NMR Characterization of the Aggregation State of the Azo Dye Sunset

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NMR Characterization of the Aggregation State of the Azo Dye Sunset Yellow in the Isotropic Phase Matthew P. Renshaw and Iain J. Day* Department of Chemistry and Biochemistry, School of Life Sciences, UniVersity of Sussex, Falmer, Brighton BN1 9QJ, U.K. ReceiVed: May 13, 2010; ReVised Manuscript ReceiVed: June 28, 2010

The azo dye sunset yellow is known to form lyotropic liquid crystals as a function of both temperature and sample composition. Numerous studies have been performed to investigate the aggregation processes in these liquid crystals; however, less attention has been paid to the nature of the aggregates in the isotropic phase. In this study we employ diffusion nuclear magnetic resonance methods to investigate the hydrodynamic properties of sunset yellow aggregates at a range of concentrations in isotropic solution. The results of these experiments are interpreted in terms of a simple thermodynamic model for aggregation and suggest that the aggregates are comprised of tens to hundreds of monomer units at the concentrations investigated. The results also demonstrate that the average number of molecules per aggregate is a factor of approximately 5 larger than previously reported. Introduction Aggregation phenomena are important across a wide range of chemical and biological applications. Understanding the interactions and processes that result in the assembly of smaller molecular fragments into larger, often noncovalent structures has applications in fields as diverse as drug-delivery,1,2 nanotechnology,3 protein aggregation,4,5 protein engineering,6 and liquid crystal science.7 The association of planar aromatic dye molecules into larger scale aggregates has attracted wide interest,8,9 for example, the association of dyes to dendrimer surfaces10 or the influence of cis/trans photoisomerism on stability.11 At high concentrations, the aggregates tend to form liquid crystal-like phases7,12 often with different polymorphic forms under different conditions.13 These associations are often driven by π-π stacking interactions between the aromatic rings.7,14 Sunset yellow (sodium (E)-6-hydroxy-5-((4-sulfonatophenyl) diazenyl) naphthalene-2-sulfonate) is a well studied monoazo dye, often used in food coloring, known to form lyotropic liquid crystal phases as a function of both temperature and composition.15-18 Several studies have appeared recently in which detailed investigations of the different liquid crystal phases of sunset yellow have been undertaken, along with the characterization of the phase diagram.15-18 These studies show that the aggregated states are stabilized by π-π stacking interactions between the aromatic rings, resulting in ring spacing between the molecules of around 0.34 nm,16 and that the ΝdN double bond is oriented perpendicular to the long axis of the aggregate, indicating the formation of rod-like structures.15-18 There is also evidence from X-ray powder diffraction and NMR spectroscopy that the molecules are ordered in a repeating AB arrangement in the stacks.15 Monoazo compounds with an R-hydroxyl group can exist in one of two tautomeric forms, either as the azo or hydrazone compounds, as shown in Figure 1. Results from the analysis of nuclear magnetic resonance (NMR) chemical shifts15 * To whom correspondence should be addressed. Telephone: +44 1273 876622. Fax: +44 1273 876687. E-mail: [email protected].

Figure 1. The azo (a) and hydrazone (b) tautomers of sunset yellow.

and X-ray scattering16 show that sunset yellow is predominantly present in the hydrazone form in solution. In this paper we describe the results of a diffusion NMR investigation into the assembled states present in the isotropic phase of sunset yellow at a variety of concentrations, and interpret the results in terms of a simple thermodynamic model for aggregation.16,19 Materials and Methods Materials. Sunset yellow FCF was purchased from Sigma Aldrich (Dorset, U.K.) and purified by ethanol precipitation, filtration, and drying overnight in vacuum oven.16,17 Deuterium oxide was purchased from Goss Scientific (Cheshire, U.K.). All other chemicals were obtained from Sigma Aldrich and used as received. NMR Spectroscopy. All NMR data were recorded on a Varian VNMRS 600 MHz spectrometer (Yarnton, Oxfordshire, U.K.) using an inverse triple resonance probe equipped with a z-gradient capable of up to 70 G cm-1. The sample temperature was regulated at 298 K. 2H NMR experiments were conducted

