Nonlocal Wigner-like correlation energy density functional: parametrization and tests on two-electron systems

Reparametrization of Wigner's correlation energy density functional yields a very close fit to the correlation energies of the helium isoelectronic sequence. However, a quite different reparametrization is required to obtain an equally close fit to the isoelectronic sequence of Hooke's atom. In an attempt to avoid having to reparametrize the functional for different choices of the one-body potential, we propose a parametrization that depends on global characteristics of the ground-state electron density as quantified by scale-invariant combinations of expectation values of local one-body operators. This should be viewed as an alternative to the density-gradient paradigm, allowing one to introduce the nonlocal dependence of the density functional on the density in a possibly more effective way. Encouraging results are obtained for two-electron systems with one-body potentials of the form r(zeta) with zeta=-12,+12,1, which span the range between the Coulomb potential (zeta=-1) and the Hooke potential (zeta=2).     

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     

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.     

Mono and Trimethine Cyanines Cyan 40 and Cyan 2 as Probes for Highly Selective Fluorescent Detection of Non-canonical DNA Structures

Two of earlier reported dsDNA sensitive cyanine dyes??monomethine Cyan 40 and meso-substituted trimethine Cyan 2 were studied for their ability to interact with non-canonical DNA conformations. These dyes were characterized by spectral-luminescent methods in the presence of G-quadruplex, triplex and dsDNA motifs. We have demonstrated that Cyan 2 binds strongly and preferentially to triple- and quadruple-stranded DNA forms that results in a strong enhancement of the dye fluorescence, as compared to dsDNA, while Cyan 40 form fluorescent complexes preferentially only with the triplex form. Highly fluorescent complexes of Cyan 2 with DNA triplexes and G-quadruplexes and Cyan 40 with DNA triplexes are very stable and do not dissociate during gel electrophoresis, leading to preferential staining of the above DNA forms in gels. The data presented point to the intercalation mechanism of the Cyan 2 binding to G4-DNA, while the complexes of Cyan 40 and Cyan 2 with triplex DNA are believed to be formed via groove binding mode. The Cyan dyes can provide a highly sensitive method for detection and quantification of non-canonical structures in genome.     

The Hardness of 3-Uniform Hypergraph Coloring

We prove that coloring a 3-uniform 2-colorable hypergraph with c colors is NP-hard for any constant c. The best known algorithm [20] colors such a graph using O(n^1/5) colors. Our result immediately implies that for any constants k ≥ 3 and c₂ > c₁ > 1, coloring a k-uniform c₁-colorable hypergraph with c₂ colors is NP-hard; the case k = 2, however, remains wide open. This is the first hardness result for approximately-coloring a 3-uniform hypergraph that is colorable with a constant number of colors. For k ≥ 4 such a result has been shown by [14], who also discussed the inherent difference between the k = 3 case and k ≥ 4. Our proof presents a new connection between the Long-Code and the Kneser graph, and relies on the high chromatic numbers of the Kneser graph [19,22] and the Schrijver graph [26]. We prove a certain maximization variant of the Kneser conjecture, namely that any coloring of the Kneser graph by fewer colors than its chromatic number, has ‘many’ non-monochromatic edges. * Research supported by NSF grant CCR-9987845. † Supported by an Alon Fellowship and by NSF grant CCR-9987845. ‡ Work supported in part by NSF grants CCF-9988526 and DMS 9729992, and the State of New Jersery.     

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     

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.