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     

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: Generalizing the recurrence relations

The formulation of a combinatorial theory of the structure of the class-algebra of the symmetric group is pursued. Recurrence relations for reduced class-coefficients involving imbedded cycles are presented. It is pointed out that for bridging cycles one can obtain elimination rules that involve symmetrization over sets of reduced class-coefficients with common cycle-structures but inequivalent index distributions. Consequently, some of these reduced class-coefficients remain individually inaccessible. © 1993 John Wiley & Sons, Inc.     

A partial recurrence relation for reduced class coefficients of the symmetric group

An expression for the product of a single-cycle class [(1)^{N - P}(p)]_N and an arbitrary class [(1)^l1(2)^l …? (N)^lN]_N of the symmetric group has recently been conjectured. This expression involves a sum over a relatively small number of reduced class sums, depending on p indices. A further conjecture is formulated and demonstrated, according to which reduced class coefficients (RCCS ) involving cycles whose length is expressed by means of a single index can be related to corresponding coefficients in the product of [(1)^{N - P+1}(p - 1)]_N with an arbitrary class sum. Consequently, the problem of evaluating the general class sum product reduces to that of obtaining a relatively small set of fundamental RCCS containing no single-index cycles. The conjectures mentioned can be used to evaluate the product [(1)^{N - p}(p)]_N · [(1)^{N - q}(q)]_N in terms of fundamental RCCS that can all be obtained from the product [(r)]_r · [(r)]_r, where r = min(p, q). For the latter product, we use a result due to Boccara.     

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     

Characters of two-row representations of the symmetric group

Characters of irreducible representations (irreps) of the symmetric group corresponding to the two-row Young diagrams, i.e., describing transformation properties of N-electron eigenfunctions of the total spin operators, have been expressed as explicit functions of the number of electrons N and of the total spin quantum number S. The formulas are useful in various areas of theory of many-electron systems, particularly in designing algorithms for evaluation of spectral density moments. © 1997 John Wiley & Sons, Inc.     

Eigenvalues of single-cycle class-sums in the symmetric group

Explicit expressions for the eigenvalues of the class sums [(p)(1)^n?p]_n, p = 2, 3,…,14, of the symmetric group S_n are presented. Partial results are given for the eigenvalues corresponding to arbitrary p.     

Optimal group symmetric localized molecular orbitals

Summary The concept and generating method of optimum group symmetric localized molecular orbitals (OSLMOs) are proposed. The OSLMOs have strong points of orthogonality, equivalence and symmetry, and they are simultaneously as close to the classical VB structure as possible. By using the OSLMOs as one-electron orbitals the multiconfigurational correlation calculations are reduced. The scheme is also a valuable popularization and development to hybridization theory.     

Eigenvalues of single-cycle class-sums in the symmetric group. II

Explicit expressions for the eigenvalues of the single-cycle class-sums [(p)(1)^n–p]_n of the symmetric group (S_n), with p ≤ 20, are constructed. A new algorithm is used that makes no use of representation-theoretic data. The expressions obtained consist of polynomials in the symmetric power-sums over the “contents” of the Young diagram specifying the irreducible representation, with coefficients that are polynomials in n. On the basis of the results obtained for p ≤ 20, a conjecture is proposed concerning the general form of the four leading terms in these polynomials. © 1993 John Wiley & Sons, Inc.     

Total solid-phase synthesis of the azathiocoraline class of symmetric bicyclic peptides

Thiocoraline is a potent antitumor agent isolated from the marine organism Micromonospora sp. This symmetric bicyclic depsipeptide binds the minor groove of DNA. Here we report two solid-phase strategies for the syntheses of azathiocoraline and its analogues. The thioester linkage was replaced by an amide bond to improve the compound's pharmacokinetic properties. The first strategy is based on a convergent (4+4) approach, whilst the second is a stepwise synthesis, cyclizations in both approaches occurring on the solid support. These two strategies were designed to overcome problems caused by the presence of consecutive noncommercial N-methyl amino acids, to avoid epimerization during cyclization and/or fragment condensation, and to form the disulfide bridge under solid-phase conditions. The heterocyclic moiety was added in the last step of the synthesis to assist the preparation of libraries of new compounds with potential therapeutic applications.     

