Expression of theSU(2) rotation matrices D( j ) in terms of normalized irreducible tensorial matrices

TheSU(2) rotation matricesD ^(j), specified in terms of axis and angle of rotation, are expressed as linear combination of normalized irreducible tensorial matricces (NITM) of rankl = 0 to 2j rotated to the polar angles of the axis. The rotated NITM are constructed from spherical harmonics of the same rank. Since this formulation requires no matrix products, it may be computationally more efficient than Euler angle formulas, particularly for largej. Rotated NITM and formulas for theD ^(j) withj = 1/2 andj = 1 are written out explicitly. A formula for the structure constants of the products of conformable NITM is also given in terms of 3-j and 6-j symbols.     

On rotations as special cases of unitary transformations with some applications to the theory of spin

Some problems in elementary geometry are approached from the point of view of linear algebra and generalized to the theory of linear spaces of finite or infinite dimensions having a positive definite binary product. The angle ω between two elements of the linear space is defined from the concept of length by means of the cosine-theorem. A rotation is then defined as a special case of a unitary transformation moving all elements the same angle ω, except that under certain circumstances, some elements may stay invariant. In the former case, one speaks of a rotation around an “external axis,” and in the latter case, of a rotation around an “internal axis” defined by the invariant elements. It is shown that the finite rotations U of both types may be expressed in the simple exponential form U = exp(iωm), where the “generator” m in the former case is an operator satisfying the relation m² = 1, and in the latter case, m³ = m. The structure of the group of finite rotations in the former case is clarified in some detail. As an illustration of the theory, some applications to the three- and two-dimensional spaces as well as to the theory of spin are given. The coupling between the ordinary three-dimensional rotations and the spinor transformations is considered in somewhat greater detail.     

Graphical representation of the state of orientation of polymer systems

The orientation distribution function for noncrystalline structural units in polymer systems cannot be determined completely from any experimental source; only the second and/or fourth moments of the distribution function, i.e., the second and/or fourth orders of the generalized orientation factors F_lm^j, can be evaluated. It is there-fore necessary to estimate the distribution function from F_2m^j and F_4m^j. In this paper, a graphical representation of the state of orientation is first discussed in terms of plots of F₄₀^j against F₂₀^j for several types of distribution functions for uniaxial orientation. These are three types of extreme concentration of the distribution at particular polar angles θ₀ given by θ₀ = 0, 0<θ₀<π/2, and θ₀ = π/2; five types of rather realistic distributions having single maxima at θ_j = 0, θ₀, π/2 and double maxima at θ_j = 0, π/2, and a single minimum at θ_j = θ₀; and four types of more realistic distributions including Kratky's floating rod model in an affine matrix. Second, estimation of the distribution function for uniaxial orientation from F₄₀^j and F₂₀^j is discussed quantitatively in terms of the mean-square error by three approximation methods: (a) expansion of the distribution function in finite series of spherical harmonics through the fourth order, (b) approximation of the distribution function as a composite of two components, random orientation and a particular orientation distribution given by N_a (cos²θ_j)^a, N_a being a constant, and (c) approximation of the distribution function by N_a (cos²θ_j)^a alone. It is concluded that when the orientation distribution is sharp, estimation by the second method of approximation gives a smaller error than the first.     

On the closed form of Wigner rotation matrix elements

The closed forms of some rotation matrix elementsd _{m m} ^j (/2) are presented. The closed forms of summation involved two binomials and some special hypergeometric functions are also obtained. The MAPLE V program which calculates d _{m m} ^j (), dm,. (/2) and the help file are given in appendix.Supported in part by the Chiu Feng-Chia research Fund. Thanks are due to the Board of Trustees Chairman Dr. Ying-Ming Liao as well as the Trustee Dr. Charles Chiu-Hsiong Huang and President Dr. Ted C. Yang of Feng-Chia University, Taichung, Taiwan, ROC.     

Common vertex matrix: A novel characterization of molecular graphs by counting

We present a novel matrix representation of graphs based on the count of equal‐distance common vertices to each pair of vertices in a graph. The element (i, j) of this matrix is defined as the number of vertices at the same distance from vertices (i, j). As illustrated on smaller alkanes, these novel matrices are very sensitive to molecular branching and the distribution of vertices in a graph. In particular, we show that ordered row sums of these novel matrices can facilitate solving graph isomorphism for acyclic graphs. This has been illustrated on all undecane isomers C₁₁H₂₄ having the same path counts (total of 25 molecules), on pair of graphs on 18 vertices having the same distance degree sequences (Slater's graphs), as well as two graphs on 21 vertices having identical several topological indices derived from information on distances between vertices. © 2013 Wiley Periodicals, Inc.     

