Determination of the rotation-diffusion tensor

MOLECULAR PHYSICS, 2002, VOL. 100, N O. 17, 2755±2761

Determination of the rotation-di€ usion tensor orientation from NMR 13C±1H cross-relaxation rates O. WALKER1 , P. MUTZENHARDT 1 , E. HALOUI 2 , J.-C. BOUBEL1 and D. CANET1 * 1

Laboratoire de me thodologie RMN (FRE CNRS 2415, INCM-FR CNRS 1742), Universie H. PoincareÂ, Nancy 1, B.P. 239, 54506 Vandoeuvre-leÁ s-Nancy Cedex, France 2 DeÂpartement de Chimie, Faculte des sciences de Tunis, Campus Universitaire, 1060-Tunis, Tunisia (Received 20 November 2001; revised version accepted 25 March 2002)

Taking advantage of the fact that ¬,¬,2,6 tetrachlorotoluene possesses only one symmetry element (the aromatic ring plane), it proved possible to measure seven di€ erent 13 C±1 H crossrelaxation rates which enable one to determine the three rotation-di€ usion coe cients (Dxx , Dyy , Dzz ), in addition to the orientation of the relevant principal axis system (PAS) with respect to a chosen molecular axis system. It turns out that molecular reorientation is strongly anisotropic and that the rotation-di€ usion PAS cannot be directly correlated with electrical molecular properties.

1. Introduction Nuclear spin relaxation is probably the most straightforward method for probing overall reorientation of rigid molecules [1, 2]. Of course it is also suitable for studying internal motions although requiring, in that case, somewhat elaborate models [3, 4]. Most of the time the overall motion is treated in the context of rotational di€ usion [5, 6], implying at most three di€ erent di€ usion coe cients Dxx , Dyy, Dzz , which are in fact the elements of the rotation-di€ usion tensor expressed in its principal axis system (PAS, where the tensor is diagonal). Conversely, one de®nes very often the socalled correlation times ½x ˆ 1=6Dxx , ½y ˆ 1=6Dyy , ½z ˆ 1=6Dzz . Both notations are strictly equivalent. It is relatively easy to determine these three correlation times when the orientation of the rotation-di€ usion tensor is perfectly de®ned by symmetry conditions [7]. In the absence of evident molecular symmetry, one may also resort to the inertia tensor (which is readily constructed from the molecular geometry) and assume that both tensors (rotation-di€ usion and inertia) coincide [8, 9]. However, this assumption is far from resting on a ®rm basis and it would be more satisfactory to be able to determine experimentally not only the three di€ erent correlation times (when full anisotropy prevails) but also the orientation of the rotation-di€ usion tensor [10]. Of * Author for correspondence. e-mail: daniel.canet@rmn. uhp-nancy.fr

course, in the general case, this would require at least ®ve independent relaxation parameters corresponding to the six unknowns which are (i) the three di€ usion coe cients and (ii) three angles de®ning the tensor orientation. Moreover, the choice of relaxation parameters is of prime importance in order to rely on a relaxation mechanism which is sensitive to overall motions and which can be unambiguously interpreted. Among the available relaxation parameters, the so-called crossrelaxation rates appear especially valuable because they depend solely on the dipolar interaction (between magnetic moments associated with nuclear spins) and therefore can be easily expressed as a function of internuclear distances and of the six unknowns de®ning the rotation-di€ usion tensor. It is di cult to ®nd a molecule amenable to the experimental determination of six crossrelaxation rates which would be di€ erentiated enough to provide, with reasonable accuracy, the six parameters characterizing the rotation-di€ usion tensor. In the present work, we shall be dealing with a simpli®ed problem since our molecule possesses a symmetry plane. As a consequence, one of the principal axes, say the z axis, is necessarily perpendicular to the symmetry plane. Thus, the only unknown concerning the rotation-di€ usion tensor orientation is the angle between the in-plane x axis and a chosen molecular axis. We shall show that, from seven 13C±1 H cross-relaxation rates (four of them being determined with a very good accuracy), it is possible to extract the three correlation times in addition to

Molecular Physics ISSN 0026±8976 print/ISSN 1362±3028 online # 2002 Taylor & Francis Ltd http://www.tandf.co.uk/journals DOI: 10.1080/00268970210141199

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the above mentioned angle. While the correlation times behave as expected, it turns out that the rotationdi€ usion tensor orientation does not appear to be related to the inertia tensor orientation as has been sometimes postulated [11, 12].

