Computer generation of matching polynimials of chemical graphs and lattices

A computer program in Pascal is developed for computing the matching polynomials of graphs and lattices. This program is based on the recursive relation for matching polynomials outlined by Hosoya [Bull. Chem. Soc. Jpn., 44 , 2332 (1971)], Gutman and Hosoya [Theor. Chim. Acta, 48 , 279 (1978)], and others. The graph whose matching polynomial is of interest is reduced recursively until the graph reduces to several trees. The characteristic polynomial of a tree is the same as the matching polynomial. The characteristic polynomials of resulting trees are computed using the computer program based on Frame's method developed by Balasubramanian [Theor. Chim. Acta, 65 , 49 (1984)]; J. Comput. Chem., 5 , 387 (1984). The resulting polynomials are then assembled to compute the matching polynomial of the initial graph. The program is especially useful in generating the matching polynomials of graphs containing a large number of vertices. The matching polynomials thus generated are potentially useful in several applications such as lattice statistics (dimer covering problem), aromaticity, valence bond methods (enumeration of perfect matchings) in the calculation of grand canonical partition functions, in the computation of thermodynamic properties of saturated hydrocarbons, and in chemical documentation.     

New analytical expressions,symmetry relations and numerical solutions for the rotational overlap integrals

In this article, extremely simple analytical formulas are obtained for rotational overlap integrals which occur in integrals over two reduced rotation matrix elements. The analytical derivations are based on the properties of the Jacobi polynomials and beta functions. Numerical results and special values for rotational overlap integrals are obtained by using symmetry properties and recurrence relationships for reduced rotation matrix elements. The final results are of surprisingly simple structures and very useful for practical applications. © 2011 Wiley Periodicals, Inc. Int J Quantum Chem, 2012     

Chemical applications of topology and group theory 13. Chirality and framework groups [1]

Pople has recently introduced the concept of a framework group to specify the full symmetry properties of a molecular structure. Furthermore, Pople has developed powerful algorithms for the use of framework groups to generate all distinguishable skeletons with a given number of sites. This paper studies the systematics of chirality arising from different framework groups. In this connection framework groups can be classified into four different types: linear, planar, achiral, and chiral. Chiral framework groups lead to chiral systems for any ligand partition including that with all ligands equivalent. Linear framework groups are never chiral even for the ligand partition with all ligands different. Planar framework groups are also never chiral since all sites are in the same plane, which therefore remains a symmetry plane for any ligand partition. However, the mirror symmetry of the molecular plane of a planar framework group can be destroyed by a process called polarization; this process can be viewed as the mathematical analogue of complexing a planar aromatic hydrocarbon to a transition metal. The chirality of four-, five-, and six-site framework groups is discussed in terms of the maximum symmetry ligand partitions resulting in removal of all of the symmetry elements corresponding to improper rotations S _n (including reflections S ₁ and inversions S ₂) from achiral and polarized planar framework groups. The Ruch-Schönhofer group theoretical algorithms for the calculation of chiral ligand partitions and pseudoscalar polynomials of lowest degree (“chirality functions”) are adapted for use with these framework groups. Other properties of framework groups relevant to a study of their chirality are also discussed: these include their transitivity (i.e. whether all sites are equivalent or not), their normality (i.e. whether proper rotations correspond to even permutations and improper rotations correspond to odd permutations), and the number of sites in their symmetry planes.     

Matrix elements of the linear Boltzmann collision operator for systems of two components at different temperatures

Matrix elements of the linearized collision operators that arise in the linearization of the Boltzmann equations for a binary gas system are calculated. The collision operators employed here differ from those usually considered in that the Maxwell—Boltzmann distribution functions which appear are parametrized by two different temperatures, one for each component. The matrix representations of the isotropic portion of the collision operators are calculated with the Sonine polynomials as basis functions, and for the hard sphere cross section, recursion relations for the matrix elements are derived which permit their efficient numerical calculation. The dependene of a few matrix elements on the mass and temperature ratios of the two components is considered. In particular, the disparate mass limit is investigated and the range of validity of the Fokker—Planck operator as an approximation to the collision operator in this limit is briefly discussed.     

