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.     

Permutational–symmetry-adapted powers of representations of the point groups deduced from the unitary group

The construction of symmetrized powers of representations of the point groups can be deduced by decomposing the appropriate representation of the unitary group into the representations of the group of the sphere, and then mapping these representations onto the point group. The method is generally simpler than the traditional method based on the character tables of the symmetric group. As an example, the symmetry-adapted powers (N) of the representations of the icosahedral group are presented for 2≤N≤5.     

On the representations of the rotation group

The matrices of the irreducible representations of the 3-dimensional rotation group are shown to be related to Krawtchouk's orthogonal polynomials of a discrete variable x = j – m', whose degrees are given by n = j + m. The relation follows directly from the recurrence formulas satisfied by the matrix elements and permits a concise development of the formal properties of the rotation matrices. In particular, an asymptotic relation for large j is developed that generalizes a formula first discussed for a special case by Wigner.     

A symmetrized Hückel analysis of carbon cages of Ih point group symmetry

Some new aspects of qualitative molecular-orbital theory are developed within the context of group theory and a symmetrized Hückel approximation. Conclusions concerning the eigenvalue sums for sets of eigenfunctions of the same symmetries, the occurrence of repetitions of symmetry-adapted projections and the relations between eigenvalues of Γ_g and Γ_u symmetries are exemplified using the ₁₂₀ cage structure of I_h point symmetry. The implications of the orbit-by-orbit approach are outlined for larger clusters. This revised version was published online in July 2006 with corrections to the Cover Date.     

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.     

Relativistic double group spinor representations of nonrigid molecules

The character theory of relativistic double group spinor representations is developed in order to represent the total rovibronic states of nonrigid molecules. It is shown that the double groups can be represented in terms of wreath products and powerful matrix cycle type generators that are used to construct their character tables. It is shown that these tables are of use when spin-orbit coupling is included in the Hamiltonian even for molecules containing lighter atoms. Applications to nonrigid molecules such as Tl2H4/Tl2H4+ are considered. It is shown that the tunneling splittings and the nuclear spin statistical weights can be obtained for such species using the character tables thus constructed. The spinor double groups of several other molecules such as hexamethyl dilead and heavy weakly bound clusters such as (PoH2)4 are also considered.     

Graphical unitary group approach to arbitrary spin representations

A graph theoretic approach to the representation of N particle systems involving arbitrary single-particle spin is presented. The method is a generalization of the Distinct Row Table (DRT ) technique employed by Shavitt in developing the graphical unitary group approach (GUGA ) to electronic spin-orbitals. A detailed analysis of the DRT and GUGA representations is presented based on several theoretical considerations and computer-tested implementations. The use of the representation to establish rules concerning the evaluation of the matrix elements of the group generators is also discussed.     

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.     

The self-consistent symmetrized OPW method with an application to crystalline selenium

A review of the nonrelativistic self-consistent symmertrized orthogonalized-plane-wave (SCSOPW ) method used for determining electronic energy bands in periodic solids is given. Working equations based on the full use of group theory at all stages are presented for elemental crystals. As an example the method is applied to the trigonal selenium crystal.     

A graphical approach to permutation group representations for many-electron systems

A simple procedure is presented for obtaining the standard Young tableaux for the representation [(N/2) + S,(N/2) − S] of the permutation group ℒ_N for an N-electron system in spin state S directly from the spin branching diagram. We redefine the coordinate axes of the branching diagram to obtain a graph in terms of the partitions of the two-rowed Young diagram and define walks in this graph which yield directly the first rows of the allowed standard Young tableaux spanning a given representation when suitable weights have been assigned to the nodes in the graph. The allowed states are in a lexically ordered form and permit going easily from an index to an array and vice versa. © 1996 John Wiley & Sons, Inc.     

Dominant representations and a markaracter table for a group of finite order

Summary The concept of markaracter is proposed to discuss marks and characters for a group of finite order on a common basis. Thus, we consider a non-redundant set of dominant subgroups and a non-redundant set of dominant representations (SDR), where coset representations concerning cyclic subgroups are named dominant representations (DRs). The numbers of fixed points corresponding to each DR are collected to form a row vecter called a dominant markaracter (mark-character). Such dominant markaracters for the SDR are collected as a markaracter table. The markaracter table is related to a subdominant markaracter table of its subgroup so that the corresponding row of the former table is constructed from the latter. The data of the markaracter table are in turn used to construct a character table of the group, after each character is regarded as a markaracter and transformed into a multiplicity vector. The concept of orbit index is proposed to classify multiplicity vectors; thus, the orbit index of each DR is proved to be equal to one, while that corresonding to an irreducible representation is equal to zero.     

The effect of the position of the methyl group on the helical twisting powers of aldol condensation products of methyl cyclohexanones

The four aldol condensation products of methylcyclohexanone isomers and 4-(4-hexyloxybenzoyloxy)benzaldehyde have been prepared. These chiral products were either obtained with high enantiomeric excess or the enantiomers were separated by means of chiral HPLC. In all cases only the E-isomer was obtained. UV irradiation was used to isomerize the E-isomers to the Z-isomers. Comparison of the helical twisting powers (HTP) of the E-isomers revealed that the derivative in which the methyl group is directly next to the double bond exhibits the highest value. Although the other three E-isomers have lower HTP values, they show large HTP changes and reversal of the helical twist sense upon irradiation. Of these compounds, the one derived from the commercially available (R)-3-methylcyclohexanone is an interesting candidate for use as an alternative to compounds derived from menthone for realizing photochemically induced twist changes in twisted nematic and cholesteric materials.     