2. Experimental The molecule which has been selected for this study is ¬,¬,2,6 tetrachlorotoluene, schematized in ®gure 1. Experiments were performed on pure liquid at ¡158 C; this relatively low temperature is such that, essentially, a single rotamer is present, the one with the CH bond (of the substituent) in the plane [13, 14]. A further proof of this statement is that we have seven di€ erent peaks in the carbon-13 spectrum (®gure 2). The other possible rotamer would involve the CH bond in a plane perpendicular to the aromatic cycle with the two chlorine atoms positioned symmetrically with respect to that plane. In that case the carbon-13 spectrum would involve only ®ve peaks. Moreover, we did not observe exchange cross-peaks in the 2D 1 H±1 H NOESY [15] spectrum at this low temperature (data not shown). NMR experiments have been performed with a Bruker Avance DSX spectrometer operating at 7.04 T with a 5 mm X/1 H probe. They include 13 C longitudinal relaxation time T C 1 measurements (carried out conventionally [16] under proton decoupling) and the determi-

Figure 2. Proton (top) and proton-decoupled carbon-13 (bottom) spectra of ¬,¬,2,6 tetrachlorotoluene.

f

Figure 1.

The ¬,¬,2,6 tetrachlorotoluene molecule with the principal axis system (PAS) of the two tensors considered in this work.

Determination of the rotation-di€ usion tensor orientation from NMR nation of 1 H±13 C cross-relaxation rates (¼CH ) by the pulse method as sketched in ®gure 3. As this sequence is aimed not only at the measurement of these quantities but also at the visualization of their e€ ect, the following experimental conditions have been employed: (i) 13C magnetization has been totally saturated at the onset and (ii) the recovery of 13 C magnetization during the mixing time is cancelled by the two-step phase cycle.

13 C±1 H

cross-relaxation rates

2757

Actually, the second experiment amounts to leaving unperturbed proton magnetization, so that 13 C relaxes only by its speci®c rate. This second experiment is subtracted from the ®rst one, yielding, for the initial part of the evolution, a build-up of carbon magnetization due exclusively to ¼CH which can thus be extracted from this initial behaviour (short values of the mixing time tm ). Alternatively, cross-relaxation rates can be deduced

Figure 3. The pulse sequence used in this work for the determination of global 1 H±13 C cross-relaxation rates (¼CH ). The notation § of the second (º=2) pulse applied to protons, concomitantly with the alternate sign of acquisition, represents a phase cycle. …¬†x …2¬†y are long radio-frequency pulses (1.5 and 3 ms respectively) which entail the saturation of all 13 C magnetization components. tm is a `mixing time’ during which cross-relaxation is active. Acquisition of the carbon-13 spectrum takes place under proton decoupling.

Figure 4. Typical results obtained according to the sequence of ®gure 3: squares for a carbon directly bonded to a proton, circles for a carbon remote from protons. The curves have been recalculated with parameters extracted from the analysis of experimental data. In the inset: details of the initial behaviour demonstrating the validity of the initial slope approach.

O. Walker et al.

2758

C from a full analysis of the evolution curve IC z =Ieq (®gure 4) according to the well-known Solomon equations, recalled below.

contributions are at most of the order of 2%). Conversely, for carbons non-directly bonded to protons, the sum of all contributions must be considered.

d H H H C C I ˆ ¡RH 1 …Iz ¡ Ieq † ¡ ¼CH …Iz ¡ Ieq †; dt z d C C C H H Iz ˆ ¡RC 1 …I z ¡ Ieq † ¡ ¼CH …Iz ¡ Ieq †; dt

3. Theory and data treatment The general form of a carbon±proton cross-relaxation rate is as follows:

…1†

with the following initial conditions: (a) step 1 of the H IH z …0† ˆ ¡Ieq ; (b) step 2 of the H IH z …0† ˆ ‡Ieq .

phase

cycle:

IC z …0† ˆ 0;

phase

cycle:

