Free Energies in the Landau and Molecular Field Approaches

An expression for the entropy as a power series in the order parameter is derived in the context of molecular field theory. The expression is valid both at and away from equilibrium. It is a unique generalization of molecular field theory to non-equilibrium situations. Discrepancies with certain expressions which have appeared in the literature are resolved. Analysis of the radius of convergence of the power series indicates that in certain cases, including the Maier-Saupe model of liquid crystals, molecular field theory and Landau theory cannot be made to agree over the entire range of possible values of the order parameter.     

The q-Zassenhaus formula

A q-deformed analogue of the Zassenhaus formula, expressing the q-exponential of a sum of two noncommuting operators in terms of an infinite product of q-exponentials, is introduced.     

A study of the weinhold-wang bounds to overlap integrals

The Weinhold-Wang extension of the Braun-Rebane formula for bounds to overlap integrals has been investigated numerically for the first and second excited s states of the hydrogen atom. The effect of the choice of basis sets is demonstrated with particular emphasis on the difference between a complete set and an incomplete set of expansion functions     

Class-sum products in the symmetric group: Combinatorial interpretation of the reduced class coefficients

An algorithm for the evaluation of the structure constants in the class algebra of the symmetric group has recently been considered. The product of the class sum [(p)]_n that consists of a cycle of length p and n−p fixed points, with an arbitrary class sum in S_n, was found to be expressible in terms of a set of reduced class coefficients (RCCs), the p-RCCs. The combinatorial significance of the p-RCCs is elucidated, showing that they are related to a well-defined enumeration problem within S_p, which has to do with a certain refinement of the corresponding class multiplication problem. This is in contrast with the representation-theoretic evaluation of the p-RCCs, which requires the evaluation of products involving [(p)]_n for several values of n > p. The combinatorial interpretation of the p-RCCs allows the derivation of some of their previously conjectured properties and of some of the “elimination rules” that specific types of p-RCCs were found to satisfy. © 1998 John Wiley & Sons, Inc. Int J Quant Chem 68: 103–118, 1998