Energy natural orbitals

We propose and numerically demonstrate that highly correlated electronic wavefunctions such as those of configuration interaction, the cluster expansion, and so on, and electron wavepackets superposed thereof can be analyzed in terms of one-electron functions, which we call energy natural orbitals (ENOs). As the name suggests, ENOs are members of the broad family of natural orbitals defined by Löwdin, in that they are eigenfunctions of the energy density operator. One of the major characteristics is that the (orbital) energies of all the ENOs are summed up exactly equal to the total electronic energy of a wavefunction under study. Another outstanding feature is that the population of each ENO varies as the chemical reaction proceeds, keeping the total population constant though. The study of ENOs has been driven by the need for new methods to analyze extremely complicated nonadiabatic electron wavepackets such as those embedded in highly quasi-degenerate excited-state manifolds. Yet, ENOs can be applied to scrutinize many other chemical reactions, ranging from the ordinary concerted reactions, nonadiabatic reactions, and Woodward-Hoffman forbidden reactions, to excited-state reactions. We here present the properties of ENOs and a couple of case studies of numerical realization, one of which is about the mechanism of nonadiabatic electron transfer.