Marco Franco-Pérez

Revisiting the definition of the electronic chemical potential, chemical hardness, and softness at finite temperatures

We extend the definition of the electronic chemical potential (μe) and chemical hardness (ηe) to finite temperatures by considering a reactive chemical species as a true open system to the exchange of electrons, working exclusively within the framework of the grand canonical ensemble. As in the zero temperature derivation of these descriptors, the response of a chemical reagent to electron-transfer is determined by the response of the (average) electronic energy of the system, and not by intrinsic thermodynamic properties like the chemical potential of the electron-reservoir which is, in general, different from the electronic chemical potential, μe. Although the dependence of the electronic energy on electron number qualitatively resembles the piecewise-continuous straight-line profile for low electronic temperatures (up to ca. 5000 K), the introduction of the temperature as a free variable smoothens this profile, so that derivatives (of all orders) of the average electronic energy with respect to the average electron number exist and can be evaluated analytically. Assuming a three-state ensemble, well-known results for the electronic chemical potential at negative (-I), positive (-A), and zero values of the fractional charge (-(I + A)/2) are recovered. Similarly, in the zero temperature limit, the chemical hardness is formally expressed as a Dirac delta function in the particle number and satisfies the well-known reciprocity relation with the global softness.

Local and linear chemical reactivity response functions at finite temperature in density functional theory

We explore the local and nonlocal response functions of the grand canonical potential density functional at nonzero temperature. In analogy to the zero-temperature treatment, local (e.g., the average electron density and the local softness) and nonlocal (e.g., the softness kernel) intrinsic response functions are defined as partial derivatives of the grand canonical potential with respect to its thermodynamic variables (i.e., the chemical potential of the electron reservoir and the external potential generated by the atomic nuclei). To define the local and nonlocal response functions of the electron density (e.g., the Fukui function, the linear density response function, and the dual descriptor), we differentiate with respect to the average electron number and the external potential. The well-known mathematical relationships between the intrinsic response functions and the electron-density responses are generalized to nonzero temperature, and we prove that in the zero-temperature limit, our results recover well-known identities from the density functional theory of chemical reactivity. Specific working equations and numerical results are provided for the 3-state ensemble model.

Electronic chemical response indexes at finite temperature in the canonical ensemble

Assuming that the electronic energy is given by a smooth function of the number of electrons and within the extension of density functional theory to finite temperature, the first and second order chemical reactivity response functions of the Helmholtz free energy with respect to the temperature, the number of electrons, and the external potential are derived. It is found that in all cases related to the first or second derivatives with respect to the number of electrons or the external potential, there is a term given by the average of the corresponding derivative of the electronic energy of each state (ground and excited). For the second derivatives, including those related with the temperature, there is a thermal fluctuation contribution that is zero at zero temperature. Thus, all expressions reduce correctly to their corresponding chemical reactivity expressions at zero temperature and show that, at room temperature, the corrections are very small. When the assumption that the electronic energy is given by a smooth function of the number of electrons is replaced by the straight lines behavior connecting integer values, as required by the ensemble theorem, one needs to introduce directional derivatives in most cases, so that the temperature dependent expressions reduce correctly to their zero temperature counterparts. However, the main result holds, namely, at finite temperature the thermal corrections to the chemical reactivity response functions are very small. Consequently, the present work validates the usage of reactivity indexes calculated at zero temperature to infer chemical behavior at room and even higher temperatures.

Average electronic energy is the central quantity in conceptual chemical reactivity theory

We show that, at both zero and nonzero temperature, the average electronic energy is the central quantity for the study of chemical reactivity using density functional theory. Specifically, we show that regardless of the temperature and the ensemble framework one chooses, the average electronic energy exhibits the functional dependence with respect to external potential and electron number of the zero-temperature case, but with the presence of an additional temperature-dependent term that vanishes in the zero-temperature limit. This reconciles the results from the finite-temperature chemical reactivity theory with those of the traditional zero-temperature approach, and resolves apparent discrepancies between the canonical and grand canonical ensembles. However, the grand canonical ensemble provides a more convenient mathematical framework for studying chemical reactivity because it regards chemical species as open quantum (sub)systems. The response functions of the average electronic energy reduce to the zero-temperature reactivity descriptors in the appropriate limit, helping to justify the use of idealized zero-temperature approximations for describing experimental results obtained at nonzero temperature. Extending this approach to the entropy, the responses of the entropy are shown to give a different set of interesting chemical reactivity response functions. The expressions for the heat capacity of an electronic system are especially elucidative.

