¹H-enhanced ¹⁰³Rh NMR spectroscopy and relaxometry of ¹⁰³Rh(acac)₃ in solution

Abstract

Recently developed polarisation transfer techniques are applied to the ${}^{\mathrm{103}}\mathrm{Rh}$ nuclear magnetic resonance (NMR) of the ${}^{\mathrm{103}}\mathrm{Rh}(\mathrm{acac}{)}_{\mathrm{3}}$ coordination complex in solution. Four-bond ${}^{\mathrm{1}}\mathrm{H}$ – ${}^{\mathrm{103}}\mathrm{Rh}$ $J$  couplings of around 0.39 $\mathrm{Hz}$ are exploited to enhance the ${}^{\mathrm{103}}\mathrm{Rh}$ NMR signal and to estimate the ${}^{\mathrm{103}}\mathrm{Rh}$ $T_{\mathrm{1}}$  and $T_{\mathrm{2}}$  relaxation times as a function of field and temperature. The ${}^{\mathrm{103}}\mathrm{Rh}$ longitudinal $T_{\mathrm{1}}$  relaxation in ${}^{\mathrm{103}}\mathrm{Rh}(\mathrm{acac}{)}_{\mathrm{3}}$ is shown to be dominated by the spin–rotation mechanism, with an additional field-dependent contribution from the ${}^{\mathrm{103}}\mathrm{Rh}$ chemical shift anisotropy.

1. Introduction

Although rhodium is one of the few chemical elements with a 100 % abundant spin- $\mathrm{1}/\mathrm{2}$ isotope, the routine nuclear magnetic resonance (NMR) of ${}^{\mathrm{103}}\mathrm{Rh}$ has been inhibited by its very small gyromagnetic ratio, which is negative and $\sim$ 31.59 times less than that of ${}^{\mathrm{1}}\mathrm{H}$ ().

While indirectly detected ${}^{\mathrm{103}}\mathrm{Rh}$ NMR has had an appreciable history (), advances in instrumentation and methodology have allowed rapid observation of ${}^{\mathrm{103}}\mathrm{Rh}$ NMR parameters on standard commercial NMR spectrometers, leading to a recent renaissance of the field ().

The $\mathrm{rhodium}\left(\mathrm{III}\right)$ acetylacetonate ( ${}^{\mathrm{103}}\mathrm{Rh}(\mathrm{acac}{)}_{\mathrm{3}}$ ) complex (see Fig. ) currently serves as the International Union of Pure and Applied Chemistry (IUPAC) ${}^{\mathrm{103}}\mathrm{Rh}$ NMR chemical-shift reference (). To the wider scientific community, ${}^{\mathrm{103}}\mathrm{Rh}(\mathrm{acac}{)}_{\mathrm{3}}$ is better known for its role in the production of thin rhodium films and nanocrystals for use in catalysis ().

Figure 1.

The molecular structure of $\mathrm{rhodium}\left(\mathrm{III}\right)$ acetylacetonate, ${}^{\mathrm{103}}\mathrm{Rh}(\mathrm{acac}{)}_{\mathrm{3}}$ , which has point group symmetry ${\mathrm{D}}_{\mathrm{3}}$ . This work exploits the long-range ${}^{\mathrm{4}}{J}_{\text{HRh}}$ scalar couplings for polarisation transfer between the ${}^{\mathrm{103}}\mathrm{Rh}$ and methine ${}^{\mathrm{1}}\mathrm{H}$ spins.

The early studies of nuclear spin relaxation in ${}^{\mathrm{103}}\mathrm{Rh}(\mathrm{acac}{)}_{\mathrm{3}}$ were greatly limited by the poor ${}^{\mathrm{103}}\mathrm{Rh}$ signal strength, and they provided somewhat conflicting conclusions for the ${}^{\mathrm{103}}\mathrm{Rh}$ relaxation mechanisms (). Recently, a two-bond ${}^{\mathrm{13}}\mathrm{C}$ – ${}^{\mathrm{103}}\mathrm{Rh}$ coupling of 1.1 $\mathrm{Hz}$ was observed in ${}^{\mathrm{103}}\mathrm{Rh}(\mathrm{acac}{)}_{\mathrm{3}}$ and was exploited for triple-resonance experiments ().

We now report the observation of a four-bond ${}^{\mathrm{1}}\mathrm{H}$ – ${}^{\mathrm{103}}\mathrm{Rh}$ $J$  coupling of ${\mathrm{|}}^{\mathrm{4}}{J}_{\text{HRh}}\mathrm{|}$ $\simeq$ 0.39 $\mathrm{Hz}$ between the central ${}^{\mathrm{103}}\mathrm{Rh}$ nucleus and each of the three methine ${}^{\mathrm{1}}\mathrm{H}$ nuclei in ${}^{\mathrm{103}}\mathrm{Rh}(\mathrm{acac}{)}_{\mathrm{3}}$ (see Fig. ). These small couplings are exploited for the ${}^{\mathrm{1}}\mathrm{H}$ -enhanced ${}^{\mathrm{103}}\mathrm{Rh}$ NMR spectroscopy of the ${}^{\mathrm{103}}\mathrm{Rh}(\mathrm{acac}{)}_{\mathrm{3}}$ complex. ${}^{\mathrm{103}}\mathrm{Rh}$ spin–lattice $T_{\mathrm{1}}$ and spin–spin $T_{\mathrm{2}}$  relaxation time constants are measured over a range of magnetic fields and temperatures. The ${}^{\mathrm{103}}\mathrm{Rh}$ $T_{\mathrm{1}}$  relaxation is found to be dominated by spin–rotation, with an additional contribution from the chemical shift anisotropy (CSA), which is significant at high fields.

2. Experimental

Experiments were performed on a saturated ( $\sim$ 140 $\mathrm{mM}$ ) solution of $\mathrm{rhodium}\left(\mathrm{III}\right)$ acetylacetonate ( ${}^{\mathrm{103}}\mathrm{Rh}(\mathrm{acac}{)}_{\mathrm{3}}$ ) dissolved in 350 $\mathrm{\mu }\mathrm{L}$ ${\mathrm{CDCl}}_{\mathrm{3}}$ . ${}^{\mathrm{103}}\mathrm{Rh}(\mathrm{acac}{)}_{\mathrm{3}}$ was purchased from Sigma-Aldrich and used as received. ${}^{\mathrm{103}}\mathrm{Rh}(\mathrm{acac}{)}_{\mathrm{3}}$ is a bright yellow powder, which is dissolved in ${\mathrm{CDCl}}_{\mathrm{3}}$ to form a solution with a deep golden colour.

