Accountability of kinetic energy densities through the lens of local temperature

Recently, there has been considerable research interest in the development and application of density-based reactivity descriptors to investigate a wide variety of physicochemical characteristics. In this regard, it is known that the unique descriptors using both electron density as well as its derivatives should be considered for the investigation of many chemical processes and transformations, while the electron density cannot be used alone. Among others, and particularly relevant to our study, local temperature (LT) based on kinetic energy density (KED) as well as electron density is noteworthy. When the LT is introduced as a density functional descriptor, KED approximations come into play, where the primary challenge lies in ensuring the accuracy and applicability of the KEDs across a diverse array of properties and systems. Hereby, in this work, we propose to use LT to evaluate the performance of the approximate KEDs, where their accountability is examined not only for predicting the total kinetic energy but also for describing the LT globally and locally as an intrinsic property of atomic and molecular systems. From the numerical data of the global analysis, it is demonstrated that the KEDs that provide reliable kinetic energy results do not necessarily lead to reasonable descriptions of the integrated LT and vice versa. Towards analyzing the reason of this issue, the local behaviors of the LT were also explored. It is unveiled that the underlying KEDs with imbalanced treatments of the related constraints cannot accurately account for the kinetic energy distribution in the regions where the electron sharing is in place and chemical bonds are formed, in contrast to the earlier efforts which focused more on satisfying the cusp and asymptotic behaviors. Putting the findings of this study together, we can offer the LT as a metric to assess the quality of KED approximations, while emphasizing that, for the development of modern KEDs, including machine-learned ones, not only the kinetic energy results but also the reactivity properties and other related propensities are of paramount importance, and the medium ranges of the kinetic energy distribution should also be taken into account.