General description of orientation factors in terms of expansion of orientation distribution function in a series of spherical harmonics

A mathematical representation of orientation distribution of structural units within the bulk polymer is given in terms of an expansion of the distribution function in a series of spherical harmonics. Each coefficient of the expanded series is discussed in general relation to the orientation factors, average degrees of orientation distribution, defined by several different authors independently. Several optical techniques to evaluate the orientation factors, the second and fourth moments of orientation distribution of crystalline and noncrystalline structural units from optical dichroic quantities, are discussed. Some graphical representations of the state of orientation are proposed, and the estimation of orientation distribution from the orientation factors of different orders is discussed quantitatively.     

General description of optical dichroic orientation factors for relating optical anisotropy of bulk polymer to orientation of structural units

The relationship between the optical anisotropy of high polymeric materials in bulk and the orientation of structural units within the materials was described in general by using several types of mean values of the orientation distribution function of three Eulerian angles, i.e., the orientation factors, under some assumptions about the symmetry of the function being applicable for the most of the industrial products. A newly defined biaxial orientation factor, F_θηⁱ = 〈sin² θ_j cos 2η_j〉, where θ_j and η_j are the polar and azimuthal angles of the jth axis within the structural unit with respect to the bulk axes, may relate the biaxial orientation of the structural units to the dichroic orientation factors, which are measurable optical anisotropic indices of the bulk materials. Some applications of the results to the birefringence and infrared and dye dichroism are also discussed.     

Internal‐to‐Cartesian back transformation of molecular geometry steps using high‐order geometric derivatives

In geometry optimizations and molecular dynamics calculations, it is often necessary to transform a geometry step that has been determined in internal coordinates to Cartesian coordinates. A new method for performing such transformations, the high‐order path‐expansion (HOPE) method, is here presented. The new method treats the nonlinear relation between internal and Cartesian coordinates by means of automatic differentiation. The method is reliable, applicable to any system of internal coordinates, and computationally more efficient than the traditional method of iterative back transformations. As a bonus, the HOPE method determines not just the Cartesian step vector but also a continuous step path expressed in the form of a polynomial, which is useful for determining reaction coordinates, for integrating trajectories, and for visualization. © 2013 Wiley Periodicals, Inc.     

Mechanical anisotropy of regenerated cellulose films in relation to orientations of crystalline and noncrystalline phases

The mechanical anisotropy of regenerated cellulose films is investigated, first, on the basis of the theory of infinitesimal elasticity. Fairly good agreement of calculated with observed results is obtained on the basis of orthogonal anisotropy with respect to the machine direction and the transverse and thickness directions of the films. The shear modulus G₂₃ along the film plane and the Poisson ratio v₃₂ are 1.5 times; 10² kg/mm² and about 0.4, respectively, in the standard dry state. Second, the mechanical anisotropy in three different dry states is analyzed in terms of the degree of biaxial orientation of two kinds of structural units, cellulose II crystallites and noncrystalline chain segments, and their mechanical anisotropy. The calculation for averaging the mechanical anisotropies of these structural units on the basis of the homogeneous strain hypothesis gives results much higher than the experimental data, whereas the calculation on the basis on the homogeneous stress hypothesis gives results rather lower than experiment. As a modification of the two extreme calculations, a different averaging gives considerably better agreement between the calculated and observed results. The mechanical anisotropy in the wet state is further analyzed primarily in terms of the degree of biaxial orientation of noncrystalline chains by a modification of Krigbaum treatment, based on application of the kinetic theory of entropy elasticity for semicrystalline polymers, to anisotropic systems. The calculation gives results, however, much lower than those obtained experimentally, unless the ratio of the end-to-end distance of the noncrystalline chain to its fully stretched length is taken as unusually large. This may be due to underestimation of the contribution of the crystalline phase to terms of the same type as appear in the Krigbaum treatment.     

Parametric analysis of infrared intensities

A parametric model of the intensities in the infrared spectra employing molecular polar parameters related with vibrational distortions of separate valence bonds is described. The parameters represent derivatives of the Cartesian components of the total dipole moment with respect to linear and angular coordinates describing the changes in length and orientation of each bond in a molecule. Matrix formulation is used throughout the mathematical procedure. The analysis results in determination of bond parametric vectors from the intensities of stretching modes, and bond parametric (3×3) matrices from the intensities of deformation modes. The application of the model in the interpretation of experimental infrared intensity data is discussed.     

