论文标题
低温的自由能扰动理论
Free Energy Perturbation Theory at Low Temperature
论文作者
论文摘要
Zwanzig引入的扰动扩张[R. W. Zwanzig，J。Chem。物理。 {\ bf 22}，1420（1954）]表达了感兴趣系统与参考系统之间的helmholtz自由能的差异，作为两种系统之间势能差的一系列累积量$κ_n$。这种扩展具有吸引人的特征，作为获得{\ it i ab intio}势能表面的绝对自由能的方法。该系列是$β= 1/t $的正式功率系列，这表明其有用性可能仅限于高温。但是，对于平稳的参考电位，$κ_n$的$ t $依赖性有助于融合。对于系统和参考势都是谐波的情况下，所有订单的$κ_N$都会得出封闭形式的表达式。在这种情况下，对于$ n \ ge 2 $，$κ_n\ propto t^n $，该系列的收敛性与温度无关。通过使用Monte Carlo Integration评估累积物至三阶，分别研究了液体Cu和固体AL的更现实的液体Cu和固体AL，分别为$ 1/r^{12} $和谐波参考电位。在所有情况下，发现扩展中的三阶与第二阶项的比率为$ \ sim 0.1 $，表明良好的收敛性。对自由能的三阶贡献通常是几个MEV/原子，并且与它们的统计错误相当。 AL的二阶自由能中的统计误差为0.4 MeV/Atom，仅100个评估{\ it it Intibil}能量。这些结果表明，扰动系列可以有效，准确地评估冷凝相的自由能。
The perturbative expansion introduced by Zwanzig [R. W. Zwanzig, J. Chem. Phys. {\bf 22}, 1420 (1954)] expresses the difference in Helmholtz free energy between a system of interest and that of a reference system as series of cumulants $κ_n$ of the potential energy difference between the two systems. This expansion has attractive features as a method for obtaining absolute free energies for {\it ab initio} potential energy surfaces. The series is formally a power series in $β=1/T$, suggesting that its usefulness may be limited to high temperature. However, for smooth reference potentials, the $T$-dependence of the $κ_n$ contributes to the convergence. A closed form expression is derived for the $κ_n$ to all orders for the case that both the system and reference potentials are harmonic. In this case $κ_n \propto T^n$ for $n \ge 2$ and the convergence of the series is independent of temperature. More realistic cases of liquid Cu and solid Al, with a $1/r^{12}$ and harmonic reference potential, respectively, are investigated numerically by evaluating the cumulants to third order using Monte Carlo integration. In all cases, the ratio of the third order to the second order term in the expansion is found to be $\sim 0.1$, indicating good convergence. Third order contributions to the free energy are typically a few meV/atom, and comparable to their statistical errors. The statistical error in the second order free energy of Al is 0.4 meV/atom with only 100 evaluations of the {\it ab initio} energy. These results suggest that the perturbation series allows for efficient and accurate evaluation of the free energy for condensed phases.
