Abstract
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.
Highlights
It is quite often encountered in quantum chemistry[1] that very accurate stationary-state electronic wavefunctions, such as those of highly correlated Configuration Interaction (CI) and its variants,[2,3,4] the method of cluster expansion,[5] Complete Active Space Multiconfiguration SCF method[6] (CASSCF) and various extensions,[7] Nakatsuji theory,[8,9] and so on, tend to seem quite different from one another in their profoundly complicated functional forms
To see how the total electronic energy is composed among the kinetic energy Te, electron–nuclei attraction energy Vne, and electron–electron repulsion energy Vee, we further study the energy variation of the representative energy natural orbitals (ENOs)
The notion of energy natural orbitals has been proposed as eigenfunctions of the energy density operator H (1), which is already a function of ρ, h, G, and the total wavefunction Ψ
Summary
It is quite often encountered in quantum chemistry[1] that very accurate stationary-state electronic wavefunctions, such as those of highly correlated Configuration Interaction (CI) and its variants,[2,3,4] the method of cluster expansion,[5] Complete Active Space Multiconfiguration SCF method[6] (CASSCF) and various extensions,[7] Nakatsuji theory,[8,9] and so on, tend to seem quite different from one another in their profoundly complicated functional forms. Intermolecular interactions, the Fukui frontier orbital (HOMO– LUMO interaction) theory,[20] and the Woodward–Hoffmann rule of conservation of orbital symmetry,[21] and so on Noting that these theories and concepts have been established in terms of the specific views and approximations, it would be instructive to translate them into a universal representation that is free of dependence on the choice of approximate expressions. In exactly the same sense as in the natural orbital theory, ENOs are invariant with respect to the choice of the functional forms of correlated electronic wavefunctions, for instance, CI, perturbation theories, cluster expansion, or others. IV, in which structural beauty and quasi-symmetry are revealed in the course of electron transfer reaction These examples should serve to clarify the similarity and difference of the ENO picture from those of the existing orbital theories. Selected properties and future perspective in applications of ENOs are discussed in Appendixes A and B
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