The Woodward–Hoffmann Rules Reinterpreted by Conceptual Density Functional Theory

In an attempt to master the overwhelming amount of data on the properties of substances and their reactions, chemists scrutinize them for underlying common patterns. In modern times quantum mechanics (QM) has played a leading role in the understanding of chemical reactivity. In the late 1960s, Woodward and Hoffmann (WH) proposed one of the most successful and elegant approaches to interpret the outcome of an important type of reaction: they could predict the allowed or forbidden character of pericyclic reactions through inspection of the phase and symmetry of the orbitals of the reactants obtained by simple extended Hückel theory. Today much more powerful computational techniques, such as density functional theory (DFT), are available that yield highly accurate results even for large systems. By focusing on the electron density, ρ(r), a fundamental carrier of information compared with the much more complicated wave function in conventional QM, DFT became the computational workhorse for systems of ever increasing complexity. However, the need for the interpretation of computational (and obviously experimental) results remains, and "conceptual DFT" has provided the answer to this challenge within the context of DFT. This branch of DFT has given precision to chemical concepts such as electronegativity, hardness, and softness and has embedded them in a perturbational approach to chemical reactivity. Previously, researchers have successfully applied conceptual DFT to generalized acid-base and, more recently, to radical and redox reactions. In this Account, we present a conceptual DFT ansatz for pericyclic reactions, a stringent test for this density-only approach, because the density has trivial symmetry and no phase. A density response function is the key quantity in a first approach: the dual descriptor f((2))(r), the second derivative of the electron density with respect to the number of electrons. Overlapping regions of the dual descriptor of the reactant(s) with different or the same sign yield pictorial representations similar to the orbital phase and symmetry-based pictures in the WH formulation. In a second approach, a key quantity is the evolution of the chemical hardness at the onset of the reaction. This quantity makes contact with Zimmerman's alternative approach to the WH rules based on the aromaticity of the transition state. Using the dual descriptor and the initial hardness response, we reinterpret the WH results for the four types of pericyclic reactions (cycloadditions, electrocyclizations, and sigmatropic and chelotropic reactions), both thermodynamically and photochemically. We demonstrate that these two approaches, which require only simple quantum chemical procedures (overlapping densities and HOMO-LUMO gap type calculations along a few points of a model reaction coordinate), are intimately related through a relation that converts the local (i.e., position-dependent) dual descriptor into the global (i.e., position-independent) (initial) hardness response. Our results show that with a density-only based approach the WH rules can be reinterpreted, pointing to the fundamental importance of the electron density as carrier of information as highlighted in the basic theorems of DFT.