Near‐exact supersymmetric partner potentials: Construction and exploitation

Precise supersymmetric (SUSY) partner potentials can be generated only for exactly solvable problems of the stationary Schrödinger equation. This is a severe restriction, as most problems are not amenable to exact solutions. We employ here a linear variational strategy to explicitly construct approximate SUSY partners of a few common, not exactly solvable potentials and subsequently examine their properties to explore the advantages in practical implementation. The efficacy of our proposed scheme is commendable. We demonstrate that, for symmetric potentials, the constructed partners may be so good that the overall recipe has the nicety of generating the whole eigenspectrum by employing only half of the full Hilbert space functions. A similar strategy is shown to work for the odd states too, with proper boundary conditions. Pilot calculations involve a number of low‐lying states of some mixed oscillator and double‐well potentials. Analysis of the results reveals a few interesting features of the problem of construction of approximate SUSY partners and their practical use. Particularly, we identify places where the operator‐level approximations are involved and how far they affect the bounding properties of energies that are obtained as eigenvalues of a matrix diagonalization problem associated with linear variations. © 2010 Wiley Periodicals, Inc. Int J Quantum Chem, 2011     

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     

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.     

The supersymmetric quantum mechanics theory and Darboux transformation for the Morse oscillator with an approximate rotational term

Anharmonic potentials with a rotational terms are widely used in quantum chemistry of diatomic systems, since they include the influence of centrifugal force on motions of atomic nuclei. For the first time the Taylor-expanded renormalized Morse oscillator is studied within the framework of supersymmetric quantum mechanics theory. The mathematical formalism of supersymmetric quantum mechanics and the Darboux transformation are used to determine the bound states for the Morse anharmonic oscillator with an approximate rotational term. The factorization method has been applied in order to obtain analytical forms of creation and annihilation operators as well as Witten superpotential and isospectral potentials. Moreover, the radial Schrödinger equation with the Darboux potential has been converted into an exactly solvable form of second-order Sturm–Liouville differential equation. To this aim the Darboux transformation has been used. The efficient algebraic approach proposed can be used to solve the Schrödinger equation for other anharmonic exponential potentials with rotational terms.     

Calculations of partial cross sections for photofragmentation processes using complex absorbing potentials

We investigate the use of complex absorbing potentials for the calculation of partial cross sections in multichannel photofragmentation processes. An exactly solvable, coupled-two-channel problem involving square-well potentials is used to compare the performance of various types of absorbing potentials. Special emphasis is given to the near-threshold regions and the conditions under which the numerical results are able to reproduce the Wigner threshold laws. It was found that singular, transmission-free absorbing potentials perform better than those of power or polynomial form.     

On fitting a gold embedded atom method potential using the force matching method

We fit a new gold embedded atom method (EAM) potential using an improved force matching methodology which included fitting to high-temperature solid lattice constants and liquid densities. The new potential shows a good overall improvement in agreement to the experimental lattice constants, elastic constants, stacking fault energy, radial distribution function, and fcc/hcp/bcc lattice energy differences over previous potentials by Foiles, Baskes, and Daw (FBD) [Phys. Rev. B 33, 7983 (1986)] Johnson [Phys. Rev. B 37, 3924 (1988)], and the glue model potential by Ercolessi et al. [Philos. Mag. A 50, 213 (1988)]. Surface energy was improved slightly as compared to potentials by FBD and Johnson but as a result vacancy formation energy is slightly inferior as compared to the same potentials. The results obtained here for gold suggest for other metal species that further overall improvements in potentials may still be possible within the EAM framework with an improved fitting methodology. On the other hand, we also explore the limitations of the EAM framework by attempting a brute force fit to all properties exactly which was found to be unsuccessful. The main conflict in such a brute force fit was between the surface energy and the liquid lattice constant where both could not be fitted identically. By intentionally using a very large number of spline sections for the pair potential, electron-density function, and embedding energy function, we eliminated a lack of functional freedom as a possible cause of this conflict and hence can conclude that it must result from a fundamental limitation in the EAM framework.     

