Kästner Johannes
Institute for Theoretical Chemistry, University of Stuttgart, Pfaffenwaldring 55, D-70569 Stuttgart, Germany.
J Chem Phys. 2009 Jul 21;131(3):034109. doi: 10.1063/1.3175798.
Umbrella integration is a method to analyze umbrella sampling simulations by calculating and integrating the mean force. Here, the method is extended to multidimensional reaction coordinates. Approximation of the probability distribution obtained from sampling by a multivariate normal distribution allows to calculate the mean force from the average and the covariance matrix of the reaction coordinate. Integration schemes of the free-energy gradient field are discussed. Integration on a real-space grid is compared to expansion of the gradient in a series of analytic functions (such as a Fourier analysis), which can be integrated, and the expansion of the gradient only at the window means in a series of analytic functions. The Fourier analysis was found particularly useful for periodic reaction coordinates, such as torsion angles. An expression is provided to calculate the Hessian of the free energy with respect to the reaction coordinates from sampling data. The utility of the method is demonstrated at the example of the free-energy surface of the alanine dipeptide in vacuum calculated with respect to the backbone torsion angles Phi and Psi. Relevance of the Jacobian term for non-Cartesian reaction coordinates is discussed.
伞形积分是一种通过计算和积分平均力来分析伞形采样模拟的方法。在此，该方法被扩展到多维反应坐标。通过多元正态分布对采样得到的概率分布进行近似，从而能够根据反应坐标的平均值和协方差矩阵来计算平均力。讨论了自由能梯度场的积分方案。将实空间网格上的积分与梯度在一系列可积分的解析函数（如傅里叶分析）中的展开以及仅在窗口均值处将梯度展开为一系列解析函数进行了比较。发现傅里叶分析对于周期性反应坐标（如扭转角）特别有用。提供了一个表达式，用于根据采样数据计算自由能相对于反应坐标的海森矩阵。以真空中丙氨酸二肽相对于主链扭转角Phi和Psi计算的自由能表面为例，展示了该方法的实用性。讨论了雅可比项对于非笛卡尔反应坐标的相关性。
