Complexity analysis of two families of orthogonal functions

We study the shape LMC (López-Ruiz, Mancini and Calvet), Fisher-Shannon (FS) and Cramér-Rao (CR) complexities of two families of orthogonal functions associated with the solutions of isospectral deformations of the Pöschl-Teller and Harmonic oscillator potentials. We have compared the behavior of these complexities for the orthogonal functions with the complexities associated with Pöschl-Teller and Harmonic oscillator potentials whose solutions are given in terms of the classical orthogonal polynomials. All these complexities are discussed in terms of the quantum number n and isospectrality parameter λ.     

Computational analysis of relative stabilities of polyazine N-oxides

There is considerable interest in polyazine N-oxides as potential frameworks for energetic compounds with relatively high enthalpies of formation and crystal densities. The N⁺→O^? linkages, if appropriately located, may diminish the destabilization associated with nitrogen catenation. We have computationally characterized 40 N-oxides of the isomeric diazines, triazines, and tetrazines in terms of their geometries, relative energies, and (for a representative selection) electrostatic potentials. The presence of N⁺→O^? linkages does partially counteract the destabilizing effects of nitrogen catenation, although the isomers with complete catenation remain the least stable. The stabilizing influence of N⁺→O^? groups, and the accompanying changes in bond lengths, can be understood in terms of resonance charge delocalization to the polyazine rings. The N(O)–N(O) bonds between nitrogens that both bear oxygens tend to be relatively weak. The electrostatic potentials above the polyazine rings become increasingly positive as there are more nitrogens and oxygens; eventually they are positive above all of the carbons and nitrogens and possibly even the oxygens, with negative regions only on the peripheries of the molecules. However, the nitrogens that bear oxygens always have more positive potentials than those that do not.     

A simple and effective technique to variationally interpret the structure of SUSY partners of mirror image potentials

Precise supersymmetric partner potentials can be generated for exactly solvable problems of the stationary Schrödinger equation. Construction of isospectral potential is not always possible for exactly solvable systems. This is a restriction, as most problems are not exactly solvable. Employment of mirror-image property can help to construct an exact isospectral partner of that potential. These potentials have chemical relevance to enantiomers. In this paper, we present a formulation as modelling to explore the form of SUSY pair of these potentials. Through polynomial fit, we correlate all possible basic SUSY partners and optimise it to best fit polynomial to present a typical energy value of N = 50.     

Random walks and their diagnostic value for characterization of atomic environment

A characterization of atomic environments based on counting random walks in a molecular skeleton is outlined. To each atom in a molecule a sequence of integers w₁, w₂, w₃,…, w_n is assigned, where w_i represents the number of self-returning walks of length k, the length being defined by the number of bonds traversed. Properties of the derived atom codes are discussed. The codes display an impressive diversity and are superior to atomic codes based on enumeration of self-avoiding walks (or paths) in discriminating atomic environments. In certain cases the codes of individual atoms are not unique and the same codes appear in different molecules or even within the same molecule. The occurrence of the nonunique codes can be related to special structural situations, associated with the occurrence of isospectral graphs. These isospectral graphs which have atoms with identical codes can generate additional isospectral structures by attaching any arbitrary group to such points. If nonequivalent atoms of a single molecule have identical random walk codes, substitution at the singular points alternatively will produce isospectral graphs. Examples of such situations are given.     

New isospectral generalized potentials

In quantum chemistry, supersymmetry, shape invariance and intertwining techniques are used to determine the class of potentials that are solvable as well as to find their isospectral and generalized partners. To do that, it is necessary to have the corresponding Witten superpotential defined by W(x)=/ where is a particular wavefunction of the Hamiltonian under study. In this work, we propose an alternative way to express the Witten superpotential in terms of reciprocal wavefunctions. Thus, when this new definition of W(x) is used as an ansatz in the Riccati equation involved, one is led to a potential identical to that resulting from the use of the standard Darboux transform, which means that it is possibly the generalization of it. Moreover, the generalization of the new Witten superpotential gives rise to a new generalized isospectral potential other than that obtained from the generalized Darboux transform. As an example of an application of the proposed approach, we found the new generalized isospectral potentials that correspond to the one-dimensional free particle, harmonic oscillator and Morse potential models. Also, owing to the fact that the proposed method is general our proposal can be used straightforwardly to obtain new, exactly solvable potentials as well as to find their isospectral generalized partners which can be used advantageously in the modeling of important quantum chemical applications.From the Proceedings of the 28th Congreso de Químicos Teóricos de Expresíon Latina (QUITEL 2002) Acknowledgement.This work was supported by CONACYT-Mexico, under scientific project No. 32762-E.     

Approximated potential functions of non-bonded interactions of methyl group

The constants of Lennard-Jones potential functionsE=A/r ¹²-C/r⁶ describing CH₃...CH₃, CH₃...H, CH₃...C, CH₃...O, CH₃...N non-bonded interactions were derived using the energy minimization procedure. The effective diameter of CH₃ group separated from the other one by four bonds is 3.7Å and corresponding constants of CH₃...CH₃ interactions areA=7.31·10⁵Ź²kcal/mol andC=570Å⁵ kcal/mol. The potentials found are compatible with those of Scott and Scheraga [1–3].     