The radio frequency channels were additionally isolated by installing a bandpass (K&L Microwave) and low-pass (Chemagnetics 30 $\mathrm{MHz}$ ) filter at the preamplifier outputs of the ${}^{\mathrm{1}}\mathrm{H}$ and ${}^{\mathrm{103}}\mathrm{Rh}$ channels, respectively. Pulse powers on the ${}^{\mathrm{1}}\mathrm{H}$ and ${}^{\mathrm{103}}\mathrm{Rh}$ channels were calibrated to give a matched nutation frequency of 2 $\mathit{\pi }$ $\times$ 4000 $\mathrm{Hz}$ , corresponding to a 90° pulse length of 62.5 $\mathrm{\mu }\mathrm{s}$ . Field-cycling experiments were performed using a motorised fast-shuttling system (). The shuttling time was kept constant at 2 $\mathrm{s}$ , in both directions.

3. Results

3.1. ${}^{\mathrm{1}}\mathrm{H}$ spectrum

The ${}^{\mathrm{1}}\mathrm{H}$ spectrum for ${}^{\mathrm{103}}\mathrm{Rh}(\mathrm{acac}{)}_{\mathrm{3}}$ in ${\mathrm{CDCl}}_{\mathrm{3}}$ (shown in Fig. a) features two resonances: a singlet at 2.170 $\mathrm{ppm}$ , corresponding to the six methyl protons on each acac ligand, and a broad, weak doublet centred at 5.511 $\mathrm{ppm}$ , corresponding to the acac methine protons. An expanded region showing just the methine resonance is presented in Fig. b. The four-bond ${}^{\mathrm{103}}\mathrm{Rh}$ – ${}^{\mathrm{1}}\mathrm{H}$ spin–spin coupling is estimated to be ${\mathrm{|}}^{\mathrm{4}}{J}_{\text{HRh}}\mathrm{|}$ $=$ 0.39 $\pm$ 0.01 $\mathrm{Hz}$ .

Figure 2.

(a)  ${}^{\mathrm{1}}\mathrm{H}$ spectrum of a $\sim$ 140 $\mathrm{mM}$ solution of ${}^{\mathrm{103}}\mathrm{Rh}(\mathrm{acac}{)}_{\mathrm{3}}$ in ${\mathrm{CDCl}}_{\mathrm{3}}$ , acquired at 9.4 $\mathrm{T}$ and 298 $\mathrm{K}$ , in a single transient. Lorentzian line broadening (1 $\mathrm{Hz}$ ) was applied. (b) Expanded view of the methine ${}^{\mathrm{1}}\mathrm{H}$ resonance. Negative Lorentzian line broadening ( $-$ 0.2 $\mathrm{Hz}$ ) was applied to enhance the resolution.

3.2. ${}^{\mathrm{103}}\mathrm{Rh}$ spectra

3.2.1. Direct ${}^{\mathrm{103}}\mathrm{Rh}$ excitation

The ${}^{\mathrm{1}}\mathrm{H}$ -decoupled ${}^{\mathrm{103}}\mathrm{Rh}$ spectrum of the ${}^{\mathrm{103}}\mathrm{Rh}(\mathrm{acac}{)}_{\mathrm{3}}$ solution, acquired with single-pulse excitation of ${}^{\mathrm{103}}\mathrm{Rh}$ transverse magnetisation, is shown in Fig. a and displays a single peak with the ${}^{\mathrm{103}}\mathrm{Rh}$ chemical shift of 8337.6 $\mathrm{ppm}$ . The signal-to-noise ratio is quite poor, even after 12 $\mathrm{h}$ of data acquisition.

Figure 3.

${}^{\mathrm{1}}\mathrm{H}$ -decoupled ${}^{\mathrm{103}}\mathrm{Rh}$ NMR spectra of a $\sim$ 140 $\mathrm{mM}$ solution of ${}^{\mathrm{103}}\mathrm{Rh}(\mathrm{acac}{)}_{\mathrm{3}}$ in ${\mathrm{CDCl}}_{\mathrm{3}}$ , in a field of 9.4 $\mathrm{T}$ and at a temperature of 295 $\mathrm{K}$ . Lorentzian line broadening (1 $\mathrm{Hz}$ ) was applied to all spectra. ${}^{\mathrm{103}}\mathrm{Rh}$ chemical shifts are referenced to the absolute frequency ( $\mathrm{\Xi }\left({}^{\mathrm{103}}\mathrm{Rh}\right)$ $=$ 3.16 %). In all spectra, ${}^{\mathrm{1}}\mathrm{H}$ decoupling was achieved using continuous wave decoupling with 0.05 $\mathrm{W}$ of power, corresponding to a nutation frequency of 1 $\mathrm{kHz}$ . (a)  ${}^{\mathrm{103}}\mathrm{Rh}$ Hdec spectrum, acquired using 300 transients, each using a single ${}^{\mathrm{103}}\mathrm{Rh}$ 90° pulse. The waiting interval between transients was 150 $\mathrm{s}$ . The total experimental duration was $\sim$ 12 $\mathrm{h}$ . (b)  ${}^{\mathrm{103}}\mathrm{Rh}$ Hdec spectrum, acquired using 16 transients and the pulse sequence shown in Fig. , with $n=\mathrm{11}$ repetitions of the DualPol (dual-channel PulsePol) sequence. The waiting interval between transients was 18 $\mathrm{s}$ . The total experimental duration was 5 $\min$ . (c)  ${}^{\mathrm{103}}\mathrm{Rh}$ Hdec spectrum, acquired using 16 transients and an optimised refocused-INEPT (Insensitive Nucleus Enhancement by Polarisation Transfer) sequence. The waiting interval between transients was 18 $\mathrm{s}$ . The total experimental duration was 5 $\min$ .

