do not augment the real information content of the overall experimental data set. Recently, two very large sets of dipolar couplings have been determined for the very interesting molecule 2,2′-bithiophene (Figure 1) dissolved in two nematic liquid crystals, I52 and ZLI1132, both of positive diamagnetic susceptibility.12 The sets include not only the partially averaged dipolar couplings between pairs of hydrogen atoms, already determined a few times in the past,13-15 but, remarkably, also all the possible 13 C-H dipolar couplings obtained from the satellite spectra in the proton LX-NMR analysis, for a total of 33 dipolar couplings for each set. In light of the above considerations, it is of interest to evaluate the performance of the ME method faced to this unprecedentedly large amount of experimental dipolar coupling data. The major objective of this work is indeed to ascertain the real information content of the new 13C-H dipolar couplings determined in ref 12 and assess their importance in shaping the conformational distribution function of a molecule in a condensed fluid media. This is an issue of importance for the field of NMR spectroscopy in aligned solvents. It is presently fortunate that this issue can be addressed for a very relevant molecule: 2,2′-bithiophene is indeed the basic unit of oligo- and polythiophenes, materials with remarkable electronic and optical properties.16 The next section provides details on the methodology used. In section III, the results are presented and discussed. Section IV summarizes the main findings of this work and outlines a few concluding comments. II. Methodology Analysis of a set of residual dipolar coupling data to extract orientational and conformational information customarily commences with assumptions on the geometry of the molecule of interest. In the present case, the geometry of the thiophene molecule as determined in a microwave spectroscopy study17 was assumed for the two thiophene rings. Different from what was done in ref 12, no attempt was pursued to determine the geometry of the thiophene subunits from the relevant dipolar coupling data. The reason is the following. Due to the symmetry of the molecule, in fact, even those dipolar couplings between two nuclei belonging to the same ring are dependent on the molecular conformation. Thus, the thiophene subunits cannot be properly isolated. The two rings were furthermore supposed

Figure 1. Labeling and numbering of the 2,2′-bithiophene constituting atoms: white, black, and gray circles represent, respectively, hydrogen, carbon, and sulfur atoms. φ is the dihedral angle that defines the internal rotation of one ring with respect to the other; in the figure, the trans conformation is depicted, which corresponds to a value of φ equal to 180°.

to be separated by a interannular distance of 1.453 Å, while the angles 516 and 1061 were set equal to 128.4°, following the indications of certain quantum-chemical calculations.12 Once the molecular geometry is frozen, the only structural parameter left to vary is the dihedral angle φ defining a conformation. The calculations then proceeded by employing the functional minimization method outlined in the Introduction. First, only the H-H dipolar couplings were considered. They were added one after the other, starting from D11,12, adding the dipolar couplings between two protons belonging to the same ring first and then those between two protons belonging to different rings; after every addition, it was verified that the functional F reached a stable minimum. The criterium of stability assumed was that the value of the minimum had to be larger than zero10,11 together with the fact that the root-mean-square (rms) deviation had to decrease after every addition. Whenever the value of F became close to zero or negative, the rms deviation was often seen to increase very steeply. In the case where F did not show a stable minimum and/or the rms deviation did not decrease, the last dipolar coupling added was discarded from the sum in eq 4. The same procedure was then applied in the treatment of 13C-H dipolar couplings. They were progressively added with the exception of those corresponding to a pair of 13C and H atoms separated by less than three bonds. In these cases, in fact, vibrational corrections cannot be disregarded. Thus, the total number of dipolar couplings considered was 25. It proved important to adopt the above-mentioned order of addition of the dipolar couplings data. In particular, it was essential to start with D11,12, i.e., the largest H-H dipolar coupling, and then add the other H-H data and finally the new 13 C-H data. While this dependence on the order of the addition of the data may seem prima facie odd, it is nevertheless comprehensible in light of the known dependence of the ME method from the quality of the data used. Except for the dipolar couplings between a 13C and a H atom separated by less than three bonds, which are however affected by vibrational corrections, the H-H data have indeed overall larger absolute values than the 13C-H data. The largest is the dipolar coupling data; the largest is the amount of information it is carrying. It is therefore of importance to start with those dipolar coupling data that carry the major amount of information on the order and structure of the molecule and which will determine the main features of the conformational distribution function and add those which will provide finer details subsequently. Trial analyses started with the 13C-H data and then followed by the addition of the H-H data confirmed this: the final rms deviations were considerably larger than those obtained via the above-