Some suggestions for improving the utilization of the functiong(α) andp(x) to identify the mechanism of a thermal transformation

The procedures allowing improvement of utilizing the known functionsg(α) andp(x) in the kinetics of non-isothermal transformations are analyzed.     

Ionization,atomization, and bond energies as functions of distances in inorganic molecules and crystals

The ionization potentials and dissociation energies of diatomic molecules are determined as functions of bond length, and the atomization energies of metals and crystalline compounds are determined as E = a/d ⁿ functions. In most cases, n ≥ 1; but for a number of metals and compounds, n < 1, as distinct from all known types of interatomic interactions. It is shown that the ratios of the bond energies and bond lengths of Group 1A and 1B metals to the respective molecular parameters have similar values, proving the identical valence states of the atoms of these metals in crystal structures.     

Isorpectral multitrees

We examined trees with one multiple edge (of multiplicityk) and report all isospectral graphs found when the number of vertices was n < 9. Tile search for isospectrul multitrees was carried out systematically by constructing the characteristic polynomials of all trees having one weighted edge. For all multitrees havingn < 7 vertices, we tabulated the coefficients of the characteristic polynomial. We restricted the analysis to trees with Me maximal valencyd = 4. The number of graphs considered exceeds 300. The smallest pair of isospectral multitrees (i.e. trees with a multiple edge) hasn = 6 vertices, There is a pair of trees whenn = 7, three pairs whenn = 8, and five pads whenn = 9. In all cases, whenk = I is assumed, isospectral multitrees reduce to the same tree. Whenk = 0 is assume(], isospectral trees produce either the same disconnected graph, or an isospectral forest.Dedicated to Basil E. Gillam, Professor emeritus of the Department of Mathematics and Computer Science at Drake University.Reported in part at the 1986 International Congress of Mathematicians, Berkeley, California, USA.Operated for the U.S. Department of Energy by the Iowa State University under Contract No. W-7405-Eng-82. This work was supported in part by the Office of R.S. Hansen, Director.     

An introduction to analysis of Rényi complexity ratio of quantum states for central potential

Rényi complexity ratio of two density functions is introduced for three and multidimensional quantum systems. Localization property of several density functions are defined and five theorems about near continuous property of Rényi complexity ratio are proved by Lebesgue measure. Some properties of Rényi complexity ratio are demonstrated and investigated for different quantum systems. Exact analytical forms of Rényi entropy, Rényi complexity ratio, statistical complexities based on Rényi entropy for integral order have been presented for solutions of pseudoharmonic and a family of isospectral potentials. Some properties of Rényi complexity ratio are verified for six diatomic molecules (CO, NO, N₂, CH, H₂, and ScH) and for other quantum systems.     

Generalized potentials: Free particle,harmonic, and Morse partner potentials

In this work, we propose a very simple procedure to find the partner to specific potentials. According to our method, partner potentials are those obtained in the generalization of standard potentials, for which they are generalized potentials whose Hamiltonian match the so‐called isospectral Hamiltonian. The proposed approach is straightforward and basically takes into consideration the use of three well‐established relationships: The first one is used to identify the particular potential; the second, to find the adjoint or modified potential; and the third, to obtain the corresponding generalized or partner potential. As a useful application of the proposed procedure, we give explicitly the generalized and modified free‐particle, harmonic oscillator, and Morse one‐dimensional potentials. As expected, it is shown that the adjoint and partner potentials are isospectral with respect to the particular or former potential. This procedure can be easily applied to the generalization of any other known potential, as well as to obtain new potentials that can be advantageously used for modeling important quantum interactions. ©1999 John Wiley & Sons, Inc. Int J Quant Chem 71: 465–472, 1999     

On electronic states and bond lengths of the truncated icosahedral C60 molecule

The electronic states and the bond lengths of the truncated icosahedral C₆₀ molecule have been calculated by the Hückel and Coulson-Golebiewski self-consistent Hückel methods. C₆₀ has a stable closed shell with a rather big energy gap (= 0.847β) between the HOMO and the LUMO. We have obtained two kinds of bond lengths r₁ = 1.434 Å and r₂ = 1.403 Å, which correspond to the edges of the regular pentagon and the edge of a hexagon not lying on a pentagon.     

Electronic properties of aluminum clusters compared with the jellium model

The experimental polarizabilities, ionization potentials and electron affinities of aluminum clusters are compared with jellium predictions. It is found that the clusters have radii and work functions which are close to the jellium model predictions for clusters with more than 13 atoms. The polarizabilities of Al _n correspond with the jellium only forn>40 and the shell structure features in the ionization potentials are anomalous up to 37. We conclude that nonjellium effects are important up ton=40.     