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using the lock channel of the spectrometer. Diffusion NMR experiments were performed using the Oneshot pulsed gradient stimulated echo sequence,20 with 12 gradient points equally spaced in g2. The diffusion delay ∆ was either 100 or 200 ms, and the resulting echo attenuation was analyzed in terms of a modified Stejskal-Tanner equation suitable for the Oneshot sequence.20,21 The data analysis was performed using a combination of NMRPipe22 and the SciPy modules23 of the Python programming language. Errors in the fitted parameters were determined using Monte Carlo methods.24 Nuclear Overhauser effect spectroscopy (NOESY) was performed on the same samples. The data were acquired with 1442 × 200 complex data points spanning a spectral width of 9.6 kHz in both dimensions. The data were apodized with a cosine bell in f2, a shifted sine bell in f1, and zero-filled to 2k × 2k data points prior to Fourier transformation. Third-order polynomial baseline correction was employed in both spectral dimensions. The data were processed using MestReNova 6.0.4 (Santiago de Compostela, Spain). Viscosity Measurements. Sunset yellow solution viscosities were measured using an Ostwald Viscometer.25 The dynamic viscosities were determined using the Poiseuille formula:

η ) AFt

(1)

following calibration of the viscometer with ethylene glycol (η ) 1.61 × 10-2 Pa s, A ) 9.38 × 10-9 m2 s-2). All measurements were performed in triplicate, at room temperature. Solution densities were determined by weighing 1 cm3 of solution, measured using a micropipet. Density Functional Therory (DFT) Calculations. Geometry optimization calculations were performed for both tautomeric forms of sunset yellow using DFT at the B3LYP 6-31G(d) level of theory, with the Gaussian03 suite of programs.26 All torsion angles and bond lengths were unconstrained and allowed to vary. Thermodynamic Model of Aggregation. The aggregate composition of the solutions was modeled using the approach outlined by Israelachvili19 and used previously by Horowitz et al.16 This approach will be outlined here for reasons of clarity. The chemical potential of a molecule in an aggregate comprising N molecules is given by

µN ) µN0 +

( )

XN kT for N ) 1, 2, 3, ... log N N

(2)

1 + 2φeR - √1 + 4φeR 2φe2R

CMM F + CMM

(5)

This model then allows the distribution of the number of aggregate comprising N molecules, XN/N, the volume fraction of aggregates containing N molecules, XN, and the average number of molecules per aggregate, 〈N〉, to be calculated. Calculation of the Diffusion Coefficient. The diffusion coefficient for each aggregate was calculated using the Einstein-Sutherland equation:21

D)

kT ∂ ln γ 1+ (1 - φ) f ∂ ln c

(

)

(6)

where f is a shape factor, the derivative term in the first set of parentheses describes the nonideality of the solution, with γ being the activity coefficient, and c being the concentration. This term was calculated by numerical differentiation of the extended Debye-Huckel formula.27 All other symbols have their usual meaning. The calculation of the shape factor f depends on the geometry of the molecule. In general, closed-form analytical expressions for f are known only for simple shapes, such as a sphere, a cylinder, and an ellipsoid. Using the assumption that the rodlike aggregated states of sunset yellow can be described as ellipsoids with semiaxes given by b and c, the appropriate shape factors are21

f ) fsphere

√1 - p2 p1/3 tan-1(√1 - p2 /p)

for p ) b/c < 1, oblate ellipsoid (7)

and

√p2 - 1 p1/3 tanh-1(√p2 - 1/p)

for p ) b/c > 1, prolate ellipsoid (8)

(3)

where R is the average interaction energy between molecules in units of kT, determined previously to be 7.25 by Horowitz et al.,16 and X1, the volume fraction of the monomer, is given by

X1 )

