The Hitchhiker's Guide to the Wave Function∥

The electronic wave function of molecules is 3N-dimensional and inseparable in the coordinates of the N electrons. Whereas molecular orbitals are often invoked to visualize the electronic structure, they are nonunique, with the same 3N-dimensional wave function being represented by an infinite number of 3-D, one-electron functions (orbitals). Furthermore, multireference wave functions cannot be described by an antisymmetrized product of a single set of occupied orbitals. What is required is a way to visualize the full dimensionality of the wave function, including the effects of correlation, as a 3N-dimensional being would be able to do. In the past 5 years, we have been developing a way to analyze and visualize highly dimensional wave functions by focusing on the structure of the repeating unit demanded by fermionic behavior. This 3N-dimensional repeating unit, the wave function "tile", can be projected onto the three dimensions of each electron, in turn, to reveal the complete electronic structure. It is found that the tile reproduces canonical chemical motifs such as core-electrons, single bonds and lone pairs. Multiple bonds emerge as the "banana" bonds favored by Pauling. As a function of the reaction coordinate, electron motions are visualized that correspond to the curly arrow notation of organic chemists. Excited states can also be inspected. Analyzing a wave function in terms of fermionic tiling allows for insight not facilitated by the inspection of orbitals or configuration interaction vectors: The wave function tiles of resonance structures reveal that electron correlation in benzene pushes opposing spin electrons to occupy alternate Kekulé structures, and in C₂, the emerging structure supports the notion of a triply bonded structure with a weak, fourth bonding contribution.