Paramagnetic Solid-State Magic-Angle Spinning NMR Spectroscopy

Abstract

A number of technical improvements have recently opened up solid-state NMR to the analysis of new classes of substrates with wide ranging implications for molecular and biological sciences, with an immediate impact on a large community of researchers. A wealth of information can be extracted from the analysis of solid-state NMR signals of paramagnetic compounds, as the changes induced by the paramagnetic center depend in a well-defined way on the structure of the molecule. Solid-state NMR is in a position to allow direct, straightforward experimental access to the fine details of the molecular electronic configuration, which is in turn a sensible reporter of the molecular geometry in small catalysts as well as in larger biomolecules.

Appendix A. More Theory

The dipolar Hamiltonian of (19) becomes

$$ {\tilde{\mathcal{H}}^{\text{D}}} = \left\{ {\frac{2}{3}{\mathcal{D}_{\text {2,0}}}{\chi_{{2,0}}} - \frac{1}{2}({\mathcal{D}_{\text {2,1}}}{\chi_{{2, - 1}}} + {\mathcal{D}_{\text {2, - 1}}}{\chi_{{2,1}}}) - \frac{{\sqrt {2} }}{3}{\mathcal{D}_{\text {2,0}}}{\chi_{{0,0}}}} \right\}{I_{\text{z}}}{B_0}. $$

(30)

In order to evaluate the transformation properties of the hyperfine Hamiltonian, this sum of products needs to be further simplified. This can be achieved by expressing the pair-wise products of the components ${\mathcal{D}_{{\text 2,\it m}}}{\chi_{{2, - m}}}$ of two tensors of rank 2 in terms of the components of a product tensor W _l,0 of rank l = 0 − 4 by using the Clebsch–Gordan expansion [114, 115]:

$$ {\mathcal{D}_{\text {2,m}}}{\chi_{\text {2, - m}}} = \sum\limits_{{l = 0}}^4 {\left( \begin{array}{lll} 2 2 l \\{ - m} m 0 \\ \end{array} \right){W_{{l,0}}}}, $$

(31)

where the term in parentheses represents the 3j symbols. Similarly, for the last term of (30):

$$ {\mathcal{D}_{\text {2,0}}}{\chi_{{0,0}}} = \left( \begin{array}{lll} \begin{array}{*{20}{c}} 0 & 2 & 2 \\ \end{array} \hfill \\ \begin{array}{*{20}{c}} 0 & 0 & 0 \\ \hfill \\ \end{array}\end{array} \right){W_{{2,0}}}. $$

(32)

When applied to the Hamiltonian of (31), the antisymmetric terms of the expansion (l = 1, 3) do not need to be considered. Therefore, the above formulas give

$$ \begin{array}{ll} {{\tilde{\mathcal{H}}}^{\text{D}}} &= \left\{ {\displaystyle\frac{2}{3}\left( {\frac{1}{{\sqrt {5} }}{W_{{0,0}}} - \sqrt {{\frac{2}{7}}} {W_{{2,0}}} + \sqrt {{\frac{{18}}{{35}}}} {W_{{4,0}}}} \right) + \left( {\frac{1}{{\sqrt {5} }}{W_{{0,0}}} - \frac{1}{{\sqrt {{14}} }}{W_{{2,0}}} - 2\sqrt {{\frac{2}{{35}}}} {W_{{4,0}}}} \right)}\right. \\ & \quad \left. - \frac{{\sqrt {2} }}{3}{W_{{2,0}}} \right\}{I_{\text{z}}}{B_0} { = \left\{ {\frac{{\sqrt {5} }}{3}{W_{{0,0}}} - \sqrt {{\frac{7}{{18}}}} W_{{2,0}}^{{({{\Delta }}\chi )}} - \frac{{\sqrt {2} }}{3}W_{{2,0}}^{{(\chi )}}} \right\}{I_{\text{z}}}{B_0}} \\& = - \left( {{\delta^{\text{PC}}} + \sqrt {{\frac{2}{3}}} {{\Delta }}{\sigma^{{({{\Delta }}\chi )}}}d_{{0,0}}^2({\beta_{\text{XL}}}) + \sqrt {{\frac{2}{3}}} {{\Delta }}{\sigma^{{(\chi )}}}d_{{0,0}}^2({\beta_{\text{DL}}})} \right){{\tilde{T}}_{{1,0}}}, \end{array}$$