Algebraic solutions for the octahedral group: Group chain O⊃C4

Using double‐induced representation and eigenfunction method, algebraic expressions are derived for irreducible matrices, projection operators, and symmetry‐adapted functions in the group chain O⊃C₄ for both single‐valued and double‐valued representations. The simplicity of these expressions lies in the fact that they are functions of the quantum numbers of the corresponding group chain (the analogy of j, m for the group chain SO₃⊃SO₂) instead of the irreducible matrix elements. The symmetries of the symmetry‐adapted functions are disclosed. ©1999 John Wiley & Sons, Inc. Int J Quant Chem 74: 7–22, 1999     

Bestimmung der Stabilit?tskonstanten von mehrkernigen Komplexen mit Hilfe einer spektrophotometrischen Methode

Zusammenfassung Es wurde eine allgemeine Gleichung für die Berechnung der Stabilitätskonstante von Komplexen L _mH_iM_nZ_j (Z=OH^–, usw.) abgeleitet und für Maximumbedingungen der Jobschen Kurve die Schwarzenbachsche graphische Methode so modifiziert, daß sie die allgemeine Bestimmung der Zusammensetzung und Stabilität von Komplexen L _mH_iM_nZ_j ermöglicht.

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Summary A general equation has been derived for the calculation of stability constants of complexes L _mH_iM_nZ_j (Z=OH^–, etc.). The graphic method of Schwarzenbach has been modified for maximum conditions of Job's curve in order to make possible the general determination of the composition and stability of complexes L _mH_iM_nZ_j.

     

Polymorphy in the stereostructure of periodic polymer backbones

We theoretically investigated the polymorphy of the stereostructures of a periodic polymer. Using the polymer's internal conformation parameters of bond lengths, bond angles, and internal rotation angles, we extended the mathematical treatment for designing polymer backbones. We considered those periodic polymers whose unit cell consists of one (m = 1), two (m = 2), or three (m = 3) kinds of atoms. Moreover, for these three types of polymers, we supposed two catenation types for the skeleton atoms; one is a “homorotatory” sequence and the other is a “heterorotatory” one. To specify the backbone's stereostructure, we introduced several conformation parameters such as the helical pitch number n, the translation distance d, and the inclination angle of the skeleton plane Θ. By combining these parameters, we can systematically understand the variety and the possible polymorphy in the stereostructure of a periodic polymer backbone. © 2003 Wiley Periodicals, Inc. J Polym Sci Part B: Polym Phys 41: 2829–2849, 2003     

A study of reorientation effects in CsF—Ar using approximate quantum methods

The EB (exponential Born) and IOS (infinite order sudden) methods are used to calculate a variety of j → j′ and jm → j′m′ integral cross sections and j → j′ differential cross sections for CsF—Ar at E_cm = 87.7 meV. Inelastic rotational cross sections are found to depend primarily on the odd long range parts of the interaction potential. The jm → j′m′ integral cross sections for the quantization axis parallel to the initial relative velocity vector are found to approximately obey the selection rule “Δj + Δm = even” and cross sections for which the orientation of the rotor is left unchanged (i.e. 11 → 11, 11 → 22, and 11 → 33 transitions) are strongly favored over those which are changed for low lying rotational states. Good agreement between the IOS and EB methods is found for most of the scattering quantities calculated.     

Construction of orthogonal subspaces

An orthogonalization procedure is presented that allows construction of at least (n?m) vectors orthogonal to {X_j}, j equals; 1, m, by linear combinations solely among {η_i}, i equals; 1, n, n>m, and 〈X_j/η_i〉≠0. An important application of the procedure is in effective core potential methods for which valence orbitals can be constructed that are orthogonal to the core orbitals and yet involve no component of the core. Thus, a separate calculation for only the valence electrons can be performed without any explicit reference to the core electrons (orbitals).     

