Many‐particle Dirac identities for arbitrary elementary spins

The eigenvalues of arbitrary conjugacy class‐sums of the symmetric group, within subspaces that contain irreducible representations with at most k rows, are considered. Explicit expressions for these eigenvalues in terms of the eigenvalues of single‐cycle class‐sums with cycle lengths up to k are obtained. © 2000 John Wiley & Sons, Inc. Int J Quant Chem 78: 407–411, 2000     

Products of arbitrary class-sums in the symmetric group

An algorithm for the evaluation of products of arbitrary conjugacy class-sums in the symmetric group is conjectured. This algorithm generalizes a procedure presented sometime ago, which deals with products in which at least one of the class-sums involved consists of a single cycle (and an appropriate number of fixed points). Let the support size of a conjugacy class be the number of indices that are not fixed points. The algorithm proposed implies that the coefficient of the class-sum C in the product of the class-sums A and B is given in terms of a well-defined enumeration problem within the symmetric group S_p, where p is the smallest of the support sizes of A, B, and C. © 1998 John Wiley & Sons, Inc. Int J Quant Chem 70: 429–440, 1998     

Products of class sums of the symmetric group: Rules of partial elimination

Progress in the formulation of a procedure for the combinatorial evaluation of the product of a single-cycle and an arbitrary class sum in the symmetric group algebra is presented. The procedure consists of a “global conjecture” concerning the representation of the product [(p)]_n·[*]_n in terms of a set of operators referred to as reduced class sums, and of an (incomplete) set of rules for the evaluation of the (n-independent!) coefficients of these operators. Two new types of index elimination rules are suggested, and some properties of the formalism are explored. These include useful sum rules as well as a certain “detailed balance” property that sheds some light on a combinatorial aspect of the global conjecture. The present results account for several new types of reduced class coefficients and suggest some feasible further developments. Some outstanding open problems are pointed out. © 1997 John Wiley & Sons, Inc. Int J Quant Chem 63: 961–979, 1997     

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     

Orbital correspondence analysis in maximum symmetry: Formulation and conceptual framework

A recently proposed method for the analysis of the course of chemical reactions, based on the maximal use of available symmetry, is formulated as a set of procedural rules. The application of these rules is illustrated with a simple prototype reaction: CH₂+C₂H₄ fcyclo-C₃H₆. They are then derived, using the formalism of time-dependent perturbation theory within the Born-Oppenheimer approximation, thus bringing out the method's underlying assumptions and its relation to the widely used Woodward-Hoffmann procedure.     

On the non-existence of maxima in variational computations containing non-linear parameters

The homogeneity properties of the kinetic and potential energy operators are used to obtain expressions for the second derivatives of the energy expectation value. These are used to demonstrate that in atoms as well as in molecules in the neighborhood of the equilibrium geometry the variational energy cannot have maxima with respect to the non-linear parameters.     

Asymptotic behavior of two-electron expectation values in two-electron excited states

Singly-excited states of the two-electron atom cease being bound when $Z\to 1$ (from above), the outer orbital becoming infinitely diffuse. The asymptotic relations$\underset{Z\to 1}{\lim }?{(Z?1)}^k〈{(1sns)}^{1,3}S\left|r_{12}^k\right|{(1sns)}^{1,3}S〉=〈(n?1)s^{(0)}\left|r^k\right|(n?1)s^{(0)}〉,$ where $k=?1,1,2,3,?$, are demonstrated to hold. Here, $(n?1)s^{(0)}$ is a hydrogenic s orbital with principal quantum number $(n?1)$. New, more nuanced light is shed on the already challenged dogma that the Pauli principle keeps the electrons further apart in the triplet than in the corresponding singlet.