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

Spin-Polarized Conceptual Density Functional Theory from the Convex Hull

We present a new, nonarbitrary, internally consistent, and unambiguous framework for spin-polarized conceptual density-functional theory (SP-DFT). We explicitly characterize the convex hull of energy, as a function of the number of electrons and their spin, as the only accessible ground states in spin-polarized density functional theory. Then, we construct continuous linear and quadratic models for the energy. The nondifferentiable linear model exactly captures the simplicial geometry of the complex hull about the point of interest and gives exact representations for the conceptual DFT reactivity indicators. The continuous quadratic energy model is the paraboloid of maximum curvature, which most tightly encloses the point of interest and neighboring vertices. The quadratic model is invariant to the choice of coordinate system (i.e., {N, S} vs {Nα, Nβ}) and reduces to a sensible formulation of spin-free conceptual DFT in the appropriate limit. Using the quadratic model, we generalize the Parr functions {P+(r), P-(r)} (and their derivatives with respect to number of electrons) to this new spin-polarized framework, integrating the Parr function concept into the context of (spin-polarized) conceptual DFT, and extending it to include higher-order effects.

Molecular interactions from the density functional theory for chemical reactivity: Interaction chemical potential, hardness, and reactivity principles

In the first paper of this series, the authors derived an expression for the interaction energy between two reagents in terms of the chemical reactivity indicators that can be derived from density functional perturbation theory. While negative interaction energies can explain reactivity, reactivity is often more simply explained using the “|dμ| big is good” rule or the maximum hardness principle. Expressions for the change in chemical potential (μ) and hardness when two reagents interact are derived. A partial justification for the maximum hardness principle is that the terms that appear in the interaction energy expression often reappear in the expression for the interaction hardness, but with opposite sign.

Molecular Interactions From the Density Functional Theory for Chemical Reactivity: The Interaction Energy Between Two-Reagents

Reactivity descriptors indicate where a reagent is most reactive and how it is most likely to react. However, a reaction will only occur when the reagent encounters a suitable reaction partner. Determining whether a pair of reagents is well-matched requires developing reactivity rules that depend on both reagents. This can be achieved using the expression for the minimum-interaction-energy obtained from the density functional reactivity theory. Different terms in this expression will be dominant in different circumstances; depending on which terms control the reactivity, different reactivity indicators will be preferred.

The electronic temperature and the effective chemical potential parameters of an atom in a molecule. A Fermi-Dirac semi-local variational approach

We developed a numerical procedure to compute the electronic temperature and the effective (local) chemical potential undergone by electrons belonging to a particular molecular species. Our strategy relies on consider atomic basins as open quantum (sub)systems within the context of the quantum theory of atoms in molecules. Each basin is represented by the two parameters, the electronic temperature and the effective chemical potential, which are determined by distributing electrons (fermions) imbedded in each atomic region, through a Fermi-Dirac semi-local variational procedure. The results obtained for 40 different chemical species show that the effective chemical potential is a useful tool to reveal the most acidic/basic atoms in a molecule while the electronic temperature is closely related to the concept of chemical hardness at the local level. Our numerical data also indicate that the electronic temperature values undergone by electrons imbedded in atomic basins are way beyond the room temperature condition, allowing to fractionally occupy several of the one-particle quantum states. In this context, we developed two new indexes useful to reveal outstanding orbitals involved in the chemical reactivity of atoms in molecules.

Reactivity of Single Transition Metal Atoms on a Hydroxylated Amorphous Silica Surface: A Periodic Conceptual DFT Investigation

The drive to develop maximal atom-efficient catalysts coupled to the continuous striving for more sustainable reactions has led to an ever-increasing interest in single-atom catalysis. Based on a periodic conceptual density functional theory (cDFT) approach, fundamental insights into the reactivity and adsorption of single late transition metal atoms supported on a fully hydroxylated amorphous silica surface have been acquired. In particular, this investigation revealed that the influence of van der Waals dispersion forces is especially significant for a silver (98 %) or gold (78 %) atom, whereas the oxophilicity of the Group 8-10 transition metals plays a major role in the interaction strength of these atoms on the irreducible SiO₂ support. The adsorption energies for the less-electronegative row 4 elements (Fe, Co, Ni) ranged from -1.40 to -1.92 eV, whereas for the heavier row 5 and 6 metals, with the exception of Pd, these values are between -2.20 and -2.92 eV. The deviating behavior of Pd can be attributed to a fully filled d-shell and, hence, the absence of the hybridization effects. Through a systematic analysis of cDFT descriptors determined by using three different theoretical schemes, the Fermi weighted density of states approach was identified as the most suitable for describing the reactivity of the studied systems. The main advantage of this scheme is the fact that it is not influenced by fictitious Coulomb interactions between successive, charged reciprocal cells. Moreover, the contribution of the energy levels to the reactivity is simultaneously scaled based on their position relative to the Fermi level. Finally, the obtained Fermi weighted density of states reactivity trends show a good agreement with the chemical characteristics of the investigated metal atoms as well as the experimental data.