3.2.2. ${}^{\mathrm{1}}\mathrm{H}$ – ${}^{\mathrm{103}}\mathrm{Rh}$ polarisation transfer by DualPol

Polarization transfer from the ${}^{\mathrm{1}}\mathrm{H}$ nuclei to the ${}^{\mathrm{103}}\mathrm{Rh}$ nuclei was performed using the previously described DualPol pulse sequence incorporating acoustic-ringing suppression (), as shown in Fig. .

Figure 4.

Pulse sequence for the acquisition of ${}^{\mathrm{1}}\mathrm{H}$ -enhanced ${}^{\mathrm{103}}\mathrm{Rh}$ spectra; an expanded view of the DualPol pulse sequence module is shown at the bottom. The black rectangles indicate symmetrised BB1 composite 180° pulses () and  $\mathit{\tau }$ indicates an interpulse delay. Phase cycles are given by the following: ${\mathit{\varphi }}_{\mathrm{1}}=[-x,x,-x,x]$ , ${\mathit{\varphi }}_{\mathrm{2}}=[x,x,-x,-x]$ , ${\mathit{\varphi }}_{\mathrm{3}}=[x,x,x,x,y,y,y,y,-x,-x,-x,-x,-y,-y,-y,-y]$ and the receiver ${\mathit{\varphi }}_{\text{rec}}=[x,-x,x,-x,y,-y,y,-y,-x,x,-x,x,-y,y$ , $-y,y]$ .

The DualPol sequence consists of two synchronised PulsePol sequences (), applied simultaneously on two radio frequency channels.

The PulsePol sequence was originally developed in the context of electron–nucleus polarisation transfer (). As discussed in , PulsePol may be interpreted as a “riffled” implementation of an R sequence, using the nomenclature of symmetry-based recoupling in solid-state NMR (). In the case of PulsePol, the R element is a composite ${\mathrm{90}}_y{\mathrm{180}}_x{\mathrm{90}}_y$ pulse, with “windows” inserted between the pulses. Furthermore, in the current implementation, the central ${\mathrm{180}}_x$ pulse of each R element is itself substituted by a BB1 composite pulse (). That substitution was previously shown to increase the robustness of the pulse sequence with respect to deviations in the radio frequency amplitudes and resonance offsets (). The total R-element duration, including all pulses and windows, is denoted using ${\mathit{\tau }}_{\mathrm{R}}$ here (see Fig. ).

For the experiments described here, the DualPol sequences used an R-element duration equal to ${\mathit{\tau }}_{\mathrm{R}}$ $=$ 70 $\mathrm{ms}$ , with pulse durations given by ${\mathit{\tau }}_{\mathrm{90}}$ $=$ 62.5 $\mathrm{\mu }\mathrm{s}$ for the 90° pulses and ${\mathit{\tau }}_{\text{BB1}}$ $=$ 10 $\times$ ${\mathit{\tau }}_{\mathrm{90}}$ $=$ 625 $\mathrm{\mu }\mathrm{s}$ for the BB1 composite 180° pulses.

The ${}^{\mathrm{103}}\mathrm{Rh}$ and methine ${}^{\mathrm{1}}\mathrm{H}$ nuclei of ${}^{\mathrm{103}}\mathrm{Rh}(\mathrm{acac}{)}_{\mathrm{3}}$ form an ${\mathrm{I}}_{\mathrm{3}}\mathrm{S}$ spin system, where the ${}^{\mathrm{103}}\mathrm{Rh}$ nucleus is the S spin and the magnetically equivalent ${}^{\mathrm{1}}\mathrm{H}$ nuclei are the I spins.

The DualPol spin dynamics are identical to those for Hartmann–Hahn J cross-polarisation (). The DualPol average Hamiltonian has the following form:

$${\stackrel{\mathrm{‾}}{H}}^{\left(\mathrm{1}\right)}=\mathit{\kappa }\times \mathrm{2}\mathit{\pi }{J}_{\text{IS}}({\mathrm{I}}_x{S}_x+{\mathrm{I}}_y{S}_y),$$ 1

where the scaling factor is given by $\mathit{\kappa }=\mathrm{1}/\mathrm{2}$ in the limit of short, ideal, radio frequency pulses. In the absence of relaxation and pulse imperfections, a DualPol sequence with a scaling factor $\mathit{\kappa }=\mathrm{1}/\mathrm{2}$ , applied to an ${\mathrm{I}}_{\mathrm{3}}\mathrm{S}$ spin system should give rise to the following enhancement of the S-spin magnetisation, relative to its thermal-equilibrium value:

$$\begin{array}{rl}{\mathit{ϵ}}_{\text{DualPol}}\left(T\right)=& {\displaystyle \frac{{\mathit{\gamma }}_{\mathrm{I}}}{\mathrm{4}{\mathit{\gamma }}_{\mathrm{S}}}}\left\{\mathrm{2}{\sin }^{\mathrm{2}}\left(\frac{\mathrm{1}}{\mathrm{2}}{J}_{\text{IS}}T\right)\right.\\ & \left.+{\sin }^{\mathrm{2}}\left(\mathit{\pi }{J}_{\text{IS}}T\right)+\mathrm{2}{\sin }^{\mathrm{2}}\left(\frac{\mathrm{1}}{\mathrm{2}}\sqrt{\mathrm{3}}\mathit{\pi }{J}_{\text{IS}}T\right)\right\}.\end{array}$$ 2

