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.     

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.     

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.      

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.      

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.     

Rayleigh–ritz method for excited quantum states via nonlinear variations without constraints: Role of supersymmetry

Quantum mechanical variation principle in the form of energy minimization is applicable only to ground states of systems, or, at best, states of lowest energies of given symmetries, provided the symmetry information is embedded in chosen trial functions. Thus, for bound quantum states with specified choices of trial functions involving nonlinear parameters, scope of the principle is severely restricted. A pedagogic way out is to enforce exact orthogonality of the chosen function with all exact lower energy states. In actual practice, this limits one to opt for linear variations where upper bound to each state is obtained in a single run. In this work, the motivation is to explore if there exists at all a way to determine optimized wave functions and energies for excited states via nonlinear variations but without any constraints, even for simple systems. Realizing that the major problem in excited‐state nonlinear variations is concerned with the variations of nodal positions, at least for problems reducible to one dimension, we seek a route via which nodes could be fixed beforehand, so that the information gained may be subsequently utilized to construct a suitable nonlinear trial function and carry out a straightforward optimization. To achieve this end, the idea of supersymmetric quantum mechanics has been used quite profitably, yielding the nodal structure of the excited states. Workability of the strategy for several excited‐state wave functions and their properties is demonstrated by choosing the problems of spherical Stark effect on hydrogen atom and anharmonic oscillator. © 2011 Wiley Periodicals, Inc. Int J Quantum Chem, 2012     

Maximum feasibility guideline in the design and analysis of protein folding potentials

Protein folding potentials are expected to have the lowest energy for the native shape. The Linear Programming (LP) approach achieves exactly that goal for a training set, or indicates that this goal is impossible to obtain. If a solution cannot be found (i.e., the problem is infeasible) two possible routes are possible: (a) choosing a new functional form for the potential, (b) finding the best potential with a feasible subset of the data, and (or) detecting inconsistent subset of the data in the training set. Here, we explore option (b). A simple heuristic for finding an approximate solution to an infeasible set of linear inequalities is outlined. An approximately feasible solution is obtained iteratively, starting from a certain initial guess, by computing a series of analytic centers of the polyhedra defined by all the inequalities satisfied at the subsequent iterations. Standard interior point algorithms for Linear Programming can be used to compute efficiently the analytic center of a polyhedron. We demonstrate how this procedure can be used for the design of folding potentials that are linear in their parameters. The procedure shows an improvement in the quality of the potentials and sometimes points to flaws in the original data.     

A class of exactly solvable matrix models

Some exactly solvable matrix models are discussed. Possible applications to problems in physical chemistry are pointed out, in particular the Hückel problem, the problem of torsional vibrations of polyatomic molecules, and of vibrations of finite polymer chains. This revised version was published online in July 2006 with corrections to the Cover Date.     

Increasing the applicability of density functional theory. II. Correlation potentials from the random phase approximation and beyond

Density functional theory (DFT) results are mistrusted at times due to the presence of an unknown exchange correlation functional, with no practical way to guarantee convergence to the right answer. The use of a known exchange correlation functional based on wave-function theory helps to alleviate such mistrust. The exchange correlation functionals can be written exactly in terms of the density-density response function using the adiabatic-connection and fluctuation-dissipation framework. The random phase approximation (RPA) is the simplest approximation for the density-density response function. Since the correlation functional obtained from RPA is equivalent to the direct ring coupled cluster doubles (ring-CCD) correlation functional, meaning only Coulomb interactions are included, one can bracket RPA between many body perturbation theory (MBPT)-2 and CCD with the latter having all ring, ladder, and exchange contributions. Using an optimized effective potential strategy, we obtain correlation potentials corresponding to MBPT-2, RPA (ring-CCD), linear-CCD, and CCD. Using the suitable choice of the unperturbed Hamiltonian, Kohn-Sham self-consistent calculations are performed. The spatial behavior of the resulting potentials, total energies, and the HOMO eigenvalues are compared with the exact values for spherical atoms. Further, we demonstrate that the self-consistent eigenvalues obtained from these consistent potentials used in ab initio dft approximate all principal ionization potentials as demanded by ionization potential theorem.     