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Figure 2. (a) Conformational distribution function of 2,2′-bithiophene. The solvent and set of dipolar couplings used are, respectively, (a) I52 and the subset of seven H-H dipolar couplings, (b) ZLI1132 and the subset of seven H-H dipolar couplings, (c) I52 and the set of, within the ME context, linearly independent H-H and 13C-H dipolar couplings, and (d) ZLI1132 and the set of, within the ME context, linearly independent H-H and 13C-H dipolar couplings.

mentioned procedure and the resulting conformational distributions were essentially and unrealistically flat. The final sets obtained by applying the above successful procedure were formed by seven H-H dipolar couplings {D11,12, D11,13, D11,14, D11,15, D11,16, D12,13, D13,16} irrespective of the solvent together with the 13C-H dipolar couplings {D2,14, D2,15, D2,16} if I52 is the solvent or the sole 13C-H dipolar coupling D2,14 if the solvent is ZLI1132. It is clear that most 13C-H data do not add new pieces of information with respect to the H-H dipolar couplings. This is understandable in view of the specific constraints provided by the molecular geometry. The effect of the few retained 13C-H data on the conformational distribution function is discussed next. III. Presentation and Discussion of the Results The conformational distribution functions, Π(φ), at each of the addition stages have been obtained by integrating P(Ω, φ) over the orientational variables.18 Figure 2 summarizes the principal results obtained during the course and at the end of the procedure described in section II. The Supporting Information provides further results. These include the values of F and rms obtained after every single addition stage, the parameters λij entering eqs 3 and 4, and the values of the experimental and calculated dipolar couplings. In the tables of the Supporting Information, the order by which the dipolar couplings are listed is that by which they were added. Figure 2a and 2b illustrates the conformational distribution function of the solute dissolved in solvent I52 and ZLI1132, respectively, obtained using only the above-mentioned subset of seven H-H dipolar couplings. The value of the functional minimum is 2.88 and 2.42, respectively. The rms deviation is 0.53 and 1.20 Hz, respectively, if all nine H-H dipolar couplings are inserted in the rms sum. These values increase to 2.22 and 2.78 Hz, respectively, if all 25 dipolar couplings are inserted in the rms sum. Figure 2c and 2d illustrates the conformational distribution function of the solute dissolved in the two solvents obtained when all of the above-listed, within the ME context linearly independent, H-H and 13C-H dipolar couplings are included.

Cinacchi The value of the functional minimum is 2.17 in the case where I52 is the solvent and 1.50 in the other case. The rms deviations of the 25 dipolar couplings are, respectively, 1.82 and 2.13 Hz. In all cases, the calculated dipolar couplings are, with very few exceptions, well within the 3% of the corresponding experimental value. The curves in Figure 2a and 2b are in qualitative agreement with the results of ref 15 and in very good agreement with the results of ref 19, indicating a cisoid and the trans conformations as the most probable. In these works, previous partially averaged H-H dipolar couplings of the present solute molecule in a nematic solvent were analyzed with, respectively, the rotational isomeric state (RIS) model and the ME method preceded by the orthogonalization of the dipolar couplings. The very good agreement of the present results with those of ref 19 further suggests that the procedure of judiciously adding experimental dipolar couplings one after the other and verifying whether F possesses a stable minimum and rms deviation decreases coincides with the determination of a set of, within the ME context, linearly independent Dij’s without the need of a preorthogonalization. The inclusion of the few relevant 13C-H dipolar couplings decisively contributes to make the conformational distribution functions more structured. Besides the increment of the most prominent peak at φ ) 180°, the above-mentioned peak corresponding to the cisoid conformation becomes resolved in two peaks, one corresponding to the cis conformation and the other corresponding to φ ≈ 90°. The populations of these conformations are as follows: cis(oid), 20%; perpendicular-like, 30%; trans(oid), 50%. These results are at odds with those obtained in ref 12, according to which 2,2′-bithiophene in a nematic phase would be frozen in two, cisoid and transoid, conformations. In ref 12 the dipolar couplings data were analyzed via the AP method assuming that the conformational distribution function is the sum of two Gaussian functions centered around two dihedral angles corresponding, respectively, to a cisoid and transoid conformation (eq 11 in ref 12). Given the conceptually different basis of the AP and ME methods, it is not surprising that, in certain cases, they can provide different results, the source of this difference being the assumptions made in the AP analysis and/or the quality and quantity of the experimental data, affecting the performance of the ME method. It may nevertheless be noticed here that eq 11 in ref 12 may not be fully consistent with situations in which the conformational distribution functions show peaks not entirely localized within the interval [0°; 180°], as, for example, in the case where the most favored conformations are exactly the cis and/or the trans conformations. The conformational distribution functions of Figure 2 are the result of a considerable orientational-conformational coupling. This can be appreciated by observing Figure 3, where the conformational second-rank orientational order parameter S2 is shown as a function of the dihedral angle. This quantity is defined as

