Generalized nuclear Fukui functions in the framework of spin-polarized density-functional theory

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.

Conceptual DFT: the chemical relevance of higher response functions

In recent years conceptual density functional theory offered a perspective for the interpretation/prediction of experimental/theoretical reactivity data on the basis of a series of response functions to perturbations in the number of electrons and/or external potential. This approach has enabled the sharp definition and computation, from first principles, of a series of well-known but sometimes vaguely defined chemical concepts such as electronegativity and hardness. In this contribution, a short overview of the shortcomings of the simplest, first order response functions is illustrated leading to a description of chemical bonding in a covalent interaction in terms of interacting atoms or groups, governed by electrostatics with the tendency to polarize bonds on the basis of electronegativity differences. The second order approach, well known until now, introduces the hardness/softness and Fukui function concepts related to polarizability and frontier MO theory, respectively. The introduction of polarizability/softness is also considered in a historical perspective in which polarizability was, with some exceptions, mainly put forward in non covalent interactions. A particular series of response functions, arising when the changes in the external potential are solely provoked by changes in nuclear configurations (the "R-analogues") are also systematically considered. The main part of the contribution is devoted to third order response functions which, at first sight, may be expected not to yield chemically significant information, as turns out to be for the hyperhardness. A counterexample is the dual descriptor and its R analogue, the initial hardness response, which turns out to yield a firm basis to regain the Woodward-Hoffmann rules for pericyclic reactions based on a density-only basis, i.e. without involving the phase, sign, symmetry of the wavefunction. Even the second order nonlinear response functions are shown possibly to bear interesting information, e.g. on the local and global polarizability. Its derivatives may govern the influence of charge on the polarizability, the R-analogues being the nuclear Fukui function and the quadratic and cubic force constants. Although some of the higher order derivatives may be difficult to evaluate a comparison with the energy expansion used in spectroscopy in terms of nuclear displacements, nuclear magnetic moments, electric and magnetic fields leads to the conjecture that, certainly cross terms may contain new, intricate information for understanding chemical reactivity.

Condensed-to-atoms electronic Fukui functions within the framework of spin-polarized density-functional theory

A simple formalism devised to calculate the condensed-to-atoms Fukui function [R. R. Contreras, P. Fuentealba, M. Galván, and P. Pérez, Chem. Phys. Lett. 304, 405 (1999)] has been further analyzed within a spin-polarized density-functional theory framework. The model is based on a frozen-core approximation to these local reactivity indices [M. Galván, A. Vela, and J. L. Gázquez, J. Phys. Chem. 92, 6470 (1988)], giving us an extended local reactivity description of systems based on the frontier spin-up and spin-down molecular orbitals. Degenerate molecular spin orbitals have been explicitly included in our model equations. Computational results for the nitric oxide (NO) and some simple carbene systems are presented in order to test the model. These quantities have been discussed in the context of changes both in charge density and spin density within the context of electron charge transfer or spin-polarization processes.

Philicity indices within the spin-polarized density-functional theory framework

The electrophilicity index is analyzed within the framework of spin-polarized density-functional theory. In this context, constrained philicities, omega(N) identical with (mu(N))(2)(2eta(NN)), are introduced in order to define the capability of a system to acquire or donate electrons in a process at constant spin number. The spin-philicity/spin-donicity indices, omega(S)(+/-) identical with (mu(S) (+/-))(2)(2eta(SS)), are examined and rationalized here as the philicity of a given system to change its spin-polarization state, as being defined through the spin potential mu(S) and spin hardness eta(SS) for a process at constant number of electrons. The local extension of these indices has been also outlined and numerical results have been discussed on the analysis of the electrophilic nature of some simple carbene systems both in the singlet and triplet states.

Hardness and softness reactivity kernels within the spin-polarized density-functional theory

Generalized hardness and softness reactivity kernels are defined within a spin-polarized density-functional theory (SP-DFT) conceptual framework. These quantities constitute the basis for the global, local (i.e., r-position dependent), and nonlocal (i.e., r and r'-position dependents) indices devoted to the treatment of both charge-transfer and spin-polarization processes in such a reactivity framework. The exact relationships between these descriptors within a SP-DFT framework are derived and the implications for chemical reactivity in such context are outlined.