具有平均场嵌入的QM/MM自由能的变分和微扰公式及其解析梯度
Variational and perturbative formulations of QM/MM free energy with mean-field embedding and its analytical gradients
摘要：
传统的量子化学溶剂化理论是基于平均场嵌入近似。也就是说，电子波函数是在存在环境平均场的情况下计算的。在本文中，一个直接的量子力学/分子力学（QM/MM）模拟这样的平均场论制定的基础上变分和微扰框架。在变分的框架中，一个适当的QM/MM自由能泛函的定义和最小化的审判波函数，最好的近似真正的QM波函数在统计平均意义上。得到了解析自由能梯度，其形式为在平均MM势下计算的有效QM能梯度。在微扰框架下，上述变分过程被证明与自洽参考场的QM能量（在精确自由能表达式中）的一阶展开是等价的。这有助于理解变分过程和精确的QM/MM自由能以及现有的QM/MM理论之间的关系。在此基础上，讨论了几种评估非平均场效应的方法（即，QM波函数的统计波动），其在平均场计算中被忽略。作为说明，该方法被应用于在水中的SN 2 Menshutkin反应，NH 3 + CH 3 CL-> NH 3 CH 3 ^{+} + CL^{-}，对于该反应，通过对自由能梯度进行积分，在HF、MP2、B3 LYP和BH&HLYP水平上获得自由能分布。使用环境的高斯涨落模型，非平均场效应被评估为< 0.5 kcal/mol，这表明这些效应对于目前在水中的反应是相当小的。
英文摘要：
Conventional quantum chemical solvation theories are based on the mean-field embedding approximation. That is, the electronic wavefunction is calculated in the presence of the mean field of the environment. In this paper a direct quantum mechanical/molecular mechanical (QM/MM) analog of such a mean-field theory is formulated based on variational and perturbative frameworks. In the variational framework, an appropriate QM/MM free energy functional is defined and is minimized in terms of the trial wavefunction that best approximates the true QM wavefunction in a statistically averaged sense. Analytical free energy gradient is obtained, which takes the form of the gradient of effective QM energy calculated in the averaged MM potential. In the perturbative framework, the above variational procedure is shown to be equivalent with the first-order expansion of the QM energy (in the exact free energy expression) about the self-consistent reference field. This helps understand the relation between the variational procedure and the exact QM/MM free energy as well as existing QM/MM theories. Based on this, several ways are discussed for evaluating non-mean-field effects (i.e., statistical fluctuations of the QM wavefunction) that are neglected in the mean-field calculation. As an illustration, the method is applied to an SN2 Menshutkin reaction in water, NH3 + CH3CL -> NH3CH3^{+} + CL^{-}, for which free energy profiles are obtained at the HF, MP2, B3LYP, and BH&HLYP levels by integrating the free energy gradient. Non-mean-field effects are evaluated to be < 0.5 kcal/mol using a Gaussian fluctuation model for the environment, which suggests that those effects are rather small for the present reaction in water.
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