Thermodynamic hardness and the maximum hardness principle

An alternative definition of hardness (called the thermodynamic hardness) within the grand canonical ensemble formalism is proposed in terms of the partial derivative of the electronic chemical potential with respect to the thermodynamic chemical potential of the reservoir, keeping the temperature and the external potential constant. This temperature dependent definition may be interpreted as a measure of the propensity of a system to go through a charge transfer process when it interacts with other species, and thus it keeps the philosophy of the original definition. When the derivative is expressed in terms of the three-state ensemble model, in the regime of low temperatures and up to temperatures of chemical interest, one finds that for zero fractional charge, the thermodynamic hardness is proportional to T-1(I-A), where I is the first ionization potential, A is the electron affinity, and T is the temperature. However, the thermodynamic hardness is nearly zero when the fractional charge is different from zero. Thus, through the present definition, one avoids the presence of the Dirac delta function. We show that the chemical hardness defined in this way provides meaningful and discernible information about the hardness properties of a chemical species exhibiting integer or a fractional average number of electrons, and this analysis allowed us to establish a link between the maximum possible value of the hardness here defined, with the minimum softness principle, showing that both principles are related to minimum fractional charge and maximum stability conditions.

Thermodynamic Justification for the Parabolic Model for Reactivity Indicators with Respect to Electron Number and a Rigorous Definition for the Electrophilicity: The Essential Role Played by the Electronic Entropy

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.

Local and nonlocal counterparts of global descriptors: the cases of chemical softness and hardness

A new strategy, recently reported by us to develop local and linear (nonlocal) counterparts of global response functions, is applied to study the local behavior of the global softness and hardness reactivity descriptors. Within this approach a local counterpart is designed to identify the most important molecular fragments for a given chemical response. The local counterpart of the global softness obtained through our methodology corresponds to the well-known definition of local softness and, in agreement with what standard conceptual chemical reactivity in density functional theory dictates, it simply reveals the softest sites in a molecule. For the case of the local hardness, we obtain two expressions that lead to different information regarding the values of the hardness at the different sites within a chemical species. The performance of these two proposal were tested by comparing their corresponding atom-condensed values to experimentally observed reactivity trends for electrophilic attack on benzene and ethene derivatives.

Chemical hardness: Temperature dependent definitions and reactivity principles

In this brief report, we show that the three different chemical hardness definitions developed in the framework of the temperature-dependent density functional theory-namely, the electronic, the thermodynamic, and the Helmholtz hardnesses-imply both the hard and soft acids and bases (HSAB) principle and the maximum hardness (MH) principle. These hardnesses are identified as the second derivative of a thermodynamic state function and avoid the somewhat arbitrary approach, based on the parabolic interpolation of the energy versus electron number, that is normally used to justify these principles. This not only leads to a more mathematically sound justification of the HSAB and MH principles in the low-temperature limit but also establishes that the HSAB and the MH principles hold at any temperature of chemical relevance.

Thermodynamic responses of electronic systems

We present how the framework of the temperature-dependent chemical reactivity theory can describe the panorama of different types of interactions between an electronic system and external reagents. The key reactivity indicators are responses of an appropriate state function (like the energy or grand potential) to the variables that determine the state of the system (like the number of electrons/chemical potential, external potential, and temperature). We also consider the response of the average electron density to appropriate perturbations. We present computable formulas for these reactivity indicators and discuss their chemical utility for describing electronic, electrostatic, and thermal changes associated with chemical processes.