How Accurate Are Approximate Methods for Evaluating Partition Functions for Hindered Internal Rotations?

The accuracy of several low-cost methods (harmonic oscillator approximation, CT-Comega, SR-TDPPI-HS, and TDPPI-HS) for calculating one-dimensional hindered rotor (1D-HR) partition functions is assessed for a test set of 644 rotations in 104 organic molecules, using full torsional eigenvalue summation (TES) as a benchmark. For methods requiring full rotational potentials, the effect of the resolution at which the rotational potential was calculated was also assessed. Although lower-cost methods such as Pitzer's Tables are appropriate when potentials can be adequately described by simple cosine curves, these were found to show large errors (as much as 3 orders of magnitude) for non-cosine curve potentials. In those cases, it is found that the TDPPI-HS method in conjunction with a potential compiled at a resolution of 60 degrees offers the best compromise between accuracy and computational expense. It can reproduce the benchmark values of the partition functions for an individual mode to within a factor of 2; its average error is just of a factor of 1.08. The corresponding error in the overall internal rotational partition functions of the molecules studied is less than a factor of 4 in all cases. Excellent cost-effective performance is also offered by CT-Comega, which requires only the geometries, energies, and frequencies of the distinguishable minima in the potential. With this method the geometric mean error in individual partition functions is 1.14, the maximum error is a modest 2.98 and the resulting error in the total 1D-HR partition function of a molecule is less than a factor of 5 in all cases.     

Transferable integrals in a deformation-density approach to crystal orbital calculations. II

A new method for calculating crystal orbitals in the Hartree-Fock-Slater approximation is proposed. The method makes use of x-ray crystallographic measurements of the deformation density, and uses transferable integrals to treat the neutral–atom potentials. Methods for evaluating matrix elements of neutral-atom potentials are discussed in detail, and in this connection, expansions of displaced Slater-type orbitals in terms of modified spherical Bessel functions and Legendre polynomials are presented. Tables of transferable integrals (moments of the neutral-atom potentials) are given for all the elements up to Z = 36, and tables of Fourier transforms of the neutral-atom potentials are also presented.     

Study of gas-liquid partitioning of alkane solutes in several organic solvents by using principal component analysis and linear solvation energy relationships

Principal component analysis (PCA) was used to extract the number of factors which can describe the 737 gas-liquid partition coefficients of five linear, four branched, and two cyclic alkanes in 67 common solvents. Based on the reconstruction of partition coefficient data matrix, we concluded that the experimental dataset could readily be reduced to two relevant factors. Using only these two factors, there were no errors larger than 3%, 7 cases had errors larger than 2%, and in 34 cases, errors were between 1 and 2%. n-Hexane and ethylcyclohexane were chosen as the test factors, and all other partition coefficients were expressed in terms of these two test factors. Prediction of the logarithmic partition coefficient of these alkanes in seven chemically different solvents, which were originally excluded from the data matrix, was excellent: the root mean square error was 0.064, only in 11 cases the errors were larger than 1%, and only 3 had errors larger than 4%.Linear solvation energy relationships (LSERs) using both theoretical and empirical solvent parameters were used to explain the molecular interactions responsible for partition. Several combinations of parameters were tried but the standard deviations were not less than 0.31. This could be attributed to the model itself, imprecisions in the data matrix or in some of the LSER parameters. Solvent cohesive parameters and surface tension in combination with polarity-polarizability or dispersion parameters perform the best.Finally, the two principal component factors were rotated onto the most relevant physicochemical parameters that control the gas-liquid partitioning phenomena.     