IC z …0† ˆ 0;

The latter analysis implies, however, the adjustment of ¼CH and of the proton longitudinal relaxation rate RH 1 (ˆ 1=T H 1 ), recognizing that such a curve depends also on C RC 1 (ˆ 1=T 1 ) for which we can use the value determined independently (see above). We have nevertheless preferred cross-relaxation rates deduced in a straightforward manner from the initial behaviour, because analysing the whole curve may involve a bias due to interproton relaxation possibly occurring at long mixing times. It can be noted that these experiments are non-selective in the sense that, in principle, we obtain the sum of all cross-relaxation rates originating from all protons and concerning a given carbon-13. An ideal experiment would consist of selecting one proton and to consider its e€ ect on all carbons. Owing to the numerous overlaps in the proton spectrum (®gure 2), such an experiment appears hopeless. However, due to the well-known 1=r6 dependency of cross-relaxation rates (r: carbon±proton distance), it can be safely assumed that the contribution from the directly bonded proton is overwhelming (other

¼CH ˆ

K ~ ‰6J…!H ‡ !C † ¡ J~…!H ¡ !C †Š r6CH

…2†

with !H ˆ 2º¸H , !C ˆ 2º¸C , ¸H and ¸C being the proton and carbon Larmor frequencies, respectively; K is a constant depending on the gyromagnetic ratios. 1=r6 should be written 1=…hr3 i†2 , where the brackets denote a vibrational average. Some care should therefore be exercised when using a given value of rCH if the determination of a given parameter requires a great accuracy. However, as detailed below, we shall be dealing here with the determination of angles which turn out to be uncritically dependent on rCH distance. For these reasons, the rCH values inserted in equation (2) are those derived from quantum chemical calculations (see table 1). J~…!† is a spectral density function, expressed as 2½c =…1 ‡ !2 ½c2 †, in its simplest form which prevails in the case of a rigid body reorienting isotropically, ½c being a correlation time. In our case, we are dealing with a quasi-planar molecule for which a strong anisotropic reorientation is expected. Moreover, because of the lack of symmetry within the aromatic cycle, three di€ erent correlation times (corresponding to three rotation-di€ usion tensor elements in its diagonal form) will probably be required. This leads to the following expression for J~…!† [17] (extreme narrowing conditions assumed: !2 ½c2 ½ 1)

Table 1. Experimental results concerning the seven carbons ¡ ¢ of ¬; ¬; 2; 6; tetrachlorotoluene. The longitudinal relaxation time of protons directly bonded to a carbon-13, T 1 1 H ¡ 13 C , has been deduced from the analysis of the whole evolution curve (see ®gure 4). The values given in the column `¼CH experimental’ have been extracted from the initial behaviours of the evolution curves (see inset of ®gure 4). Cross-relaxation rates have been recalculated with the values corresponding to the best ®t: ¿ ˆ 47:9 § 58; ½x ˆ 12:0 § 2:5 ps; ½y ˆ 32:1 § 4 ps; ½z ˆ 4:1 § 1:0 ps (see text). The rCH values are derived from quantum chemical calculations. Carbon number (®gure 1) 1 2 6 3 5 4 7

Chemical shift/ ppm

rCH /A

T 1 …13 C†/s

T 1 …1 H ¡ 13 C†

133.5 132.6 136.5 132.3 128.8 131.8 66.5

± ± ± 1.071 1.073 1.075 1.068

12.00 10.90 10.47 4.76 4.71 3.83 5.43

± ± ± 3.56 3.24 3.17 4.31

¯

¼CH experimental

¼CH recalculated

0.0016 § 0.0005 0.0026 § 0.0005 0.0025 § 0.0005 0.0826 § 0.0041 0.0860 § 0.0043 0.1000 § 0.0050 0.0733 § 0.0037

0.001 923 0.002 338 0.001 899 0.082 637 0.085 981 0.100 00 0.073 3449

Determination of the rotation-di€ usion tensor orientation from NMR ¯ J~…!† ˆ 32 sin2 …2’† …4Dz ‡ 2D‡ † ‡

