Orientation-Selective DEER Using Rigid Spin Labels, Cofactors, Metals, and Clusters

Abstract

The dipolar interaction between two paramagnetic centres depends upon their spin–spin distance and relative orientation. Generally most experiments are carried out under conditions where the DEER signal only reports on the spin–spin distances and, for this type of data, sophisticated analysis methods for obtaining distance distributions have been developed. Recently there have been an increasing number of studies on systems where the DEER signals depend upon both distance and spin pair orientation. These investigations have relied on the use of rigid spin labels (those with a well-defined spatial position) and/or spectrometers operating at Q-band frequencies and above capable of performing DEER experiments with high resolution and sensitivity. In this article we discuss in detail orientation-selective DEER experiments for which the modulation depth and the dipolar frequencies depend on the relative orientation of the two paramagnetic centres and the distance. Analysis of the data in the presence of distance and orientation distributions is discussed, and representative examples from the literature are given for systems containing spin labels, organic cofactors, metals, and metal clusters.

Appendices

Appendix I: Hamiltonians in Explicit Vector and Matrix Form

$$ \begin{array}{lll} {{{\bf D}}_{\mathrm{ A}\mathrm{ B}}} =\frac{{{\mu_0}{\beta_e}^2}}{{4\pi \hbar }}\frac{{{{{\boldsymbol{g}}}_{\mathrm{ A}}}{\boldsymbol{g}}_{\mathrm{ B}}^{\mathrm{ T}}-3({{{\boldsymbol{g}}}_A}{\bf n}_{\mathrm{ A}\mathrm{ B}}^{\mathrm{ T}}){{{({{{\boldsymbol{g}}}_{\mathrm{ B}}}{\boldsymbol{n}}_{\mathrm{ A}\mathrm{ B}}^{\mathrm{ T}})}}^{\mathrm{ T}}}}}{{r_{\mathrm{ A}\mathrm{ B}}^3}} \\= \frac{{{\mu_0}{\beta_e}^2}}{{4\pi \hbar }}\frac{1}{{r_{\mathrm{ A}\mathrm{ B}}^3}} {{\left\{ {\left[ {\begin{array}{lllll} {g_{\mathrm{ x}\mathrm{ x}}^{\mathrm{ A}}} & {g_{\mathrm{ x}\mathrm{ y}}^{\mathrm{ A}}} & {g_{\mathrm{ x}\mathrm{ z}}^{\mathrm{ A}}} \\{} & {g_{\mathrm{ y}\mathrm{ y}}^{\mathrm{ A}}} & {g_{\mathrm{ y}\mathrm{ z}}^{\mathrm{ A}}} \\ {} & {} & {g_{\mathrm{ z}\mathrm{ z}}^{\mathrm{ A}}}\end{array}} \right]\left[ {\begin{array}{lllll} {g_{\mathrm{ x}\mathrm{ x}}^{\mathrm{ B}}} & {g_{\mathrm{ x}\mathrm{ y}}^{\mathrm{ B}}} & {g_{\mathrm{ x}\mathrm{ z}}^{\mathrm{ B}}} \\ {} & {g_{\mathrm{ y}\mathrm{ y}}^{\mathrm{ B}}} & {g_{\mathrm{ y}\mathrm{ z}}^{\mathrm{ B}}} \\ {} & {} & {g_{\mathrm{ z}\mathrm{ z}}^{\mathrm{ B}}}\end{array}} \right]}\right.}} \\ \left. {-3\left( {\left[ {\begin{array}{llllllll} {g_{\mathrm{x}\mathrm{x}}^{\mathrm{A}}} & {g_{\mathrm{ x}\mathrm{y}}^{\mathrm{ A}}} & {g_{\mathrm{ x}\mathrm{ z}}^{\mathrm{ A}}} \\ {} & {g_{\mathrm{ y}\mathrm{y}}^{\mathrm{ A}}} & {g_{\mathrm{ y}\mathrm{ z}}^{\mathrm{ A}}} \\ {} & {} & {g_{\mathrm{ z}\mathrm{ z}}^{\mathrm{ A}}}\end{array}} \right]\left[ {\begin{array}{llllllll}{ {n_{\mathrm{ x}}^{\mathrm{ A}\mathrm{ B}}}} \\ {n_{\mathrm{ y}}^{\mathrm{ A}\mathrm{ B}}} \\ {n_{\mathrm{ z}}^{\mathrm{ A}\mathrm{ B}}} \\ \end{array}} \right]} \right){{{\left( {\left[ {\begin{array}{lllllll} {g_{\mathrm{ x}\mathrm{ x}}^{\mathrm{ B}}}& {g_{\mathrm{ x}\mathrm{y}}^{\mathrm{ B}}} & {g_{\mathrm{ x}\mathrm{ z}}^{\mathrm{ B}}} \\ {} & {g_{\mathrm{ y}\mathrm{ y}}^{\mathrm{ B}}} & {g_{\mathrm{ y}\mathrm{ z}}^{\mathrm{ B}}} \\ {} & {} & {g_{\mathrm{ z}\mathrm{ z}}^{\mathrm{ B}}}\end{array}} \right]\left[ {\begin{array}{lllll} {n_{\mathrm{ x}}^{\mathrm{ A}\mathrm{ B}}} \\ {n_{\mathrm{ y}}^{\mathrm{ A}\mathrm{ B}}} \\ {n_{\mathrm{ z}}^{\mathrm{ A}\mathrm{ B}}}\end{array}} \right]} \right)}}^{\mathrm{ T}}}} \right\} \end{array} $$