Construction and applications of symmetrized valence bond wave functions

A method constructing symmetry-adapted bonded Young tableau bases is proposed, based on the symmetry properties of bonded tableaus and the projection operator associated with a point group. Several examples including the ground states and π excited states of O₃⁻, O₃, O₃⁺, and C₃⁻ are shown for instruction to construct the symmetrized valence bond (VB) wave function. Excitation energies of transitions from the ground states to π excited states of O₃⁻, C₃H₅, and C₃⁻ are calculated with an optimized symmetrized valence bond wave function in the σ–π separation approximation. Good agreement between the VB and experimental excitation energies is observed. The bonding features of the ground state and the first π excited singlet and triplet states for S₃ are discussed according to bonding populations from VB calculations. Both the singlet-biradical and the dipole structures have significant contributions to the ground state X ¹A₁ of S₃, while the excited state 1 ¹B₂ is essentially composed of the dipole structures, and the 1 ³B₂ excited state is comprised from a triplet-biradical structure. © 1998 John Wiley & Sons, Inc. Int J Quant Chem 66 : 1–7, 1998     

The effect of the position of the methyl group on the helical twisting powers of aldol condensation products of methyl cyclohexanones

The four aldol condensation products of methylcyclohexanone isomers and 4-(4-hexyloxybenzoyloxy)benzaldehyde have been prepared. These chiral products were either obtained with high enantiomeric excess or the enantiomers were separated by means of chiral HPLC. In all cases only the E-isomer was obtained. UV irradiation was used to isomerize the E-isomers to the Z-isomers. Comparison of the helical twisting powers (HTP) of the E-isomers revealed that the derivative in which the methyl group is directly next to the double bond exhibits the highest value. Although the other three E-isomers have lower HTP values, they show large HTP changes and reversal of the helical twist sense upon irradiation. Of these compounds, the one derived from the commercially available (R)-3-methylcyclohexanone is an interesting candidate for use as an alternative to compounds derived from menthone for realizing photochemically induced twist changes in twisted nematic and cholesteric materials.     

Transformation matrices for representations of the spin permutation group. A graphical approach

Transformation matrices for the representations of the spin permutation group may be easily derived using graphical methods of angular momentum theory. The transformation from the geneological spin basis to the Jahn-Serber basis is considered as an illustrative example.     

Characters for symmetric and antisymmetric higher powers of representations: Application to the number of anharmonic force constants in symmetrical molecules

Explicit formulas are given for the characters of symmetric and antisymmetric powers of an arbitrary representation up to the sixth, and a general method for obtaining the higher ones is described. The results allow, among others, the determination of nonvanishing higher force constants in symmetrical molecules. The benzene molecule, for instance, has 237 nonvanishing cubic and 1890 quartic force constants. Other potential applications are a general method for the symmetry species of vibrational overtones, the determination of the number of independent centrifugal distortion constants, and the symmetry classification of vibrational multiplets.     

A labeling scheme for young tableaux spanning representations of permutation group S(N)

A structure‐dependent labeling scheme for the Standard Young Tableaux spanning the representations of the permutation group is outlined in the present work. This scheme is used to generate the representations of a select class of permutations such as dense cycles and general transpositions of the group using minimal storage requirements. Two distinct approaches are outlined for generating the tableaux in the present labeling scheme. Detailed application has been made to two‐column Young diagram representations that are extremely useful in electron correlation studies in molecules. © 2000 John Wiley & Sons, Inc. J Comput Chem 21: 185–190, 2000     

Path integral based calculations of symmetrized time correlation functions. I

In this paper, we examine how and when quantum evolution can be approximated in terms of (generalized) classical dynamics in calculations of correlation functions, with a focus on the symmetrized time correlation function introduced by Schofield. To that end, this function is expressed as a path integral in complex time and written in terms of sum and difference path variables. Taylor series expansion of the path integral's exponent to first and second order in the difference variables leads to two original developments. The first order expansion is used to obtain a simple, path integral based, derivation of the so-called Schofield's quantum correction factor. The second order result is employed to show how quantum mechanical delocalization manifests itself in the approximation of the correlation function and hinders, even in the semiclassical limit, the interpretation of the propagators in terms of sets of guiding classical trajectories dressed with appropriate weights.     

Path integral based calculations of symmetrized time correlation functions. II

Schofield's form of quantum time correlation functions is used as the starting point to derive a computable expression for these quantities. The time composition property of the propagators in complex time is exploited to approximate Schofield's function in terms of a sequence of short time classical propagations interspersed with path integrals that, combined, represent the thermal density of the system. The approximation amounts to linearization of the real time propagators and it becomes exact with increasing number of propagation legs. Within this scheme, the correlation function is interpreted as an expectation value over a probability density defined on the thermal and real path space and calculated by a Monte Carlo algorithm. The performance of the algorithm is tested on a set of benchmark problems. Although the numerical effort required is considerable, we show that the algorithm converges systematically to the exact answer with increasing number of iterations and that it is stable for times longer than those accessible via a brute force, path integral based, calculation of the correlation function. Scaling of the algorithm with dimensionality is also examined and, when the method is combined with commonly used filtering schemes, found to be comparable to that of alternative semiclassical methods.