Substitution methyl group structures in symmetric top molecules

In the absence of a direct experimental determination of A₀ for a symmetric top methyl group molecule, the relations used to determine the off-axis hydrogen coordinates by the mono-substitution technique are shown to lead to the prediction of abnormally long CH bonds. The molecules methyl iodide and methane, for which A₀ constants for the CH₃I and CH₃D species have been determined directly, and for which the ground and equilibrium state geometries are well known, are used as test cases. Methyl chloride, methyl cyanide, and methyl acetylene, for which reliable ground state geometries are known, are also considered.In order to obtain the A₀ rotational constant for CH₂DI required in this study, an analysis of the rovibration spectrum, of CH₂DI was carried out. The results of this work are given in the Appendix.     

Some properties of the bivariational Hartree–Fock scheme for complex symmetric many-particle operators

The bivariational Hartree–Fock scheme for a general many-body operator T is discussed with particular reference to the complex symmetric case: T^? = T*. It shown that, even in the case when the complex symmetric operator T is real and hence also self-adjoint, the complex symmetric Hartree–Fock scheme does not reduce to the conventional real form, unless one introduces the constraint that the N-dimensional space spanned by the Hartree–Fock functions ? should be stable under complex conjugation, so that ?* = ?α. If one omits this constraint, one gets a complex symmetric formulation of the Hartree–Fock scheme for a real N-electron Hamiltonian having the properties H = H* = H^?, in which the effective Hamiltonian H_eff (1) may have complex eigenvalues ε_k. By using the method of complex scaling, it is indicated that these complex eigenvalues—at least for certain systems—may be related to the existence of so-called physical resonance states, and a simple example is given. Full details will be given elsewhere.     

A multireference direct CI program based on the symmetric group graphical approach

Specific features of an SGGA-based multireference direct CI program working in large internal spaces are discussed. In particular, advantages resulting from the explicit separation of the orbital and the spin spaces are explored. Concepts allowing for the efficient creation of a flexible and symmetry-adapted CI basis, for the high-speed generation of the coupling coefficients and for structuring a simple permutation driven algorithm to handle the orbital space are briefly discussed.Alexander von Humboldt Fellow 1985/87     

Symmetric group approach to relativistic CI. IV. Representations of one-electron spin operators and their products in a symmetric group-adapted basis of N-electron spin functions

A detailed study on representations of one-electron spin operators and of their products in an N-electron S_N-adapted spin space is presented. Some conclusions relevant for their evaluation and implementation in relativistic two-component SGA-CI calculations are derived. © 1997 John Wiley & Sons, Inc.     

Cu-catalyzed synthesis of symmetric group 6 (Fischer) bis-carbene complexes

The efficient copper oxidative homocoupling (CuCl/TMEDA/O2) of alkynyl Fischer carbene complexes yields bis-carbene complexes having pi-tethers. The bimetallic complexes are exceptional templates to prepare diverse organic symmetric structures connected by pi-extended fragments.     

Representations of the symmetric group generated by projected spin functions: A graphical approach

The representation matrices generated by the projected spin functions have some very interesting properties. All the matrix elements are integers and they are quite sparse. A very efficient algorithm is presented for the calculation of these representation matrices based on a graphical approach and a new indexing scheme for representation of primitive spin functions is introduced. Test calculations show that the method is very fast and suited for calculations on vector computers. © 1996 John Wiley & Sons, Inc.     

