Calculations of free energy profile, aka potential of mean force (PMF), along a chosen collective variable (CV) are now routinely applied to the studies of chemical processes, such as enzymatic reactions and chemical reactions in condensed phases. However, if the ab initio QM/MM level of accuracy is required for the PMF, it can be formidably expensive even with the most advanced enhanced sampling methods, such as umbrella sampling. To ameliorate this difficulty, we developed a novel method for the computation of free energy profile based on the reference-potential method recently, in which a low-level reference Hamiltonian is employed for phase space sampling and the free energy profile can be corrected to the level of interest (the target Hamiltonian) by energy reweighting in a nonparametric way. However, when the reference Hamiltonian is very different from the target Hamiltonian, the calculated ensemble averages, including the PMF, often suffer from numerical instability, which mainly comes from the overestimation of the density-of-states (DoS) in the low-energy region. Stochastic samplings of these low-energy configurations are rare events. If a low-energy configuration has been sampled with a small sample size N, the probability of visiting this energy region is ~ 1/N (shall be exactly 1/N for a single ensemble), which can be orders-of-magnitude larger than the actual DoS. In this work, an assumption of Gaussian distribution is applied to the DoS in each CV bin, and the weight of each configuration is rescaled according to the accumulated DoS. The results show that this smoothing process can remarkably reduce the ruggedness of the PMF and increase the reliability of the reference-potential method.

中文翻译：

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通过半经验参考电势从头算QM / MM精度加速计算自由能曲线。三，状态密度的高斯平滑

现在，通常将自由能曲线，也就是平均力势（PMF）沿着选定的集体变量（CV）的计算应用于化学过程的研究，例如酶促反应和缩合阶段的化学反应。但是，如果PMF需要从头开始QM / MM精度，那么即使使用最先进的增强采样方法（例如伞状采样），也可能会非常昂贵。为了缓解这一困难，我们最近在参考电位方法的基础上开发了一种计算自由能分布的新方法，其中将低级参考哈密顿量用于相空间采样，并且可以将自由能分布校正为通过以非参数方式进行能量加权来确定感兴趣的水平（目标汉密尔顿）。然而，当参考哈密顿量与目标哈密顿量有很大不同时，包括PMF在内的计算集合平均数通常会出现数值不稳定性，这主要是由于低能量区域的状态密度（DoS）过高估计造成的。这些低能耗配置的随机采样是罕见的事件。如果已经以较小的样本量N采样了低能量配置，则访问该能量区域的概率约为1 / N（对于单个集合，恰好是1 / N），这可能是数量级大。比实际的DoS。在这项工作中，将高斯分布的假设应用于每个CV bin中的DoS，并根据累积的DoS重新调整每个配置的权重。