Here, $T$  is the overall duration of the DualPol sequence. As the magnetogyric ratios of ${}^{\mathrm{103}}\mathrm{Rh}$ and ${}^{\mathrm{1}}\mathrm{H}$ have opposite signs, the function ${\mathit{ϵ}}_{\text{DualPol}}\left(T\right)$ is negative for all values of  $T$ . The blue curve in Fig.  shows a plot of $\mathrm{|}{\mathit{ϵ}}_{\text{DualPol}}\mathrm{|}$ against  $T$ , for a $J$  coupling of $\mathrm{|}{J}_{\text{IS}}\mathrm{|}$ $\simeq$ 0.39 $\mathrm{Hz}$ . The maximum value of $\mathrm{|}{\mathit{ϵ}}_{\text{DualPol}}\mathrm{|}$ is given in the absence of relaxation by

$$\left|{\mathit{ϵ}}_{\text{DualPol}}^{\text{max}}\right|=\mathrm{|}\mathrm{17}{\mathit{\gamma }}_{\mathrm{I}}/\mathrm{16}{\mathit{\gamma }}_{\mathrm{S}}\mathrm{|}\simeq \mathrm{33.56}$$ 3

for the case of I $=$ ${}^{\mathrm{1}}\mathrm{H}$ and S $=$ ${}^{\mathrm{103}}\mathrm{Rh}$ . As Eq. () is quasi-periodic (), the value of  $T$ which maximises $\mathrm{|}{\mathit{ϵ}}_{\text{DualPol}}\mathrm{|}$ is indeterminate. The first maximum may be found numerically and occurs at the duration $T_{\mathrm{1}\text{st max}}$ $\simeq$ $\mathrm{0.6098}{J}_{\text{IS}}^{-\mathrm{1}}$ , at which point the theoretical enhancement is given by $\mathrm{|}{\mathit{ϵ}}_{\text{DualPol}}\mathrm{|}$ $\simeq$ $\mathrm{1.052}\mathrm{|}{\mathit{\gamma }}_{\mathrm{I}}/{\mathit{\gamma }}_{\mathrm{S}}\mathrm{|}$ $\simeq$ 33.23 for the case of I $=$ ${}^{\mathrm{1}}\mathrm{H}$ and S $=$ ${}^{\mathrm{103}}\mathrm{Rh}$ . Hence, for the estimated ${}^{\mathrm{1}}\mathrm{H}$ – ${}^{\mathrm{103}}\mathrm{Rh}$ $J$  coupling of $\mathrm{|}{J}_{\text{IS}}\mathrm{|}\simeq$ 0.39 $\mathrm{Hz}$ , assuming $\mathit{\kappa }=\mathrm{1}/\mathrm{2}$ , the ${}^{\mathrm{103}}\mathrm{Rh}$ signal enhancement is expected to reach its first maximum at a DualPol duration of $T_{\mathrm{1}\text{st max}}$ $\simeq$ 1.563 $\mathrm{s}$ , in the absence of relaxation.

Figure 5.

${}^{\mathrm{103}}\mathrm{Rh}$ signal enhancement factor for ${}^{\mathrm{103}}\mathrm{Rh}(\mathrm{acac}{)}_{\mathrm{3}}$ as a function of DualPol sequence duration  $T$ , normalised against thermal-equilibrium ${}^{\mathrm{103}}\mathrm{Rh}$ polarisation. Black diamonds: experimental data points; solid blue line: the theoretical enhancement factor $\mathrm{|}{\mathit{ϵ}}_{\text{DualPol}}\left(T\right)\mathrm{|}$ for an ${\mathrm{I}}_{\mathrm{3}}\mathrm{S}$ spin system in the absence of relaxation, as given by Eq. () for $\mathrm{|}{J}_{\text{IS}}\mathrm{|}$ $=$ 0.39 $\mathrm{Hz}$ .

In the experiments described here, the optimum duration of the DualPol sequence was found for a repetition number of $n=\mathrm{11}$ . For an R-element duration of ${\mathit{\tau }}_{\mathrm{R}}$ $=$ 70 $\mathrm{ms}$ , this corresponds to a total DualPol sequence duration of $T$ $=$ 1.54 $\mathrm{s}$ , which is in good agreement with the theoretical value.

The DualPol-enhanced ${}^{\mathrm{103}}\mathrm{Rh}$ spectrum is shown in Fig. b and displays an experimental signal enhancement of $\sim$ 23 over the directly excited ${}^{\mathrm{103}}\mathrm{Rh}$ spectrum in Fig. a.

Figure  shows the experimental ${}^{\mathrm{103}}\mathrm{Rh}$ signal enhancement factor as a function of the DualPol sequence duration  $T$ . Although the maximum of the experimental enhancement occurs at a similar position to the maximum of the theoretical curve, there is clearly a strong damping of the enhancement with respect to the duration  $T$ , leading to a loss of intensity at the theoretical maximum. This damping may be associated with transverse relaxation of the ${}^{\mathrm{1}}\mathrm{H}$ and ${}^{\mathrm{103}}\mathrm{Rh}$ transverse magnetisation during the polarisation transfer process.

3.2.3. ${}^{\mathrm{1}}\mathrm{H}$ – ${}^{\mathrm{103}}\mathrm{Rh}$ polarisation transfer by refocused INEPT

Polarisation transfer from ${}^{\mathrm{1}}\mathrm{H}$ to ${}^{\mathrm{103}}\mathrm{Rh}$ may also be conducted by the standard refocused-INEPT pulse sequence . In this case, the theoretical enhancement of the S-spin magnetisation, due to transfer from the I spins, is given for the ${\mathrm{I}}_{\mathrm{3}}\mathrm{S}$ case, in the absence of relaxation and other imperfections, by :

$$\begin{array}{rl}{\mathit{ϵ}}_{\text{RI}}({\mathit{\tau }}_{\mathrm{1}},{\mathit{\tau }}_{\mathrm{2}})=& {\displaystyle \frac{\mathrm{3}{\mathit{\gamma }}_{\mathrm{I}}}{\mathrm{4}{\mathit{\gamma }}_{\mathrm{S}}}}\sin \left(\mathit{\pi }{J}_{\text{IS}}{\mathit{\tau }}_{\mathrm{1}}\right)\\ & \times \left(\sin \left(\mathit{\pi }{J}_{\text{IS}}{\mathit{\tau }}_{\mathrm{2}}\right)+\sin \left(\mathrm{3}\mathit{\pi }{J}_{\text{IS}}{\mathit{\tau }}_{\mathrm{2}}\right)\right),\end{array}$$ 4