Rotational symmetry of the molecular potential energy in the Cartesian coordinates

We consider the molecular Born-Oppenheimer potential energy as a function of atomic Cartesian coordinates and discuss the non-stationary Hessian properties arising due to rotational symmetry. A connection with the extended Hessian theory is included. New applications of Cartesian representation for examining and correcting raw numerical Hessian data and a simple formulation of harmonic vibrational analysis of partially optimized systems are proposed. Exemplary calculations for the porphyrin molecule with an internal proton transfer are reported. We also develop the normal transformation method to incorporate the rotational symmetry into the approximate analytical potentials, which are parametrized in the Cartesian coordinates. The transformation converts the coordinates from the space fixed frame to the frame which translates and rotates with the molecule and is determined by the Eckart conditions. New simple analytical formulas for the first and second derivatives of the transformed potential are derived. This fast method can be used to calculate the potential and its derivatives in the simulations of chemical reaction dynamics in the space fixed Cartesian frame without the need to constrain the molecular rotation or to define the local non-redundant internal coordinates.     

First and second derivative matrix elements for the stretching,bending, and torsional energy

Matrix elements for the first and second derivatives of the internal coordinates with respect to Cartesian coordinates are reported for stretching, linear, nonlinear, and out-of-plane bending and torsional motion. Derivatives of the energy with respect to the Cartesian coordinates are calculated with the chain rule. Derivatives of the energy with respect to the internal coordinates are straightforward, but the calculation of the derivatives of the internal coordinates with respect to the Cartesian coordinates can be simplified by the following two steps outlined in this article. First, the number of terms in the analytical functions can be reduced or will vanish when the derivatives of the bond length, bond angle, and torsion angle are reported in a local coordinate system in which one bond lies on an axis and an adjacent bond lies in the plane of two axes or is projected onto perpendicular planes for linear and out-of-plane bending motion. Second, a simple rotation transforms these derivatives to the appropriate orientation in the space-fixed molecular coordinate system. Functions of the internal coordinates are invariant with respect to translation and rotation. The translational invariance and the symmetry of the second derivatives for a system with L atoms are used to select L-1- and L(L-1)/2-independent first and second derivatives, respectively, of which approximately half of the latter vanish in the local coordinate system. The rotational invariance permits the transformation of the simplified derivatives in the local coordinate system to any orientation in space. The approach outlined in this article simplifies the formulas by expressing them in a local coordinate system, identifies the most convenient independent elements to compute, from which the dependent ones are calculated, and defines a transformation to the space-fixed molecular coordinate system.     

Techniques for geometry optimization: A comparison of cartesian and natural internal coordinates

A comparison is made between geometry optimization in Cartesian coordinates, using an appropriate initial Hessian, and natural internal coordinates. Results on 33 different molecules covering a wide range of symmetries and structural types demonstrate that both coordinate systems are of comparable efficiency. There is a marked tendency for natural internals to converge to global minima whereas Cartesian optimizations converge to the local minimum closest to the starting geometry. Because they can now be generated automatically from input Cartesians, natural internals are to be preferred over Z-matrix coordinates. General optimization strategies using internal coordinates and/or Cartesians are discussed for both unconstrained and constrained optimization. © John Wiley & Sons, Inc.     

Evaluation of multicenter one-electron integrals of noninteger u screened Coulomb type potentials and their derivatives over noninteger n Slater orbitals

Multicenter integrals over noninteger n Slater type orbitals with integer and noninteger values of indices u of screened Coulomb type potentials, f(u)(eta,r)=r(u-1)e(-etar), and their first and second derivatives with respect to Cartesian coordinates of the nuclei of a molecule are described. Using complete orthonormal sets of Psi(alpha) exponential type orbitals and rotation transformation of two-center overlap integrals, these integrals are expressed through the noncentral potential functions depending on the molecular auxiliary functions A(k) and B(k). The series expansion formulas derived for molecular integrals of screened Coulomb potentials and their derivatives are especially useful for the computation of multicenter electronic attraction, electric field, and electric field gradient integrals. The convergence of series is tested for arbitrary values of parameters of potentials and orbitals.     