Errors in molecular integrals: An influence on RHF energy values

The influence of errors in molecular integrals on the calculated RHF energy values is considered for two models which correspond to round-off and shift errors. The energy variations induced by errors in elements of one- and two-electron matrices and the overlap matrix are represented in a quadratic approximation and the same degree of accuracy is maintained for mean values and standard deviations. The formulas given point out that mean values are less than zero when some of the errors are non-shifted. Special care is required when Gaussian lobe functions are used.     

Comparison of several background compensation methods useful for evaluation of energy-dispersive X-ray fluorescence spectra

A number of background estimation and modelling strategies suitable for evaluating energy-dispersive X-ray spectra by means of non-linear least squares fitting are evaluated and intercompared. As background modelling functions, exponential and linear combinations of mutually orthogonal polynomials are considered. These functions allow the shape of the background to be determined together with the photopeak intensities. As background estimation algorithms, an iterative stripping algorithm and a background channel selection procedure which is also based on the use of orthogonal polynomials are studied. The last two methods calculate the spectral background prior to the actual fitting process. For the intercomparison, the various methods were incorporated in the software package AXIL (Analysis of X-ray spectra by means of Iterative Least Squares). By using simulated spectra in which the intensity of all lines is a priori known, the accuracy and noise-sensitivity of the different background compensation strategies are evaluated. The method in which the background is modelled as a linear combination of orthogonal polynomials is identified as being the most robust and yielding the most accurate results.     

Chemical applications of topology and group theory. XXI: Chirality in transitive skeletons and qualitative completeness [1]

This paper unifies the following ideas for the study of chirality polynomials in transitive skeletons: (1) Generalization of chirality to permutation groups not corresponding to three-dimensional symmetry point groups leading to the concepts of signed permutation groups and their signed subgroups; (2) Determination of the total dimension of the chiral ligand partitions through the Frobenius reciprocity theorem; (3) Determination of signed permutation groups, not necessarily corresponding to three-dimensional point groups, of which a given ligand partition is a maximum symmetry chiral ligand partition by the Ruch-Schönhofer partial ordering, thereby allowing the determination of corresponding chirality polynomials depending only upon differences between ligand parameters; such permutation groups having the point group as a signed subgroup relate to qualitative completeness. In the case of transitive permutation groups on four sites, the tetrahedron and polarized square each have only one chiral ligand partition, but the allene and polarized rectangle skeletons each have two chiral ligand partitions related to their being signed subgroups of the tetrahedron and polarized square, respectively. The single transitive permutation group on five sites, the polarized pentagon, has a degenerate chiral ligand partition related to its being a signed subgroup of a metacyclic group with 20 elements. The octahedron has two chiral ligand partitions, both of degree six; a qualitatively complete chirality polynomial is therefore homogeneous of degree six. The cyclopropane (or trigonal prism or trigonal antiprism) skeleton is a signed subgroup of both the octahedron and a twist group of order 36; two of its six chiral ligand partitions come from the octahedron and two more from the twist group. The polarized hexagon is a signed subgroup of the same twist group but not of the octahedron and thus has a different set of six chiral ligand partitions than the cyclopropane skeleton. Two of its six chiral ligand partitions come from the above twist group of order 36 and two more from a signed permutation group of order 48 derived from the P₃[P ₂] wreath product group with a different assignment of positive and negative operations than the octahedron.     

Generating function for rotation matrix elements

The rotation matrix elements are expressed in terms of the Jacobi, Hypergeometric, and Legendre polynomials in the literature. In this study, the generating function is presented for rotation matrix elements by using properties of Jacobi polynomials. In addition, some special values and Rodrigues' formula of rotation matrix elements are obtained by using the generating function. © 2011 Wiley Periodicals, Inc. Int J Quantum Chem, 2012     

Analytical method for the representation of atoms-in-molecules densities

We present analytic refinements and applications of the deformed atomic densities method [Fernández Rico, J.; López, R.; Ramírez, G. J Chem Phys 1999, 110, 4213-4220]. In this method the molecular electron density is partitioned into atomic contributions, using a minimal deformation criterion for every two-center distributions, and the atomic contributions are expanded in spherical harmonics times radial factors. Recurrence relations are introduced for the partition of the two-center distributions, and the final radial factors are expressed in terms of exponential functions multiplied by polynomials. Algorithms for the practical implementation are developed and tested, showing excellent performances. The usefulness of the present approach is illustrated by examining its ability to describe the deformation of atoms in different molecular environments and the relationship between these atomic densities and some chemical properties of molecules.     