where ${\mathit{\tau }}_{\mathrm{1}}$ and ${\mathit{\tau }}_{\mathrm{2}}$  refer to the total echo durations including two inter-pulse intervals, as shown in Fig. 1 of . The maximum of this function is found at ${\mathit{\tau }}_{\mathrm{1}}=(\mathrm{2}{J}_{\text{IS}}{)}^{-\mathrm{1}}$ and ${\mathit{\tau }}_{\mathrm{2}}=\text{arcsin}\left({\mathrm{3}}^{-\mathrm{1}/\mathrm{2}}\right)/\left(\mathit{\pi }{J}_{\text{IS}}\right)$ , giving an enhancement of $\mathrm{|}{\mathit{ϵ}}_{\text{RI}}\mathrm{|}=\mathrm{|}\mathrm{2}{\mathit{\gamma }}_{\mathrm{I}}/\sqrt{\mathrm{3}}{\mathit{\gamma }}_{\mathrm{S}}\mathrm{|}$ (). Therefore, the maximum theoretical enhancement by refocused INEPT is

$$\left|{\mathit{ϵ}}_{\text{RI}}^{\text{max}}\right|=\mathrm{|}\mathrm{2}{\mathit{\gamma }}_{\mathrm{I}}/\sqrt{\mathrm{3}}{\mathit{\gamma }}_{\mathrm{S}}\mathrm{|}\simeq \mathrm{1.155}\mathrm{|}{\mathit{\gamma }}_{\mathrm{I}}/{\mathit{\gamma }}_{\mathrm{S}}\mathrm{|}\simeq \mathrm{36.48}$$ 5

for the case of I $=$ ${}^{\mathrm{1}}\mathrm{H}$ and S $=$ ${}^{\mathrm{103}}\mathrm{Rh}$ . Hence, in the absence of relaxation, refocused INEPT can give a slightly greater enhancement than DualPol in an ${\mathrm{I}}_{\mathrm{3}}\mathrm{S}$ system. However, the maximum enhancements by both DualPol and INEPT are less than the theoretical bound on the enhancement of S-spin magnetisation by polarisation transfer from the I spins in a permutation-symmetric ${\mathrm{I}}_{\mathrm{3}}\mathrm{S}$ spin system, which is equal to $\mathrm{|}\mathrm{3}{\mathit{\gamma }}_{\mathrm{I}}/\mathrm{2}{\mathit{\gamma }}_{\mathrm{S}}\mathrm{|}$ ().

The theoretical advantage of INEPT over DualPol is not realised in practice for the case of ${}^{\mathrm{103}}\mathrm{Rh}(\mathrm{acac}{)}_{\mathrm{3}}$ . The maximum enhancement by refocused INEPT was realised for durations of ${\mathit{\tau }}_{\mathrm{1}}$ $=$ 920 and ${\mathit{\tau }}_{\mathrm{2}}$ $=$ 500 $\mathrm{ms}$ , which yielded a ${}^{\mathrm{103}}\mathrm{Rh}$ enhancement factor of $\sim$ 17 over thermal polarisation, i.e. less than the maximum DualPol enhancement, which was $\sim$ 23. The experimentally optimised interval  ${\mathit{\tau }}_{\mathrm{1}}$ is significantly shorter than the optimum theoretical value in the absence of relaxation, which is ${\mathit{\tau }}_{\mathrm{1}}^{\text{theor}}$ $=$ 1.28 $\mathrm{s}$ , assuming a ${}^{\mathrm{103}}\mathrm{Rh}$ – ${}^{\mathrm{1}}\mathrm{H}$ spin–spin coupling of $J_{\text{HRh}}$ $=$ 0.39 $\mathrm{Hz}$ . The optimum value of  ${\mathit{\tau }}_{\mathrm{2}}$ , on the other hand, is very similar to the theoretical value, which is ${\mathit{\tau }}_{\mathrm{2}}^{\text{theor}}$ $=$ 503 $\mathrm{ms}$ .

The ${}^{\mathrm{1}}\mathrm{H}$ -enhanced ${}^{\mathrm{103}}\mathrm{Rh}$ spectrum, produced by an optimised refocused-INEPT sequence, is shown in Fig. c. It shows a significantly lower enhancement than the DualPol result of Fig. b, despite the fact that the theoretical enhancement by refocused INEPT is higher than that of DualPol in the absence of relaxation (see Eqs.  and ). The loss of amplitude relative to the theoretical values may be attributed to transverse ${}^{\mathrm{1}}\mathrm{H}$ relaxation during the polarisation transfer process. It is known that Hartmann–Hahn-style cross-polarisation sequences such as DualPol can outperform INEPT in the presence of transverse relaxation ().

4. ${}^{\mathrm{1}}\mathrm{H}$ -detected ${}^{\mathrm{103}}\mathrm{Rh}$ $T_{\mathrm{2}}$

The ${}^{\mathrm{103}}\mathrm{Rh}$ $T_{\mathrm{2}}$  relaxation time constant for ${}^{\mathrm{103}}\mathrm{Rh}(\mathrm{acac}{)}_{\mathrm{3}}$ in ${\mathrm{CDCl}}_{\mathrm{3}}$ was measured via the methine ${}^{\mathrm{1}}\mathrm{H}$ signals using a variant of a previously described indirectly detected $T_{\mathrm{2}}$  DualPol pulse sequence (), which is shown in Fig. . The pulse sequence starts with a DualPol sequence of duration $T$ $=$ 1.54 $\mathrm{s}$ to transfer thermal-equilibrium longitudinal ${}^{\mathrm{1}}\mathrm{H}$ magnetisation to ${}^{\mathrm{103}}\mathrm{Rh}$ . The ${}^{\mathrm{103}}\mathrm{Rh}$ longitudinal magnetisation is converted into ${}^{\mathrm{103}}\mathrm{Rh}$ transverse magnetisation by a 90° pulse on the ${}^{\mathrm{103}}\mathrm{Rh}$ channel. The ${}^{\mathrm{103}}\mathrm{Rh}$ transverse magnetisation evolves under a Carr–Purcell–Meiboom–Gill (CPMG) train of $m$  spin echoes, each with an echo duration ${\mathit{\tau }}_{\text{echo}}$ $=$ 45 $\mathrm{ms}$ . The Carr–Purcell sequence suppresses the confounding effects of translational diffusion and mixing with antiphase spin operators (). The ${}^{\mathrm{103}}\mathrm{Rh}$ transverse magnetisation is converted to ${}^{\mathrm{103}}\mathrm{Rh}$ longitudinal magnetisation by a second 90°  ${}^{\mathrm{103}}\mathrm{Rh}$ pulse. A ${}^{\mathrm{1}}\mathrm{H}$ “D-filter” module is applied to destroy any residual ${}^{\mathrm{1}}\mathrm{H}$ magnetisation, before another DualPol sequence of duration $T$ $=$ 1.54 $\mathrm{s}$ transfers the ${}^{\mathrm{103}}\mathrm{Rh}$ longitudinal magnetisation to ${}^{\mathrm{1}}\mathrm{H}$ longitudinal magnetisation. A ${}^{\mathrm{1}}\mathrm{H}$ “z-filter” module is applied to destroy any other ${}^{\mathrm{1}}\mathrm{H}$ magnetisation components, followed by a 90°  ${}^{\mathrm{1}}\mathrm{H}$ pulse which excites ${}^{\mathrm{1}}\mathrm{H}$ transverse magnetisation whose precession induces a ${}^{\mathrm{1}}\mathrm{H}$ NMR signal which is detected in the following interval. The ${}^{\mathrm{1}}\mathrm{H}$ D-filter and z-filter modules are described in Figs. 3 and 4 of , respectively.