(32)

$$ \begin{array}{lll} \hat{H}_{\mathrm{ dd}}^{\mathrm{ A}\mathrm{ B}} ={{{\hat{{\bf S}}}}_{\mathrm{ A}}}{{{\bf D}}_{\mathrm{ A}\mathrm{ B}}}\hat{{\bf S}}_{\mathrm{ B}}^{\mathrm{ T}} \hfill \cr = \left[ {\matrix{ {\hat{S}_{\mathrm{ x}}^{\mathrm{ A}}} & {\hat{S}_{\mathrm{ y}}^{\mathrm{ A}}} & {\hat{S}_{\mathrm{ z}}^{\mathrm{ A}}} \cr }<!end array>} \right]\left[ {\matrix{ {D_{\mathrm{ x}\mathrm{ x}}^{\mathrm{ A}\mathrm{ B}}} & {D_{\mathrm{ x}\mathrm{ y}}^{\mathrm{ A}\mathrm{ B}}} & {D_{\mathrm{ x}\mathrm{ z}}^{\mathrm{ A}\mathrm{ B}}} \cr {} & {D_{\mathrm{ y}\mathrm{ y}}^{\mathrm{ A}\mathrm{ B}}} & {D_{\mathrm{ y}\mathrm{ z}}^{\mathrm{ A}\mathrm{ B}}} \cr {} & {} & {D_{\mathrm{ z}\mathrm{ z}}^{\mathrm{ A}\mathrm{ B}}} \cr }<!end array>} \right]\left[ {\matrix{ {\hat{S}_{\mathrm{ x}}^{\mathrm{ B}}} \cr {\hat{S}_{\mathrm{ y}}^{\mathrm{ B}}} \cr {\hat{S}_{\mathrm{ z}}^{\mathrm{ B}}} \cr }<!end array>} \right] , \end{array} $$

(33)

$$ \begin{array}{lll} {\hat{H}} =\hat{H}_{\mathrm{ eZ}}^{\mathrm{ A}}+\hat{H}_{\mathrm{ eZ}}^{\mathrm{ B}}+\hat{H}_{\mathrm{ dd}}^{\mathrm{ A}\mathrm{ B}} \hfill \cr = \frac{{{\beta_e}}}{\hbar }{{{\bf B}}_0}{{{\bf g}}_{\mathrm{ A}}}\hat{{\bf S}}_{\mathrm{ A}}^{\mathrm{ T}}+\frac{{{\beta_e}}}{\hbar }{{{\bf B}}_0}{{{\bf g}}_{\mathrm{ B}}}\hat{{\bf S}}_{\mathrm{ B}}^{\mathrm{ T}}+{{{\hat{{\bf S}}}}_{\mathrm{ A}}}{{{\bf D}}_{\mathrm{ A}\mathrm{ B}}}\hat{{\bf S}}_{\mathrm{ B}}^{\mathrm{ T}}, \end{array} $$

(34)

where

$$ \hat{H}_{\mathrm{ eZ}}^k=\frac{{{\beta_e}}}{\hbar}\left[ {\matrix{ {{B_{0x }}} & {{B_{0y }}} & {{B_{0z }}} \cr }<!end array>} \right]\left[ {\matrix{ {g_{\mathrm{ x}\mathrm{ x}}^k} & {g_{\mathrm{ x}\mathrm{ y}}^k} & {g_{\mathrm{ x}\mathrm{ z}}^k} \cr {} & {g_{\mathrm{ y}\mathrm{ y}}^k} & {g_{\mathrm{ y}\mathrm{ z}}^k} \cr {} & {} & {g_{\mathrm{ z}\mathrm{ z}}^k} \cr }<!end array>} \right]\left[ {\matrix{ {\hat{S}_{\mathrm{ x}}^k} \cr {\hat{S}_{\mathrm{ y}}^k} \cr {\hat{S}_{\mathrm{ z}}^k} \cr }<!end array>} \right] ,\;\;k=\mathrm{ A}\ \mathrm{ or}\ \mathrm{ B}. $$

