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     

Estimating bounds on the highest and lowest eigenvalues of any matrix

A method for estimating the bounds for the highest and lowest eigenvalues of a finite-dimensional matrix is presented. The method is tested for the Hamiltonian matrix for several particle-in-a-box-like systems. We also provide results for the well-studied benchmark case of the ro-vibrational states of H₃ ⁺, and consider bounds obtained for a completely random non symmetric matrix. Finally, we discuss how the error in a Chebychev expansion solution of quantum scattering depends on the error made in estimating the highest eigenvalue of the Hamiltonian matrix. Received: 7 May 1999 / Accepted: 30 July 1999 / Published online: 15 December 1999     

Molecular anharmonicity: A computer-aided treatment

We present a new software package for the theoretical treatment of anharmonic vibrational spectra of nonlinear polyatomic molecules. The package, called “B&D,” computes vibrational energies starting from sets of force constants defined as potential energy derivatives. The method employed allows us to combine experimental rotation-vibration data with any information made available from ab initio calculations. The package follows the natural procedure in which a molecular problem is solved, both in the symbolic construction of Hamiltonian operator and basis functions and in the numerical computation of the Hamiltonian matrix elements. The novelty consists in making the entire procedure fully automatic, so that the occurrence of errors is greatly reduced and the laborious process involved in deriving and implementing the Hamiltonian is dramatically simplified. © 1999 John Wiley & Sons, Inc. J Comput Chem 20: 1716–1730, 1999     

Geminal approach to vibronic models of superconductivity in crystals. I. The Jahn–Teller effect

Electronic geminals constructed as linear combinations of binary products of site functions are used to formulate a vibronic model of superconductivity in crystals that is based upon the approximation of independent correlated electron pairs obtained variationally from an electron‐pair Hamiltonian and the Jahn–Teller effect. The cyclic symmetry of the system is taken into account and the geminals are sorted into doubly degenerate pairs. The Herzberg–Teller expansion of the pair Hamiltonian in terms of vibrational modes leads directly to the Jahn–Teller effect. A contact transformation of the vibronic Hamiltonian containing only linear terms lowers the energy of the system by a second‐order term associated with the Jahn–Teller stabilization energy. A possible model for superconductivity in solids is proposed on the basis of the Jahn–Teller effect. © 2003 Wiley Periodicals, Inc. Int J Quantum Chem, 2003     

Regioregularity in Poly(3‐alkoxythiophene)s: Effects on the Faraday Rotation and Polymerization Mechanism

Summary: Poly(3‐alkoxythiophene)s with different degrees of regioregularity were prepared using three different methodologies. It is shown that their Faraday rotation is highly dependent on the degree of regioregularity. The origin of the differences in regiospecificity of the methodologies is discussed.

Structure of the polymer pol 3 .     

Non‐Conformity with Faraday Law by Magnetoelectrolysis with Aluminum Electrodes

The present paper presents a study on the anode dissolution and corrosion of aluminum electrodes by electrolysis of aqueous solutions of sodium chloride in a magnetic field. The electrolysis was carried out under a galvanostatic regime with direct current and a uniform electric field. The magnetic field was generated by permanent magnets placed along the axis between an anode and cathode. The parameter current efficiency was used as a measure of the non‐conformity with the generalized Faraday law observed.     

Do Carbon Nano-onions Behave as Nanoscopic Faraday Cages? A Comparison of the Reactivity of C60, C240, C60@C240, Li+@C60, Li+@C240, and Li+@C60@C240

From the analysis of the polarizability of carbon nano-onions (CNOs), it was concluded that CNOs behave as near perfect nanoscopic Faraday cages. If CNOs behave as ideal Faraday cages, the reactivity of the C₂₄₀ cage should be the same in Li⁺@C₂₄₀ and Li⁺@C₆₀@C₂₄₀. In this work, the Diels–Alder reaction of cyclopentadiene to the free C₂₄₀ cage and the C₆₀@C₂₄₀ CNO together with their Li⁺-doped counterparts were analyzed using DFT. It was found that in all cases the preferred cycloaddition is on bond [6,6] of type B of C₂₄₀. Encapsulation of Li⁺ results in lower enthalpy barriers due to the decrease of the energy of the LUMO orbital of the C₂₄₀ cage. When the Li⁺ is placed inside the CNO C₆₀@C₂₄₀, the decrease in enthalpy barrier is similar to that of Li⁺@C₂₄₀. However, the location of Li⁺ in Li⁺@C₂₄₀ (off-centered) and Li⁺@C₆₀@C₂₄₀ (centered) is quite different. When Li⁺ was placed in the center of the C₂₄₀ cage in Li⁺@C₂₄₀, the barriers increased significantly. Taking into account this effect, the barriers in Li⁺@C₂₄₀ and Li⁺@C₆₀@C₂₄₀ differ by about 4 kcal mol⁻¹. This result can be attributed to the shielding effect of C₆₀ in Li⁺@C₆₀@C₂₄₀. As a result, we conclude that this CNO does not act as a perfect Faraday cage.     

