Random fluctuations in the \(\tilde{g}\) and \(\tilde{A}\) matrices of a spin system due to thermal motion of a molecule are specifically considered to calculate the relaxation matrix for the four-level electron-nuclear spin-coupled system (\(S = 1/2\); \(I = 1/2\)) in a malonic acid crystal, using the formalism outlined by Lee et al. (J Chem Phys 98:3665–3689, 1993). The correlation time, τ_c, and the value of the parameter λ, characterizing the harmonic-oscillator restoring potential of the small-amplitude fluctuation of the director of the malonic-acid molecule due to thermal motion are estimated from the knowledge of the experimental values of (τ_c, λ). The four electronic, \(T_{2e} ,\) and the two nuclear, \(T_{{2{\text{n}}}}\), spin relaxation times are calculated to be functions of (τ_c, λ) governing the fluctuations. The values of (τ_c, λ), evaluated with these expressions, when fitted to the experimental values of \(T_{2e}\) and \(T_{2n}\), assuming the molecule to be in the ground state (n = 0) in the harmonic-oscillator potential, a rather narrow region of (τ_c, λ) values about τ_c = 0.081 μs and \(\lambda = 4.4\) is found. These values are then used to calculate the time-dependent echo-ELDOR signal by the relevant Liouville von-Neumann (LVN) equation. The resulting Fourier transform is found to be in excellent agreement with the experimental data. The (τ_c, λ) values for the excited states described by \(n = 1,2\) have also been calculated, although these states are unlikely to be populated at room temperature.

自旋系统的\（\ tilde {g} \）和\（\ tilde {A} \）矩阵中由于分子的热运动而引起的随机波动被特别考虑为计算四能级电子的弛豫矩阵丙二酸晶体中的核自旋耦合系统（\（S = 1/2 \） ; \（I = 1/2 \）），使用Lee等人概述的形式。（J Chem Phys 98：3665-3689，​​1993）。相关时间，τ _Ç，并且该参数的值λ，表征谐波振荡器恢复丙二酸分子的导向器的小振幅波动的电位由于热运动从实验值的知识估计的（τ _ç，λ）。四个电子，\（T_ {2E}，\）和两个核，\（T _ {{2- {\文本{N}}}} \） ，自旋弛豫时间计算为的（函数τ _Ç，λ）控制波动。的值（τ _Ç，λ），用这些表达式求值，当装配到的实验值\（T_ {2E} \）和\（T_ {2N} \） ，假设分子为处于基态（ñ  = 0）在谐波振荡器电位的相当窄的区域（τ _ç，λ）值大约τ _ç= 0.081μs，并且找到\（\ lambda = 4.4 \）。然后，将这些值用于通过相关的Liouville von-Neumann（LVN）方程来计算时间相关的回声ELDOR信号。发现所得的傅里叶变换与实验数据非常吻合。的（τ _Ç，λ），用于通过所述的激发态值\（N = 1,2 \）也被计算出，尽管这些状态是不太可能在室温下进行填充。