Subduction of dominant representations for combinatorial enumeration

Summary A new method for giving cycle indices is presented for combinatorial enumeration. Thus, cyclic groups are characterized by markaracter tables, the elements of which are determined by the orders of their subgroups. A set of such cyclic groups (defined as dominant subgroups) is used to characterize a group G of finite order, where the markaracter table for the group G is constructed with respect to dominant representations (DRs), which are defined as coset representations corresponding to the dominant subgroups. By starting from the markaracter table, we propose an essential set of subdominant markaracter tables and a magnification set for the group G; the latter concept clarifies the relationship between each subdominant markaracter table and the markaracter table of a dominant subgroup. The subduction of DRs is obtained by the markaracter table to produce a dominant subduction table and a dominant USCI (unit-subduced cycle index) table. The latter is used to evaluate a cycle index to be applied to combinatorial enumeration. The cycle index is shown to be equivalent to the couterpart of our previous approach concerning both cyclic and non-cyclic subgroups. The latter, in turn, has been proved to be equivaltent to the cycle index obtained by the Redfield-Pólya theorem.     

Direct subduction of Q-conjugacy representations to give characteristic monomials for combinatorial enumeration

Characteristic monomials for a finite group are obtained by direct subductions of Q-conjugate representations. They are shown to give a generating function that is capable of solving enumeration problems. Received: 1 May 1998 / Accepted: 16 July 1998 / Published online: 18 September 1998     

Subduction of coset representations

Summary Enumerations of compounds based on a parent skeleton with and without the influence of obligatory minimum valency (OMV) are reported. The effect of the OMV is formulated by assigning different weights to the respective orbits of the parent skeleton. This type of enumeration requires introduction of several new concepts that are derived from the subduction of coset representations, e.g., a unit subduced cycle index, a subduced cycle index and the number of suborbits.     

A simplified proof of a theorem about group representations

It is a standard theorem of group representation theory that the dimension of an irreducible representation is a divisor of the order of the group. This paper gives a new, relatively simple proof, intended to make the theorem understandable to readers unfamiliar with algebraic integers.     

对称群标准表示矩阵计算新方法

在多体问题的对称群方法中,群表示矩阵的计算是关键性问题。Young-Yamanouchi规则给出了标准正交表示的计算方法,然而该法相当繁琐,颇难使用,本文将2列Young表标准表示的计算方法[1]推广到任意不可约表示,给出对称群标准表示矩阵计算新方法.     

Analysis and applications of a group contribution sPC-SAFT equation of state

A group contribution (GC) method for estimating pure compound parameters for the molecular-based perturbed-chain statistical associating fluid theory (PC-SAFT) equation of state (EoS) is proposed in a previous work [A. Tihic, G.M. Kontogeorgis, N. von Solms, M.L. Michelsen, L. Constantinou, Ind. Eng. Chem. Res. 47 (2008) 5092–5101]. In this paper, an investigation of the predictive capability of the GC sPC-SAFT EoS through comparison of the method’s predictions for compounds with high molecular weights and several selected binary mixtures of industrial significance with experimental data such as thiols, sulphides and polynuclear aromatics is presented. Additionally, predictions of activity coefficient at infinite dilution for athermal systems are compared with the results using existing activity coefficient models. The results show that calculated pure compound parameters using the proposed GC method allow satisfactory representation of experimental data of investigated systems with the sPC-SAFT EoS. Moreover, the variety of functional groups in the available GC scheme ensures broad applications of the GC sPC-SAFT EoS.     

