text: Recently, the Woodward–Hoffmann rules have been reinterpreted using conceptual density functional theory (DFT).   The key to the analysis is the dual descriptor function, proposed by Christophe Morell, André Grand and Alejandro Toro-Labbé  , the second derivative of the electron density  with respect to the number of electrons .  This response function is important as the reaction of two components A and B involving a transfer of electrons will depend on the responsiveness of the electron density to electron donation or acceptance, i.e. the derivative of the Fukui function .  In fact, from a simplistic viewpoint, the dual descriptor function gives a readout on the electrophilicity or nucleophilicity of the various regions of the molecule.  For , the region is electrophilic, and for , the region is nucleophilic.  Using the frontier molecular orbital assumption and a finite difference approximation of the Fukui function, one may write the dual descriptor as
  This makes intuitive sense as if a region is better at accepting electrons than donating, then the LUMO must dominate and dual descriptor function will be positive.  Conversely, if a region is better at donating electrons then the HOMO term will dominate and the descriptor will be negative.  Notice that although the concept of phase and orbitals are replaced simply by the notion of electron density, this function still takes both positive and negative values.
  The Woodward–Hoffmann rules are reinterpreted using this formulation by matching favorable interactions between regions of electron density for which the dual descriptor has opposite signs.  This is equivalent to maximizing predicted favorable interactions and minimizing repulsive interactions.  For the case of a [4+2] cycloaddition, a simplified schematic of the reactants with the dual descriptor function colored (red=positive, blue=negative) is shown in the optimal supra/supra configuration to the left.  This method correctly predicts the WH rules for the major classes of pericyclic reactions.