S2(Φ) )

∫ dΩ[ 23 cos2 β - 21 ]P(Ω, Φ) ∫ dΩP(Ω, Φ)

(5)

with β being the angle that the vector joining the carbon atoms numbered 1 and 6 in Figure 1 forms with the director. The cisoid and transoid conformations bear a considerable positive value

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Figure 3. Second-rank orientational order parameter, S2 (eq 5), as a function of the dihedral angle φ. The letters a-d refer to the cases in Figure 2 with the same label.

of S2. The latter drops to zero or even to negative values for perpendicular-like conformations. This means that, in the bulk nematic phase, 2,2′-bithiophene molecules in the cisoid or transoid conformations tend to orient parallel to the director, whereas the orientation of 2,2′-bithiophene molecules with the two thiophene rings reciprocally almost orthogonal is more isotropically distributed. The evaluation of the average orientational order parameter defined by

¯ S2 )

∫0π dφΠ(φ)S2(φ)

(6)

provides a value of Sj2 )0.39 for 2,2′-bithiophene dissolved in I52 and Sj2 )0.45 for this solute in ZLI1132. These numbers, typical of an elongated molecule in a calamitic nematic phase, have been obtained irrespective of whether the sole subset of H-H dipolar couplings is used or the relevant 13C-H dipolar couplings are also included. They are consistent with the fact that the curves of Figure 2 related to 2,2′-bithiophene in ZLI1132 are less flat than those related to the solute in I52; 2,2′bithiophene is in fact more orientationally ordered in the former solvent. It is well established that the more orientationally ordered the solute is the less flat its ME conformational distribution function. Moreover, the insensitivity of the value of Sj2 upon addition of the linearly independent, within the ME context, 13C-H dipolar couplings points to the fact that the latter dipolar coupling data uniquely act to mold more precisely the conformational distribution functions, as can be appreciated by looking at the top and bottom panels of Figure 2, and, consequently, the orientational order of each conformation, S2(φ), leaving the overall orientational order of the solute molecule, as measured by Sj2, unaltered. From an electron diffraction experiment20 and quantumchemical calculations,21-24 it seems quite settled that the torsional potential-energy curve of the isolated 2,2′-bithiophene molecule has two minima, the lowest located at =145° and the other at =35°. The energy difference between them was estimated to be 0.75 kJ/mol in ref 20. Quantum-chemical methods21-24 predict this difference to be larger, =2 kJ/mol. In addition, these calculations predict that the cis, perpendicular, and trans conformations are transition states, although the value of their barriers is quite sensitive to the level of theory and basis set employed. The results of Figure 2 suggest that 2,2′-bithiophene is a rather flexible molecule, with torsional potential-energy barriers of a

