Jacob Katriel

Theoretical Interpretation of Hund's Rule

Publisher Summary This chapter discusses a comprehensive treatment of some recent developments in the interpretation of Hunds rule. This rule can be considered as an ordering principle for the ground configuration, or at best, as a criterion for the lowest term of any configuration. The maximal formulation attempts to use the Hunds rule as a general ordering principle for any configuration. Experimentally, Hunds rule is highly reliable so far as the ground state is concerned. It is, however, not nearly as dependable for the ordering of higher terms in the configuration. The most restrictive formulation of Hunds rule states that in a configuration of equivalent electrons the deepest lying term corresponds to the highest possible value of the total spin. Generalizations of this rule to the ground state corresponding to configurations of nonequivalent electrons, as well as using it to predict the ordering of all the terms corresponding to a given configuration have been discussed. In order to illustrate quantitatively the results of scaling with reference to the interpretation of Hunds rule, the ground configuration of the carbon atom using Slater-type orbitals are reviewed.

Intramolecular electronic energy transfer via exchange interaction in bichromophoric molecules

Abstract A theory is presented for intramolecular electronic energy transfer in bichromophoric molecules. Expressions are given for the donor moiety fluorescence (phosphorescence) decay and for its fluorescence (phosphorescence) quantum yield in terms of the average distance between the donor and acceptor moieties and the donor—acceptor bridge flexibility. Comparison with available experimental data supports the predictions of the analysis.

On q-squeezed states

The authors present the results of an analysis of the squeezing of components of the (conventional) electromagnetic field in quantum group analogues of the Heisenberg-Weyl (HW) coherent state and SU(1,1) squeezed state. They find that squeezing occurs for all finite-q values not equal to unity in the HW q-coherent state, in contrast to the usual case; and also in the SUq(1,1) case, although here less than in the usual (q=1) SU(1,1) squeezed state.

Multiphoton squeezed states

The multiphoton squeezed states defined in this paper are generalizations of the conventional coherent (Glauber) and squeezed (Yuen) states previously discussed by many authors. We define multiphoton generalizations of the latter by a unified class of states that includes the Holstein–Primakoff realizations of SU(2) and SU(1, 1) as well as the standard harmonic oscillator coherent states (Weyl–Heisenberg group) and squeezed states in a general framework that allows also non-Hermitian realizations. We determine the squeezing properties of these states in a unified formalism and study numerically their dependence on the parameter classifying the states.

ORDERING RELATIONS FOR Q-BOSON OPERATORS, CONTINUED FRACTION TECHNIQUES AND THE Q-CBH ENIGMA

Ordering properties of boson operators have been very extensively studied, and q-analogues of many of the relevant techniques have been derived. These relations have far reaching physical applications and, at the same time, provide a rich and interesting source of combinatorial identities and of their g-analogues. An interesting exception involves the transformation from symmetric to normal ordering, which, for conventional boson operators, can most simply be effected using a special case of the Campbell-Baker-Hausdorff (CBH) formula. To circumvent the lack of a suitable q-analogue of the CBH formula, two alternative procedures are proposed, based on a recurrence relation and on a double continued fraction, respectively. These procedures enrich the repertoire of techniques available in this field. For conventional bosons they result in an expression that coincides with that derived using the CBH formula.

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+C2H4fcyclo-C3H6. They are then derived, using the formalism of time-dependent perturbation theory within the Born-Oppenheimer approximation, thus bringing out the methods underlying assumptions and its relation to the widely used Woodward-Hoffmann procedure.

Coefficients of fractional parentage in the L–S coupling scheme

An efficient procedure for the evaluation of the coefficients of fractional parentage (cfp’s) for L–S coupled wave functions is presented. The cfp’s are calculated separately for N particles, each with angular momentum l (s), coupled into a total angular momentum L (S). The N‐particle states formed can belong to any permutational symmetry. The procedure for the evaluation of the L and the S cfp’s for arbitrary permutational symmetry is a generalization of the procedure proposed by Bayman and Lande [Nucl. Phys. 77, 1 (1966)] for symmetric and antisymmetric states. It involves the construction and diagonalization of the matrices representing the quadratic Casimir operators for the appropriate special unitary and symplectic (or orthogonal) groups. The cfp’s of the antisymmetric L–S coupled states are obtained in terms of products of cfp’s for L and S corresponding to conjugate representations of the symmetric group. This method is demonstrated to provide cfp’s for L–S states for systems with a considerably l...

Effect of the one-body potential on interelectronic correlation in two-electron systems

The correlation energies of the helium isoelectronic sequence (IS) and of Hookes IS are very similar and are both weakly increasing upon increasing the nuclear charge/force constant, respectively. However, their separation into radial and angular correlations shows interesting differences. First, for intermediate (and high) values of the force constant radial correlation in Hookes IS is surprisingly low. Second, both systems exhibit a decrease in the relative contribution of radial versus angular correlation upon strengthening the one-body attractive potential; however, unlike the helium IS, in Hookes IS the radial correlation energy increases in absolute value upon strengthening the attractive one-body potential. The contribution of radial correlation to the Coulomb hole is examined and the asymptotic behavior at both strong and weak attractive potentials is considered. Radial correlation in Hookes IS is found to constitute about 9.3% of the total correlation energy when the spring constant approaches the limit k-->infinity, but 100% of the total correlation energy for k-->0. Our results highlight both the similarities and the differences between the helium and Hookes ISs.

Explicit expressions for the central characters of the symmetric group

Abstract A conjecture concerning the construction of explicit expressions for the central characters of the symmetric group Sn is presented. The expression for the central characters corresponding to a class with a given cycle structure (1) l 1 (2) l 2 … (n) l n is a polynomial in the symmetric power sums over the “contents” of the Young diagram specifying the irreducible representation of interest. Each term in this polynomial is a product of symmetric power sums multiplied by a polynomial in n. Both the degrees of the polynomials in n and the powers of the various symmetric power sums appearing in each term are specified by an algorithm involving sets of partitions of integers associated with the lengths of the nontrivial cycles specifying the class of interest and combinations thereof. The coefficients of the polynomials in n are obtained by solving a system of linear equations which arise from the evaluation of the proposed expression for the central characters with respect to Young diagrams whose number of boxes is less than ∑ni=2ibi,, and equating to zero.

Stirling number identities: interconsistency of q-analogues

q-analogues of Stirling number identities are formulated, and the interconsistency among the q-analogues of the Stirling numbers and of the binomial coefficients is investigated. The close relation with the normal ordering problem for Arik-Coon-type q-bosons plays a central role in the derivations presented.