φ)

f ) fsphere

where µN0 is the mean interaction free energy per molecule, and XN is the volume fraction of molecules in aggregates of N species. T is the thermodynamic temperature, and k is Boltzmann’s constant. At thermal equilibrium, µN is equal for all N, therefore allowing the following expression to be derived for rod-like aggregates:19

XN ) N(X1eR)Ne-R

with φ being the volume fraction of the solute, which can be related to the molar concentration CM, molecular weight M, and solution density F as follows:

(4)

where fsphere ) 6πηa, with a being equal to the radius of the sphere with the same volume as the ellipsoid, i.e., a3 ) bc2. The average diffusion coefficient, corresponding to that measured experimentally, was calculated as the weighted average of the diffusion coefficients for aggregates of N monomers:

Dav )

∑ i

XN(i)Di N

(9)

This approach is valid provided that the interchange of monomer units between aggregates and between free monomer and aggregate can be considered to be in the fast exchange limit.21,28 That is, exchange is rapid on the time scales defined by both the difference in chemical shifts and the diffusion labeling time ∆.21,28

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Figure 2. (a) Example Oneshot pulsed field gradient stimulated echo spectra for an 8.2 mM sample of sunset yellow in D2O recorded for a diffusion time of 100 ms, and (b) the signal intensity as a function of gradient strength with the fit to the modified Stejskal-Tanner equation shown as a dashed line. Similar results were obtained at each concentration investigated.

Results and Discussion As a starting point to investigate the aggregation behavior of sunset yellow in the isotropic phase, the average diffusion coefficient was measured using pulsed field gradient NMR spectroscopy for a series of concentrations within the isotropic phase. Confirmation that the samples had not undergone a phase transition and remained in isotropic solution was obtained by observation of a singlet in the 2H NMR spectrum (data not shown). The presence of an anisotropic phase would be evident by the splitting of the solvent resonance due to the deuterium nuclear quadrupole interaction.15 Isotropic solutions were observed for all sample concentrations below approximately 1 M. An example of the Oneshot echo attenuation data obtained for the 8.2 mM sample is shown in Figure 2a. The 12 spectra are spaced equally in the square of the applied gradient strength g2. Figure 2b shows the result of fitting the intensity of the doublet at 6.44 ppm to the Stejskal-Tanner equation, appropriately modified for the Oneshot sequence.20 Similar results were obtained for all 17 samples, with concentrations in the range 1.64 mM to 0.98 M. The excellent fit to a single exponential decay for all samples, combined with a single set of resonances, indicates that the analysis of the diffusion coefficient as population-weighted average diffusion coefficient over all the aggregates present in the solution is valid.21 Figure 3a shows the measured diffusion coefficient as a function of concentration for the 17 samples investigated. There is clearly a steady decrease in the measured diffusion coefficient from around 2.5 × 10-10 m2 s-1 for the 1.64 mM sample to less than 2.0 × 10-11 m2 s-1 as the sample concentration approaches 1 M, close to the transition to the nematic phase at

Renshaw and Day

Figure 3. (a) Measured diffusion coefficients as a function of sunset yellow concentration (solid points: ∆ ) 100 ms, open points: ∆ ) 200 ms) with the thermodynamic aggregation models shown as lines. The interaction of Horowitz et al.16 is shown as a solid line, while the fitted model is drawn as dashed (∆ ) 100 ms) or dot-dash (∆ ) 200 ms). (b) Dynamic viscosity of sunset yellow solutions (in H2O) as a function of concentration. The dashed line is the fit to a fourth-order polynomial.

