By introducing an electron bath that represents the chemical environment in which a chemical species is immersed, and by making use of the second-order Taylor series expansions of the energy as a function of the number of electrons in the intervals between N - 1 and N, and N and N + 1, we show that the electrodonating (omega-) and the electroaccepting (omega+) powers may be defined as omega-/+ = (mu-/+)2/2eta-/+, where mu-/+ are the chemical potentials and eta-/+ are the chemical hardnesses, in their corresponding intervals. Approximate expressions for omega- and omega+ in terms of the ionization potential I and the electron affinity A are established by assuming that eta- = eta+ = eta = mu+ - mu-. The functions omega-/+(r) = omega-/+f -/+(r), where f -/+(r) are the directional Fukui functions, derived from a functional Taylor series for the energy functional truncated at second order, represent the local electrodonating and electroaccepting powers.

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Density-functional-theory-based chemical reactivity indicators are formulated for degenerate and near-degenerate ground states. For degenerate states, the functional derivatives of the energy with respect to the external potential do not exist, and must be replaced by the weaker concept of functional variation. The resultant reactivity indicators depend on the specific perturbation. Because it is sometimes impractical to compute reactivity indicators for a specific perturbation, we consider two special cases: point-charge perturbations and Dirac delta function perturbations. The Dirac delta function perturbations provide upper bounds on the chemical reactivity. Reactivity indicators using the common used "average of degenerate states approximation" for degenerate states provide a lower bound on the chemical reactivity. Unfortunately, this lower bound is often extremely weak. Approximate formulas for the reactivity indicators within the frontier-molecular-orbital approximation and special cases (two or three degenerate spatial orbitals) are presented in the supplementary material. One remarkable feature that arises in the frontier molecular orbital approximation, and presumably also in the exact theory, is that removing electrons sometimes causes the electron density to increase at the location of a negative (attractive) Dirac delta function perturbation. That is, the energetic response to a reduction in the external potential can increase even when the number of electrons decreases.

Fluctuation formulas for the external potential v(r) are introduced in a modified Legendre-transformed representation of the density functional theory of electronic structure (isomorphic ensemble). A new (nuclear/geometric) reactivity index h(r), having the same status as the electronic Fukui function in the canonical ensemble, is thereby identified, h(r)=(1/N)[δμ/δσ(r)]N,T=(1/kT) [〈μ⋅v(r)〉−〈μ〉〈v(r)〉], where μ is the electronic chemical potential, σ is the shape factor of the electron density distribution, N is the number of electrons, 〈...〉 denotes the ensemble average of a quantity, and 〈v(r)〉 is the ensemble averaged external potential. This new local quantity is shown to be an inverse of the local softness, and to provide a useful definition of a local hardness.

The Fukui function for a neutral atom is expressed as its LDA approximation plus a one-parameter gradient correction, and the resultant formula is numerically tested. Expressing hardness as a density functional involving this Fukui function, global hardness values are determined for several atoms. Estimates also are made of the covalent radii of neutral atoms. Calculated Fukui functions exhibit characteristics similar to those reported in the literature. Calculated hardnesses compare favorably with experimental values, and predicted covalent radii are in agreement with existing theoretical values and experimental data. No information other than the electron densities of the neutral species enter in the calculations. An exact nuclear cusp condition on the Fukui function is derived.

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We introduce and test a nucleophilicity index as a new descriptor of chemical reactivity. The index is derived from a perturbation model for the interaction between the nucleophile and a positive test charge. The computational implementation of the model uses an isoelectronic process involving the minimum values of the electronic part of the perturbed molecular electrostatic potential. The working expression defining the nucleophilicity index encompasses both the electrostatic contributions and the second-order polarization effects in a form which is consistent with the empirical scales previously proposed. The index is validated for a series of neutral nucleophiles in the gas phase for which the nucleophilicity pattern has been experimentally established within a spectroscopic scale.

The electrophilic addition of HCl to a series of asymmetric alkeness propene, 2-methyl-2-butene, styrene, 2-phenylpropene, and 1-cyanopropenesis used as a model system to study the regioselectivity Markovnikov rule using density functional theory reactivity descriptors. The results show that this rule may be interpreted on the basis of a site activation model that goes beyond the Li-Evans model of selectivity if both the fluctuations in global softness and Fukui functions at the active site are taken into account. A local static analysis based on the condensed Fukui function at the ground state of alkenes was also performed. For all the systems considered, the Markovnikov carbon (M) atom (i.e., the less substituted one) displays electrophilic Fukui function values that are larger than those associated with the more substituted anti-Markovnikov (AM) carbon atom at the double bond. In most cases, they are also larger than the corresponding nucleophilic Fukui function values at both carbon centers of the ethylenic functionality. Site activation at the nucleophilic and electrophilic centers of the alkenes considered was probed by changes in regional softness with reference to the transition state structures. The results are consistent with the empirical Markovnikov rule. A global analysis of involved structures in the electrophilic addition of HCl shows that while the ground state and transition state structures display relative values of the energy and molecular hardness ordered in a way that is consistent with the maximum hardness principle (MHP), the comparison between the Markovnikov and anti-Markovnikov transition state structures do not: the Markovnikov channel presents a transition state which is lower in energy and softer than the one corresponding to the anti-Markovnikov addition.