Quantum dynamics of dissociative adsorption of H2 and D2: The time-dependent wave packet method

A time-dependent quantum wave packet method was used to study the dynamics of dissociative adsorption of H₂ and D₂ on a flat and static surface. The molecule-surface interaction is described using a modified London-Eyring-Polanyi-Sato (LEPS) type potential for the H₂/Ni(100) system. The three-dimensional (3-D) dissociation probabilities were calculated for different initial rovibrational states as a function of initial incident energies. Our results show that the dissociation of the diatomic rotational states whose quantum numbers satisfyj+m = odd is forbidden at low energies for the homonuclear H_z and D₂ due to the selection rule. The effect of the rotational orientation of diatoms on adsorption predicts that the in-plane rotation (m = j) is more favorable for dissociation than the out-of-plane rotation (m = 0). Enhanced dissociation for vibrationally excited molecules and the significant enhancement of the dissociation probability of H2 when compared to D₂ were explained reasonably in terms of quantum mechanical zero-point energies, the tunneling effect and the reflection from an activation barrier. Project supported by the National Natural Science Foundation of China (Grant No. 19694033) and partially by the Science Foundation for Overseas Chinese Scholars and Students, administered by the State Education Commission of China (Grant No. 1992), by the State Key Laboratory of Theoretical and Computational Chemistry of Jilin University at Changchun (Grant No. 98011, and by the Natural Science Foundation of Shandong Province (Grant No. Y96B03022)     

Computation of electronic molecular polarizabilities by a variational method at the CISD level

The variational method proposed earlier has been generalized, using a trial function of the form: ψ = (Λ₀ + Σ_sΛ_sm_s)ψ₀ in which m_s = r^2p+1C^m_l, s standing for a triplet (p, l, m) and implemented into the program Hondo-8.4. The second-order density matrices are used to take into account the mono and bi-excited states (DM1 and DM2 matrices, GUGA Algorithm). This allows us to compute the dipole (α), dipole-quadrupole (A) and quadrupole (C) polarizability tensors at the CISD level. The results obtained for a series of test molecules: CO, HF, NH₃, and methane with various gaussian basis sets are compared with experimental results (when available for A and C) and those obtained with other theoretical methods. The correlation is found to lower the values of the dipole polarizability which was generally too high when computed by the variational method at the RHF level and the values obtained here are in good agreement with the experimental ones. © 1996 John Wiley & Sons, Inc.     

V-Coupling coefficients from SO(3) to the octahedral group

The properties of V-coupling coefficients from SO(3) to the octahedral group are discussed and, for the purpose of practical use in ligand field theory, the numerical values from j = 1/2 to j = 12 1/2 have been worked out through an Elliott-503 computer. The values for j ≤ 4 1/2 are listed.     

An experimental investigation of electromechanical coupling in cholesteric liquid crystals

Abstract

Cholesteric liquid crystals which have a helical arrangement of oriented chiral molecules are expected to show novel cross couplings between fluxes and forces. The most convincing demonstration of these couplings is through a dynamical effect on the structure. Though Lehmann found a rotation of the structure under a temperature gradient in 1900, there has been no subsequent experiment confirming the same. We argue that it is very difficult to obtain a sufficiently weak anchoring of the director at the solid-cholesteric interface which is a necessary condition for the occurrence of Lehmann rotation. In order to achieve a practically zero anchoring energy at the surface, we have devised a simple technique of floating essentially flat cholesteric drops in the isotropic phase. Using this configuration we study the electromechanical coupling which produces a rotation of the structure under the action of a DC electric field. Using measurements on samples with different values of the pitch the relevant electromechanical coefficient of the materials investigated is found to satisfy the relation v _E = ?0·6 × 10^?12(q/m^?1)J m^?2, where q is the wavevector of the helix, whose sign is positive (negative) for a right (left) handed structure, confirming that v _E is hydrodynamic in origin.     

Degeneracy-averaged and polarized cross sections for rotational scattering in (Ar,LiH)

Close-coupling calculations at 500 cm^?1 for the system (Ar,LiH) are used to obtain scattering amplitudes and cross sections for the rotational transitionsj = 1 to j'= 0, 2, 3, 4, 5,j = 2 to j' = 3 and j = 3 to j'= 4. The degeneracy-averaged cross sections are in reasonable agreement with experiment for two different simple model potentials. Values for m-dependent cross sections with respect to two frames of reference are compared with various rules for Δm and with the previous results for (Ar,LiH).     

