The Lagrange-mesh method

Baye, Daniel Jean

doi:10.1016/j.physrep.2014.11.006

Cited by 213 publications

(354 citation statements)

References 232 publications

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“…This nonorthogonal basis will be treated as orthogonal as is usual in the Lagrange-mesh method [18]. The Lagrange basis has the enormous advantage that the matrix elements can be approximated with the consistent Gauss quadrature.…”

Section: Lagrange-mesh Methods Over the Internal Regionmentioning

confidence: 99%

“…In the Schrödinger case, a further simplification keeping a good convergence is obtained by combining the Lagrange-mesh method [17,18] with the R matrix [19,12]. The resulting method does not require any analytical or numerical calculation of matrix elements.…”

Section: Introductionmentioning

confidence: 99%

“…Because of the use of a Gauss quadrature associated with the basis, the matrix elements of the potential are approximated by values of this potential at a set of mesh points. This method can be very accurate [12,18] and is particularly interesting for large coupled-channel calculations [20,21]. The Lagrange-mesh method is applied here to the Dirac case with the same basis in the internal region as in the Schrödinger case.…”

Section: Introductionmentioning

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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Section: Lagrange-mesh Methods Over the Internal Regionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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CalculableR-matrix method for the Dirac equation

Baye

2015

Phys. Rev. A

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“…This term is treated as an inhomogeneous one when solving numerically the FY equations. The core part of the FY partial amplitudes is expanded on the basis of Lagrange-Laguerre mesh functions, employing Lagrange-mesh method [73]:…”

Section: Faddeev-yakubovsky Equations In Configuration Spacementioning

confidence: 99%

Benchmark calculation ofp−H3andn−He3

et al. 2017

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p-3 H and n-3 He scattering in the energy range above the n-3 He but below the d − d thresholds is studied by solving the 4-nucleon problem with a realistic nucleon-nucleon interaction. Three different methods -Alt, Grassberger and Sandhas, Hyperspherical Harmonics, and Faddeev-Yakubovskyhave been employed and their results for both elastic and charge-exchange processes are compared. We observe a good agreement between the three different methods, thus the obtained results may serve as a benchmark. A comparison with the available experimental data is also reported and discussed.

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“…The Schrödinger equation is solved by the Lagrange-mesh method [5][6][7], an approximate variational approach taking the form of a system of mesh equations by computing the Hamiltonian and overlap matrix elements with a Gauss quadrature. Using the coordinates (u, v, w) is essential for an easy treatment of the confinement and an high accuracy of the Gauss quadrature.…”

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confidence: 99%

Helium atom under pressure

Dohet-Eraly

Baye

2016

EPJ Web of Conferences

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Abstract. Hard-sphere confinement is used to study helium atoms under pressure. The confined-helium Schrödinger equation is solved with a high accuracy by a Lagrangemesh method.The effects of high pressure on a helium gas can be estimated by studying the helium atom in a hard confinement, i.e. confined at the centre of an impenetrable spherical cavity, for different cavity radii. Contrary to the free helium atom, the confined helium has not been described with a very high accuracy until recently, when we have developed a Lagrange-mesh method to study this system [1]. This method improves by several order of magnitudes the accuracy of previous approaches [2][3][4].The outlines of the model are the following. The assumed-infinite-mass nucleus is fixed and the electrons are characterized by coordinates r 1 and r 2 with respect to this nucleus. In atomic units, the Hamiltonian of the helium atom readswhere r 12 = r 1 − r 2 and Δ 1 and Δ 2 are the Laplacians with respect to r 1 and r 2 . The confinement is introduced by forcing the wave function into some spherical cavity of radius R (r 1 , r 2 ≤ R). The wave function ψ(r 1 , r 2 , r 12 ) of an S state must thus verify the Schrödinger equationand vanishes at r 1 = R and r 2 = R. The coordinates (r 1 , r 2 , r 12 ) are advantageously replaced by the coordinates (u, v, w) defined over [0,1] byThe confinement implies that the wave function ψ(u, v, w) vanishes at u = 1, v = 1, and w = 1. The Schrödinger equation is solved by the Lagrange-mesh method [5][6][7], an approximate variational approach taking the form of a system of mesh equations by computing the Hamiltonian and overlap matrix elements with a Gauss quadrature. Using the coordinates (u, v, w) is essential for an easy treatment of the confinement and an high accuracy of the Gauss quadrature. The ground-state energy and wave function are obtained by diagonalizing a rather large (matrix dimension≈ 10 3 ∼ 2×10 4 ) but sparse matrix. Ground-state energies and mean interparticle distances for several confinement radii R are given in table 1. The pressure acting on the confined helium atom is also given in table 1. It is a

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The Lagrange-mesh method

Cited by 213 publications

References 232 publications

CalculableR-matrix method for the Dirac equation

CalculableR-matrix method for the Dirac equation

Benchmark calculation ofp−H3andn−He3

Helium atom under pressure

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