(33)

where the superscripts χ and Δχ denote the contributions to the anisotropy that originate from the isotropic and anisotropic parts of the X tensor, respectively. It is interesting to note that the final form of the hyperfine interaction is again in all respects analogous to that of a diamagnetic chemical shift. Notably, the W _4,0 term vanishes, in contrast to, for example, the second-order quadrupolar effect where the product of two second-rank components contributes to the W _4,0. The rank-zero interaction is the isotropic shift and the rank-two components represent a shift anisotropy analogous to the diamagnetic CSA (the “dipolar shift anisotropy” or DSA).

The rank-zero portion of the dipolar interaction is the PCS δ ^PC. Its full expression can be obtained expanding back the W _0,0 term as a product of two second-rank tensors according to the inverse of (31):

$$ \begin{array}{lll} {W_{{0,0}}} = \sum\limits_{{m = - 2}}^2 \left( \begin{array}{*{20}{c}} 2 & 2 & 0 \\m & { - m} & 0 \\\end{array} \right){\mathcal{D}_{\text {2,\it m}}}{\chi_{{2, - m}}} \\ \qquad{ = \sum\limits_{{m = - 2}}^2 {\frac{{{{( - 1)}^m}}}{{\sqrt {5} }}{\mathcal{D}_{\text {2,\it m}}}{\chi_{{2, - m}}}}. } \end{array} $$

(34)

After including the transformation Ω _DX to move from the PAS of the D tensor to that of the X tensor, we get

$$ {W_{{0,0}}} = \frac{1}{{\sqrt {5} }}{\mathcal{D}_{\text {2,0}}}\sum\limits_{{m^{\prime} = - 2}}^2 {\mathcal{D}_{{0{\it m^{\prime}}}}^{2}{\chi_{{2,{\it m^{\prime}}}}}} $$

(35)

so that we obtain

$$\begin{array}{lll} {{\delta^{\text{PC}}} = \frac{1}{{3\hbar {\gamma_{\text{I}}}}}{\mathcal{D}_{{{2},{0}}}}\sum\limits_{{m^{\prime} = - 2}}^2 {\mathcal{D}_{{0 \it m^{\prime}}}^{2}{\chi_{{2,\it m^{\prime}}}}} } \\ { \qquad= \frac{{\sqrt {6} }}{{12\pi {r^3}}}\left[ {\left( {\frac{{3{{\cos }^2}\theta - 1}}{2}\sqrt {{\frac{2}{3}}} {{\Delta }}{\chi_{\text{ax}}}} \right) + \left( {\sqrt {{\frac{3}{8}}} {{\sin }^2}\theta {{\text{e}}^{{ - {\text{i}}2\varphi }}}\frac{1}{2}{{\Delta }}{\chi_{\text{rh}}}} \right) + \left( {\sqrt {{\frac{3}{8}}} {{\sin }^2}\theta {{\text{e}}^{{{\text{i}}2\varphi }}}\frac{1}{2}{{\Delta }}{\chi_{\text{rh}}}} \right)} \right]} \\ {\qquad = \frac{1}{{12\pi {r^3}}}\left\{ {{{\Delta }}{\chi_{\text{ax}}} (3{{\cos }^2} \theta - 1) + \frac{3}{2}{{\Delta }}{\chi_{\text{rh}}}{{\sin }^2}\theta \cos 2\varphi } \right\}} \\. \end{array} $$

(36)

The first component of the shift anisotropy is

$$ {{\Delta }}{\sigma^{{(\chi )}}} = \frac{1}{{{\mu_0}{\mu_{\text{B}}}{g_{\text{e}}}}}\frac{{\sqrt {2} }}{{3\hbar {\gamma_{\text{I}}}}}{\mathcal{D}_{\text {2,0}}}{\chi_{{0,0}}}, $$

(37)

which, in the PAS of the dipolar interaction, becomes

$$ {{\Delta }}{\sigma^{{(\chi )}}} = \frac{1}{{2\pi {\mu_0}{\mu_{\text{B}}}{g_{\text{e}}}}}\frac{{{\chi_{\text{iso}}}}}{{{r^3}}}. $$

(38)