this temperature (298 K). The decrease in diffusion coefficient with increasing solute concentration indicates that the molecular species being probed has a reduced mean-square displacement during the diffusion period (∆) of the stimulated echo pulse sequence. This can be attributed to both an increase in molecular size, i.e., the formation of larger aggregated species, and/or a variation in the solution hydrodynamic properties, such as viscosity, as a function of solute concentration. The presence of large-scale aggregates in the solution has the potential to complicate the analysis if these aggregates cause restricted diffusion on a time scale comparable to the diffusion labeled period ∆. In order to check for the presence of restricted diffusion, two experiments were performed with different values of ∆, either 100 or 200 ms. The determined diffusion coefficients are shown as the solid or shaded symbols in Figure 3a. The overlay of the data points for the two experiments clearly demonstrates that the sunset yellow molecules do not experience any restricted diffusion on these time scales.21 Above the phase transition to the nematic phase, the 1H NMR spectra are dominated by extensive line broadening arising from dipolar couplings, which are no longer averaged to zero by isotropic tumbling. This results in spectra with line widths of over 10 kHz as has been previously observed by Edwards et al. (data not shown).15 The diffusion coefficient of a molecule in solution is related to the hydrodynamic properties of the solute and solvent. This relationship can be modeled, in part, using the EinsteinSutherland equation21 (eq 6). Specifically, this provides a link between the diffusion coefficient and the dynamic viscosity of the solution. Therefore in order to fully interpret the diffusion

Aggregation State of Sunset Yellow in the Isotropic Phase NMR data in terms of the molecular aggregation state, knowledge of the dynamic viscosity was required. A series of sunset yellow solutions of varying concentration were prepared in deionized water, and their dynamic viscosity was measured using an Ostwald viscometer,25 calibrated using the known viscosity of ethylene glycol as described above. The results of these measurements are presented in Figure 3b and show that the viscosity increases dramatically and nonlinearly with increasing sunset yellow concentration. The dashed line shows the fit to a fourth-order polynomial. The returned coefficients have no physical significance; however, they allow the viscosity to be modeled across the range of concentrations investigated in the diffusion NMR experiments. This polynomial was used to determine the solution viscosity in the subsequent calculations of average diffusion coefficient presented below. Correction for the viscosity differences between H2O and D2O solutions was not included, and is expected to be small, especially at higher solute concentrations. The calculation of the average diffusion coefficient through the Einstein-Sutherland equation also requires that proper account be taken of the shape of the diffusing species, via the shape factor f in eq 6. This corrects for the fact that ellipsoids will have subtly different diffusion properties compared with spheres of similar sizes. These so-called Perrin shape factors21 can only be expressed analytically for relatively simple geometries such as spheres and ellipsoids.21,29 In order to calculate these factors for the sunset yellow aggregates, the overall molecular dimensions are required. Unsuccessful attempts were made to grow crystals suitable for investigation by single-crystal X-ray diffraction, therefore a geometry optimization calculation using DFT was performed at the B3LYP 6-31G(d) level of theory using the Gaussian 03 suite.26 The optimized geometries obtained for the azo and hydrazone tautomers are shown in Figure 4, with the atomic coordinates (z-matrix format) available in the Supporting Information. The result of this geometry optimization reveals that the overall molecular dimensions are very similar for the two tautomers, with both structures being highly planar. The only deviations from the plane are at the tetrahedral sulfonate groups. The longest heavy-atom distance in both tautomers is the S-S distance between the sulfonate groups on the naphthol and phenyl rings, being 1.254 and 1.273 nm for the azo and hydrazone tautomers, respectively. In order to account for the hydration of the sulfonate groups in aqueous solution, an additional contribution of approximately 0.23 nm was added to the S-S distance, yielding a maximal overall length, 2c, used in subsequent calculations of 1.5 nm. The thickness of a monomer unit, b, was taken to be 0.17 nm, being half the ring stacking distance reported by Edwards et al.15 This semiaxis of the rod-like ellipse is therefore given by the product of the number of monomers in the aggregate and the thickness of the monomer unit. The distribution of aggregates present in solution as a function of concentration was modeled using the simple thermodynamic aggregation of Israelachvili.19 This approach has been used previously by Horowitz et al.16 to calculate the distributions of aggregates comprising 1 to 100 molecules, assuming the formation of rod-like aggregates. Figure 5 shows the number of aggregates comprising N molecules (Figure 5a) and the volume fraction of those aggregates (Figure 5b), both a function of the number of molecules per aggregate. These distributions were generated using eqs 3 and 4, with the intermonomer interaction energy R ) 7.25, as determined by Horowitz et al.16 These distributions clearly show that as the sample concentration increases, more aggregates are composed of larger numbers of

