The temperature-dependence of the Helmholtz free energy with respect to the number of electrons is analyzed within the framework of the Grand Canonical Ensemble. At the zero-temperature limit, the Helmholtz free energy behaves as a Heaviside function of the number of electrons; however, as the temperature increases, the profile smoothens and exhibits a minimum value at noninteger positive values of the fractional electronic charge. We show that the exact average electronic energy as a function of the number of electrons does not display this feature at any temperature, since this behavior is solely due to the electronic entropy. Our mathematical analysis thus indicates that the widely used parabolic interpolation model should not be viewed as an approximation for the average electronic energy, but for the dependence of the Helmholtz free energy upon the number of electrons, and this analysis is corroborated by numerical results. Finally, an electrophilicity index is defined for the Helmholtz free energy showing that, for a given chemical species, there exists a temperature value for which this quantity is equivalent to the electrophilicity index defined within the parabolic interpolation of the electronic energy as a function of the number of electrons. Our formulation suggests that the convexity property of the energy versus the number of electrons together with the entropic contribution does not allow for an analogous nucleophilicity index to be defined.

中文翻译：

---

关于电子数和对亲电性的严格定义的反应性指标抛物线模型的热力学证明：电子熵起的基本作用

亥姆霍兹自由能相对于电子数的温度依赖性在大正则合奏的框架内进行了分析。在零温度极限下，亥姆霍兹自由能表现为电子数量的Heaviside函数。但是，随着温度的升高，轮廓会变得平滑，并在分数电荷的非整数正值处显示最小值。我们表明，确切在任何温度下，平均电子能量随电子数量的变化都不会显示此特征，因为这种行为完全是由于电子熵引起的。因此，我们的数学分析表明，不应将广泛使用的抛物线插值模型视为平均电子能量的近似值，而应视为亥姆霍兹自由能对电子数量的依赖性，并且此分析得到了数值结果的证实。最后，为亥姆霍兹自由能定义了亲电性指数，表明对于给定的化学物质，存在一个温度值，该温度值等于在电子的抛物线插值中定义的亲电性指数，是该函数的函数。电子数。