Molecular Fragments in Density Functional Theory Jose ´ L. Ga ´ zquez,* ,² Andre ´ s Cedillo, ² Badhin Go ´ mez, ‡ and Alberto Vela § Departamento de Quı ´mica, DiVisio ´ n de Ciencias Ba ´ sicas e Ingenierı ´a, UniVersidad Auto ´ noma Metropolitana-Iztapalapa, A. P. 55-534, Me ´ xico, D. F. 09340, Me ´ xico, Instituto Mexicano del Petro ´ leo, Eje Central La ´ zaro Ca ´ rdenas, Me ´ xico, D. F. 07730, Me ´ xico, and Departamento de Quı ´mica, Centro de InVestigacio ´ n y de Estudios AVanzados, A. P. 14-740, Me ´ xico, D. F. 07000, Me ´ xico ReceiVed: NoVember 7, 2005; In Final Form: January 9, 2006 The second-order density functional approach to the partitioning of the molecular density of Cedillo, Chattaraj, and Parr (Int. J. Quantum Chem. 2000, 77, 403-407) is used, together with a local assumption for the function that projects the total density into its components, to show that the distribution function adopts a stockholders form, in terms of the local softness of the isolated fragments, and that the molecular Fukui function is distributed in the molecular fragments in the same proportion as the electronic density. Introduction The description of atoms or functional groups in molecules has always been a desirable goal in chemistry. Knowledge of how the atoms or the functional groups change, with respect to their structure when they are isolated, due to the polarization and charge transfer that occurs on bond formation, allows one to understand different aspects of the chemical behavior of a molecule or a family of molecules and how will they interact with different reactants. In this context, Bader and co-workers 1 developed a theory in which the molecular density is divided into nonoverlapping regions separated by surfaces on which the flux of the density gradient is zero. The atoms thus obtained have several important properties. However, the absence of overlap between the atoms could not be as adequate to describe the chemical bonds. Parr and co-workers 2-6 established a definition of atoms in molecules by introducing the concept of promotion energy, which is the change in the energy of each atom from its isolated ground state to its state in the molecule. This way, by making use of the chemical potential equalization principle and through the minimization of the total promotion energy, one finds a unique set of densities for the atoms in the molecule, whose sum is equal to the molecular density, and that are not disjoint. That is, this approach leads to fuzzy overlapping atoms. Later, by taking into account that a fragment in a molecule is an open system that can exchange energy and electrons with the rest of the molecule, Cedillo, Chattaraj, and Parr 7 defined a partitioning of the molecular density, through the minimization of the molecular grand potential with respect to the densities of the fragments, subject to the constraint that they add up to the molecular density. Another definition of molecular fragments, proposed by Hirshfeld, 8 is based on the assumption that the molecular density at each point may be divided among the fragments, in proportion to their respective contributions to the promolecular density at that point. The promolecular density is the sum of the isolated fragment densities at the actual positions of the nuclei. Thus where F i H (r) is the density of the ith fragment in the molecule, the superscript H indicates a Hirshfeld fragment, F i 0 (r) is the density of the ith isolated fragment, F m (r) is the molecular ground-state density is the promolecular density, and w i H (r) ) {F i 0 (r)/F pm 0 (r)} is the Hirshfeld stockholders distribution function. The sum of all the fragment densities, F i H (r), is equal to the molecular ground- state density, F m (r). Recently, there has been a renewed interest in the Hirshfeld stockholder partitioning, motivated by the important demonstra- tions of the information-theoretic basis of this division scheme 9-15 and the thermodynamic-like properties of the Hirshfeld subsystems. 16 In addition, the Hirshfeld partitioning has also been applied to calculate condensed Fukui functions, leading to very reasonable values of these reactivity criteria that have been used to explain several aspects about the chemical behavior of a wide variety of chemical systems. 17-22 However, the calculation of the condensed Fukui functions, with the Hirshfeld distribution function, implies the assumption of using the distribution function that is employed in the neutral system for the cases in which the molecule has a net positive or negative charge. A situation that also implies that the molecular Fukui function is distributed in the molecular fragments in the same proportion as the electronic density. That is, through this approximation one has that w i H (r) ) {F i 0 (r)/F pm 0 (r)} ) {F i H (r)/F m (r)} ) {f i H (r)/f m (r)}, where f m (r) ) ({∂F m (r)/∂N}) V is the ground-state molecular Fukui function, and f i H (r) ) w i H (r)f m (r) such that ∑ i f i H (r) ) f m (r). It is important to note that Ayers, Morrison, and Roy 23 have established a formal mathematical and physical basis for the condensed Fukui functions, originally introduced by Yang and * Author for correspondence. E-mail address: jlgm@xanum.uam.mx. ² Universidad Auto ´noma Metropolitana. ‡ Instituto Mexicano del Petro ´leo. § Centro de Investigacio ´n y de Estudios Avanzados. F i H (r) ) F i 0 (r) F pm 0 (r) F m (r) ) w i H (r) F m (r) (1) F pm 0 (r) ) ∑i F i 0 (r) (2) 4535 J. Phys. Chem. A 2006, 110, 4535-4537 10.1021/jp056421q CCC: $33.50 © 2006 American Chemical Society Published on Web 03/14/2006