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Figure 4. Optimized geometries obtained using DFT (B3LYP 6-31G(d) level of theory) for the (a) azo and (b) hydrazone tautomers of sunset yellow. The coordinates (z-matrix) are available in the Supporting Information.

monomer units. Using the distribution of the number of aggregates comprising N molecules allows the average number of molecules per aggregate to be determined, along with the corresponding Perrin shape factor. These functions are plotted as the solid and dashed lines in Figure 5c, respectively. This shows that under these conditions, a spherical description of the sunset yellow molecules is only valid around concentrations of 20 mM, where aggregates comprise, on average, five or six molecules. Below this concentration, the aggregates in the distribution are typically oblate ellipsoids in shape, whereas at higher concentrations the formation of rod-like structures dominates, described as prolate ellipsoids. Given the calculated distributions of aggregates, it is possible to model the average diffusion coefficient and allow comparison with that determined in the diffusion NMR experiments (see Materials and Methods). The average diffusion coefficient as a function of sample concentration, calculated using the interaction energy R ) 7.25 of Horowitz et al.,16 generates the solid line in Figure 3a. This model shows good agreement with the experimental data points only for higher concentrations of sunset yellow, above around 0.6 M up to the phase transition to the nematic phase. Below this concentration, the model predicts

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Renshaw and Day

Figure 5. Aggregation distributions calculated using the Israelachvili model.19 (a) The number distribution (XN/N) calculated using the interaction energy (R ) 7.25) of Horowitz et al.16 (b) The volume fractions (XN) calculated as a function of the number of molecules in the aggregate. (c) Plots of the average number molecules per aggregate (〈N〉) and the associated Perrin shape factor as a function of concentration. The plots in panels d, e, and f correspond to a, b, and c, but calculated for the average (fitted) interaction energy, R ) 11.10.

species with larger diffusion coefficients than observed experimentally. This suggests that the aggregated species present in the NMR tube are larger, and comprise more monomer units than reported previously.16 Fitting the aggregation model of Israelachvili19 to the experimental data yields a larger interaction energy R ) 11.10 ( 0.18, averaged over the two diffusion data sets. This indicates that the interaction between monomer units, i.e., the sunset yellow molecules, is stronger than reported previously.15-17 The fitted thermodynamic model does appear to underestimate the diffusion coefficients slightly at higher solute concentrations; however, the deviation over the whole set of experimental data is greatly reduced. More significant deviation of the model from the experimental results is apparent in the region of 0.1-0.4 M sample concentration. In this region, the simple thermodynamic aggregation model appears to predict the presence of larger aggregates than the average diffusion coefficient would suggest. The reasons for this deviation from the model in this region are not clear; however, it may be that aggregates with other geometries, such as pairwise assembly of the rods are formed, which are not accounted for in this

simple model. This is similar to effects observed in small-angle X-ray scattering studies of sunset yellow, around the phase transition points.30 The difference between the interaction energy determined here and that obtained previously can be attributed to the fact that the various physical techniques probe interactions over different length scales. Similar values for R have been obtained using optical absorption (R ) 7.25 ( 0.01)16 or X-ray scattering data (R ) 7.4).17 At higher concentrations another study suggests lower values for the interaction energy (R ) 4.3 ( 0.3 or 3.5 ( 0.2), depending on the phase present in the solution.30 A recent molecular dynamics study of small-scale (n ) 3-8) sunset yellow aggregates reached a similar conclusion regarding the interaction energy (R ≈ 7).31 Investigation of a structurally similar dye, Bordeaux, yields a larger value for the interaction energy (R ) 9.2) derived from X-ray scattering data.32 The apparent discrepancy between these previous results and the interaction energy determined by diffusion NMR can be understood in terms of the length scales probed by the various techniques. Diffusion NMR measurements are, in principle,