How many dimensions are required to approximate the potential energy landscape of a model protein?

A scheme to approximate the multidimensional potential energy landscape in terms of a minimal number of degrees of freedom is proposed using a linear transformation of the original atomic Cartesian coordinates. For one particular off-lattice model protein the inherent frustration can only be reproduced satisfactorily when a relatively large number of coordinates are employed. However, when this frustration is removed in a Go-type model, the number of coordinates required is significantly lower, especially around the global potential energy minimum. To aid our interpretation of the results we consider modified disconnectivity graphs where a measure of the structural diversity and a metric relation between the stationary points are incorporated.     

Green's Function Determination of Force Constants-H2O as Example

Treating isotopically substituted molecule as a perturbed system, Green's function for the perturbation are constructed and related to the force field of vibration. By spectral representation, Green's function is diagonalized in the normal coordinates. Then transforming back to the Cartesian coordinates, the Cartesian force constants are generated without solving the secular equation directly. The relations between the internal force constants and the Cartesian force constants ate given and complete internal force field can be obtained. The results for H₂O are discussed.     

铈对2090铝锂合金织构和力学性能的影响

为了解Ｃｅ对铝锂合金力学性能各向异性的影响,对不同Ｃｅ含量２０９０铝锂合金织构进行了ＯＤＦ（取向分布函数）测算,对合金不同方向性能进行了测量,分析了Ｃｅ对合金 构和力学性能的影响。２０９０铝锂合金主织构类型为（２２５〈５５４〉,Ｃｅ不改变合金主织构类型,但造成主织构类型分布漫散;由于Ｃｅ降低了蛔屈服强度各向异性织构成分的含量,故Ｃｅ可改变２０００铝锂合金理想织构成分的体积分数,减小屈服强度各向异性     

The location of transition states: A comparison of Cartesian,Z-matrix,and natural internal coordinates

A comparison is made between geometry optimization in Cartesian coordinates, in Z-matrix coordinates, and in natural internal coordinates for the location of transition states. In contrast to the situation with minima, where all three coordinate systems are of comparable efficiency if a reliable estimate of the Hessian matrix is available at the starting geometry, results for 25 different transition states covering a wide range of structural types demonstrate that in practice Z-matrix coordinates are generally superior. For Cartesian coordinates, the commonly used Hessian update schemes are unable to guarantee preservation of the necessary transition state eigenvalue structure, while current algorithms for generating natural internal coordinates may have difficulty handling the distorted geometries associated with transition states. The widely used Eigenvector Following (EF) algorithm is shown to be extremely efficient for optimizing transition states. © 1996 by John Wiley & Sons, Inc.     

Isomers of C(20): an energy profile

Semiempirical calucaltions, at the PM 3 level, are used to geometrically optimize and determine the absolute energies (heats of formation) of a variety of C(20) isomers. Based on the geometrically optimized Cartesian coordinates of the ring and the bowl isomers, and the subsequent saddle-point calculation, a two-dimensional energy profile between these two isomers is generated. Performing geometry optimization on the Cartesian coordinates that correspond to energy minima within the ring-bowl profile, we have been able to identify several more isomers of C(20) that are predicted to be energitically stable. With these additional stable structures, we have identified pairs of isomers that lie adjacent to one another on the potential energy surface, as is evidenced by the form of their respective energy profiles. These adjacent pairs of isomers establish a step-wise transformation between the ring and the bowl. This process, which extends out over the three-dimensional surface, is predicted to require less energy than that of the direct, two-dimensional transformation predicted in the ring-bowl profile.     

High-performance transformation of protein structure representation from internal to Cartesian coordinates

We present a highly parallel algorithm to convert internal coordinates of a polymeric molecule into Cartesian coordinates. Traditionally, converting the structures of polymers (e.g., proteins) from internal to Cartesian coordinates has been performed serially, due to an inherent linear dependency along the polymer chain. We show this dependency can be removed using a tree-based concatenation of coordinate transforms between segments, and then parallelized efficiently on graphics processing units (GPUs). The conversion algorithm is applicable to protein engineering and fitting protein structures to experimental data, and we observe an order of magnitude speedup using parallel processing on a GPU compared to serial execution on a CPU.     

