Geometry of the canonical Van Vleck transformation

Bloch's transformation from the zeroth‐order space for a perturbation problem to the corresponding space of exact eigenvectors, was found as a geometrically defined alternative to the algebraically constructed Van Vleck transformation. Klein's theorem of uniqueness transferred some of this geometrical interpretation to its canonical form . Quite recently Kvaal has taken a large step further by writing as a product of commuting planar rotations, obtained by describing and in terms of certain principal vectors and canonical angles. Kvaal's approach is now developed further, using a new commutation relation which simplifies algebraic manipulations substantially. It allows for a simple definition of an operator for the angle between and which has Kvaal's vectors and angles as eigenvectors and eigenvalues. Klein's theorem is refined in various ways. The impact of the approach on a number of previous results is considered. © 2015 Wiley Periodicals, Inc.     

Chemical reaction rates using the semiclassical Van Vleck initial value representation

A semiclassical initial value representation formulation using the Van Vleck [Proc. Natl. Acad. Sci. U.S.A. 14, 178 (1928)] propagator has been used to calculate the flux correlation function and thereby reaction rate constants. This Van Vleck formulation of the flux-flux correlation function is computationally as simple as the classical Wigner [Trans. Faraday Soc. 34, 29 (1938)] model. However, unlike the latter, it has the ability to capture quantum interference/coherence effects. Classical trajectories are evolved starting from the dividing surface that separates reactants and products, and are evolved negatively in time. This formulation has been tested on model problems ranging from the Eckart barrier, double well to the collinear H+H2.     

Van der Waals supercritical fluid: exact formulas for special lines

In the framework of the van der Waals model, analytical expressions for the locus of extrema (ridges) for heat capacity, thermal expansion coefficient, compressibility, density fluctuation, and sound velocity in the supercritical region have been obtained. It was found that the ridges for different thermodynamic values virtually merge into single Widom line only at T < 1.07T(c), P < 1.25P(c) and become smeared at T < 2T(c), P < 5P(c), where T(c) and P(c) are the critical temperature and pressure. The behavior of the Batschinski lines and the pseudo-Gruneisen parameter γ of a van der Waals fluid were analyzed. In the critical point, the van der Waals fluid has γ = 8/3, corresponding to a soft sphere particle system with exponent n = 14.     

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.     

Closed‐form formulas for Kirchhoff index

We find closed‐form expressions for the resistance, or Kirchhoff index, of certain connected graphs using Foster's theorems, random walks, and the superposition principle. © 2001 John Wiley & Sons, Inc. Int J Quant Chem 81: 135–140, 2001     

Numerical-analytic implementation of the higher-order canonical Van Vleck perturbation theory for the interpretation of medium-sized molecule vibrational spectra

Anharmonic vibrational states of semirigid polyatomic molecules are often studied using the second-order vibrational perturbation theory (VPT2). For efficient higher-order analysis, an approach based on the canonical Van Vleck perturbation theory (CVPT), the Watson Hamiltonian and operators of creation and annihilation of vibrational quanta is employed. This method allows analysis of the convergence of perturbation theory and solves a number of theoretical problems of VPT2, e.g., yields anharmonic constants y(ijk), z(ijkl), and allows the reliable evaluation of vibrational IR and Raman anharmonic intensities in the presence of resonances. Darling-Dennison and higher-order resonance coupling coefficients can be reliably evaluated as well. The method is illustrated on classic molecules: water and formaldehyde. A number of theoretical conclusions results, including the necessity of using sextic force field in the fourth order (CVPT4) and the nearly vanishing CVPT4 contributions for bending and wagging modes. The coefficients of perturbative Dunham-type Hamiltonians in high-orders of CVPT are found to conform to the rules of equality at different orders as earlier proven analytically for diatomic molecules. The method can serve as a good substitution of the more traditional VPT2.     