Aggregation State of Sunset Yellow in the Isotropic Phase sensitive to the overall physical size of the aggregate, and therefore its total length. X-ray scattering and optical techniques are more sensitive to the “correlation length”, which is likely to be significantly shorter than the overall length of the aggregate, especially in the case of large aggregates comprised of hundreds of monomer units, due to defects and discontinuities in the aggregated structure.17 Using the increased interaction energy suggested by fitting the experimental diffusion data allows the distribution functions that describe the aggregated species to be recalculated. The results of these calculations are shown in Figure 5d,e and show that, due to the increased strength of the intermolecular interaction within the aggregates, much larger molecular assemblies are present in solution, with assemblies comprising tens to hundreds of monomer units at the concentrations investigated, as shown by the average number of molecules per aggregate (Figure 5f). On average, the increased interaction energy suggests that the average number of molecules per aggregate is a factor of 5 larger than that determined with the smaller R value. The determination of the Perrin shape factor from the NMR diffusion data also shows that elongated rodlike species, characterized as prolate ellipsoids, predominate at all sample concentrations investigated. Attempts were also made to investigate the intermonomer contacts within the aggregates using NOESY, with the aim of confirming the proposed “AB” stacking arrangement within the aggregates.15 Spectra were recorded using a relatively short mixing time of 100 ms in order to reduce the potential for spin diffusion effects. Three representative spectra are shown in Figure 6. At the lowest concentrations studied (Figure 6a), there is an absence of NOE cross peaks at this mixing time. Spectra recorded with longer mixing times (τm ) 500 ms or 1 s) also do not show the present of any significant cross peaks. This can be understood if the molecules are in the intermediate tumbling regime in which the magnitude of the NOE undergoes a zero crossing, hence resulting in the lack of observable cross peaks. The critical rotational correlation time at which this would occur at 600 MHz is approximately 300 ps, which would correspond to a sphere of around 7 Å undergoing isotropic rotation in a nonviscous solution at room temperature. At higher concentrations, cross peaks with the same phase as the diagonal are clearly observed between all resonances with a short mixing time τm ) 100 ms. This is indicative of the system being in the slow tumbling or spin-diffusion limit, characteristic of large molecules with long rotational correlation times. These observations are in reasonable agreement with the diffusion measurements described above. Unfortunately, further information as to the nature of the stacking could not be obtained from these NOESY experiments. Conclusions Sunset yellow is well-known to form liquid crystals as a function of both temperature and sample composition. In this study we have used diffusion NMR methods to investigate the aggregation state of sunset yellow in the isotropic phase, below the transition to a liquid crystal phase. We find that the system is well described in terms of aggregates formed of tens to hundreds of monomer units, as a function of concentration. Using a simple thermodynamic model of aggregation, we determined that the interaction energy R ) 11.10 ( 0.18, between monomer units is significantly larger than that described previously,16,17,30,31 indicating the formation of larger aggregated species. Attempts to further characterize the nature of the stacking interactions using NOESY proved unsuccessful due

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Figure 6. NOESY spectra for (a) 8.2 mM, (b) 0.4 M, and (c) 0.9 M samples of sunset yellow in D2O. The spectra were acquired and processed with identical parameters as described in the Materials and Methods. The mixing time was 100 ms. The relevant 1H NMR spectra are shown as the horizontal and vertical traces.

to the large rotational correlation times of the aggregated species leading to efficient spin diffusion within the aggregates. The approach described in this paper should be directly applicable to a number of aggregating systems in solution, including other dye systems and the self-association of drug-like molecules.33,34 Acknowledgment. M.P.R. thanks the University of Sussex for a Junior Research Associate Bursary. The authors thank Mick Henry for the loan of the Ostwald viscometer and the EPSRC for financial support (EP/H025367/1). Supporting Information Available: Molecular coordinates from DFT geometry optimizations; the output is in z matrix form. This material is available free of charge via the Internet at http://pubs.acs.org.