Figure 6.

Sequence used for the indirect measurement of rhodium $T_{\mathrm{2}}$ through ${}^{\mathrm{1}}\mathrm{H}$ detection. The phase cycles are given by the following: ${\mathit{\varphi }}_{\mathrm{1}}=[x,x,-x,-x]$ , ${\mathit{\varphi }}_{\mathrm{2}}=[-x,x,-x,x]$ , ${\mathit{\varphi }}_{\mathrm{3}}=[x,x,x,x,y,y,y,y,-x,-x,-x,-x,-y,-y,-y,-y]$ and the receiver ${\mathit{\varphi }}_{\text{rec}}=[x,-x,-x,x,y,-y,-y,y,-x,x,x,-x,-y,y,y,-y]$ . The echo interval ${\mathit{\tau }}_{\text{echo}}$ was 45 $\mathrm{ms}$ . The black rectangle indicates a symmetrised BB1 composite 180° pulse (). An MLEV-64 supercycle was applied to the phases of the 180° pulses (). The ${}^{\mathrm{1}}\mathrm{H}$ “D-filter” and “z-filter” modules are described in Figs. 3 and 4 of , respectively.

Repetition of the experiment with increasing values of  $m$ leads to the ${}^{\mathrm{103}}\mathrm{Rh}$ $T_{\mathrm{2}}$  decay curve shown in Fig. . This fits well to a single exponential decay with the time constant $T_{\mathrm{2}}{(}^{\mathrm{103}}\text{Rh})$ $=$ 18.36 $\pm$ 0.92 $\mathrm{s}$ .

Figure 7.

Decay curve for the ${}^{\mathrm{103}}\mathrm{Rh}$ transverse magnetisation of ${}^{\mathrm{103}}\mathrm{Rh}(\mathrm{acac}{)}_{\mathrm{3}}$ in solution at a field of 9.4 $\mathrm{T}$ , measured by the indirectly detected multiple spin-echo scheme in Fig. . The experimental duration was 15 $\min$ .

5. ${}^{\mathrm{1}}\mathrm{H}$ -detected ${}^{\mathrm{103}}\mathrm{Rh}$ $T_{\mathrm{1}}$

The ${}^{\mathrm{103}}\mathrm{Rh}$ $T_{\mathrm{1}}$  relaxation time constant for ${}^{\mathrm{103}}\mathrm{Rh}(\mathrm{acac}{)}_{\mathrm{3}}$ in ${\mathrm{CDCl}}_{\mathrm{3}}$ was measured indirectly using the methine ${}^{\mathrm{1}}\mathrm{H}$ signals, by means of the pulse sequence shown in Fig.  (). The pulse sequence starts with a DualPol sequence of duration $T$ $=$ 1.54 $\mathrm{s}$ to transfer thermal-equilibrium longitudinal ${}^{\mathrm{1}}\mathrm{H}$ magnetisation to ${}^{\mathrm{103}}\mathrm{Rh}$ . For variable-field experiments, the sample is shuttled out of the high-field magnet into a low-field region. The nuclear magnetisation is allowed to relax for an interval ${\mathit{\tau }}_{\text{relax}}$ . If necessary, the sample is shuttled back into high field, and residual ${}^{\mathrm{1}}\mathrm{H}$ magnetisation is destroyed by a ${}^{\mathrm{1}}\mathrm{H}$ D-filter module. A pair of phase-cycled 90° ${}^{\mathrm{103}}\mathrm{Rh}$ pulses are applied to select for ${}^{\mathrm{103}}\mathrm{Rh}$ z-magnetisation before a second DualPol sequence of duration $T$ $=$ 1.54 $\mathrm{s}$ transfers the partially relaxed longitudinal ${}^{\mathrm{103}}\mathrm{Rh}$ magnetisation to ${}^{\mathrm{1}}\mathrm{H}$ magnetisation. A ${}^{\mathrm{1}}\mathrm{H}$ z-filter module is applied to destroy any other ${}^{\mathrm{1}}\mathrm{H}$ magnetisation components, followed by a 90°  ${}^{\mathrm{1}}\mathrm{H}$ pulse which excites ${}^{\mathrm{1}}\mathrm{H}$ transverse magnetisation whose precession induces a ${}^{\mathrm{1}}\mathrm{H}$ NMR signal that is detected in the following interval. The ${}^{\mathrm{1}}\mathrm{H}$ D-filter and z-filter modules are described in Figs. 3 and 4 of , respectively. During ${\mathit{\tau }}_{\text{relax}}$ , ${}^{\mathrm{1}}\mathrm{H}$ -enhanced ${}^{\mathrm{103}}\mathrm{Rh}$ magnetisation decays toward thermal ${}^{\mathrm{103}}\mathrm{Rh}$ magnetisation, which is very small; hence, at large values of ${\mathit{\tau }}_{\text{relax}}$ , resulting ${}^{\mathrm{103}}\mathrm{Rh}$ -derived ${}^{\mathrm{1}}\mathrm{H}$ signals are very weak and close to zero, even for measurements performed at higher magnetic field strengths.

Figure 8.

