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\begin{document}
\title{Supplemental Material for: Two-dimensional NMR Measurement and Point Dipole Model Prediction of Paramagnetic Shift Tensors in Solids}
\maketitle
\section{Double-quantum experiments}
\subsection{Discussion}
Well-established double-quantum techniques correlating quadrupolar and shift evolution for \mbox{$I=1$} nuclei exploit the symmetry property of \mbox{$\transition{d}_I(-m,m) = 0$}, indicating suppression of the first-order quadrupolar broadening for symmetric $\ketbra{m}{-m}$ transitions. Thus, a period of double-quantum evolution allows one to obtain a spectrum devoid of first-order quadrupolar broadening; however, since \mbox{$\transition{p}_I(\mp1,\pm1) = \pm2$}, the broadening due to isotropic and anisotropic shift components will be twice as large during double-quantum evolution.
The two-pulse sequence\cite{prl_37_43_1976} for indirectly detecting double-quantum coherence can, in principle, be used to construct a 2D spectrum that correlates these anisotropies and is capable of producing 2D absorption mode lineshapes. Doing so requires the \mbox{$p_I = -2 \rightarrow -1$} anti-path be collected in addition to the \mbox{$p_I = +2 \rightarrow -1$} path in order to sample adjacent 2D signal quadrants in $t_1$\cite{pnmr_59_121_2011}. In practice, however, the resulting spectrum will still be distorted because of the significant loss of signal during receiver dead time. Once again, the simple solution is to generate a shifted-echo signdal\cite{jmra_103_72_1993}, where the large degree of inhomogeneous broadening can be turned into an advantage through whole-echo acquisition. The best way to achieve this is by appending another non-selective $\pi/2$ pulse at a time $\tau$ after the mixing pulse to generate the transition pathways:
\begin{align}
\small
\left\{\!\!
\begin{array}{c}
I =1\\ \text{path}
\end{array}
\!\!\right\}:
\left\{\!
\begin{array}{l}
\!\left[z_I\right]
\!\rightarrow\!
\ketbra{1}{-1}_{t_1} \!\rightarrow\! \ketbra{0}{-1}_\tau \rightarrow \ketbra{-1}{0}_{t_2}, \\
\!\left[z_I\right]
\!\rightarrow\! \ketbra{1}{-1}_{t_1} \!\rightarrow\! \ketbra{1}{0}_\tau \rightarrow \ketbra{0}{1}_{t_2},
\end{array}
\right. \label{eq:dqpath}
\\
\left\{\!\!
\begin{array}{c}
I =1 \\ \text{anti}
\end{array}
\!\!\right\}:
\small
\left\{\!
\begin{array}{l}
\!\left[z_I\right]
\!\rightarrow\!
\ketbra{-1}{1}_{t_1} \!\rightarrow\! \ketbra{0}{-1}_\tau \rightarrow \ketbra{-1}{0}_{t_2}, \\
\!\left[z_I\right]
\!\rightarrow\! \ketbra{-1}{1}_{t_1} \!\rightarrow\! \ketbra{1}{0}_\tau \rightarrow \ketbra{0}{1}_{t_2},
\end{array}
\right. \label{eq:dqanti}
\end{align}
also illustrated in Fig.~\ref{fg:FigureS1}. Interestingly, few attempts to generate shifted-double-quantum echoes in this manner are found in the literature\cite{ssnmr_10_25_1997}.
By defining the $d_I$ echo top, occurring $\tau$ after the final pulse, as the $t_2$ origin of a $t_2$-$t_1$ coordinate system, clear parallels to the single-quantum correlation experiments of the main text emerge. The only major difference in comparison to the shifting-$p$ echo experiment evident from the symmetry pathway analysis is that the $t_2'$ coordinate defining pure quadrupolar evolution advances along the line \mbox{$t_2=2t_1$}, so that the affine transformation required to segregate and orient the interactions of differing symmetries is characterized by \mbox{$\kappa^{(t_1)}=-\frac{1}{2}$} and \mbox{$\varsigma^{(t_1)}=-2$}, assuming a pathway signal as defined by Eq.~\eqref{eq:dqpath}. Processing of the anti-pathways defined by Eq.~\eqref{eq:dqanti} proceeds similarly, utilizing an affine transformation characterized by \mbox{$\kappa^{(t_1)}=\frac{1}{2}$} and \mbox{$\varsigma^{(t_1)}=2$}.