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Renshaw and Day (22) Delaglio, F.; Grzesiek, S.; Vuister, G. W.; Zhu, G.; Pfeifer, J.; Bax, A. J. Biomol. NMR 1995, 6, 277. (23) Jones, E.; Oliphant, T.; Peterson, P. SciPy: Open Source Scientific Tools for Python, 2001 (http://www.scipy.org/). (24) Press, W. H.; Flannery, B. P.; Teukolsky, S. A.; Vetterling, W. T. Numerical Recipes in Fortran 77; Cambridge University Press: Cambridge, U.K., 1992. (25) Champion, F. C.; Davy, N. Properties of Matter, 3rd ed.; Blackie and Son Ltd.: London, 1959. (26) Frisch, M. J.; Trucks, G. W.; Schlegel, H. B.; Scuseria, G. E.; Robb, M. A.; Cheeseman, J. R.; Montgomery, J. A., Jr.; Vreven, T.; Kudin, K. N.; Burant, J. C.; Millam, J. M.; Iyengar, S. S.; Tomasi, J.; Barone, V.; Mennucci, B.; Cossi, M.; Scalmani, G.; Rega, N.; Petersson, G. A.; Nakatsuji, H.; Hada, M.; Ehara, M.; Toyota, K.; Fukuda, R.; Hasegawa, J.; Ishida, M.; Nakajima, T.; Honda, Y.; Kitao, O.; Nakai, H.; Klene, M.; Li, X.; Knox, J. E.; Hratchian, H. P.; Cross, J. B.; Bakken, V.; Adamo, C.; Jaramillo, J.; Gomperts, R.; Stratmann, R. E.; Yazyev, O.; Austin, A. J.; Cammi, R.; Pomelli, C.; Ochterski, J. W.; Ayala, P. Y.; Morokuma, K.; Voth, G. A.; Salvador, P.; Dannenberg, J. J.; Zakrzewski, V. G.; Dapprich, S.; Daniels, A. D.; Strain, M. C.; Farkas, O.; Malick, D. K.; Rabuck, A. D.; Raghavachari, K.; Foresman, J. B.; Ortiz, J. V.; Cui, Q.; Baboul, A. G.; Clifford, S.; Cioslowski, J.; Stefanov, B. B.; Liu, G.; Liashenko, A.; Piskorz, P.; Komaromi, I.; Martin, R. L.; Fox, D. J.; Keith, T.; Al-Laham, M. A.; Peng, C. Y.; Nanayakkara, A.; Challacombe, M.; Gill, P. M. W.; Johnson, B.; Chen, W.; Wong, M. W.; Gonzalez, C.; Pople, J. A. Gaussian03; Gaussian, Inc.: Wallingford, CT, 2004. (27) Atkins, P. W. Physical Chemistry, 5th ed.; Oxford University Press: Oxford, U.K., 1997. (28) Bain, A. D. Prog. Nucl. Magn. Reson. Spectrosc. 2003, 43, 60. (29) Jennings, B. R.; Parslow, K. Proc. R. Soc. London, A 1988, 419, 137. (30) Joshi, L.; Kang, S.-W.; Agra-Kooijman, D. M.; Kumar, S. Phys. ReV. E. 2009, 80, 041703. (31) Chami, F.; Wilson, M. R. J. Am. Chem. Soc. 2010, 132, 7794. (32) Tomasik, M. R.; Collings, P. J. J. Phys. Chem. B 2008, 112, 9883. (33) Davies, D. B.; Djimant, L. N.; Veselkov, A. N. J. Chem. Soc., Faraday Trans. 1996, 92, 383. (34) Heyn, M. P.; Bretz, R. Biophys. Chem. 1975, 3, 33.

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