2 ‡ d2¡

12D2¡

µ £ 9D2¡ cos 2 …2’† ‡ …3D¡ d¡ †

¶¿ d2¡ £ cos …2’† ‡ …d‡ ‡ 6D‡ † 4 µ 2 ‡ 9D2¡ cos2 …2’† 12D2¡ ‡ d2‡ ¶¿ d2‡ ‡ …3D¡ d¡ † cos …2’† ‡ …d¡ ‡ 6D‡ †; 4

…3†

where the x, y principal axes (of the rotation-di€ usion tensor) are as shown in ®gure 1, z being of course perpendicular to the aromatic plane. The angle ’ is de®ned with respect to a known molecular frame (we have chosen for this reference frame the PAS of the inertia tensor: X, Y, Z) and can be decomposed as ’ ˆ ¿ ‡ À, where ¿ is the angle between the x and X axes; À stands for the angle between the CH bond and the X axis. The quantities D§ , d§ are de®ned below. Dz stands for Dzz (same convention for x and y). ¡ ¢ …4† D§ ˆ Dx § Dy =2; 2

d§ ˆ 2…Dz ¡ D‡ † ¨ ‰4…Dz ¡ D‡ † ‡

12D2¡ Š1=2 :

…5†

It can be noted that J~…!† can be expressed as J~…!† ˆ ax2 ‡ bx ‡ c;

…6†

where x ˆ cos …2’†. a, b and c are complicated expressions but an interesting feature is that they depend exclusively on the dynamical parameters (D§ , d§ ) 18D2¡ 3 ¡ aˆ 2 2 …12D¡ ‡ d¡ †…d‡ ‡ 6D‡ † 2…4Dz ‡ 2D‡ † ‡

18D2¡ ; …12D2¡ ‡ d2‡ †…d¡ ‡ 6D‡ †

3 d2¡ ‡ 2 2 2…4D 2…12D¡ ‡ d¡ †…d‡ ‡ 6D‡ † z ‡ 2D‡ † d2¡ : 2…12D2¡ ‡ d2‡ †…d¡ ‡ 6D‡ †

2759

rates must be available. It can be noted that the unknowns (Dx , Dy , Dz ) can be substituted by (a, b, c), the advantage of the latter being that they appear in J~…!† (see (6)) in a linear form. Thus, for a given value of the angle ¿, the three unknowns a, b, c can be determined by a linear least squares procedure as detailed below. Let Y be a column vector containing the experimental cross-relaxation rates ¼1 , ¼2 . . . 0 1 ¼1 B C ¼ C YˆB …10† @ 2 A: .. . ¼CH can be expressed as (see (2) and (6)) Á ! Á ! X x2i X xi ¼Ci Hi ˆ 5K a ‡ 5K b 6 6 i r Ci Hi i r Ci Hi Á ! X 1 ‡ 5K c 6 i r Ci Hi ˆ ui1 a ‡ ui2 b ‡ ui3 c;

…11†

where the summation runs over all protons in the molecule (only those which are not too remote are evidently liable to contribute). De®ning the column vector A as 0 1 a B C A ˆ @bA …12† c

we can write

…13†

with …7†

…8†

‡

cross-relaxation rates

UA ˆ Y

6D¡ d¡ 6D¡ d¡ ‡ bˆ ; …12D2¡ ‡ d2¡ †…d‡ ‡ 6D‡ † …12D2¡ ‡ d2‡ †…d¡ ‡ 6D‡ †

cˆ

13 C±1 H

…9†

With the help of the above equations, we must determine the four unknowns Dx , Dy , Dz and ’ (or rather ¿, as the molecular geometry is supposed to be known). This means that at least four di€ erent cross-relaxation

0

u11

B u UˆB @ 21 .. .

u12 u22 .. .

u13

1

C u23 C: A .. .

…14†

The above matrix is easily established because the quantities ui1 , ui2 and ui3 can be calculated if the molecular geometry is available. Equation (13), which corresponds to an overdeterminated linear system, is then solved according to the well-known linear algebra procedures. The above calculations are repeated by varying ¿ until a Pn exp 2 1=2 minimum is reached for ‰… iˆ1 …¼calc . It C i Hi ¡ ¼ Ci Hi † †=nŠ can be mentioned that Fushman and co-workers [18] have recently presented a similar approach based on a singular value decomposition which leads to the determination of the rotation-di€ usion tensor.