The calculation of vibrational energy levels of polyatomic molecules including anharmonic effect using contact transformation perturbation method

Using contact transformation perturbation method based on the Taylor expansion of the potential energy function in terms of dimensionless normal coordinates up to sixth‐order, the vibrational energy levels in terms of force constants are derived. The contact transformation theory has been applied to simplify the calculation of perturbation effects. To calculate the second‐order vibrational energy correction, the third and fourth‐order terms of potential function have been placed in the first‐order perturbation Hamiltonian and the second‐order Hamiltonian contains hexatic ones. We present expressions which give relations between the fourth‐ and sixth‐order terms in dimensionless normal coordinates of the potential and the anharmonicity coefficients. For illustration, a set of vibrational energies levels of SO₂, and H₂O molecules including anharmonic effects has been calculated. © 2013 Wiley Periodicals, Inc.     

Dynamical Jahn-Teller effect in the first excited

Jahn-Teller effect of C₆₀ monoanion in the first electronically excited states was theoretically investigated. The orbital vibronic coupling parameters for t_1g next lowest unoccupied molecular orbitals were derived from the Kohn-Sham orbital levels calculated using a frozen phonon approach with both hybrid B3LYP and CAM-B3LYP functionals, which take long-range interaction correction into consideration. With these coupling parameters, the vibronic states of first excited were derived by exactly diagonalizing dynamical Jahn-Teller Hamiltonian. The results showed that dynamical Jahn-Teller effects are more significant in the first excited than those in the ground electronic states. This work also clarified that CAM-B3LYP gives results closer to experimental data than B3LYP.     

On an application of the intermediate Hamiltonian method for molecular systems

An application of the intermediate Hamiltonian method is reported in estimation of the lower bounds to the potential energy curve of the hydrogen molecule ion. An improvement of the method and its limitation are also discussed. ©1999 John Wiley & Sons, Inc. Int J Quant Chem 72: 101–107, 1999     

Solvatomagnetism-induced Faraday effect in a cobalt hexacyanochromate-based magnet

Solvent exchange caused reversible variations in color, magnetic properties, and the Faraday spectra of Co(II)(1.5)[Cr(III)(CN)(6)].7.5H2O (1) prepared in water. Compound 1 turned from peach to deep blue, which was due to a change in the coordination geometry on Co(II) ion from six-coordinate pseudo-octahedral (OhCo(II)) to four-coordinate pseudo-tetrahedral (TdCo(II)) geometries, when it was immersed in EtOH. The confirmed formula for the deep blue powder was Co(II)(1.5)[Cr(III)(CN)(6)].2.5H2O.2.0EtOH. The magnetic properties also changed; that is, the magnetic critical temperature, saturation magnetization, and coercive field went from 25 to 18 K, from 7.0 to 5.5 micro(B), and from 240 to 120 G, respectively. This solvatomagnetism is because the ferromagnetic magnetic coupling between OhCo(II) (S = 3/2) and Cr(III) (S = 3/2) is replaced by the antiferromagnetic coupling between TdCo(II) (S = 3/2) and Cr(III) (S = 3/2). Accompanying the solvatochromism and solvatomagnetism, the Faraday spectra drastically changed. The Faraday ellipticity (FE) spectrum of 1 had a distorted dispersive peak (A), which is due to the 4T1g --> 4T1g, 2T1g transitions of OhCo(II) ion, around 480 nm, but the FE spectra of 2 showed a new dispersive-shaped band (B) at 580 nm. The observed B band was assigned to the 4A2 --> 4T2 transition of the TdCo(II) ion. The Faraday spectra were well reproduced by a simulation that considers the ligand field splitting, spin-orbital coupling, and the ferromagnetic ordering. These solvatochromic effects were repeatedly observed.     

Quantum statistical analysis of superconductivity,fractional quantum Hall effect,and aromaticity

