Symmetry Principles for Periodic Systems of Molecules Jr-Phys-Sc/Spbu 1

Extensive research, most of it inspired by fundamental-particle physics, has been done on the group-dynamic foundations of the classification of atoms known as the periodic system. Also, a very great deal of work has been carried out for a century on varieties of ways to classify molecules and to predict the properties of new ones in a systematic way. Much of this work has been motivated by the need for small-molecule astrophysical data and for pharmaceutical data. It is the purpose of this report to employ the methods of group dynamics, as they have been applied to atoms, for the molecular classification problem. We begin with known bosonic creation and annihilation operators, since any number of identical atoms may be present in a molecule. Then we define state vectors in the space () of -atomic molecules. We require that this space be decomposed into irreducible representations (IRs) of a group with demonstrated applicability to the classification of atoms; the basis vectors of these IR’s form multiplets of molecules which bear a great deal of similarity to multiplets of atoms. Finally, we show how to formulate observable operators of various orders and in several approximations; these operators provide expectation values expressed as functions of state quantum numbers. Tabulated experimental data for various properties, substituted into these expectation values and plotted one multiplet at a time, produce surfaces which are periodic, are smooth, and show an approximate invariance for isoelectronic molecules. In this report we list the symmetry groups to be treated and describe the role which each of them plays in classifications of atoms. We take up compact multiplets in the following different conformal symmetries: (3), (3)x(2), (3)x(2), (3)x(2)x(2), (4)x(2), and (2); for each we give the group chains, their quantum numbers, the operators and their relationships in Lie algebras, and examples of molecular multiplets. Then we consider noncompact multiplets in the conformal symmetry (2,1), and multiplets in the unitary symmetries (υ) and (2), following a similar outline. Many of the molecular state vectors are composed of linear superpositions of molecular symbols; this situation is analogous to the linear superpositions of quark combinations in some hadron states of earlier standard models.