O. Walker et al.

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4. Results and discussion As explained in the previous section, all possible values of the angle ¿ in the range 0±908 (in 0.18 steps) were considered (symmetrical solutions were found for the 908±1808, 1808±2708 and 2708±3608 ranges; they simply arise from the labelling and orientation of axes which are obviously non-relevant here). For each value of ¿, the three parameters a, b and c were calculated according to the linear method detailed in the previous section. This ®tting procedure involved either the four Ci Hi cross-relaxation rates (i ˆ 4, 5, 6, 7; carbons directly bonded to a proton), or the whole set of the seven cross-relaxation rates available. Almost identical results have been found for both data sets. The root mean squares deviation ³

rmsd:

µ³ X n iˆ1

exp 2 …¼calc Ci Hi ¡ ¼Ci Hi †

´¿ ¶1=2 ´ n

is plotted in ®gure 5 as a function of angle ¿. Besides two local minima which seem to be related to a mathematical singularity, the deepest minimum occurs for ¿ ˆ 47:9 § 58 corresponding to the following correlation time values: ½x ˆ 12:0 § 2:5 ps, ½y ˆ 32:1 § 4:0 ps, ½z ˆ 4:1 § 1:0 ps. The existence of local minima prompted us to further check these results by using a standard nonlinear ®tting routine (Simplex). In this case, local minima disappeared (®gure 5); an absolute minimum was found close to the one deduced from the linear procedure and similar values as above were obtained as

well for ½x , ½y , ½z . However, the agreement between recalculated and experimental cross-relaxation rates is not so good; this is re¯ected by a higher rmsd whatever the value of ¿. The di€ erence between these two ®tting procedures arises from the linear nature of the former which therefore should lead to more reliable results, as happens here. All experimental results, in addition to the recalculated values of the cross-relaxation rates, are given in table 1. The agreement between experimental and recalculated cross-relaxation rates is seen to be excellent except, perhaps, for carbon 3. The slight inconsistency could be due to a small fraction of other rotamers but most probably to an overestimation of the C3 ±H7 distance. The correlation time values can be easily rationalized according to the hindrance of reorientation about the considered axis. For instance it is well known that the in-plane reorientation is the fastest motion for substituted aromatic rings [19, 20] and we observe indeed here that ½z has the smallest value. Now, the fact that ½y is roughly three times larger than ½x can be understood on the following basis: it appears from ®gure 1 that reorientation about the y axis implies displacements of the chlorine atoms which should be more di cult than about the x axis. We have still to discuss the orientation of the rotation-di€ usion tensor. For this goal, we have computed several molecular properties with the methods of quantum chemistry. Some of the most important properties that a quantum chemistry calculation can provide are the electric multipole moments which re¯ect

Pn exp 2 1=2 Figure 5. Root mean square deviation ‰… iˆ1 …¼calc as a function of the angle ¿ (between X and x, see ®gure 1). Ci Hi ¡ ¼Ci Hi † †=nŠ For each value of ¿ varied by a step of 0:18, ½x , ½y and ½z are ®tted according to a multilinear regression analysis or to the Simplex algorithm.

Determination of the rotation-di€ usion tensor orientation from NMR the charge distribution in a molecule. However, only the lowest order non-zero multipole moment (here the dipole moment) will be of concern because it is the only one which is frame independent. We must also keep in mind that factors such as the basis set can have a signi®cant impact on the accuracy of the calculated properties. As a result, the molecular geometry, as well as the multipole moments, were computed using electronic correlation theory (Mùller±Plesset to the second order method) and a minimum 6-31G** basis set. All the calculations were carried out using the Gaussian 98 package [21] on a PC computer. As expected, the di€ usion tensor has nothing to do with the inertia tensor [22]; moreover none of its principal axes seem to be correlated with the direction of the dipole moment (®gure 1). A qualitative interpretation of this feature may rest on the charge distribution within the considered molecule. This charge distribution can be thought as governing its reorientation via intermolecular electrostatic interactions. Suppose that we are dealing with an elongated polar molecule (rod). In that case, of course, the dipole moment should be the dominant feature for electrostatic interactions in a pure liquid. Now, if the molecule cannot be seen as a rod but rather as a rectangle, the charge distribution within this rectangle should be substituted for the previous dipolar moment. However, as the PAS of the molecular quadrupolar tensor cannot be de®ned (the molecule possesses a dipole moment), it appears illusory to try to establish a ®rm correlation between the tensor orientation and other electrical molecular properties. 5. Conclusion The determination of the rotation-di€ usion tensor orientation is not common. Only a few examples can be found in the literature. In this work, we were able to obtain this information with a reasonable accuracy (and without any ambiguity) because the reorientation of the quasi-planar molecule under investigation is strongly anisotropic. Once reliable results have been obtained, we have to interpret them on a molecular basis. As already noted in several instances [22], it appears that in molecules without enough symmetry no correlation exists with the inertia tensor. The same conclusion applies to the dipole moment. We have thus to invoke the local charge distribution within the mol-