The phenomena of superconductivity and fractional quantum Hall effect (FQHE) as well as the well‐known chemical concepts of aromaticity and antiaromaticity are analyzed on the basis of quantum statistical considerations. We suggest that the superconducting transition is caused by a first‐order interaction between the charge carriers which does not necessarily involve a second‐order coupling of the electron–phonon type. For molecular model systems it is demonstrated that the formation of superconducting Cooper pairs can lead to an attenuation of destabilizing quantum constraints of the intersite type, i.e., constraints due to the Pauli antisymmetry principle (PAP). We suggest that this attenuation is the driving force for the superconducting transition. Such a reduction of the PAP influence on the quantum ensemble is also the key element of the present explanation of the FQHE. Analogies between the superconducting transition and the plateaus in the Hall conductance are emphasized. Both phenomena can be interpreted in terms of an electronic phase transition which shifts the original fermionic (fe) system towards a hard core bosonic (hcb) boundary. hcb ensembles are characterized by on‐site anticommutativity and intersite commutativity. The collective solid‐state phenomena superconductivity and FQHE are correlated with the popular chemical concepts of aromaticity and antiaromaticity. Numerical results for the superconducting pairing are derived by the two‐parameter Hubbard Hamiltonian. In order to express physically transparent interrelations between fe and hcb ensembles, the so‐called statistical transmutation is adopted. Arguments on the basis of experimental results are summarized which support the present PAP‐driven superconducting pairing formalism. © 2000 John Wiley & Sons, Inc. Int J Quant Chem 79: 125–162, 2000     

Diagrammatic formulation of the second‐order many‐body multipartitioning perturbation theory

The second‐order multireference perturbation theory employing multiple partitioning of the many‐electron Hamiltonian into a zero‐order part and a perturbation is formulated in terms of many‐body diagrams. The essential difference from the standard diagrammatic technique of Hose and Kaldor concerns the rules of evaluation of energy denominators which take into account the dependence of the Hamiltonian partitioning on the bra and ket determinantal vectors of a given matrix element, as well as the presence of several two‐particle terms in zero‐order operators. The novel formulation naturally gives rise to a “sum‐over‐orbital” procedure of correlation calculations on molecular electronic states, particularly efficient in treating the problems with large number of correlated electrons and extensive one‐electron bases. ©1999 John Wiley & Sons, Inc. Int J Quant Chem 73: 395–401, 1999     

Enhanced probability for the photoelectric effect in a Faraday cage

The cross section for the photoelectric effect is calculated for the case where the photoionization takes place within a Faraday cage held at a constant electrostatic potential. When the geometry has spherical symmetry, and when the potential is repulsive, the cross section is enhanced by a factor E/(E ? V₀). The predicted effect is a purely quantum mechanical one, since according to classical electrodynamics, no forces due to the potential of the cage will act on a charged particle in the fieldfree region inside it. In quantum theory, the effect arises because the continuum states extend to the region outside the cage.     

A general treatment of vibration-rotation coordinates for triatomic molecules

An exact, within the Born–Oppenheimer approximation, body-fixed Hamiltonian for the nuclear motions of a triatomic system is presented. This Hamiltonian is expressed in terms of two arbitrarily defined internal distances and the angle between them. The body-fixed axis system is related to these coordinates in a general fashion. Problems with singularities and the domain of the Hamiltonian are discussed using specific examples of axis embedding. A number of commonly used coordinate systems including Jacobi, bond-length-bond-angle, and Radau coordinates are special cases of this Hamiltonian. Sample calculations on the H₂S molecule are presented using all these and other coordinate systems. The possibility of using this Hamiltonian for reactive scattering calculations is also discussed.     

Hamiltonian for the spin quantum Hall effect solution and derivation based on an SU (2) gauge field

A Hamiltonian to describe a spin quantum Hall effect with two types of spin‐orbit coupling is introduced and the eigenfunctions and eigenvalues are obtained for it. It is shown that this Hamiltonian also results by gauging a kinetic energy Hamiltonian by an SU (2) gauge field. The Berry phase is obtained from the model wavefunction for the model and used to define a filling factor. This allows for the calculation of the spin Hall conductivity. © 2009 Wiley Periodicals, Inc. Int J Quantum Chem, 2010     

Graphical approach to correlations in high‐spin systems

We present a graphical technique for generating and indexing spin monomials of high‐spin systems. The procedure consists of developing a graph with at most n line segments from each node in a given row to the one immediately lower, where n is the multiplicity of single‐particle spin function. The paths lead to monomials with definite M values at each node. This technique has been used to generate and diagonalize the model spin Hamiltonian. ©1999 John Wiley & Sons, Inc. Int J Quant Chem 73: 389–393, 1999     

Group of symmetry of the periodic system of chemical elements

The concurrence of the group of symmetry of the periodic system of elements with the group of dynamical symmetry of a hydrogenlike atom is employed in the theoretical investigation of atoms. The character of the degeneracy of the eigenvalues of a hydrogenlike atom Hamiltonian, without changing its eigenfunctions, was changed by introducing into this Hamiltonian a term which violates the symmetry in relation to transformations from the subgroup O(4) of the group SO(4, 2). In consequence, it was realized that such “reorganization” of the states of a hydrogenlike atom, which form a representation of the group SO(4, 2), effects the splitting of this representation into finite‐dimensional multiplets, first of which are in full agreement with the experimentally observable composition of electron shells of atoms, and retains the physical meaning of quantum numbers that define electron states. ©1999 John Wiley & Sons, Inc. Int J Quant Chem 72: 499–508, 1999