Department of Chemistry and Francis Bitter Magnet Laboratory, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139, USA.
J Chem Phys. 2011 Oct 7;135(13):134509. doi: 10.1063/1.3635374.
Proton-driven spin diffusion (PDSD) experiments in rotating solids have received a great deal of attention as a potential source of distance constraints in large biomolecules. However, the quantitative relationship between the molecular structure and observed spin diffusion has remained obscure due to the lack of an accurate theoretical description of the spin dynamics in these experiments. We start with presenting a detailed relaxation theory of PDSD in rotating solids that provides such a description. The theory applies to both conventional and radio-frequency-assisted PDSD experiments and extends to the non-Markovian regime to include such phenomena as rotational resonance (R(2)). The basic kinetic equation of the theory in the non-Markovian regime has the form of a memory function equation, with the role of the memory function played by the correlation function. The key assumption used in the derivation of this equation expresses the intuitive notion of the irreversible dissipation of coherences in macroscopic systems. Accurate expressions for the correlation functions and for the spin diffusion constants are given. The theory predicts that the spin diffusion constants governing the multi-site PDSD can be approximated by the constants observed in the two-site diffusion. Direct numerical simulations of PDSD dynamics via reversible Liouville-von Neumann equation are presented to support and compliment the theory. Remarkably, an exponential decay of the difference magnetization can be observed in such simulations in systems consisting of only 12 spins. This is a unique example of a real physical system whose typically macroscopic and apparently irreversible behavior can be traced via reversible microscopic dynamics. An accurate value for the spin diffusion constant can be usually obtained through direct simulations of PDSD in systems consisting of two (13)C nuclei and about ten (1)H nuclei from their nearest environment. Spin diffusion constants computed by this method are in excellent agreement with the spin diffusion constants obtained through equations given by the relaxation theory of PDSD. The constants resulting from these two approaches were also in excellent agreement with the results of 2D rotary resonance recoupling proton-driven spin diffusion (R(3)-PDSD) experiments performed in three model compounds, where magnetization exchange occurred over distances up to 4.9 Å. With the methodology presented, highly accurate internuclear distances can be extracted from such data. Relayed transfer of magnetization between distant nuclei appears to be the main (and apparently resolvable) source of uncertainty in such measurements. The non-Markovian kinetic equation was applied to the analysis of the R(2) spin dynamics. The conventional semi-phenomenological treatment of relxation in R(2) has been shown to be equivalent to the assumption of the Lorentzian spectral density function in the relaxatoin theory of PDSD. As this assumption is a poor approximation in real physical systems, the conventional R(2) treatment is likely to carry a significant model error that has not been recognized previously. The relaxation theory of PDSD appears to provide an accurate, parameter-free alternative. Predictions of this theory agreed well with the full quantum mechanical simulations of the R(2) dynamics in the few simple model systems we considered.
质子驱动的自旋扩散（PDSD）实验在旋转固体中受到了极大的关注，因为它可能是生物大分子中距离约束的潜在来源。然而，由于缺乏对这些实验中自旋动力学的精确理论描述，分子结构与观察到的自旋扩散之间的定量关系仍然不清楚。我们首先提出了一种详细的旋转固体中 PDSD 的弛豫理论，该理论提供了这样的描述。该理论适用于传统和射频辅助的 PDSD 实验，并扩展到非马尔可夫区域，包括旋转共振（R(2)）等现象。该理论在非马尔可夫区域的基本动力学方程采用记忆函数方程的形式，其中记忆函数由相关函数扮演。该方程的推导中使用的关键假设表达了在宏观系统中相干不可逆耗散的直观概念。给出了相关函数和自旋扩散常数的精确表达式。该理论预测，控制多点 PDSD 的自旋扩散常数可以通过在两点扩散中观察到的常数来近似。通过可逆李代数-冯·诺依曼方程对 PDSD 动力学进行的直接数值模拟支持并补充了该理论。值得注意的是，在由仅 12 个自旋组成的系统中，可以在这种模拟中观察到差异磁化的指数衰减。这是一个真实物理系统的独特示例，其典型的宏观和显然不可逆的行为可以通过可逆的微观动力学来追踪。通过直接模拟由两个（13）C 核和大约十个（1）H 核及其最近环境组成的系统中的 PDSD，通常可以获得自旋扩散常数的精确值。通过这种方法计算的自旋扩散常数与 PDSD 弛豫理论给出的方程计算的自旋扩散常数非常吻合。通过这两种方法得到的常数与在三个模型化合物中进行的二维旋转共振再耦质子驱动自旋扩散（R(3)-PDSD）实验的结果非常吻合，其中在 4.9 Å 以内的距离上发生了磁化交换。通过所提出的方法，可以从这些数据中提取出高度精确的核间距离。在这些测量中，似乎主要（且显然可分辨）的不确定性来源是远距离核之间的磁化传递。非马尔可夫动力学方程被应用于 R(2)自旋动力学的分析。已经表明，R(2)中松弛的传统半唯象处理与 PDSD 弛豫理论中的洛伦兹谱密度函数的假设等效。由于该假设在实际物理系统中是一个较差的近似，因此传统的 R(2)处理可能存在尚未认识到的显著模型误差。PDSD 弛豫理论似乎提供了一种准确、无参数的替代方法。该理论的预测与我们考虑的少数几个简单模型系统中 R(2)动力学的全量子力学模拟吻合良好。