On closed derivation formulas of the Nirmala indices from the M-polynomial of a graph

A new category of polymeric materials is dendrimers. These are monodisperse macromolecules with a high branching level. The physical and chemical characteristics of these materials are strongly influenced by their structure. Dendrimers are ideal for a wide variety of biological and industrial applications due to their distinctive behaviour. Very recently, in 2021, the Nirmala index, and the first and second inverse Nirmala indices are proposed and calculated their values for four standard dendrimers. In this present article, we propose closed derivation formulas for finding the above variations of Nirmala indices of a graph in terms of its M-polynomial. We also determine the M-polynomials and their geometrical natures for some families of dendrimers. Finally, we compute the Nirmala indices for each of the considered dendrimers using the M-polynomial approach based on the proposed closed derivation formulas and get the same numeric results as originally calculated.     

Connections between perturbation theory and the Van Vleck transformation: Illustrative calculations on the perturbed harmonic oscillator

We continue to examine the connection between perturbation theory and the Van Vleck unitary transformation. Here we illustrate the formalism derived earlier by applying it to compute the stationary states of the perturbed harmonic oscillator. We find that each solution of the traditional Brillouin-Wigner perturbation theory equations gives rise to a different unitary transformation which, when operating on the unperturbed ground state, produces one or the other of the perturbed eigenstates. With any of the perturbed states able to be reached by a unitary transformation on the unperturbed ground state, we advise caution in using approximate solutions of the perturbation equations in general cases, lest an unexpected stationary state be obtained.     

Inversion of configuration in the closed form of spirothiopyran molecules

Conclusions The Gvalue for the inversion of configuration of the closed form of spirothiopyrans is significantly greater than for their oxygen analogs due to steric hindrance for rotation of the rotation of the benzopyran fragment relative to the indoline fragment.Translated from Izvestiya Akademii Nauk SSSR, Seriya Khimicheskaya, No. 7, pp. 1666–1668, July, 1987.     

On the use of asymptotic expansions

Divergent asymptotic expansions in quantum chemistry often must be evaluated on Stokes lines, where the form of the expansion changes discontinuously and might appear to be ambiguous. Towards clarifying the use of asymptotic expansions on Stokes lines we discuss by numerical example the Airy function Bi(x) for real, positive x. Two physical problems to which this example is relevant, among others, are the Rayleigh-Schrödinger perturbation theory for the LoSurdo-Stark effect in hydrogen and the JWKB connection-formula problem, for which real series are associated with complex sums. The various roles of partial summation, Padé approximants, and Borel summation are compared. In addition, a derivation is given for an integral that occurs in a simple proof of the Borel summability of asymptotic expansions for the confluent hypergeometric function, which function is fundamental to certain quantum chemistry problems, and which integral is given incorrectly in several standard references.Dedicated to Professor J. Koutecký on the occasion of his 65th birthday     

On the properties of cumulant expansions

Cumulants represent a natural language for expressing macroscopic properties of a solid. We show that cumulants are subject to a nontrivial geometry. This geometry provides an intuitive understanding of a number of cumulant relations which have been obtained so far by using algebraic considerations. We give general expressions for their infinitesimal and finite transformations and represent a cumulant wave operator through an integration over a path in the Hilbert space. Cases are investigated where this integration can be done exactly. An expression of the ground-state wave function in terms of the cumulant wave operator is derived. In the second part of the article, we derive the cumulant counterpart of Faddeev's equations and show its connection to the method of increments. © 1998 John Wiley & Sons, Inc. Int J Quant Chem 66 : 377–389, 1998     

On the theory for closed shell intermolecular forces

The electron gas model of Gordon and Kim is applied to the interaction of inert gases with a hydrogen molecule. Comparison of the isotropic potential wells with available experimental data shows clearly that modification of the exchange potential along the lines suggested by Rae produces greatly improved agreement when simple power series representations of the dispersion energy are included.     