Sequence used for the indirect measurement of the rhodium  $T_{\mathrm{1}}$ through the ${}^{\mathrm{1}}\mathrm{H}$ NMR signal. Phase cycles are given by the following: ${\mathit{\varphi }}_{\mathrm{1}}=[x,x,-x,-x]$ , ${\mathit{\varphi }}_{\mathrm{2}}=[-x,x,-x,x]$ , ${\mathit{\varphi }}_{\mathrm{3}}=[x,x,x,x,y,y,y,y,-x,-x,-x,-x,-y,-y,-y,-y]$ and the receiver ${\mathit{\varphi }}_{rec}=[x,-x,-x,x,y,-y,-y,y,-x,x,x,-x,-y,y,y,-y]$ . The optional shuttling of the sample to low field, and back again, during the interval ${\mathit{\tau }}_{\text{relax}}$ is indicated. The ${}^{\mathrm{1}}\mathrm{H}$ “D-filter” and “z-filter” modules are described in Figs. 3 and 4 of , respectively.

The trajectory of indirectly detected ${}^{\mathrm{103}}\mathrm{Rh}$ z-magnetisation in a field of 9.4 $\mathrm{T}$ and at a temperature of 295 $\mathrm{K}$ is shown in Fig. a. The trajectory fits well to a single exponential decay with time constant $T_{\mathrm{1}}\left({}^{\mathrm{103}}\mathrm{Rh}\right)$ $=$ 41.8 $\pm$ 0.9 $\mathrm{s}$ . A trajectory in the low magnetic field of 10 $\mathrm{mT}$ and at a temperature of 295 $\mathrm{K}$ is shown in Fig. b. This was produced by shuttling the sample to a low magnetic field, and back again, during the interval ${\mathit{\tau }}_{\text{relax}}$ . The relaxation process is somewhat slower in a low magnetic field, with a time constant of $T_{\mathrm{1}}\left({}^{\mathrm{103}}\mathrm{Rh}\right)$ $=$ 57.8 $\pm$ 1.7 $\mathrm{s}$ .

Figure 9.

Trajectories of longitudinal ${}^{\mathrm{103}}\mathrm{Rh}$ magnetisation for ${}^{\mathrm{103}}\mathrm{Rh}(\mathrm{acac}{)}_{\mathrm{3}}$ in solution at a temperature of 295 $\mathrm{K}$ , measured indirectly through the methine ${}^{\mathrm{1}}\mathrm{H}$ signals, using the pulse sequence in Fig. . (a) Filled symbols: ${}^{\mathrm{1}}\mathrm{H}$ signal amplitudes at a magnetic field of 9.4 $\mathrm{T}$ . The data were acquired in $\sim$ 2 $\mathrm{h}$ . The integrals are normalised against the ${}^{\mathrm{1}}\mathrm{H}$ spectrum obtained by a single ${}^{\mathrm{1}}\mathrm{H}$ 90° pulse applied to a system in thermal equilibrium at 9.4 $\mathrm{T}$ and at 295 $\mathrm{K}$ . Dotted line: fitted exponential decay with time constant $T_{\mathrm{1}}\left({}^{\mathrm{103}}\mathrm{Rh}\right)$ $=$ 41.8 $\pm$ 0.9 $\mathrm{s}$ . Panel (b) is the same as panel (a) but the sample is shuttled to a field of 10 $\mathrm{mT}$ during the relaxation delay ${\mathit{\tau }}_{\text{relax}}$ . Dotted line: fitted exponential decay with time constant $T_{\mathrm{1}}\left({}^{\mathrm{103}}\mathrm{Rh}\right)$ $=$ 57.8 $\pm$ 1.7 $\mathrm{s}$ .

The observed field dependence of the ${}^{\mathrm{103}}\mathrm{Rh}$ relaxation rate constant $T_{\mathrm{1}}^{-\mathrm{1}}$ is shown in Fig. a. The magnetic field dependence of $T_{\mathrm{1}}^{-\mathrm{1}}$ is quite weak in this range of fields. The relaxation rate constant increases slightly with increasing magnetic field at the high-field end, suggestive of a weak relaxation contribution from the ${}^{\mathrm{103}}\mathrm{Rh}$ chemical shift anisotropy. The blue curve in Fig. a shows the best-fit quadratic function $T_{\mathrm{1}}^{-\mathrm{1}}\left(B\right)=T_{\mathrm{1}}^{-\mathrm{1}}\left(\mathrm{0}\right)+a{B}^{\mathrm{2}}$ , where $T_{\mathrm{1}}^{-\mathrm{1}}\left(\mathrm{0}\right)$ $=$ (167 $\pm$ 7) $\times$ 10 ${}^{-\mathrm{4}}$ ${\mathrm{s}}^{-\mathrm{1}}$ and $a$ $=$ (7 $\pm$ 2) $\times$ 10 ${}^{-\mathrm{5}}$ ${\mathrm{s}}^{-\mathrm{1}}{\mathrm{T}}^{-\mathrm{2}}$ .

Figure 10.

(a)  ${}^{\mathrm{103}}\mathrm{Rh}$ relaxation rate constant $T_{\mathrm{1}}^{-\mathrm{1}}$ for ${}^{\mathrm{103}}\mathrm{Rh}(\mathrm{acac}{)}_{\mathrm{3}}$ in solution, as a function of magnetic field strength at a temperature of 295 $\mathrm{K}$ . The blue line shows the best-fit quadratic function $T_{\mathrm{1}}^{-\mathrm{1}}\left(B\right)=T_{\mathrm{1}}^{-\mathrm{1}}\left(\mathrm{0}\right)+a{B}^{\mathrm{2}}$ , where $T_{\mathrm{1}}^{-\mathrm{1}}\left(\mathrm{0}\right)$ $=$ (167 $\pm$ 7) $\times$ 10 ${}^{-\mathrm{4}}$ ${\mathrm{s}}^{-\mathrm{1}}$ and $a$ $=$ (7 $\pm$ 2) $\times$ 10 ${}^{-\mathrm{5}}$ ${\mathrm{s}}^{-\mathrm{1}}{\mathrm{T}}^{-\mathrm{2}}$ . (b)  ${}^{\mathrm{103}}\mathrm{Rh}$ relaxation rate constant $T_{\mathrm{1}}^{-\mathrm{1}}$ for ${}^{\mathrm{103}}\mathrm{Rh}(\mathrm{acac}{)}_{\mathrm{3}}$ in solution, as a function of temperature at a magnetic field strength of 9.4 $\mathrm{T}$ .

