(ii) the use of complex mathematical models for the analysis of the experimental data

Conclusions

The potential offered by Solid State NMR has been exploited in order to get detailed and quanti- tative information about the dynamic processes occurring in crystalline solid phases. This work is mainly characterized by the combination of (i) the application of advanced SSNMR experi- ments, performed over an unconventional range of temperatures and at different instrumental frequencies; (ii) the use of complex mathematical models for the analysis of the experimental data; (iii) the exploitation of DFT calculations as a support to the experimental work.

The systems to which the developed SSNMR approaches were applied are four widely used non-steroidal anti-inflammatory drugs: Ibuprofen, Sodium Ibuprofen, Naproxen and Sodium Naproxen. These systems exhibit a quite similar chemical and crystal structure, for instance the presence of an isopropionic group (either acid of salified) bonded to an hydrophobic moiety, and the crystal structures which in all the cases presents regions with strong electrostatic interactions alternated with regions dominated by the hydrophobic fragments. In spite of this, as carried out from the thesis work, their dynamic behaviors were found to be quite different.

The system with the largest interconformational mobility was the di-hydrated form of Sodium Ibuprofen, where methyl rotations about their symmetry axes and the interconformational mo- tion of the isobuthyl group were found to be in the fast regime (frequencies larger than 10⁶ Hz) and the π-flip of the aromatic ring was found to be in the intermediate regime (frequency between 10³ and 10⁶ Hz) at room temperature. All the molecular motions were quantitatively characterized by means of variable temperature¹³C CP-MAS spectra,¹³C chemical shift tensors ad ¹H and ¹³C relaxation times, with the use of models for spectral simulation and fitting of relaxation data. It is worth noticing that for a system with many fast motions like this the contribution of Cryogenic NMR measurements was particularly relevant.

The acid form of Ibuprofen showed a different dynamic behavior, in particular the fast interconformational motions found were the rotation of the methyl groups and the π-flip of the cyclic dimeric structure formed by the acidic groups, while the interconformational motion of the isobuthyl group and the π-flip of the aromatic ring were found to be frozen and in the slow

regime, respectively. The rigidity, in particular of the aromatic fragment, allowed the effects of small-amplitude motions on ¹³C chemical shift tensors to be identified and analyzed. For this purpose, a combined SSNMR and DFT approach was developed and applied to Ibuprofen.

The validity of this solid state NMR/DFT method was checked also by variable temperature

13C chemical shift tensors measurements, and the results obtained suggest that in comparing experimental and calculated chemical shift tensors the role of vibrations should always be taken into account.

Naproxen and Sodium Naproxen crystalline forms show the most restricted interconforma- tional mobility. For both these samples only the rotations of methyl and methoxyl groups were identified and quantitatively characterized by the fitting of relaxation time data. The very rigid structure of the two compounds allowed interesting static interactions to be highlighted. In par- ticular, noticeable intermolecular ring current effects on ¹H chemical shifts have been observed for both samples. The extent of these effects is different for different nuclei and for the same nucleus in the two forms. The observed ring current effects have been quantified by comparing the¹H chemical shifts calculated in the crystal structures and for isolated molecules (“in vacuo”).

The three studies presented show how SSNMR approaches can be devised for the study of systems with different dynamic properties and that, with decreasing the degree of interconfor- mational mobility, other aspects, such as vibrational motions and ring current effects, can be highlighted. In all cases the wide range of temperature investigated, the combination of different experiments and the use of suitable mathematical models together with the support of DFT calculations led to a very detailed characterization of the systems studied.

Appendix A Euler Angles

In NMR, it is common to define rotation operators using the Euler angles between two axes frames, for example (X, Y, Z) and (x, y, z). Euler angles are generally labelled Ω = (α, β, γ).

Figure A.1: Operative definition of Euler angles [13]. The transformation of frame (X, Y, Z) into (x, y, z) is described by a rotation of (X, Y, Z) (a) by angle α about Z. This takes the (X, Y, Z) frame into (X1, Y1, Z). Then (b) a rotation of an angle β, about the Y2 axis resulting from the previous rotation, takes the (X1, Y1, Z) into (X2, Y1, Z2). Finally (c) there is a rotation of an angle γ about the new Z2axis, which takes (X2, Y1, Z2) into the (x, y, z) frame

Following the definition of ref. [13], the transformation of frame (X, Y, Z) into (x, y, z) is described by a rotation of (X, Y, Z) by angle α about Z. This takes the (X, Y, Z) frame into (X1, Y1, Z). Then a rotation of an angle β, about the Y2 axis resulting from the previous rotation, takes the (X1, Y1, Z) into (X2, Y1, Z2). Finally there is a rotation of an angle γ about the new

170 Euler Angles

Z2 axis, which takes (X2, Y1, Z2) into the (x, y, z) frame. The process is described in Figure A.1.

The relation between Euler angles and polar angles is often useful:

• (α, β) are the polar angles (θ, φ) of the z-axis in the (X, Y, Z) frame.

• (β, 180^◦− γ) are the polar angles (θ, φ) of the Z-axis in the (x, y, z) frame.

Rotation of Cartesian axis frames can be performed through the rotation matrix R, which describes how to rotate an axis frame (x, y, z) fixed on an object within a frame (X, Y, Z), and therefore rotate the object in the process. The rotation matrix that performs this transformation is:

RZ(γ) =







cos γ − sin γ 0 sin γ cos γ 0

0 0 1





 (A.1)

The rotation matrix describing the following rotation of β about Y is of the form:

RY(β) =







cos β 0 sin β

0 1 0

− sin β 0 cos β





 (A.2)

Finally the last rotation of α about Z is described by:

RZ(α) =







cos α − sin α 0 sin α cos α 0

0 0 1





 (A.3)

The total transformation matrix R(α, β, γ), to go from (X, Y, Z) to the final orientation (x, y, z), by multiplying the three matrices of Equations A.1, A.2, A.3:

R(α, β, γ) = RZ(α)RY(β)RZ(γ)=







cos α cos β cos γ − sin α sin γ − cos α cos β sin γ − sin α cos γ cos α sin β sin α cos β cos γ + cos α sin γ − sin α cos β sin γ + cos α cos γ sin α sin β

− sin β cos γ sin β sin γ cos β





 (A.4)

At last, for the transformation of a cartesian tensor T from being expressed in frame (X, Y, Z) to being expressed in a frame (x, y, z), i.e. a passive rotation, the rotation matrix R(α, β, γ) of equation A.4 can be used as follows:

171

T (x, y, z) = R⁻¹(α, β, γ) T (X, Y, Z) R(α, β, γ) (A.5)

172 Euler Angles

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