Parity symmetry in a number of problems of quantum and structural chemistry

A synthetic review and new results are given of the alternant symmetry theory and its applications within a unified approach. It is based on J–symmetry (parity) operators. Unlike usual commutation rules, these symmetry operators anticommute with Hamiltonians or other relevant quantities. In the J–symmetry terms we treat a variety of problems and topics, mainly related to π-shells of conjugated molecules. In particular, various orbital theories are outlined with a systematic use of block-matrix technique (density matrices, operator functions etc.). Noval π‑models and their J–symmetry are studied within the current context of single-molecule conductance and the relevant problems concerning Green’s function and electron transmission evaluation. We stress on the key importance of account for π-electron correlation for describing correctly transmission π-spectra. We discuss electron-structure peculiarities of alternant radical states and the validity of the Lieb-Ovchinnikov spin rule resulting from the J–symmetry and electron correlation effects. It is shown how the simplified (based on Hückel’s MOs) spin-polarized theory provides a correct number of effectively unpaired electrons in polyradicaloid alternant molecules. Another type of problems is concerned with chirality (generllly, structural asymmetry) problems. By spectral analysys of the previously defined chirality operator we could reinterpret the problem in terms of J–symmetry. It allowed us to construct here the noval chirality operator which is nonnegative definite and vanishes on achiral structures. Its simplest invariant, the matrix trace, surves us as a quantitative measure of the structural (electronic) chirality. Preliminary calculations tell us that the new chirality index behaves reasonably even for the difficult (high-symmetry) chiral systems.