Position-dependent effective mass Schrödinger equations for PT-symmetric potentials

We use the method of point canonical transformations and choose the Rosen-Morse-type potential as the reference potential to study exact solutions of the position-dependent effective mass Schr?dinger equations. Choosing three position-dependent mass distributions, we construct seven exactly solvable target potentials with PT symmetry. The energy spectra of the bound states and corresponding wavefunctions for the PT-symmetric potentials are given in the exact closed forms. We also discuss the isospectrality of different Schr?dinger equations with the same mass distribution or different mass distributions for different PT-symmetric potentials.      

Effective potentials and the Gel''fand—Levitan equation

Moses et al, have derived an algorithm using the Gel'fand-Levitan equation for generating exactly solvable potentials for a particle in a box, harmonic oscillator, and Coulomb potentials by adding or subtracting a finite number of eigenvalues. We propose that their algorithm can be used to evaluate effective potentials for non exactly soluble molecular model Hamiltonians. We show that the algorithm can be used to remove bound states from the spectrum and to obtain an effective potential which supports predissociation resonances only. It can also be used to remove a specific resonance state from the spectrum, and to facilitate evaluations of excited states.     

Approximate ℓ-bound state solutions of q-deformed exponential-type potentials

In this work, the exactly solvable Schrödinger equation for s-states of a class of multiparameter exponential-type potential (E-tP) is used to obtain the approximate ℓ ≠ 0 bound state solutions for the corresponding q-deformed radial potentials. To deal with the effective potential, the Pekeris approximation for the centrifugal term is applied. The proposal has the advantage that, depending on the choice of the parameters involved in the E-tP, several q-deformed exponential potentials are obtained here as particular cases. That is, our proposition indicates that it is not necessary to use specialized methods to solve the Schrödinger equation for specific q-deformed exponential potentials. At this regard, in order to show the usefulness of the proposed method, we consider the ℓ ≠ 0 bound state solutions of the q-deformed Tietz, Hulthén, Manning-Rosen, Shioberg, quadratic exponential, Wei and Hua among other potential models. Furthermore, with the example of the Hua potential we are contributing to solve the recent controversy on the energy spectra of this q-deformed model. Moreover, in some applications considered in this work the results can be used to correct similar findings already published. Besides, the proposal is useful when considering new q-deformed potentials with hypergeometric wavefunctions to be used in quantum chemical applications of diatomic molecules.     

Correlation hole and physical properties: A model calculation

The correlation hole of Coulson and Nielson and its extension to momentum space by Banyard and Reed is studied by using an exactly solvable model. For this model all relevant quantities pertaining to the correlation hole have been calculated exactly. We use this model to study the relationship between the fit to the correlation hole for an approximate wave function and the closeness of the approximate energies to the exact ones. We show that, although in general the better the fit the closer are the approximate physical quantities to the exact ones, there are exceptions where that is not the case. Also, we present a convenient method for the calculation of the two particle distribution in momentum space and generalize the concept of the correlation hole by defining it in the pseudophase space of position and momentum.     

Generalized Pöschl-teller potential

From the point of view of the theory of differential equations, we present a four-parameter exactly solvable generalized Pöschl-Teller potential, related to the Jacobi polynomials, using the previously unconsidered equations.     

Bound state solutions of D‐dimensional schrödinger equation with exponential‐type potentials

In this article, using an exactly‐solvable multiparameter exponential‐type potential we propose a unified treatment of the analytical bound—state solutions of the Schrödinger equation for exponential‐type potentials in D‐dimensions. Our proposal accepts different approximations to the centrifugal term; however, its usefulness is exemplified in the frame of the Green and Aldrich approach. This fact enables us to compare our results with specific potentials found in the literature and that are obtained here as particular cases of our proposal. That is, instead of solving a specific exponential‐type potential, by resorting each time to a specialized method, the energy spectra and wavefunctions are derived straightforward from the proposed approach. Furthermore, our proposal can be used as an alternative way in the search of solutions to new exponential‐type potentials besides that one can study different approximations to the term . © 2014 Wiley Periodicals, Inc.     