Quantum trajectories in complex space: one-dimensional stationary scattering problems

One-dimensional time-independent scattering problems are investigated in the framework of the quantum Hamilton-Jacobi formalism. The equation for the local approximate quantum trajectories near the stagnation point of the quantum momentum function is derived, and the first derivative of the quantum momentum function is related to the local structure of quantum trajectories. Exact complex quantum trajectories are determined for two examples by numerically integrating the equations of motion. For the soft potential step, some particles penetrate into the nonclassical region, and then turn back to the reflection region. For the barrier scattering problem, quantum trajectories may spiral into the attractors or from the repellers in the barrier region. Although the classical potentials extended to complex space show different pole structures for each problem, the quantum potentials present the same second-order pole structure in the reflection region. This paper not only analyzes complex quantum trajectories and the total potentials for these examples but also demonstrates general properties and similar structures of the complex quantum trajectories and the quantum potentials for one-dimensional time-independent scattering problems.     

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.     

Survey of exact N-body decompositions of stochastic scalar-Jastrow–Hartree form

The quantum N-body dynamics problem with pairwise interactions can be exactly decomposed into the average of N stochastically evolving 1-body problems, thereby eliminating the usual exponential scaling of computational costs. Unfortunately, the variance in such averages can be large leading to slow Monte Carlo convergence. In addition, norm preserving decompositions are available only for identical fermions or bosons. Here we introduce a family of decompositions of scalar-Jastrow–Hartree form which can be applied to electronic structure and many molecular dynamics problems. We also discuss their convergence properties and test a few such methods on the vibrational stretching mode dynamics of CH₄. Finally, we explain how the Monte Carlo convergence problem can be completely eliminated via the introduction of a perfect control variate.     

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.     

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.     

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.     

Solution of periodic Poisson's equation and the Hartree–Fock approach for solids with extended electron states: Application to linear augmented plane wave method

We formulate a Hartree–Fock‐LAPW method for electronic band structure calculations. The method is based on the Hartree–Fock–Roothaan approach for solids with extended electron states and closed core shells where the basis functions of itinerant electrons are linear augmented plane waves. All interactions within the restricted Hartree–Fock approach are analyzed and in principle can be taken into account. In particular, we obtained the matrix elements for the exchange interactions of extended states and the crystal electric field effects. To calculate the matrix elements of exchange for extended states, we first introduce an auxiliary potential and then integrate it with an effective charge density corresponding to the electron exchange transition under consideration. The problem of finding the auxiliary potential is solved by using the strategy of the full potential LAPW approach, which is based on the general solution of periodic Poisson's equation. Here, we use an original technique for the general solution of periodic Poisson's equation and multipole expansions of electron densities. We apply the technique to obtain periodic potentials of the face‐centered cubic lattice and discuss its accuracy and convergence in comparison with other methods. © 2002 Wiley Periodicals, Inc. Int J Quantum Chem, 2002     

Quantum properties of complete solutions for a new noncentral ring‐shaped potential

We propose a new exactly solvable ring‐shaped potential V(r,θ) = ?(α/r) + (σ/r²) + β cos²θ/(r²sin²θ). The exact bound‐state solutions are presented explicitly. The creation and annihilation operators are established directly from the normalized radial wave functions. We present two important recurrence relations among the diagonal matrix elements and obtain some explicit expressions of mean values of r^k (8 ≥ k ≥ ?11). The exact form of continuum states is also obtained analytically. Comments are made on the calculation formula of phase shifts and the analytical properties of the scattering amplitude. It is interesting to find that the exact form of continuum states will reduce to that of the bound states at the poles of the scattering amplitude. © 2005 Wiley Periodicals, Inc. Int J Quantum Chem, 2005     

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.