The conservation of generalized parity: molecular symmetry and conservation rules in chemistry

Abstract

Last century, one of the most important achievements in theoretical chemistry was the conservation principle of orbital symmetry, i.e. the Woodward–Hoffmann rules (W–H rules). Owing to the greatness of the accomplishment, some defects in connection with the rules have been neglected. For more than 20 years ago, we have paid attention to these problems. Our early works were somewhat crude and all of the relevant papers and book were announced in China. This work could not be easily known by most scientists in the world. Now, we have improved the papers and rewrite this article from a more fundamental basis. The main points of this article are as follows. (1) According to quantum field theory, there is an important theorem (Noether's theorem), which relates the selection rules of different processes with their symmetries. There are two different concepts—symmetry and an invariant, but they are not distinguished in W–H rules. Using Noether's theorem to analyze the W–H rules, these two concepts may be described more clearly. (2) For some more important chemical systems given the point symmetry, we can analyze them using Noether's theorem, and get the corresponding conservation rule. The invariant is a kind of generalized parity. (3) In chemistry, there are other selection rules as for the spectral transition processes. All of the rules may be related to the corresponding conservation rules of generalized parity. (4) By means of a quantum field method, we can analyze the rules according to operator methods. To different processes (such as a chemical reaction, a spectral transition, etc.), we can obtain the corresponding operators. If the process is allowed by symmetry, the corresponding operator may transform the initial state functional to the final one. If the process is forbidden, the corresponding process operator will perish and no process operator can act on the initial state. (5) For photochemical reactions, there are two consecutive steps: excitation with the absorption of a photon and chemical reaction of the excited molecule. Both would be considered. The mechanism of photochemical dimerization of ethylene, according to the common view of W–H rule is uncertain. (6) For extended system with the space group symmetry, we can analyze them by means of the Noether's theorem, too. Selection rules in connection with physical processes in solids can be obtained. (7) For some chemical reaction systems, such as sigmatropic reactions, there is no symmetry in connection with the point group operation of the reaction system. However, there are some special symmetry properties contained in such reaction system. (8) To describe such special symmetry, we introduce a new transformation. It is the reaction-reversal transformation in the intrinsic reaction space. The corresponding special symmetry is the group from the union of the reaction-reversal and point symmetry transformation. Using Noether's theorem and such special symmetry, the selection rules of sigmatropic reactions may be obtained. (9) As for such symmetry, not only sigmatropic reaction, but also some other reactions, such as the disproportionation of an alkene, may be analyzed. (10) The molecular local field method had been introduced. We may analyze a reaction involving chirality or open-shell molecules.