Recurrence relations and closed formulas connecting Franck—Condon factors and squeezed states

Recurrence relations and closed formulas which are common to Franck-Condon factors and squeezed states are presented in a unified formulation. Both concepts present a certain parallelism that is explained by formally showing their becoming particular cases of more general concepts. This treatment sets forth the mathematical structure that unifies the two concepts. © 1997 John Wiley & Sons, Inc. Int J Quant Chem 63: 229–232, 1997     

Effective closed form starting point determination for kinetic model interpreting NR vulcanized with sulphur

In this paper, a closed form analytical approach for a recently presented kinetic model proposed in Milani and Milani (Polym Test, 2013, under review) to interpret NR sulphur vulcanization in presence of either experimental or surrogate rheometer curves is proposed. The model has kinetic base and is aimed at predicting, by means of a very refined approach, the vulcanization degree of NR vulcanized with sulphur. It needs as input only rheometer curves to fit and provides as output kinetic constants of the single reactions occurring during the crosslink process. In Milani and Milani (Polym Test, 2013, under review) a cure chemical scheme constituted by five reactions occurring in series and parallel was adopted. The chemical scheme, translated mathematically into a differential equations system, was suitably re-arranged and a single analytical equation was derived, representing rubber crosslink degree evolution upon time. The main drawback of such procedure is that the five kinetic constants corresponding to each reaction were determined through a standard non-linear least squares procedure, trying to minimize the deviation of the analytical cure curve from experimental data. Such a limitation is here superseded and a major improvement is proposed utilizing (1) a closed form solution which does not require any optimization algorithm and (2) finding analytically a starting point for the unknown kinetic constants, very near to the actual solution and thus very convenient for a successive least squares minimization. In the model, it is shown how the analytical condition deduced from the scorch point (second derivative of the rheometer curve equal to zero) and two further conditions, e.g. the time at 90 % of vulcanization and the reversion percentage, allow the simple direct evaluation of kinetic constants, providing a closed form analytical formula to predict well the state of cure of the rubber under consideration. To assess the results obtained with the model proposed, several examples on two different NRs are discussed. The approach proved to be extremely robust and much faster when compared with the model proposed by Milani and Milani (Polym Test, 2013, under review).     

On topological form in structures

This paper begins with a review of the Euler relation for the polyhedra and presents the corresponding Schläfli relation in n, the polygonality, and p, the connectivity of the polyhedra. The use of ordered pairs as given by (n, p), the Schläfli symbols, to organize the mapping of the polyhedra and its extension into the two-dimensional (2D) and three-dimensional (3D) networks is described. The topological form index, represented by l, is introduced and is defined as the ratio of the polygonality, n, to the connectivity, p, in a structure, it is given by l = n/p. Next a discussion is given of establishing a conventional metric of length in order to compare topological properties of the polyhedra and networks in 2D and 3D. A fundamental structural metric is assumed for the polyhedra. The metric for the polyhedra is, in turn, used to establish a metric for tilings in the Euclidean plane. The metrics for the polyhedra and 2D plane are used to establish a metric for networks in 3D. Once the metrics have been established, a conjecture is introduced, based upon the metrics assumed, that the area of the elementary polygonal circuit in the polyhedra and 2D and 3D networks is proportional to a function of the topological form index, l, for these structures. Data of the form indexes and the corresponding elementary polygonal circuit areas, for a selection of polyhedra and 2D and 3D networks is tabulated, and the results of a least squares regression analysis of the data plotted in a Cartesian space are reported. From the regression analysis it is seen that a quadratic in l, the form index, successfully correlates with the corresponding elementary polygonal circuit area data of the polyhedra and 2D and 3D networks. A brief discussion of the evident rigorousness of the Schläfli indexes (n, p) over all the polyhedra and 2D and 3D networks, based upon the correlation of the topological form index with elementary polygonal circuit area in these structures, and the suggestion that an Euler–Schläfli relation for the 2D and 3D networks, is possible, in terms of the Schläfli indexes, concludes the paper.     

On the accuracy of correlation-energy expansions in terms of local increments

The incremental scheme for obtaining the energetic properties of extended systems from wave-function-based ab initio calculations of small (embedded) building blocks, which has been applied to a variety of van der Waals-bound, ionic, and covalent solids in the past few years, is critically reviewed. Its accuracy is assessed by means of model calculations for finite systems, and the prospects for applying it to delocalized systems are given.