Algebraic solutions for point groups: The octahedral group for the group chain O⊃T⊃C3

Using the right‐induced technique and the eigenfunction method, concise algebraic expressions of the projection operators for both single‐valued and double‐valued representations are found for the group chain O?T?C₃ in terms of the projection operators of T?C₃. Extremely simple relations are discovered between the symmetry adapted functions (SAFs) of the groups T and O; namely the SAFs of the subgroup T which have proper symmetry are the SAFs of the group O. The projection operators and SAFs are functions of only the quantum numbers of the group chain [the analogy of ( j,m) for the group chain SO₃?SO₂], without involving any irreducible matrix elements. © 2001 John Wiley & Sons, Inc. Int J Quant Chem 83: 259–270, 2001     

Prediction of Antoine constants using a group contribution method

Vapor pressure is one of the important properties necessary for process design. Antoine equation has been widely used to evaluate the vapor pressures. This article deals with the proposal of a method for predicting the Antoine constants using a group contribution method. The 1817 compounds treated here are grouped into six classes. The group contribution parameters for each class have been determined using a regression analysis. The group contribution method with help of experimental normal boiling point has good results for predicting the vapor pressures.     

A theory for the construction of the irreducible representations of finite groups

It is shown how the irreducible representations of a finite group can be calculated from the irreducible characters (the latter can be calculated exactly by using Dixon's method). All elements of the matrix, representing a group element, lie in the rational field of polynomials of ξ = exp (2πi/e), where e is the exponent of the group.     

9-Triptycenyl complexes of group 13 and 15 halides and hydrides

The reactions of 9-lithiotriptycene with AlCl₃, GaCl₃ and InBr₃ have been carried out and have yielded the complexes, [(tript)AlCl₂(OEt₂)], [(tript)GaCl₂(THF)] and [(tript)InBr(μ-Br)₂Li(OEt₂)₂], tript=9-triptycenyl. The latter two complexes have been structurally characterised. The corresponding reactions of 9-lithiotriptycene with ECl₃ (E=P, As, Sb or Bi) afforded the complexes, [(tript)ECl₂], of which all but the phosphorus compound have been crystallographically authenticated. In addition, the high yield syntheses of the thermally stable primary pnictanes [(tript)EH₂] (E=As, Sb) have been achieved by reaction of the relevant halide complex with LiAlH₄. The X-ray crystal structure of the antimony hydride complex is reported.     

Fast procedure for reconstruction of full-atom protein models from reduced representations

We introduce PULCHRA, a fast and robust method for the reconstruction of full-atom protein models starting from a reduced protein representation. The algorithm is particularly suitable as an intermediate step between coarse-grained model-based structure prediction and applications requiring an all-atom structure, such as molecular dynamics, protein-ligand docking, structure-based function prediction, or assessment of quality of the predicted structure. The accuracy of the method was tested on a set of high-resolution crystallographic structures as well as on a set of low-resolution protein decoys generated by a protein structure prediction algorithm TASSER. The method is implemented as a standalone program that is available for download from http://cssb.biology.gatech.edu/skolnick/files/PULCHRA.     

A necessary and sufficient condition for the existence of real coupling coefficients for a finite group

We establish a theorem which gives a necessary and sufficient condition for a set of matrix irreps of a finite group to admit real coupling (Clebsch–Gordan) coefficients. The proof is based on the method used by Feit to prove that a full set of coupling coefficients for a finite group determines the group up to isomorphism. A consequence of the theorem is that a finite group with real coupling coefficients is necessarily quasiambivalent. The theorem is used to demonstrate that real coupling coefficients do not exist for the point-group hierarchies T ? D₂ and I ? T or for the double-group hierarchies I* ? D₃*, I* ? D₅*, and O* ? D₃*.     

Optimal grids for generalized finite basis and discrete variable representations: definition and method of calculation

The method of optimal generalized finite basis and discrete variable representations (FBR and DVR) generalizes the standard, Gaussian quadrature grid-classical orthonormal polynomial basis-based FBR/DVR method to general sets of grid points and to general, nondirect product, and/or nonpolynomial bases. Here, it is shown how an optimal set of grid points can be obtained for an optimal generalized FBR/DVR calculation with a given truncated basis. Basis set optimized and potential optimized grids are defined. The optimized grids are shown to minimize a function of grid points derived by relating the optimal generalized FBR of a Hamiltonian operator to a non-Hermitian effective Hamiltonian matrix. Locating the global minimum of this function can be reduced to finding the zeros of a function in the case of one dimensional problems and to solving a system of D nonlinear equations repeatedly in the case of D>1 dimensional problems when there is an equal number of grid points and basis functions. Gaussian quadrature grids are shown to be basis optimized grids. It is demonstrated by a numerical example that an optimal generalized FBR/DVR calculation of the eigenvalues of a Hamiltonian operator with potential optimized grids can have orders of magnitude higher accuracy than a variational calculation employing the same truncated basis. Nevertheless, for numerical integration with the optimal generalized FBR quadrature rule basis optimized grids are the best among grids of the same number of points. The notions of Gaussian quadrature and Gaussian quadrature accuracy are extended to general, multivariable basis functions.     