Extending conceptual DFT to include additional variables: oriented external electric field

The extension of the E = E[N, v] functional for exploring chemical reactivity in a conceptual DFT context to include external electric fields is discussed. Concentrating on the case of a homogeneous field the corresponding response functions are identified and integrated, together with the conventional response functions such as permanent dipole moment and polarizability, in an extended response function tree associated with the E = E[N, v, ε] functional. In a case study on the dihalogens F2, Cl2, Br2, I2 the sensitivity of condensed atomic charges (∂q/∂ε) is linked to the polarizability of the halogen atoms. The non-integrated (∂ρ(r)/∂ε) response function, directly related to the field induced density change, is at the basis of these features. It reveals symmetry breaking for a perpendicular field, not detectable in its atom condensed counterpart, and accounts for the induced dipole moment directly related to the molecular polarizability. The much higher sensitivity of the electronic chemical potential/electronegativity as compared to the chemical hardness is highlighted. The response of the condensed Fukui functions to a parallel electric field increases when going down in the periodic table and is interpreted in terms of the extension of the outer contours in the non-condensed Fukui function. In the case of a perpendicular field the (∂f(r)/∂ε) response function hints at stereoselectivity with a preferential side of attack which is not retrieved in its condensed form. In an application the nucleophilic attack on the carbonyl group in H2CO is discussed. Similar to the dihalogens, stereoselectivity is displayed in the Fukui function for nucleophilic attack (f+) in the case of a perpendicular electric field, and opposite to the one that would arise based on the induced density. Disentangling the expression for the evolution of the Fukui function in the presence of an electric field reveals that this difference can be traced back to local differences in the polarization or induced density between the anionic and the neutral system. This difference may be exploited, e.g. for an appropriately substituted H2CO, to generate enantioselectivity.

Temperature-Dependent Approach to Electronic Charge Transfer

A charge transfer model is developed within the framework of the grand canonical ensemble through the analysis of the behavior of the fractional charge as a function of the chemical potential of the bath when the temperature and the external chemical potential are kept fixed. Departing from the fact that, before the interaction between two species, each one has a zero fractional charge, one can identify two situations after the interaction occurs where the fractional charge of at least one of the species is different from zero, indicating that there has been charge transference. One of them corresponds to the case when one of the species is immersed in a bath conformed by the other one, while the other is related to the case in which both species are present in equal amounts (stoichiometric proportion). Correlations between the fractional charges and average energies, thus obtained with experimental equilibrium constants, kinetic rate constants, hydration constants, and bond enthalpies, indicate that, although at the experimental temperatures, they are very small quantities, they have chemically meaningful information. Additionally, in the stoichiometric case, one also finds a rather good correlation between the equalized chemical potential and the one obtained from experimental information for a test set of diatomic and triatomic molecules.

Establishing the Relationship between Quantum Capacitance and Softness of N-Doped Graphene/Electrolyte Interfaces within the Density Functional Theory Grand Canonical Kohn-Sham Formalism

The joint density functional theory (JDFT) is applied in the context of the grand canonical Kohn-Sham theory to calculate the global and local softness of pristine and N-substituted graphene structures. A comparison is established between the different theoretical approaches to evaluate total capacitance, revealing that the JDFT approach presents the closest result of this property with experimental data. A model of series capacitors is used to determine the quantum and nonquantum contributions of total capacitance, which enables us to determine the limitations of the rigid band approximation for the studied systems. It is found that global chemical softness is proportional to the total capacitance measured in the experiments, when the geometry relaxation is neglected. In this context, it is possible to obtain quantum and total capacitance (and consequently softness) from an average number of electrons vs applied potential plots and the model of series capacitors. Likewise, the relation of capacitance and softness gives rise to a new definition of local capacitance within the JDFT formalism. The evaluation of global and local softness paves the way to analyze electrochemical surface reactivity as a function of applied potential for a solid-electrolyte interface.

Electronegativities of Pauling and Mulliken in Density Functional Theory

Electronegativity is a fundamental concept in chemistry that allows one to infer important aspects about the interactions between chemical species. In the present work we make use of the framework provided by the density functional theory of chemical reactivity, to discuss in a unified way the approaches to the concept of electronegativity developed by Pauling and by Mulliken. Our analysis starts by making use of the identification of the electronegativity of Mulliken with the chemical potential of density functional theory, and continues to show that the ionic correction proposed by Pauling can be derived, with certain approximations, from the quadratic smooth interpolation of the energy as a function of the number of electrons in terms of the chemical potentials and the hardnesses of the interacting species, from which one can infer the close qualitative relationship between Pauling's electronegativity and the electrophilicity concept.

An electronic temperature definition for the reactive electronic species: Conciliating practical approaches in conceptual chemical reactivity theory with a rigorous ensemble formulation

By working under the framework of the Helmholtz potential as a functional of the equilibrium density matrix, in this contribution, we provide theoretical evidence about a particular thermodynamic situation, where electronic species display their highest susceptibility to exchange electrons to or from surroundings. This situation is denominated as the electronic temperature condition. Neutral chemical species display their lowest possible hardness value at the electronic temperature condition, and remarkably, under this circumstance, the exchange of any amount of electronic charge will necessarily be translated into a net increase in the corresponding chemical hardness. Chemical response functions defined as partial derivatives of the Helmholtz potential with respect to the (average) number of electrons and evaluated at the electronic temperature condition provide comparable results than those obtained from the coarse quadratic approximation to the exact dependence of the electronic energy vs the number of electrons, including composite quantities as the electrophilicity index. In this context, we show that the exact Helmholtz potential dependence with respect to the number of electrons can accurately be approximated by "temperature dependent" polynomial fits (up to fourth order), evaluated at the electronic temperature condition.

New advances in conceptual-DFT: an alternative way to calculate the Fukui function and dual descriptor

An alternative way of calculating the Fukui function and the partial derivative of second order of the electronic density with respect to the number of electrons N is presented, the new formulas agree with the usual ones but only in cases without degeneracy. The new operative formulas are more general than the previous ones and are the right ones for those problematic cases where one or both of the frontier molecular orbitals are degenerate. Finally, we present a new way of applying the finite difference approximation that leads to more realistic results than the usual formulas. Graphical abstract A new way of calculating the Fukui function is presented that results in a new operative formula of the function. It has also been obtained the partial derivative of second order of the electronic density with respect to the number of electrons N, and it agree with the usual formula of the dual descriptor function but only in cases without degeneration.