Arbitrary l‐wave bound states of the Schrödinger equation for the hyperbolical molecular potential

Within the framework of supersymmetric quantum mechanics method, we study by an algebraic method the arbitrary l‐wave bound states of the Schrödinger equation for the hyperbolical molecular potential by a proper approximation to nonlinear centrifugal term. The explicitly analytical formula of energy levels is derived, and the corresponding bound state wave functions are presented. The function analysis method is used to rederive the same energy levels of the quantum system under consideration to check the validity of this algebraic method. In addition, it is shown from numerical results of energy levels that above certain α parameter depending on the choices of potential parameters V₁ and V₂ the hyperbolical molecular potential cannot trap a particle as it becomes weaker and the energy level starts to turn positive when the potential parameter α becomes larger. © 2014 Wiley Periodicals, Inc.     

Applications of automatic differentiation to the time‐dependent Schrödinger equation

A new approach based upon the Taylor series method is proposed for propagating solutions of the time‐dependent Schrödinger equation. Replacing the spatial derivative of the wave function with finite difference formulas, we derive a recursive formula for the evaluation of Taylor coefficients. The automatic differentiation technique is used to recursively calculate the required Taylor coefficients. We also develop an implicit scheme for the recursive evaluation of these coefficients. We then advance the solution in time using a Taylor series expansion. Excellent computational results are obtained when this method is applied to a one‐dimensional reflectionless potential and a two‐dimensional barrier transmission problem. © 2010 Wiley Periodicals, Inc. Int J Quantum Chem, 2010     

Exact solution of multidimensional hyper‐radial Schrödinger equation for many‐electron quantum systems

In quantum theory, solving Schrödinger equation analytically for larger atomic and molecular systems with cluster of electrons and nuclei persists to be a tortuous challenge. Here, we consider, Schrödinger equation in arbitrary N‐dimensional space corresponding to inverse‐power law potential function originating from a multitude of interactions participating in a many‐electron quantum system for exact solution within the framework of Frobenius method via the formulation of an ansatz to the hyper‐radial wave function. Analytical expressions for energy spectra, and hyper‐radial wave functions in terms of known coefficients of inverse‐power potential function, and wave function parameters have been obtained. A generalized two‐term recurrence relation for power series expansion coefficients has been established. © 2016 Wiley Periodicals, Inc.     

Symplectic algorithm for use in computing the time‐independent Schrödinger equation

The structure of benzoic acid as monomer was studied by semiempirical, ab initio, and density functional methods using several basis sets. The performance of these methods in calculating and describing the vibrational frequencies of benzoic acid and several derivatives was determined. The cyclic dimer form of benzoic acid was also reproduced. Two new procedures of scaling the frequencies were presented. For the ring modes, specific scale equations and scale factors were used from benzene molecule. For the carboxylic group, scaling equations and specific scale factors at different levels were also determined to be used in benzoic acid derivatives. A reassignment of several bands was done. A comparison of the cost/effective method and procedure of scaling was carried out. A significant reduction of the error in the predicted frequencies was obtained over the one‐factor standard scaling procedure. © 2002 Wiley Periodicals, Inc. Int J Quantum Chem, 2002     

Approximate analytical versus numerical solutions of Schrödinger equation under molecular Hua potential

We solve the D‐dimensional Schrödinger equation under the Hua potential by using a Pekeris‐type approximation and the supersymmetry quantum mechanics. The reliability of the spectrum is checked via a comparison with the finite difference method. This interaction resembles Eckart, Morse, and Manning–Rosen potentials. Some useful quantities are reported via the Hellmann–Feynman Theorem. © 2012 Wiley Periodicals, Inc.     

The Rayleigh‐Schrödinger perturbation series of quasi‐degenerate systems

We describe the first representation of the general term of the Rayleigh‐Schrödinger series for quasidegenerate systems. Each term of the series is represented by a tree and there is a straightforward relation between the tree and the analytical expression of the corresponding term. The combinatorial and graphical techniques used in the proof of the series expansion allow us to derive various resummation formulas of the series. A relation with several combinatorial objects used for special cases (degenerate or non‐degenerate systems) is established. © 2011 Wiley Periodicals, Inc. Int J Quantum Chem, 2011     

Analytical Solutions of the Schrödinger Equation with Power Potentials

Conditions for the existence of polynomial solutions of the Schrödinger equation with potentials which are linear combinations of powers of r ranging from ?2 to 2m, where m is a non‐negative integer, are derived and explicit expressions for the coefficients are given. Potentials with m = 0, 1, 2 are discussed in detail.     

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.