Organometallic chemistry in a conventional microwave oven: the facile synthesis of group 6 carbonyl complexes

Syntheses proceeding by reflux may be improved, accelerated and simplified, by carrying out the reaction in a modified conventional microwave oven. To demonstrate the potential of this method, the synthesis of over 20 group 6 organometallic compounds is reported. Hexacarbonyls, most notably Mo(CO)₆, react with a range of mono, and bi, and tridentate ligands in a modified conventional microwave oven. They generally proceed without an inert atmosphere, yields are high and reaction times are short. For example, cis-[Mo(CO)₄(dppe)] is prepared in >95% yield in 20 min. Reaction of Mo(CO)₆ with dicyclopentadiene affords a simple one-step synthesis of [CpMo(CO)₃]₂ in >90% yield, which reacts further with alkynes in toluene to produce dimetallatetrahedrane derivatives, [Cp₂Mo₂(CO)₄(μ-RC₂R)]; presumably via the in situ formation of air-sensitive [CpMo(CO)₂]₂. Dimolybdenum tetra-acetate is also prepared in 48% yield in 45 min, however, this reaction requires an inert atmosphere. While W(CO)₆ reacts rapidly with amines to give cis diamine adducts in high yields, direct reactions with phosphines are not so clean. Bis(phosphine) complexes are, however, cleanly formed when a small amount of piperidine is added to the reaction mixture, presumably via the bis(piperidine) complex cis-[W(CO)₄(pip)₂]. Reactions with Cr(CO)₆ generally require an inert atmosphere and proceed less cleanly, although the important synthon [Cr(CO)₅Cl][NEt₄] was prepared in 30 min (74% yield), while [(η⁶-C₆H₅OMe)Cr(CO)₃] can be prepared in 45% after 4 h.     

Bases for the irreducible representations of O(3) symmetry adapted to a crystallographic point group

We construct bases for the irreducible representations of the rotation group O(3) which are symmetry adapted to a Crystallographic point group. We obtain explicit expressions for the cubic groups, which are valid for arbitrary values of the angular momentum quantum number l. Our method yields an efficient algorithm for both analytical and numerical work. An explicit formula for the multiplicities of an irreducible representation for the cubic groups in an arbitrary angular momentum term l is also derived.     

A group additivity approach for the estimation of heat capacities of organic liquids and solids at 298 K

A group additivity method is described which provides heat capacity estimates of the condensed phase. The data base consists of 810 liquids and 446 solids. Group values for carbon in various common substitution and hybridization states and for 47 functional groups are provided. The standard error of estimation using this approach on this data base is 19.5 (liquids) and 26.9 J/ (mole K) (solids). This can be compared to typical experimental uncertainties of 8.12 and 23,4 J/ (mole K) associated with these measurements, respectively. Experimental uncertainties were estimated from the numerical differences obtained for a given substance from multiple independent literature reports.     

A simple proof for the formula to get symmetrized powers of group representations

A general formula to decompose the p-power of irreducible representations of an arbitrary space group into sum of sets of irreducible representations of such a group, having identical permutational symmetry, is presented. Its proof is based upon a straightforward application of the properties of the generalized projection (shift) operators. © 1993 John Wiley & Sons, Inc.     

Calculation of normal boiling points for alkane isomers by a second order group contribution method

Second order groups are introduced into the group contribution method proposed by Tu to calculate the normal boiling points of alkanes from C₁ to C₁₀. The present method can well distinguish between the normal boiling points of alkane isomers. The absolute average error is 1.24 K.