Similarities in transition symmetry separation strategy do not carry over as similarities between the final single and double-quantum spectra. Since double-quantum excitation is contingent on the tilting of the rotating frame eigenstates by the first-order quadrupolar splitting, the correlation spectrum exhibits a marked dependence on $\omega_q$. This is seen strikingly in Fig.~\ref{fg:FigureS1}C, where the node of an antiphase powder pattern is observed at precisely zero frequency in the quadrupolar dimension. Extracting tensor parameters from these 2D spectra is not straightforward, since any transfer profile has a value of zero when the first-order quadrupolar splitting goes to zero. Needless to say, when shift anisotropies are significant, the nearly uniform excitation profile offered up by the single quantum correlation experiments make them superior sequences when accurate interaction parameters are desired.
\begin{figure*}
\centering
\includegraphics{FigureS1.pdf}
\caption{(A) Spin transition diagrams for the $I=1$ shifted double-quantum echo experiment, displaying the observable transition pathways given in Eq.~\eqref{eq:dqpath}. For clarity, the anti-pathways have been omitted. (B) Pulse sequence and spin transition pathways for the experiment. Signal acquisition begins immediately after the final pulse where \mbox{$t_2 = -\tau$}. In lieu of hypercomplex acquisition in the manner of States\cite{jmr_48_286_1982}, $\phi_1$, $\phi_2$, and the receiver phase are cycled for the desired coherence changes, while $\phi$ is incremented in steps of $2\pi/8$ as an extra dimensional parameter. FT of this phase dimension allows for easy extraction of the signal along \mbox{$\Delta p = 3$} and \mbox{$\Delta p = -1$}. (C) Shifting double-quantum echo $^2$H NMR spectrum of polycrystalline \mbox{CuCl$_2 \cdot 2$D$_2$O} composed from path and antipath signal, exhibiting a strong antiphase profile along the quadrupolar dimension (\mbox{$\Delta t_2 = 2 \; \mu$s}, \mbox{$\Delta t_1 = 6 \; \mu$s}, \mbox{$\tau = 250 \; \mu$s}).}
\label{fg:FigureS1}
\end{figure*}
\subsection{Experimental}
Due to calculations by Vega and Pines \cite{jcp_66_5624_1977} that appreciable excitation of double-quantum coherence occurs even when \mbox{ $|\gamma_I B_1| > |\omega_q|$}, we used the highest pulse powers possible \mbox{($\nu_1 = 200$ kHz)} for not only the mixing pulses but the excitation pulse as well. In this way, the pulse durations can be kept short, and inflections of the double-quantum to observable coherence transfer profile over the broad pattern are minimized. The optimum duration of the double-quantum excitation pulse at this power level was found to be 4.8 $\mu$s. The conversion and echo shift pulse durations were 1.24 $\mu$s. The double-quantum echo experiment in Fig.~\ref{fg:FigureS1} was performed with 32 $t_1$ points and an effective 36864 scans per slice. The total experiment time was \mbox{13 h 25 min}.
\section{2D Tenting}
The frequency-domain simulation of a two-dimensional powder pattern correlating anisotropies with an appropriate interpolation scheme can yield an accurate spectrum with fewer crystallite orientations much more quickly than an equivalent time-domain simulation. For one-dimensional anisotropic powder patterns, an effective interpolation scheme is the POWDER method created by Alderman and Grant\cite{jcp_84_3717_1986}. In this approach a sphere of $\left(\theta,\phi\right)$ angle pairs are constructed as a large number of triangles over the faces of an octahedron inscribed in the unit sphere. The frequency at any given point is determined by interpolating across the triangle vertices via a process called "tenting". In this way, the number of angle pairs required for a smooth powder average is reduced by one or two orders of magnitude.
In the two-dimensional case, the problem is more complicated because vertices of the triangles span two frequency dimensions instead of one. The simplest approach, in which the the vertices are directly binned into the spectrum without interpolation, fails to produce a smooth spectrum output unless a very large number of angle pairs is used. In our algorithm, the Alderman and Grant
``tenting" algorithm is modified to allow for an interpolation across both frequency dimensions and bins the signal in the two-dimensional frequency domain spectrum appropriately, as illustrated in Fig.~\ref{fg:FigureS2}. This strategy was previously used by Charpentier and co-workers to develop a frequency-domain simulation of an MQMAS experiment\cite{jcp_109_3116_1998}; however, the algorithm they developed suffered from a sub-optimal binning of intensities into the two-dimensional spectrum and produced unsatisfactory patterns. This problem is corrected in the algorithm that we have developed. These details are beyond the scope of the present work and will be given in an upcoming manuscript.
\begin{figure*}
\centering
\includegraphics{FigureS2.pdf}
\caption{Contribution of the $\left(\theta,\phi\right)$ angle pairs inside the spherical triangle with vertices $\left(P_1,P_2,P_3\right)$ to the two-dimensional correlation spectrum. The spectral intensities are binned into the grid based upon the overlap of each grid square with the triangle formed by interpolating the two-dimensional frequency-domain representation of $\left(P_1,P_2,P_3\right)$.}
\label{fg:FigureS2}
\end{figure*}
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