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cross-relaxation rates

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ecule and, as a consequence, speci®c intermolecular interactions. Of course this latter feature could be assessed by further studies involving, for example, di€ erent solvents. References [1] Python, H., Mutzenhardt, P., and Canet, D., 1997, J. phys. Chem., 101, 1793. [2] Witt, R., Sturz, L., DoÈ lle, A., and MuÈ llerPlathe., F., 2000, J. phys. Chem., 104, 5716. [3] Bluhm, T., 1982, Molec. Phys., 47, 475. [4] Kowalewski, J., MuÈ ller, L., and Widmalm, G., 1998, J. molec. Liq., 78, 255. [5] Woessner, D. E., 1962, J. chem. Phys., 37, 647. [6] Hubbard, P. S., 1970, J. chem. Phys., 52, 563. [7] Grant, D. M., Pugmire, R. J., Black, E. P., and Christensen, K. A., 1973, J. Am. chem. Soc., 95, 8465. [8] Mutzenhardt, P., Walker, O., Canet, D., Haloui, E., and Furo, I., 1998, Molec. Phys., 94, 565. [9] Huntress, W. T., 1968, J. chem. Phys., 48, 3524. [10] Dais, P., 1994, Carbohydr. Res., 263, 13. [11] Kratochwill, A., and Vold, R., 1980, J. magn. Reson., 40, 197. [12] Amato, M. E., Grassi, A., and Perly, B., 1990, Magn. reson. Chem., 28, 779. [13] Schaeffer, T., Schwenk, R., and Macdonald, C. J.,, and Reynolds, W. F., 1968, Can. J. Chem., 46, 2187. [14] Schaeffer, T., and Takeguchi, C., 1989, Can. J. Chem., 67, 1022. [15] Jeener, J., Meyer, B. H., Bachman, P., and Ernst, R. R., 1979, J. chem. Phys., 71, 4546. [16] Vold, R. L., Waugh, J. S., Klein, M. P., and Phelps, D. E., 1968, J. chem. Phys., 48, 3831. [17] Canet, D., 1998, Concepts magn. Reson., 10, 291. [18] Ghose, R., Fushman, D., and Cowburn, D., 2001, J. magn. Reson., 149, 204. [19] Stark, R. E., Vold, R. L., and Vold, R. R., 1979, J. magn. Reson., 33, 421. [20] Sturz, L., and DoÈ lle, A., 2001, J. phys. Chem., 105, 5055. [21] Frisch, M. J., Trucks, G. W., Schlegel, H. B., Gill, P. M. W., Johnson, B. G., Robb, M. A., Cheeseman, J. R., Keith, T., Petersson, G. A., Montgomery, J. A., Raghavachari, K., Al-Laham, M. A., Zakrzewski, V. G., Ortiz, J. V., Foresman, J. B., Peng, C. Y., Ayala, P. Y., Chen, W., Wong, M. W., Andres, J. L., Replogle, E. S., Gomperts, R., Martin, R. L., Fox, D. J., Binkley, J. S., Defrees, D. J., Baker, J., Stewart, J. P., Head-Gordon, M., Gonzalez, C., and Pople, J. A., 1995, Gaussian 98 (Pittsburgh, PA: Gaussian Inc.). [22] DoÈ lle, A., and Bluhm, T., 1989, Prog. NMR Spectrosc., 21, 175.