few kJ/mol, readily modified by intermolecular interactions. In this respect, it is worth noticing how largely influenced the internal rotation of 2,2′-bithiophene is predicted to be by the presence of an either aqueous or organic solvent.25 The fact that the trans conformation is the most populated in the anisotropic liquid phase is in accord with X-ray diffraction data in the crystalline phase,26 which are consistent with the 2,2′-bithiophene molecule existing solely in the trans conformation, as well as with fluorescence spectroscopy data,27 which are consistent with a barrier between the transoid absolute minimum and the trans saddle point of only 0.3 kJ/mol. In this respect, B3LYP density functional theory gives the better agreement among the various quantum-chemical methods investigated. The other quantum-chemical methods give an energy difference between the trans saddle point and the transoid absolute minimum of =2 kJ/mol.23,24 However, the B3LYP method tends to overestimate the barrier at φ ) 90° and underestimate that at φ ) 0°.23,24 These two barriers should be of comparable magnitude, with that at φ ) 0° being slightly smaller. This is stated because it has been found that in the bulk nematic phase the populations of the cisoid and perpendicular-like conformations are not so dissimilar and because the larger value of the population of the perpendicularlike conformations is presumably correlated to the larger orientational entropy of these conformations, which are more isotropically distributed. These conjectures are consistent with the results of quantumchemical methods like MP2, MP4(SDQ), CCSD, and CCSD(T), employing the cc-pVDZ or larger basis sets.23,24 The torsional potential-energy curves calculated at these levels of theory and basis set seem overall the most realistic. IV. Conclusions The H-H and 13C-H partially averaged dipolar couplings of 2,2′-bithiophene dissolved in two nematic solvents have been analyzed by means of the maximum entropy method. Within this context, the majority of the new 13C-H data result in being linearly dependent on a subset of the seven H-H data, but the very few which are not contribute significantly to the form of the conformational distribution function. The latter provides evidence of a considerable orientational-conformational coupling. This, in turn, suggests that the resulting form of the conformational distribution function is the product of a combined process in which both the internal torsional potentialenergy and intermolecular interaction terms play an active role. The comparison of the conformational distribution functions obtained in this work with corresponding quantum-chemical calculations indicates that ab initio correlated methods with a cc-pVDZ or larger basis set should provide the most realistic torsional potential-energy function of the isolated molecule. The major advantage of the maximum entropy method and its recent extensions28,29 is that it does not require any assumption on the shape of the torsional potential-energy function. Therefore, it can be applied to complex, multirotor molecular systems for which residual dipolar couplings have been determined. Biologically active molecules are certainly among those of much importance. However, despite the fact that partially averaged dipolar couplings are widely recognized as an extremely valid source of information for the structure of biomolecules,30 their analyses have, almost invariably, been performed with methodologies which appear rather rudimentary if compared to the techniques, such as the maximum entropy method and extensions thereof, used to analyze the conformational features of small molecules. One important exception in

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this respect is the work of ref 31 on certain di- and trisaccharides. How might the detailed entire methodology to analyze partially averaged dipolar couplings developed for small molecules be profitably extended to such much larger molecules as oligonucleotides, -peptides, and -saccharides? Presently, the response to this question appears overambitious. One nevertheless hopes that soon this will be no longer the case. Acknowledgment. The author acknowledges the financial support of the European Commission via a Marie Curie Research Fellowship (project number PIEF-GA-2007-220557). He thanks Profs. R. Levine (Hebrew University of Jerusalem) for useful correspondence and G. Celebre (University of Calabria) for the critical reading of the manuscript. Supporting Information Available: This material is available free of charge via the Internet at http://pubs.acs.org. References and Notes (1) Saupe, A.; Englert, G. Phys. ReV. Lett. 1963, 11, 462–464. (2) Nuclear Magnetic Resonance of Liquid Crystals; Emsley, J. W., Ed.; Reidel: Dordrecht, 1985. (3) NMR in Oriented Phases; Burnell, E. E., de Lange, C. A. Eds.; Kluwer: Dordrecht, 2003. (4) Celebre, G.; De Luca, G.; Longeri, M.; Emsley, J. W. J. Phys. Chem. 1992, 96, 2466–2470. Emsley, J. W.; De Luca, G.; Celebre, G.; Longeri, M. Liq. Cryst. 1996, 20, 569–575. Celebre, G.; De Luca, G.; Longeri, M.; Pileio, G.; Emsley, J. W. J. Chem. Phys. 2004, 120, 7075– 7084. (5) Jaynes, E. T. Phys. ReV. 1957, 106, 620–630. (6) Zannoni C. , Nuclear Magnetic Resonance of Liquid Crystals; Emsley, J. W., Ed.; Reidel: Dordrecht, 1985; Chapter 2. (7) Di Bari, L.; Forte, C.; Veracini, C. A.; Zannoni, C. Chem. Phys. Lett. 1988, 143, 263–269. Catalano, D.; Di Bari, L.; Veracini, C. A.; Shilstone, G. N.; Zannoni, C. J. Chem. Phys. 1991, 94, 3928–3935. (8) Berardi, R.; Spinozzi, F.; Zannoni, C. J. Chem. Soc., Faraday Trans. 1992, 88, 1863–1873. (9) Cinacchi, G.; Veracini, C. A. UniVersality and diVersity in science. Festschrift in honor of Naseem K. Rahman’s 60th birthday; World Scientific: Singapore, 2004; pp 39-59. (10) Alhassid, Y.; Agmon, N.; Levine, R. D. Chem. Phys. Lett. 1978, 53, 22–26.