Theory of relaxation properties of two‐dimensional polymer networks, 1. Normal modes and relaxation times

The local relaxation properties of polymer networks with a two‐dimensional connectivity are considered. We use the mesh‐like network model in which the average positions of junctions form the regular spatial structure consisting of square repeating units (network cells). The two‐dimensional polymer network consisting of “bead and spring” Rouse chains and the simplified coarse‐grained network model describing only the large‐scale collective relaxation of a network are studied. For both dynamic network models the set of relaxation times and the transformation from Cartesian coordinates of network elements to normal modes are obtained. Using the normal mode transformation obtained, in Part 2 of this series the exact analytical expressions for various local dynamic characteristics of the polymer network having a two‐dimensional connectivity will be calculated.     

Isomers of C(20): an energy profile II

Semi-empirical calculations, at the PM3 level provided within the Winmopac v2.0 software package, are used to geometrically optimize and determine the absolute energies (heats of formation) of a variety of C(20) isomers that are predicted to exist in and around the bowl and cage isomers. Using the optimized Cartesian coordinates for the bowl and the cage isomers, a saddle-point calculation was performed. The output file generated, containing energy, distance, and geometry information, is then organized into a graphical format. The resulting graph, which plots the energy of the 20-atom system as a function of the distance from the geometric midpoint, is a two-dimensional energy profile. This profile illustrates an estimation of the contours on the potential energy surface, showing energy minima and maxima that are encountered as the bowl evolves into the cage structure, or vice-versa. To expand the surface into three dimensions, geometry optimizations were performed on the sets of Cartesian coordinates that correspond to energy minima in the bowl-cage profile. Based on these optimizations, eight additional isomers of C(20) have been identified and are predicted to be energetically stable. These additional isomers were subsequently subjected to saddle-point calculations in order to identify those isomers that lie adjacent to one another on the three-dimensional surface. Two isomers that are adjacent to each other will exhibit an energy profile that progresses smoothly from the potential well of each isomer up to the saddle point separating them. Consequently, these adjacent pairs of isomers establish a step-wise transformation between the bowl and the cage. This process, which extends out over the three-dimensional surface, is predicted to require less energy than that of the direct, two-dimensional transformation predicted in the bowl-cage profile.     

Modeling of the optical anisotropy of a dye polarizer

A new model has been developed to account for the dependence of the optical anisotropy of a dye polarizer on the dye concentration. The effect of the dye concentration has been studied through an examination of the changes in the orientation distribution of the polymer. The model takes into account the intrinsic optical anisotropy of the dichroic dye, the polymer orientation, the polymer orientation distribution, and the dye orientation with respect to the polymer. It is assumed that (1) the orientation distribution function of the polymer segments can be expressed as an elliptical distribution function and that (2) the free rotation of each dye molecule on its axis is suppressed because of the attraction force between the dye molecules and the polymer chains. The pseudo‐order parameter, which takes into account the aforementioned assumptions, determines the relation between single‐piece transmittance and polarizing efficiency. The orientation distribution of the polymer molecules in the experiment and its effect on the optical performance of a polarizer are quantitatively determined. The model predicts that the effect of the orientation distribution becomes more significant as the polymer chains are oriented more highly. © 2002 Wiley Periodicals, Inc. J Polym Sci Part B: Polym Phys 40: 1363–1370, 2002     

A unified scheme for the calculation of differentiated and undifferentiated molecular integrals over solid-harmonic Gaussians

Utilizing the fact that solid-harmonic combinations of Cartesian and Hermite Gaussian atomic orbitals are identical, a new scheme for the evaluation of molecular integrals over solid-harmonic atomic orbitals is presented, where the integration is carried out over Hermite rather than Cartesian atomic orbitals. Since Hermite Gaussians are defined as derivatives of spherical Gaussians, the corresponding molecular integrals become the derivatives of integrals over spherical Gaussians, whose transformation to the solid-harmonic basis is performed in the same manner as for integrals over Cartesian Gaussians, using the same expansion coefficients. The presented solid-harmonic Hermite scheme simplifies the evaluation of derivative molecular integrals, since differentiation by nuclear coordinates merely increments the Hermite quantum numbers, thereby providing a unified scheme for undifferentiated and differentiated four-center molecular integrals. For two- and three-center two-electron integrals, the solid-harmonic Hermite scheme is particularly efficient, significantly reducing the cost relative to the Cartesian scheme.