Characterization of the Chemical Behavior of the Low Excited States through a Local Chemical Potential Christophe Morell,* ,† Vanessa Labet, † Andre ´ Grand, † Paul W. Ayers, ‡ Frank De Proft, § Paul Geerlings, § and Henry Chermette ⊥ Commissariat a ` L’E ´ nergie Atomique Grenoble, 17, Rue des Martyrs, F-38054 Grenoble Cedex 9, France, Department of Chemistry, McMaster UniVersity, Hamilton, Ontario, L8S 4M1, Eenheid Algemene Chemie, Vrije UniVersiteit Brussel, Faculteit Wetenschappen Pleinlaan 2, 1050 Brussels, Belgium, Member of the QCMM Alliance Ghent-Brussels Belgium, and UniVersite ´ de Lyon, UniVersite ´ Lyon 1, Sciences Analytiques CNRS UMR 5180 Chimie Physique The ´orique, F-69622 Villeurbanne Cedex, France Received May 17, 2009 Abstract: Exploiting the locality of the chemical potential of an excited state when it is evaluated using the ground-state density functional theory (DFT), a new descriptor for excited states has been proposed. This index is based on the assumption that the relaxation of the electronic density drives the chemical reactivity of excited states. The sign of the descriptor characterizes the electrophilic or nucleophilic behavior of the atomic regions. A relation between the new descriptor and the dual descriptor is derived and provides a posteriori justification of its use to rationalize the Woodward-Hoffmann rules for photochemical reactions within the conceptual DFT. Finally, the descriptor is successfully applied to some [2 + 2] photocycloadditions, like Paterno-Bu ¨ chi reactions. 1. Introduction During the past two decades, conceptual density functional theory (DFT) 1-4 has been a fruitful paradigm for the analysis of chemical processes. Through successive derivatives of the energy, with respect to either the number of electrons or the external potential, different reactivity and selectivity descrip- tors have been designed to account for the outcome of chemical reactions. 5,6 More importantly, descriptors previ- ously proposed on an intuitive basis found a sharp definition within conceptual DFT. For instance, the electronegativity defined by Mulliken 7 has been identified as the opposite of the first derivative of the energy with respect to the number of electrons. 8 Another example is the proposal to measure the chemical hardness through the second derivative of the energy with respect to the number of electrons. 9 Besides, different older theories, such as the Frontier Molecular Orbital (FMO) initiated by Fukui 10-15 and the Hard and Soft Acids and Bases (HSAB) proposed by Pearson, 16-22 have been unified within conceptual DFT. However, some chemical processes are still out of the reach of conceptual DFT. Indeed, due to the lack of a suitably formulated excited-state DFT, all the chemical reactions that involve an excited molecule are difficult to rationalize. Even though some reactions have been investigated by trans- posing, 23-25 the traditional local descriptors for the ground state to the reactivity of the excited state, no formal theory has been designed to support their use. It must be noticed that a preliminary study of time-dependent DFT to design local descriptors has been proposed by Chattaraj and co-workers. 26-28 The purpose of this paper is then to provide some insight into both the reactivity and selectivity of low excited states using the locality of the excited state’s chemical potential when evaluated using the energy density functional for ground states. The paper is organized as follows. In Section 2, using the true density of an excited state as a trial function for the ground-state density, a local descriptor is proposed to characterize the electrophilicity or nucleophilicity of a region * Corresponding author. E-mail: Christophe.morell@ujf-grenoble.fr. † Commissariat a L’Energie Atomique Grenoble. ‡ McMaster University. § Vrije Universiteit Brussel. ⊥ Universite ´ de Lyon. J. Chem. Theory Comput. 2009, 5, 2274–2283 2274 10.1021/ct900248a CCC: $40.75  2009 American Chemical Society Published on Web 08/05/2009 Downloaded by christophe morell on September 8, 2009 | http://pubs.acs.org Publication Date (Web): August 5, 2009 | doi: 10.1021/ct900248a