The observed temperature dependence of the ${}^{\mathrm{103}}\mathrm{Rh}$ relaxation rate constant $T_{\mathrm{1}}^{-\mathrm{1}}$ is shown for a field of $B$ $\simeq$ 9.4 $\mathrm{T}$ in Fig. b. The rhodium $T_{\mathrm{1}}^{-\mathrm{1}}$ increases monotonically with increasing temperature over the relevant temperature range. At 315 $\mathrm{K}$ , relaxation occurs with a time constant of $T_{\mathrm{1}}\left({}^{\mathrm{103}}\mathrm{Rh}\right)$ $=$ 30.6 $\pm$ 1.1 $\mathrm{s}$ .

A positive dependence of the ${}^{\mathrm{103}}\mathrm{Rh}$ $T_{\mathrm{1}}^{-\mathrm{1}}$ on temperature was reported previously for ${}^{\mathrm{103}}\mathrm{Rh}(\mathrm{acac}{)}_{\mathrm{3}}$ in solution ().

6. Discussion

The temperature dependence of the ${}^{\mathrm{103}}\mathrm{Rh}$ $T_{\mathrm{1}}^{-\mathrm{1}}$ , as shown in Fig. b, indicates a dominant spin–rotation relaxation mechanism. For small molecules with a short rotational correlation time relative to the nuclear Larmor period, spin–rotation is the only mechanism that leads to a positive correlation of $T_{\mathrm{1}}^{-\mathrm{1}}$ with temperature (). This is because the amplitudes of the local magnetic fields generated by the spin–rotation interaction are proportional to the root-mean-square rotational angular momentum of the participating molecules – a quantity that is linked to the mean rotational kinetic energy of the molecules, which increases linearly with temperature. For other mechanisms, the decrease in the rotational correlation time ${\mathit{\tau }}_c$ with increasing temperature leads to a decrease in the relaxation rate with increasing temperature, in the fast-motion limit.

The field dependence of the ${}^{\mathrm{103}}\mathrm{Rh}$ $T_{\mathrm{1}}^{-\mathrm{1}}$ , as shown in Fig. a, displays a modest increase in relaxation rate with increasing magnetic field at high field, which suggests an additional contribution from the rotational modulation of the ${}^{\mathrm{103}}\mathrm{Rh}$ CSA tensor. A finite ${}^{\mathrm{103}}\mathrm{Rh}$ CSA tensor is allowed with respect to symmetry under the ${\mathrm{D}}_{\mathrm{3}}$ point group of ${}^{\mathrm{103}}\mathrm{Rh}(\mathrm{acac}{)}_{\mathrm{3}}$ (). Indeed, solid-state NMR data indicate a ${}^{\mathrm{103}}\mathrm{Rh}$ shielding anisotropy of $\mathrm{\Delta }\mathit{\sigma }$ $\simeq$ 460 $\mathrm{ppm}$ , with relativistic quantum chemistry calculations in reasonable agreement (). The magnitude of this CSA tensor is modest by ${}^{\mathrm{103}}\mathrm{Rh}$ standards. For example, the ${}^{\mathrm{103}}\mathrm{Rh}$ nuclei in Rh paddlewheel complexes have a shielding anisotropy of $\mathrm{|}\mathrm{\Delta }\mathit{\sigma }\mathrm{|}$ $\sim$ 9900 $\mathrm{ppm}$ ().

In summary, we have demonstrated the successful transfer of polarisation between the central ${}^{\mathrm{103}}\mathrm{Rh}$ nucleus and the three methine ${}^{\mathrm{1}}\mathrm{H}$ nuclei in ${}^{\mathrm{103}}\mathrm{Rh}(\mathrm{acac}{)}_{\mathrm{3}}$ , through the very small four-bond ${}^{\mathrm{1}}\mathrm{H}$ – ${}^{\mathrm{103}}\mathrm{Rh}$ couplings. The polarisation transfer is more efficient for DualPol than for refocused INEPT, even though the theoretical efficiency of DualPol is slightly less than that of refocused INEPT, for the relevant ${\mathrm{I}}_{\mathrm{3}}\mathrm{S}$ spin system. We have successfully exploited ${}^{\mathrm{1}}\mathrm{H}$ – ${}^{\mathrm{103}}\mathrm{Rh}$ polarisation transfer to study the longitudinal and transverse relaxation of ${}^{\mathrm{103}}\mathrm{Rh}$ for ${}^{\mathrm{103}}\mathrm{Rh}(\mathrm{acac}{)}_{\mathrm{3}}$ in solution. The ${}^{\mathrm{103}}\mathrm{Rh}$ $T_{\mathrm{1}}$  relaxation is dominated by the spin–rotation mechanism, with a significant additional contribution from the ${}^{\mathrm{103}}\mathrm{Rh}$ CSA at a high magnetic field.

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Code availability

The software code for the graphics shown in this paper is available from the authors on reasonable request.

Data availability

The dataset can be accessed at 10.5258/SOTON/D3209 ().

Author contributions

HHC: conceptualisation (equal), data curation (equal), formal analysis (equal), investigation (equal), methodology (equal), software (equal), validation (equal), visualisation (equal), and writing (original draft and review and editing) (equal). MS: conceptualisation (equal), data curation (equal), formal analysis (equal), investigation (equal), methodology (equal), software (equal), visualisation (equal), and writing (original draft and review and editing) (equal). ML: conceptualisation (equal), funding acquisition (equal), investigation (equal), validation (equal), and writing (review and editing) (equal). MHL: conceptualisation (equal), formal analysis (equal), funding acquisition (equal), investigation (equal), project administration (equal), resources (equal), supervision (lead), and writing (original draft and review and editing) (equal).

Competing interests

At least one of the (co-)authors is a member of the editorial board of Magnetic Resonance. The peer-review process was guided by an independent editor, and the authors also have no other competing interests to declare.

Review statement

This paper was edited by Patrick Giraudeau and reviewed by two anonymous referees.