Theodorou Doros N
School of Chemical Engineering, National Technical University of Athens, 9 Heroon Polytechniou Street, Zografou Campus, 157 80 Athens, Greece.
J Chem Phys. 2006 Jan 21;124(3):034109. doi: 10.1063/1.2138701.
A general method is introduced for the calculation of the free-energy difference between two systems, 0 and 1, with configuration spaces omega(0), omega(1) of the same dimensionality. The method relies upon establishing a objective mapping between disjoint subsets gamma(i)(0) of omega(0) and corresponding disjoint subsets gamma(i)(1) of omega(1), and averaging a function of the ratio of configurational integrals over gamma(i)(0) and gamma(i)(1) with respect to the probability densities of the two systems. The mapped subsets gamma(i)(0) and gamma(i)(1) need not span the entire configuration spaces omega(0) and omega(1). The method is applied for the calculation of the excess chemical potential mu(ex) in a Lennard-Jones (LJ) fluid. In this case, omega(0) is the configuration space of a (N-1) real molecule plus one ideal-gas molecule system, while omega(1) is the configuration space of a N real molecule system occupying the same volume. Gamma(i)(0) and gamma(i)(1) are constructed from hyperspheres of the same radius centered at minimum-energy configurations of a set of "active" molecules lying within distance a from the ideal-gas molecule and the last real molecule, respectively. An algorithm is described for sampling gamma(i)(0) and gamma(i)(1) given a point in omega(0) or in omega(1). The algorithm encompasses three steps: "quenching" (minimization with respect to the active-molecule degrees of freedom), "mutation" (gradual conversion of the ideal-gas molecule into a real molecule, with simultaneous minimization of the energy with respect to the active-molecule degrees of freedom), and "excitation" (generation of points on a hypersphere centered at the active-molecule energy minimum). These steps are also carried out in reverse, as required by the bijective nature of the mapping. The mutation step, which establishes a reversible mapping between energy minima with respect to the active degrees of freedom of systems 0 and 1, ensures that excluded volume interactions emerging in the process of converting the ideal-gas molecule into a real molecule are relieved through appropriate rearrangement of the surrounding active molecules. Thus, the insertion problem plaguing traditional methods for the calculation of chemical potential at high densities is alleviated. Results are presented at two state points of the LJ system for a variety of radii a of the active domain. It is shown that the estimated values of mu(ex) are correct in all cases and subject to an order of magnitude lower statistical uncertainty than values based on the same number of Widom [J. Chem. Phys. 39, 2808 (1963)] insertions at high fluid densities. Optimal settings for the new algorithm are identified and distributions of the quantities involved in it [number of active molecules, energy at the sampled minima of systems 0 and 1, and free-energy differences between subsets gamma(i)(0) and gamma(i)(1) that are mapped onto each other] are explored.
介绍了一种计算两个系统（0和1）自由能差的通用方法，这两个系统具有相同维度的构型空间ω(0)和ω(1)。该方法依赖于在ω(0)的不相交子集γ(i)(0)与ω(1)的相应不相交子集γ(i)(1)之间建立一个目标映射，并对γ(i)(0)和γ(i)(1)上构型积分之比的函数相对于两个系统的概率密度进行平均。映射的子集γ(i)(0)和γ(i)(1)不必覆盖整个构型空间ω(0)和ω(1)。该方法应用于计算 Lennard-Jones（LJ）流体中的过量化学势μ(ex)。在这种情况下，ω(0)是一个（N - 1）个真实分子加一个理想气体分子系统的构型空间，而ω(1)是占据相同体积的N个真实分子系统的构型空间。γ(i)(0)和γ(i)(1)分别由以距离理想气体分子和最后一个真实分子距离为a的一组“活性”分子的最低能量构型为中心、半径相同的超球面构建而成。描述了一种在给定ω(0)或ω(1)中的一个点时对γ(i)(0)和γ(i)(1)进行采样的算法。该算法包括三个步骤：“淬火”（相对于活性分子自由度进行最小化）、“突变”（将理想气体分子逐渐转化为真实分子，同时相对于活性分子自由度最小化能量）和“激发”（在以活性分子能量最小值为中心的超球面上生成点）。根据映射的双射性质，这些步骤也按相反顺序进行。突变步骤在系统0和1的活性自由度的能量最小值之间建立了可逆映射，确保在将理想气体分子转化为真实分子的过程中出现的排除体积相互作用通过周围活性分子的适当重排得到缓解。因此，困扰传统高密度化学势计算方法的插入问题得到了缓解。给出了LJ系统在两个状态点下各种活性域半径a的结果。结果表明，在所有情况下，μ(ex)的估计值都是正确的，并且与基于相同数量的维登（Widom）[《化学物理杂志》39, 2808 (1963)]在高流体密度下插入得到的值相比，统计不确定性低一个数量级。确定了新算法的最佳设置，并探索了其中涉及的量的分布[活性分子数、系统0和1采样最小值处的能量以及相互映射的子集γ(i)(0)和γ(i)(1)之间的自由能差]。