Identification and thermodynamic treatment of several types of large-amplitude motions

We present a partially automated method for the thermodynamic treatment of large-amplitude motions. Starting from the molecular geometry and the Hessian matrix, we evaluate anharmonic partition functions for selected vibrational degrees of freedom. Supported anharmonic vibration types are internal rotation and inversion (oscillation in a double-well potential). By heuristic algorithms, we identify internal rotations in most cases automatically from the Hessian eigenvectors, and we also estimate the parameters of anharmonic partition functions (e.g., potential barrier, periodicity, and symmetry number) with thermodynamically sufficient precision. We demonstrate the validity of our schemes by comparison to pointwise calculated ab initio potential curves.     

A new approach to the exact and approximate anharmonic vibrational partition function of diatomic and polyatomic molecules utilizing Morse and Rosen–Morse oscillators

Exact closed forms of the equilibrium partition functions in terms Jacobi elliptic functions are derived for a particle in a box and Rosen–Morse (Poschl–Teller) oscillator (perfect for modeling bending vibrational modes). An exact form of the equilibrium partition function of Morse oscillator is reported. Three other approximate forms of Morse partition function are presented. Having an exact closed‐form for the vibrational partition function can be very helpful in evaluating thermodynamic state functions, e.g., entropy, internal energy, enthalpy, and heat capacity. Moreover, the herein presented closed forms of the vibrational partition function can be used for obtaining spectroscopic and dynamical information through evaluating the two‐ and four‐point dipole moment time correlation functions in anharmonic media. Finally, a closed exact form of the rotational partition function of a particle on a ring in terms of the first kind of complete elliptic integral is derived. © 2010 Wiley Periodicals, Inc. Int J Quantum Chem, 2011     

Master formulas for two‐ and three‐center one‐electron integrals involving Cartesian GTO,STO, and BTO

As a first application of the shift operators method we derive master formulas for the two‐ and three‐center one‐electron integrals involving Gaussians, Slater, and Bessel basis functions. All these formulas have a common structure consisting in linear combinations of polynomials of differences of nuclear coordinates. Whereas the polynomials are independent of the type (GTO, BTO, or STO) of basis functions, the coefficients depend on both the class of integral (overlap, kinetic energy, nuclear attraction) and the type of basis functions. We present the general expression of polynomials and coefficients as well as the recurrence relations for both the polynomials and the whole integrals. Finally, we remark on the formal and computational advantages of this approach. © 2000 John Wiley & Sons, Inc. Int J Quant Chem 78: 83–93, 2000     

Extended force field method and statistical mechanical model applied to large polyatomic molecules : Part 1. Adenosine

The equilibrium structure, the 3N—6 vibrational frequencies and diagonalized inertia tensor molecular elements and consequently translational, vibrational and rotational partition functions were evaluated in order to calculate thermodynamic properties of adenosine. A detailed analysis of conformational features in terms of different valency coordinates energy contributions is also given.     

Basis-set extensions for two-component spin-orbit treatments of heavy elements

The accuracy of standard basis sets of quadruple-zeta and lower quality for the use in two-component self-consistent field procedures including spin-orbit coupling is investigated for the elements In-I and Au-At. Spin-orbit coupling leads to energetic and spatial splittings of inner shells, which are not described accurately with standard basis sets optimized for scalar relativistic calculations. This results in large errors in total atomic energies and significant errors in atomization energies of compounds containing these atoms. We show how these errors can be corrected by adding just a few steep sets of basis functions and demonstrate the quality of the resulting extended basis sets.