On the equilibrium structures and the IR active bending vibrations of linear C13 and C15: results of large-scale coupled cluster calculations

Accurate equilibrium (r_e) structures (ca. 0.0005 Å accuracy in bond lengths) have been established for linear C₁₃ and C₁₅ by applying a uniform correction to the results of CCSD(T) calculations with the cc-pVQZ basis set. The equilibrium bond lengths cover a small range between 1.2690 and 1.2928 Å and are indicative of strong carbon–carbon double bonds. Equilibrium structures of still longer chains may be obtained by taking the recommended r_e structure for C₁₅ and inserting one or more C₂ links with R_e = 1.277 Å in the middle of the molecule. Both linear C₁₃ and C₁₅ exhibit no sign of floppiness and appear to behave like fairly normal semi-rigid molecules. Diagonal potentials for the IR active bending vibrations of both molecules have large correlation contributions between -33 and 47%, with MP2 strongly overshooting with respect to CCSD(T). Harmonic cis-bending vibrational wavenumbers and their absolute IR intensities are predicted. Since the latter are rather small, the chances of an interstellar detection of linear C₁₃ or C₁₅ in the far IR may be poor.     

Vibrational analyses employing cartesian coordinates

Geometry-optimised ab initio (STO-3G minimal basis set) calculations have been carried out for S-trans and orthogonal forms of 1,3-butadiene. The central C-C bond is shorter (1.482 Å) in the S-trans form than the orthogonal form (1.504 Å). Calculations on the orthogonal form indicate that the additional bond length decrease destabilizes the σ bond. the stabilization of the S-trans form being associated with conjugative (or π delocalization) effects.     

Rotational barrier and structure of aminoborane determined by AB initio calculations

Ab initio calculations have been used to determine the structural parameters of both the planar and the orthogonal form of H₂BNH₂. The BN bond lengths were found to be 1.378 Å and 1.469 Å, respectively. The arrangement about nitrogen is pyramidal in the orthogonal form. The best estimate of the barrier to internal rotation is 33.3 kcal/mole.     

Experimental determination of the potential for Na(32P)+ Ar from differential scattering cross sections

We have measured the total differential cross section for the interaction of laser excited Na(3²P) interacting with Ar at thermal energies. From these data, which show a well resolved oscillatory pattern, the respective interaction potentials have been determined. The best fit to the experimental data is found with ε = 2.55 × 10^?3 au, r_m = 5.75 au for the well depth parameter of the ²II_{$\text{1}\text{2}$} potentials.     

Alternating furanylene-meta-phenylene oligomers: synthesis and photophysics

A range of alternating furanylene-meta-phenylene oligoaryls of different chain lengths is synthesized by a convergent/divergent protocol from the annulation of a propargylic dithioacetal and an aldehyde with a propargylic dithioacetal moiety as a substituent. The emission properties of these oligomers showed chain-length dependent in poor solvent such as cyclohexane. Time resolved fluorescence spectroscopic analyses indicate that there might be intramolecular interaction between the chromophores in 2c-e and such interaction became more prominent as the chain-lengths increased from 9 to 21. Since oligomers 2 have meta-phenylene and 2,5-furnaylene linkages, the oligomers 2e, may likely be folded to enable intrachain chromophore-chromophore interactions. In addition, these oligomers are electrochemically active and the first oxidation potentials are chain-length dependent.     

Model potentials suitable for calculations with slater-type basis for Sc through Zn

Model potentials appropriate for molecular calculations with Slater-type (ST) basis sets have been generated for the first-row transition-metal atoms. Two sets of model potentials are presented. The first one has been optimized using standard 2ζ ST basis sets. The second is consistent with a reduced ST basis set. The reduced bases have been obtained by means of a new algorithm, whose results are compared to those found with the method of Y. Sakai and S. Huzinaga (J. Chem. Phys.76, 2537 (1982)). The comparison shows that the new approach leads to significant improvements in the overall results. Two different valence shells have been investigated, one formed by the 3s, 3p, 3d, and 4s AOs (SPDS), and the other one formed by the 3d and 4s AOs (DS). The model potentials presented here describe these valence shells in good agreement with the all-electron calculations taken as reference, the SPDS calculations being uniformly more accurate. Special attention has been paid to the transferability of the model potentials to electronic states with different orbital occupation. It is shown that the reported potentials have a wide transferability of this sort, being thus suitable for calculating the electronic structure of transition-metal compounds with STO 2ζ quality.     

Modified <Emphasis Type="Italic">ℓ</Emphasis>-states of diatomic molecules subject to central potentials plus an angle-dependent potential

We present modified ?-states of diatomic molecules by solving the radial and angle-dependent parts of the Schrödinger equation for central potentials, such as Morse and Kratzer, plus an exactly solvable angle-dependent potential V _θ (θ)/r ² within the framework of the Nikiforov–Uvarov (NU) method. We emphasize that the contribution which comes from the solution of the Schrödinger equation for the angle-dependent potential modifies the usual angular momentum quantum number ?. We calculate explicitly bound state energies of a number of neutral diatomic molecules composed of a first-row transition metal and main-group elements for both Morse and Kratzer potentials plus an angle-dependent potential. We also compare the bound state energies for both potentials, taking into account spectroscopic parameters of diatomic molecules and arbitrary values of potential constants.