A simple method of identifying π orbitals for non-planar systems and a protocol of studying π electronic structure

The characteristic of pi electrons has a crucial role in determining various properties of chemical systems, such as reactivity, aromaticity and spectroscopy. There are a large number of methods could be used for investigating pi electronic structure, for example, the well-known electron localization function and multicenter bond order. For completely planar systems, the pi molecular orbitals can be unambiguously identified and thus studying their pi electronic structure is easy. However, for non-planar systems, identification of pi orbitals and then analysis of pi electrons are often not trivial. In this work, based on localized molecular orbitals (LMOs), we propose a conceptually simple and easy way to automatically identify pi orbitals for any kind of systems, which makes subsequent analyses of pi electrons straightforward. In addition, we show that the identified pi LMOs can also be used to reliably estimate pi component of molecular orbitals or other kinds of orbitals. The method proposed in this work has been implemented into our wavefunction analysis code Multiwfn as a key ingredient of standard analysis protocol for pi electrons. Application examples given in this article illustrated that this protocol makes analysis of pi electronic structure for a wide variety of chemical systems unprecedentedly convenient and reliable.