Appendix II: Density Matrix Simulations

If a fundamental description of the DEER time trace with respect to details of the mw pulses is required, then density matrix calculations can be employed. For 4-pulse DEER, this simulation involves an evolution Hamiltonian in a frame rotating with either the detection or pump pulse frequency:

$$ \begin{array}{lll} {{\hat{H}}^{\mathrm{ rot}}} ={\Omega_{\mathrm{ A}}}\hat{S}_{\mathrm{ z}}^{\mathrm{ A}}+{\Omega_{\mathrm{ B}}}\hat{S}_{\mathrm{ z}}^{\mathrm{ B}}+{{{\hat{{\bf S}}}}_A}{{{\bf D}}_{\mathrm{ A}\mathrm{ B}}}\hat{{\bf S}}_{\mathrm{ B}}^{\mathrm{ T}} \hfill \cr \cong {\Omega_{\mathrm{ A}}}\hat{S}_{\mathrm{ z}}^{\mathrm{ A}}+{\Omega_{\mathrm{ B}}}\hat{S}_{\mathrm{ z}}^{\mathrm{ B}}+{\omega_{\mathrm{ dd}}}(1-3{\cos^2}\theta )\hat{S}_{\mathrm{ z}}^{\mathrm{ A}}\hat{S}_{\mathrm{ z}}^{\mathrm{ B}}, \end{array} $$

where ${\Omega_{\mathrm{ A}}}={\omega_{\mathrm{ A}}}-{\omega_{\det }}$ and ${\Omega_{\mathrm{ B}}}={\omega_{\mathrm{ B}}}-{\omega_{\det }}$ are the frequency offsets of the A- and B-spins from the detection pulse frequency, respectively. The offsets in the pump pulse frame are defined similarly. The mw pulse Hamiltonian, in the frame of rotation of either the detection pulse mw frequency or pump pulse mw frequency, is

$$ {{\hat{H}}_1}={\omega_1}{{\hat{S}}_{\mathrm{ x}}}, $$

where ω ₁ is the strength of the mw pulse, in radians. The density matrix ρ(t) can then be propagated from the Boltzmann equilibrium $\rho (0)=\hat{S}_{\mathrm{ z}}^{\mathrm{ A}}+\hat{S}_{\mathrm{ z}}^{\mathrm{ B}}$ to the time of the echo,

$$ \rho (\mathrm{ echo})=U_{\mathrm{ ev}4}U_{\det 3}U_{\mathrm{ ev}3}U_{\mathrm{ coh}}U_{\mathrm{ p}}U_{\mathrm{ ev}2}U_{\det 2}U_{\mathrm{ ev}1}U_{\det 1}\rho (0)\times U_{\det 1}^{\mathrm{ T}}U_{\mathrm{ ev}1}^{\mathrm{ T}}U_{\det 2}^{\mathrm{ T}}U_{\mathrm{ ev}2}^{\mathrm{ T}}U_{\mathrm{ p}}^{\mathrm{ T}}U_{\mathrm{ coh}}^{\mathrm{ T}}U_{\mathrm{ ev}3}^{\mathrm{ T}}U_{\det 3}^{\mathrm{ T}}U_{\mathrm{ ev}4}^{\mathrm{ T}}. $$

The exponential operators U which propagate ρ(t) are given by

$$ \begin{array}{lll} {U_{{\det 1}}} =\exp (-i{{\hat{H}}_{\det }}{t_{{\uppi /2}}}) \hfill \cr {U_{{\mathrm{ ev}1}}} =\exp (-i{{\hat{H}}_0}{\tau_1}) \hfill \cr {U_{{\det 2}}} =\exp (-i{{\hat{H}}_{\det }}{t_{\uppi}}) \hfill \cr {U_{\mathrm{ ev}2}} =\exp (-i{{\hat{H}}_0}t) \hfill \cr {U_{\mathrm{ p}}} =\exp (-i{{\hat{H}}_{\mathrm{ p}\mathrm{ ump}}}t_{\uppi}^{\mathrm{ p}\mathrm{ ump}}) \hfill \cr {U_{\mathrm{ coh}}} =\exp (-i{{\hat{H}}_{\mathrm{ coh}\mathrm{ erence}}}t_{\uppi}^{\mathrm{ p}\mathrm{ ump}}) \hfill \cr {U_{\mathrm{ ev}3}} =\exp (-i{{\hat{H}}_0}({\tau_1}+\tau +t_{\uppi}^{\mathrm{ p}\mathrm{ ump}}-t)) \hfill \cr {U_{{\det 3}}} =\exp (-i{{\hat{H}}_{\det }}{t_{\uppi}}) \hfill \cr {U_{\mathrm{ ev}4}} =\exp (-i{{\hat{H}}_0}\tau ) \end{array} $$

The propagator U _coh with ${{\hat{H}}_{\mathrm{ coherence}}}=({\omega_{\det }}-{\omega_{\mathrm{ pump}}})(\hat{S}_{\mathrm{ z}}^{\mathrm{ A}}+\hat{S}_{\mathrm{ z}}^{\mathrm{ B}})$ corrects for the relative phase accumulated in the pump pulse frame during the pump pulse to keep track consistently of the coherence evolution in the detection frame during the whole simulation. The components of the magnetisation M _x , M _y , and M _z are given by

$$ {M_i}=\mathrm{ trace}(\rho \hat{S}_i),\;\mathrm{ with}\;\;i=x,y,z. $$

Numerical implementation of the theory is straightforward on a computer. A useful example is given in [2] where a calculation to determine the lower distance limit of applicability of the DEER sequence due to the bandwidth of rectangular mw pulses was presented.

Appendix III: MATLAB Code for a 4-Pulse DEER Density Matrix Simulation