Accurate solution of the Dirac equation on Lagrange meshes

Baye, Daniel Jean; Filippin, Livio; Godefroid, Michel

doi:10.1103/physreve.89.043305

Cited by 21 publications

(52 citation statements)

References 30 publications

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“…The potential matrix is then diagonal and only involves values of the potential at mesh points. Recently, we have shown that numerically exact solutions of the CoulombDirac equation can be obtained with this method [12,13]. More generally, the method is very accurate for most central potentials as illustrated with Yukawa potentials in Ref.…”

Section: Introductionmentioning

confidence: 99%

Relativistic two-photon decay rates with the Lagrange-mesh method

2016

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Relativistic two-photon decay rates of the 2s 1/2 and 2p 1/2 states towards the 1s 1/2 ground state of hydrogenic atoms are calculated by using numerically exact energies and wave functions obtained from the Dirac equation with the Lagrange-mesh method. This approach is an approximate variational method taking the form of equations on a grid because of the use of a Gauss quadrature approximation. Highly accurate values are obtained by a simple calculation involving different meshes for the initial, final, and intermediate wave functions and for the calculation of matrix elements. The accuracy of the results with a Coulomb potential is improved by several orders of magnitude in comparison with benchmark values from the literature. The general requirement of gauge invariance is also successfully tested, down to rounding errors. The method provides high accuracies for two-photon decay rates of a particle in other potentials and is applied to a hydrogen atom embedded in a Debye plasma simulated by a Yukawa potential.

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Section: Introductionmentioning

confidence: 99%

Relativistic two-photon decay rates with the Lagrange-mesh method

2016

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“…This can be improved by replacing the Lagrange-Legendre basis in the internal region by an appropriate Lagrange-Jacobi basis [18], in the same spirit as done for bound states in Ref. [27]. With a Laguerre mesh in the external region, no restriction is needed on the potential in bound-state calculations.…”

Section: Resultsmentioning

confidence: 99%

“…This nonorthogonal basis is treated as orthonormal [18]. Using the Gauss-Legendre quadrature for the calculation of integrals [18,27]…”

Section: Lagrange-mesh Methods Over the Internal Regionmentioning

confidence: 99%

“…Potential (111) can be studied with the regularized Lagrange-Laguerre basis over (0, ∞) encountered in Refs. [27,18], with α = 0, i.e. with standard Laguerre polynomials.…”

Section: Regularized Coulomb Potentialmentioning

confidence: 99%

“…The CODATA 2010 value 1/α = 137.035999074 [29] is used for the fine-structure constant α like in Ref. [27]. The exact energies are given by with the effective principal quantum number N = [(n r + γ)…”

Section: Coulomb Potentialmentioning

confidence: 99%

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CalculableR-matrix method for the Dirac equation

Baye

2015

Phys. Rev. A

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An efficient version of the calculable R-matrix method, a technique of determination of scattering and bound-state properties, is extended to the Dirac equation. The configuration space is divided into internal and external regions at the channel radius. In both regions, the introduction of a Bloch operator allows restoring the Hermiticity. The most general Bloch operator contains three free parameters. With a basis without constraint at the channel radius in the internal region, the phase shifts converge to the same value for any choice of these parameters. Nevertheless, some choices provide a faster convergence than others. The determination of the bound-state energies is performed with an extension of the method using a second set of basis functions in the external region. Neither the knowledge of asymptotic expressions nor a large channel radius are required. These R-matrix methods are particularly simple and very accurate when combined with the Lagrange-mesh method. No analytical or numerical evaluation of matrix elements is then necessary. Very accurate phase shifts are obtained with a Legendre mesh for various short-range potentials. A combination of Legendre and Laguerre meshes provides accurate energies for the bound states even for potentials with a Coulomb-like asymptotic behavior.

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Confinement of hydrogen atom with Dirac equation

Baye

2019

Int J of Quantum Chemistry

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The problem of a particle confined in a spherical cavity is studied with the Dirac equation. A hard confinement is obtained by forcing the large component to vanish at the cavity radius. It is shown that the small component cannot vanish simultaneously at this radius. In the case of a confined hydrogen atom, the energies are given by an implicit equation. For some values of the radius, explicit analytical expressions of the energy exist like in the nonrelativistic case. Very accurate energies and wave functions are obtained with the Lagrange‐mesh method with few mesh points. To this end, two differently regularized Lagrange‐Jacobi bases associated with the same mesh are used for the large and small components. The importance of relativistic effects is discussed for hydrogen‐like ions. The validity of this definition of hard confinement is discussed with a soft‐confinement model studied with the R‐matrix method.

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Accurate solution of the Dirac equation on Lagrange meshes

Cited by 21 publications

References 30 publications

Relativistic two-photon decay rates with the Lagrange-mesh method

Relativistic two-photon decay rates with the Lagrange-mesh method

CalculableR-matrix method for the Dirac equation

Confinement of hydrogen atom with Dirac equation

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