Distorted-wave calculations for the three dimensional chemical reaction H + H2(v ⩽ 2, j = 0) → H2(v′ ⩽ 2, j′, mj) + H

Calculations of the vibrational—rotational product state population distributions and differential cross sections for the chemical reaction H + H₂(v ? 2, j = 0) → H₂(v′ ? 2, j′, m_j) + H have been carried out on the Porter—Karplus potential energy surface. The vibrationally-adiabatic-distorted-wave (VADW) method has been used. The relative rotational product distributions, differential cross sections and the helicity m_j, dependences of these quantities for the v = 0 reaction agree well with accurate close-coupling results. The absolute integral cross sections are considerably smaller than the accurate quantum values, however. The calculations for the v = 1 reaction agree with the findings of previous sudden quantum, limited close-coupling and quasiclassical theoretical studies and experiments that product H₂(v′ = 1) is more likely to be produced than H₂(v′ = 0). For the reaction with v = 2, it is found that at high translational energies product H₂(v′ = 2) is favoured over H₂(v′ = 1) or H₂(v′ = 0). The VADW differential cross sections for the v = 1 and v = 2 reactions have a similar shape to those of the v = 0 reaction, with backward peaking when summed over all m_j states. The relative rotational distributions for the v = 2, j = 0 → v′ = 2, j and v = 1, j = 0 → v′ = 1, j reactions are also similar to those obtained for the v = 0, j = 0 → v′ = 0, j reaction, with low rotational excitation.     

Symmetrization of eigenfunctions of angular momentum in point groups

The eigenfunctions |jm〉 of angular momentum can combine linearly to make basis functions of irreducible representations of point groups. We surmount the projection operator and find a new method to calculate the combination coefficients. It is proven that these coefficients are components of eigenvectors of some hermitian matrices, and that for all pure rotation point groups, the coefficients can be made real numbers by properly choosing the azimuth angles of symmetry elements of point groups in the coordinate system. We apply the coupling theory of angular momentum to obtain the general formulas of the basis functions of point groups. By use of our formulas, we have calculated the basis functions with half‐integers j from 1/2 to 13/2 of double‐valued irreducible representations for the icosahedral group. © 2001 John Wiley & Sons, Inc. Int J Quant Chem 83: 286–302, 2001     

Azeotropy in Terpolymerization

This work deals with the determination of the various azeotropies in radical terpolymerization: unitary azeotropy (m_i = M_i); binary azeotropy (m_i/m_j. = M_i/M_j), and ternary or true azeotropy (M_j = m_j for j = 1, 2, 3). General analytical solutions are given which are easily handled by a computer. In addition, these calculations show the existence of pseudo-azeotropy where the deviation between polymer and monomer feed compositions is very small, not only at the beginning of the reaction, but also with increasing conversion.     

Rheo-optical studies on the deformation mechanism of semicrystalline polymers. XIV. Alpha and beta mechanical dispersions of spherulitic high-density polyethylene and the dynamic orientation distribution function of crystallites

The dynamic tensile deformation mechanism of spherulitic high-density polyethylene was investigated by dynamic x-ray diffraction at various temperatures and frequencies in order to assign the α and β mechanical dispersions explicity. The uniaxial orientation distribution function q(ζ_j,0) of the jth crystal plane and its dynamic response Δq′_j(ζ_j,0) in phase with dynamic strain were observed for the (110), (200), (210) and (020) crystal planes. Then the orientation distribution function w(ζ,0,η) of crystallites (crystal grains) and its dynamic response Δw′(ζ,0,η), also in phase with the dynamic strain, were determined by a mathematical transformation procedure proposed by Roe and Krigbaum on the basis of the Legendre addition theorem. The temperature and frequency dependences of w′(ζ,0,η) were analyzed in terms of the model parameters for dynamic spherulite deformation combining affine orientation of crystal lamellae with several types of preferential reorientation of the crystal grains within the orienting lamellae. The following assignments are made: (i) The α mechanical dispersion must be assigned to the dynamic orientation dispersion of crystal grains within the crystal lamellae, involving two types of preferential rotations of the grains about their own crystal b and a axes. The rotation about the b axis is associated with lamellar detwisting, mostly in the equatorial zone of uniaxially deformed spherulites; the rotation about the a axis is associated with intralamellar shearing, mostly in the polar zone of the spherulites. Thus both rotations are intralamellar grain-boundary phenomena. (ii) The β mechanical dispersion must be assigned to the dynamic orientation dispersion of the crystal lamellae behaving as rigid bodies. It is not accompanied by reorientation of the crystal grains, but is associated with orientation dispersion of noncrystalline material between the lamellae. Thus it is an interlamellar grain-boundary phenomena.