Origin of threefold symmetric torsional potential of methyl group in 4-methylstyrene

To understand the effect of the para position vinyl group substitution in toluene on methyl torsion, we investigated 4-methylstyrene, a benchmark molecule with an extended pi conjugation. The assignment for a 33 cm(-1) band in the excitation spectrum to the 3a(2) torsional transition, in addition to the assignments suggested previously for the other bands in the excitation spectrum, leads to the model potentials for the ground as well as excited states with V(3) (")=19.6 cm(-1), V(6) (")=-16.4 cm(-1) and V(3) (')=25.6 cm(-1), V(6) (')=-30.1 cm(-1), respectively. These potentials reveal that both in ground and excited states, the methyl group conformations are staggered with a 60 degrees phase shift between them. MP2 ab initio calculations support the ground state conformations determined from experiments, whereas Hartree-Fock calculations fail to do so. The origin of the modified ground state potential has been investigated by partitioning the barrier energy using the natural bond orbital (NBO) theoretical framework. The NBO analysis shows that the local delocalization (bond-antibond hyperconjugation) interactions of the methyl group with the parent molecule is sixfold symmetric. The threefold symmetric potential, on the other hand, stems from the interaction of the vinyl group and the adjacent ring pi bond. The threefold symmetric structural energy arising predominantly from the pi electron contribution is the barrier forming term that overwhelms the antibarrier contribution of the delocalization energy. The observed 60 degrees phase shift of the excited state potential is attributed to the pi(*)-sigma(*) hyperconjugation between out of plane hydrogens of the methyl group and the benzene ring.     

The liquid crystal properties of symmetric and non-symmetric dimers based on the azobenzene mesogenic group

A novel system of symmetric and non-symmetric dimers containing azobenzene groups has been synthesized and studied in an attempt to understand further the molecular origins of the intercalated smectic phases. For the non-symmetric dimers, the lack of symmetry was derived solely from the differences in length of the two terminal alkyl chains. Both the spacer and terminal chain lengths were varied. The spacer length was found to exert a profound influence on the clearing temperatures of these materials and a large odd-even effect was observed for the series. The smectic A phase stability was observed to increase with the terminal chain length, yet decrease with increasing spacer length. X-ray diffraction has revealed the structure of the smectic A phase of both the symmetric and non-symmetric azobenzene dimers to be of the monolayer type and not intercalated. The existence of the intercalated phase has previously been explained in terms of either a charge-transfer interaction, or by an electrostatic quadrupolar interaction. However, it has been thought that it may also be the result of an excluded volume or space filling constraint. For the non-symmetric liquid crystal dimers described here, a charge-transfer interaction should be minimal, as should the stabilization from the quadrupolar interaction between the two mesogens. However, it appears that some sort of specific interaction is required to stabilize the intercalated structure.     

The liquid crystal properties of symmetric and non-symmetric dimers based on the azobenzene mesogenic group

A novel system of symmetric and non-symmetric dimers containing azobenzene groups has been synthesized and studied in an attempt to understand further the molecular origins of the intercalated smectic phases. For the non-symmetric dimers, the lack of symmetry was derived solely from the differences in length of the two terminal alkyl chains. Both the spacer and terminal chain lengths were varied. The spacer length was found to exert a profound influence on the clearing temperatures of these materials and a large odd-even effect was observed for the series. The smectic A phase stability was observed to increase with the terminal chain length, yet decrease with increasing spacer length. X-ray diffraction has revealed the structure of the smectic A phase of both the symmetric and non-symmetric azobenzene dimers to be of the monolayer type and not intercalated. The existence of the intercalated phase has previously been explained in terms of either a charge-transfer interaction, or by an electrostatic quadrupolar interaction. However, it has been thought that it may also be the result of an excluded volume or space filling constraint. For the non-symmetric liquid crystal dimers described here, a charge-transfer interaction should be minimal, as should the stabilization from the quadrupolar interaction between the two mesogens. However, it appears that some sort of specific interaction is required to stabilize the intercalated structure.