Accurate rotational correlation times (\({\tau }_{\text{c}}\)) are critical for quantitative analysis of fast timescale NMR dynamics. As molecular weights increase, the classic derivation of \({\tau }_{c}\) using transverse and longitudinal relaxation rates becomes increasingly unsuitable due to the non-trivial contribution of remote dipole–dipole interactions to longitudinal relaxation. Derivations using cross-correlated relaxation experiments, such as TRACT, overcome these limitations but are erroneously calculated in 65% of the citing literature. Herein, we developed an algebraic solutions to the Goldman relationship that facilitate rapid, point-by-point calculations for straightforward identification of appropriate spectral regions where global tumbling is likely to be dominant. The rigid-body approximation of the Goldman relationship has been previously shown to underestimate TRACT-based rotational correlation time estimates. This motivated us to develop a second algebraic solution that employs a simplified model-free spectral density function including an order parameter term that could, in principle, be set to an average backbone S² ≈ 0.9 to further improve the accuracy of \({\tau }_{\text{c}}\) estimation. These solutions enabled us to explore the boundaries of the Goldman relationship as a function of the H–N internuclear distance (\(r\)), difference of the two principal components of the axially-symmetric ¹⁵N CSA tensor (\(\Delta {\delta }_{N}\)), and angle of the CSA tensor relative to the N–H bond vector (\(\theta\)). We hope our algebraic solutions and analytical strategies will increase the accuracy and application of the TRACT experiment.

准确的旋转相关时间 ( \({\tau }_{\text{c}}\) ) 对于快速时间尺度 NMR 动力学的定量分析至关重要。随着分子量的增加，\({\tau }_{c}\)的经典推导由于远程偶极 - 偶极相互作用对纵向弛豫的重要贡献，使用横向和纵向弛豫率变得越来越不合适。使用交叉相关松弛实验（如 TRACT）的推导克服了这些限制，但在 65% 的引用文献中计算错误。在这里，我们为 Goldman 关系开发了一种代数解决方案，该解决方案有助于快速、逐点计算，以便直接识别全局翻滚可能占主导地位的适当光谱区域。Goldman 关系的刚体近似先前已被证明低估了基于 TRACT 的旋转相关时间估计。² ≈ 0.9 以进一步提高\({\tau }_{\text{c}}\)估计的准确性。这些解决方案使我们能够探索作为 H-N 核间距离 ( \(r\) ) 函数的 Goldman 关系的边界，轴对称¹⁵ N CSA 张量 ( \(\Delta {\delta }_{N}\)），以及 CSA 张量相对于 N-H 键矢量的角度（\(\theta\)）。我们希望我们的代数解决方案和分析策略能够提高 TRACT 实验的准确性和应用性。