Equivalence of the Wei potential model and Tietz potential model for diatomic molecules

By employing the dissociation energy and the equilibrium bond length for a diatomic molecule as explicit parameters, we generate improved expressions for the well-known Rosen-Morse, Manning-Rosen, Tietz, and Frost-Musulin potential energy functions. It is found that the well-known Tietz potential function that is conventionally defined in terms of five parameters [T. Tietz, J. Chem. Phys. 38, 3036 (1963)] actually only has four independent parameters. It is shown exactly that the Wei [Phys. Rev. A 42, 2524 (1990)] and the well-known Tietz potential functions are the same solvable empirical function. When the parameter h in the Tietz potential function has the values 0, +1, and -1, the Tietz potential becomes the standard Morse, Rosen-Morse, and Manning-Rosen potentials, respectively.     

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.     

Effective mass schrödinger equation for exactly solvable class of one-dimensional potentials

We deal with the exact solutions of Schr?dinger equation characterized by position-dependent effective mass via point canonical transformations. The Morse, P?schl-Teller, and Hulthén type potentials are considered, respectively. With the choice of position-dependent mass forms, exactly solvable target potentials are constructed. Their energy of the bound states and corresponding wavefunctions are determined exactly.      

Effect of interactions on molecular fluxes and fluctuations in the transport across membrane channels

Transport of molecules across membrane channels is investigated theoretically using exactly solvable one-dimensional discrete-state stochastic models. An interaction between molecules and membrane pores is modeled via a set of binding sites with different energies. It is shown that the interaction potential strongly influences the particle currents as well as fluctuations in the number of translocated molecules. For small concentration gradients, the attractive sites lead to largest currents and fluctuations, while the repulsive interactions yield the largest fluxes and dispersions for large concentration gradients. Interaction energies that lead to maximal currents and maximal fluctuations are the same only for locally symmetric potentials, where transition states are equally distant from the neighboring binding sites, while they differ for the locally asymmetric potentials. The conditions for the most optimal translocation transport with maximal current and minimal dispersion are discussed. It is argued that, in this case, the interaction strength is independent of local symmetry of the potential of mean forces. In addition, the effect of the global asymmetry of the interaction potential is investigated, and it is shown that it also strongly affects the particle translocation dynamics. These phenomena can be explained by analyzing the details of the particle entering and leaving the binding sites in the channel.     

Information theoretic spreading measures of orthogonal functions

We calculate information theoretic spreading measures of orthogonal functions associated with solutions of quantum mechanical isospectral potentials. In particular, Shannon, Renyi and Fisher lengths have been evaluated for potentials isospectral to the linear harmonic oscillator and the symmetric Rosen-Morse potentials. We have also compared the behaviour of different lengths for the orthogonal functions and the associated orthogonal polynomials.     

The factorisation of chemical graphs and their polynomials: A systematic study of certain trees

Characteristic polynomials of the set of 284 trees which have 1 – 12 vertices of valency 1 – 3 have been examined for possible factors (divisors) using polynomial division. Twenty of these trees are prime in the sense that they contain no other trees in the set as factors. The remaining 264 trees can all be constructed from a subset of 5 trees and a at of 152 non-graphical polynomials. Some of these polynomials exhibit iso- or sub-spectral relationships with acyclic (matching) polynomials of certain cyclic structures. A few cyclic factors of trees are noted briefly. Twenty pairs and one triad of the trees examined are isospectral.     

The experimental design for computing the harmonic and anharmonic force constant matrices of polyatomic molecules

In the present work we propose a numerical approach to estimate the harmonic and anharmonic force constant matrices, supposing we are able to compute analytically the first order derivative vector of the potential energy surface with respect to the internal coordinates. We use a polynomial least square fit to interpolate this gradient in the stationary point region. The structure of the regression matrix shows that the harmonic force constant matrix may be obtained even for large molecules; the evaluation of the anharmonic contributions request slightly more labor but is possible for 5 to 7 atoms. The present work is applicable even at the CI level and the number of computations remains small. We use the experimental planification to select the geometries to be computed in order to improve the estimation of the regression coefficients i.e. this means to lower their variance.chercheur qualifié du Fonds National Belge de la Recherche Scientifique.