Cinacchi (11) Agmon, N.; Alhassid, Y.; Levine, R. D. J. Comput. Phys. 1979, 30, 250–258. (12) Concistre´, M.; De Lorenzo, L.; De Luca, G.; Longeri, M.; Pileio, G.; Raos, G. J. Phys. Chem. A 2005, 109, 9953–9963, 2005). (13) Bucci, P.; Longeri, M.; Veracini, C. A.; Lunazzi, L. J. Am. Chem. Soc. 1974, 96, 1305–1309. (14) Khetrapal, C. L.; Kunwar, A. C. Mol. Phys. 1974, 28, 441–446. (15) Ter Beek, L. C.; Zimmerman, D. S.; Burnell, E. E. Mol. Phys. 1991, 74, 1027–1035. (16) Roncali, J. Chem. ReV. 1992, 92, 711–738. Tour, J. M. Chem. ReV. 1996, 96, 537–554. Martin, R. E.; Diederich, F. Angew. Chem., Int. Ed. 1999, 38, 1350–1377. Kertesz, M.; Choi, C. H.; Yang, S. Chem. ReV. 2005, 105, 3448–3481. Coropceanu, V.; Cornil, J.; da Silva Filho, D. A.; Olivier, Y.; Silbey, R.; Bre´das, J. L. Chem. ReV. 2007, 107, 926–952. Mishra, A.; Ma, C. Q.; Ba¨urle, P. Chem. ReV. 2009, 109, 1141–1276. (17) Bak, B.; Christensen, D.; Hansen-Nygaard, L.; Rastrup-Andersen, J. J. Mol. Spectrosc. 1961, 7, 58–63. (18) The normalization condition adopted for the conformational distribution functions has been ∫0π dφΠ(φ) ) 1. (19) Berardi, R.; Spinozzi, F.; Zannoni, C. Liq. Cryst. 1994, 16, 381– 397. (20) Samdal, S.; Samuelsen, E. J.; Volden, H. V. Synth. Met. 1993, 59, 259–265. (21) Ortı´, E.; Viruela, P. M.; Sa´nchez-Marı´n, J.; Toma´s, F. J. Phys. Chem. 1995, 99, 4955–4963. (22) Karpfen, A.; Choi, C. H.; Kertesz, M. J. Phys. Chem. A 1997, 101, 7426–7433. (23) Duarte, H. A.; Dos Santos, H. F.; Rocha, W. R.; De Almeida, W. B. J. Chem. Phys. 2000, 113, 4206–4215. (24) Raos, G.; Famulari, A.; Marcon, V. Chem. Phys. Lett. 2003, 379, 364–372. (25) Rodrı´guez-Ropero, F.; Casanovas, J.; Ale´man, C. Chem. Phys. Lett. 2005, 416, 331–335. (26) Visser, G. J.; Heeres, G. J.; Wolters, J.; Vos, A. Acta Crystallogr., Sect. B 1968, 24, 467–473. (27) Takayanagi, M.; Gejo, T.; Hanozaki, T. J. Phys. Chem. 1994, 98, 12893–12898. (28) Stevensson, B.; Sandstro¨m, D.; Maliniak, A. J. Chem. Phys. 2003, 119, 2738–2746. (29) Celebre, G.; Cinacchi, G. J. Chem. Phys. 2006, 124, 176101. (30) Tjandra, N.; Bax, A. Science 1997, 278, 1111–1114. Prestegard, J. H.; Al Hashimi, H. M.; Tolman, J. R. Q. ReV. Biophys. 2000, 33, 371– 424. Prestegard, J. H.; Bougault, C. M.; Kishore, A. I. Chem. ReV. 2004, 104, 3519–3540. Bax, A.; Grishaev, A. Curr. Opin. Struct. Biol. 2005, 15, 563–570. Blackledge, M. Prog. NMR Spectrosc. 2005, 46, 23–61. (31) Stevensson, B.; Landersjo¨, C.; Widmalm, G.; Maliniak, A. J. Am. Chem. Soc. 2002, 124, 5946–5947.

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