A numerical analysis of motion in symmetric double-well harmonic potentials using pseudospectral methods

Kowari, Ken-ichi

doi:10.1016/j.cplett.2019.136941

Cited by 2 publications

(2 citation statements)

References 26 publications

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“…Numerical solution of differential equations can also be obtained with the pseudospectral (PS) method which is based on a combination of the spectral and real-space representations. It has been found to be very accurate in solving onedimensional Schrödinger equation [35][36][37][38][39][40], in particular the Kohn-Sham (KS) equation for atoms [41,42] described with the density functional theory (DFT). The KS equation includes only the local (multiplicative) effective potential which makes the application of the PS method straightforward once suitable scaling is applied.…”

Section: Introductionmentioning

confidence: 99%

Highly accurate numerical solution of Hartree–Fock equation with pseudospectral method for closed-shell atoms

Cinal

2020

J Math Chem

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The Hartree-Fock (HF) equation for atoms with closed (sub)shells is transformed with the pseudospectral (PS) method into a discrete eigenvalue equation for scaled orbitals on a finite radial grid. The Fock exchange operator and the Hartree potential are obtained from the respective Poisson equations also discretized using the PS representation. The numerical solution of the discrete HF equation for closed-(sub)shell atoms from He to No is robust, fast and gives extremely accurate results, with the accuracy superior to that of the previous HF calculations. A very moderate number of 33 to 71 radial grid points is sufficient to obtain total energies with 14 significant digits and occupied orbital energies with 12 to 14 digits in numerical calculations using the double precision (64-bit) of the floating-point format.The electron density at the nucleus is then determined with 13 significant digits and the Kato condition for the density and s orbitals is satisfied with the accuracy of 11 to 13 digits. The node structure of the exact HF orbitals is obtained and their asymptotic dependence, including the common exponential decay, is reproduced very accurately. The accuracy of the investigated quantities is further improved by performing the PS calculations in the quadruple precision (128-bit) floating-point arithmetic which provides the total energies with 25 significant digits while using only 80 to 130 grid points.

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Section: Introductionmentioning

confidence: 99%

Highly accurate numerical solution of Hartree–Fock equation with pseudospectral method for closed-shell atoms

Cinal

2020

J Math Chem

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“…For this reason, a numerical solution of the Schrödinger equation for the DWP is still urgently needed. Recently, various techniques are competing, including pseudospectral methods [34], asymptotic iteration method (AIM) [35], combination of energy factorization approach and stabilization approach [36], Lagrangian description [37], fractional Schrödinger equation [38] and Hill determinant method [39].…”

Section: Introductionmentioning

confidence: 99%

Estimation of Energy Separation in Tunelling States in Symmetric-Hyperbolicus Double-Well Potential Problem using Filter Method

Abdurrouf,

Saroja,

Nurhuda

2024

Trends Sci

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Double well potential (DWP) plays an important role in quantum physics but its analytical exact solutions are incomplete, except for some limited quasi-exact solvable (QES) and therefore numerical solutions are still required. This paper is aimed to obtain numerical solution of the recently proposed symnetric-hyperbolicus DWP and by using our newly developed filter method. The filter method is able to produce eigen-energies and eigenfunctions those are in accordance with the exact analytical results. Next, we obtain the dependence of the ground state energy and the energy separation between the 2 lowest states on the DWP parameter k. For large k, the 2 lowest energy levels are below the potential barrier so the eigenfunctions penetrate the barrier, so the 2 lowest energy levels can be considered as the result of a single energy split. In this case, we observe that the energy separation is an exponential function k, as opposed to a linear function for smaller k in the non tunelling region. The exponential trend of the numerical results is in good agreement with the theoretical approach and allows one to estimate the 2 lowest energies for a given k. HIGHLIGHTS For small , the two lowest energy levels lie above the barrier potential, where both ground state and the energy separation is a linier function of For large , the two lowest energy levels lie below the barrier potential so that the eigenfunction penetrates the barrier. As a result, the two lowest energy levels can be thought of as splitting energy states. In this case, we observe that the energy separation is an exponential function . Since also depends on , it is possible to estimate the two lowest energy levels The exponential trend of the numerical results for large agrees well with the theoretical approach, shown by the linear relationship between and the classical actions in the classical forbidden regions, GRAPHICAL ABSTRACT

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A numerical analysis of motion in symmetric double-well harmonic potentials using pseudospectral methods

Cited by 2 publications

References 26 publications

Highly accurate numerical solution of Hartree–Fock equation with pseudospectral method for closed-shell atoms

Highly accurate numerical solution of Hartree–Fock equation with pseudospectral method for closed-shell atoms

Estimation of Energy Separation in Tunelling States in Symmetric-Hyperbolicus Double-Well Potential Problem using Filter Method

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