1
|
Baye D. Klein-Gordon equation on a Lagrange mesh. Phys Rev E 2024; 109:045303. [PMID: 38755927 DOI: 10.1103/physreve.109.045303] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 02/05/2024] [Accepted: 03/25/2024] [Indexed: 05/18/2024]
Abstract
The Lagrange-mesh method is an approximate variational method which provides accurate solutions of the Schrödinger equation for bound-state and scattering few-body problems. The stationary Klein-Gordon equation depends quadratically on the energy. For a central potential, it is solved on a Lagrange-Laguerre mesh by iteration. Results are tested with the Coulomb potential for which exact solutions are available. A high accuracy is obtained with a rather small number of mesh points. For various potentials and levels, few iterations provide accurate energies and mean values in short computer times. Analytical expressions of the wave functions are available.
Collapse
Affiliation(s)
- Daniel Baye
- Nuclear Physics and Quantum Physics, C.P. 229, Université Libre de Bruxelles (ULB), B-1050 Brussels Belgium
| |
Collapse
|
2
|
Rodríguez-Arcos M, Bermúdez-Montana M, Lemus R, Arias JM, Gómez-Camacho J. Configuration localised states from orthogonal polynomials for effective potentials in 3D systems vs. algebraic DVR approaches. Mol Phys 2022. [DOI: 10.1080/00268976.2022.2044082] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 10/18/2022]
Affiliation(s)
- M. Rodríguez-Arcos
- Instituto de Ciencias Nucleares, Universidad Nacional Autónoma de México, Ciudad de Mèxico, Mexico
| | - M. Bermúdez-Montana
- Instituto de Ciencias Nucleares, Universidad Nacional Autónoma de México, Ciudad de Mèxico, Mexico
| | - R. Lemus
- Instituto de Ciencias Nucleares, Universidad Nacional Autónoma de México, Ciudad de Mèxico, Mexico
| | - J. M. Arias
- Departamento de Física Atómica, Molecular y Nuclear, Facultad de Física, Universidad de Sevilla, Sevilla, Spain
- Instituto Carlos I (iCI) de Física Teórica y Computacional, Universidad de Sevilla, Sevilla, Spain
| | - J. Gómez-Camacho
- Departamento de Física Atómica, Molecular y Nuclear, Facultad de Física, Universidad de Sevilla, Sevilla, Spain
- CN de Aceleradores (U. Sevilla, J. Andalucía, CSIC), Sevilla, Spain
| |
Collapse
|
3
|
Algebraic DVR Approaches Applied to Piecewise Potentials: Symmetry and Degeneracy. Symmetry (Basel) 2022. [DOI: 10.3390/sym14030445] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/17/2022] Open
Abstract
Algebraic discrete variable representation (DVR) methods that have been recently proposed are applied to describe 1D and 2D piecewise potentials. First, it is shown that it is possible to use a DVR approach to describe 1D square well potentials testing the wave functions with exact results. Thereafter, Morse and Pöschl-Teller (PT) potentials are described with multistep piecewise potentials in order to explore the sensibility of the potential to reproduce the transition from a pure square well energy pattern to an anharmonic energy spectrum. Once the properties of the different algebraic DVR approaches are identified, the 2D square potential as a function of the potential depth is studied. We show that this system displays natural degeneracy, accidental degeneracy and systematic accidental degeneracy. The latter appears only for a confined potential, where the symmetry group is identified and irreducible representations are constructed. One particle confined in a rectangular well potential with commensurate sides is also analyzed. It is proved that the systematic accidental degeneracy appearing in this system is removed for finite potential depth.
Collapse
|
4
|
Rodríguez-Arcos M, Bermúdez-Montana M, Lemus R. Algebraic discrete variable representation approach applied to Lennard-Jones and H 2 potentials. Mol Phys 2021. [DOI: 10.1080/00268976.2021.1957169] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 10/20/2022]
Affiliation(s)
| | | | - R. Lemus
- Instituto de Ciencias Nucleares, UNAM, Mexico, CDMX, Mexico
| |
Collapse
|
5
|
Rodríguez-Arcos M, Bermúdez-Montaña M, Arias JM, Gómez-Camacho J, Orgaz E, Lemus R. Algebraic discrete variable representation approaches: application to interatomic effective potentials. Mol Phys 2021. [DOI: 10.1080/00268976.2021.1876264] [Citation(s) in RCA: 2] [Impact Index Per Article: 0.7] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 10/22/2022]
Affiliation(s)
- M. Rodríguez-Arcos
- Instituto de Ciencias Nucleares, Universidad Nacional Autónoma de México, Coyoacan, México, DF, México
| | - M. Bermúdez-Montaña
- Instituto de Ciencias Nucleares, Universidad Nacional Autónoma de México, Coyoacan, México, DF, México
- Facultad de Química, Universidad Nacional Autónoma de México, Coyoacan, México, DF, México
| | - J. M. Arias
- Departamento de Física Atómica, Molecular y Nuclear, Facultad de Física, Universidad de Sevilla, Sevilla, Spain
- Instituto Carlos I (iCI) de Física Teórica y Computacional, Universidad de Sevilla, Sevilla, Spain
| | - J. Gómez-Camacho
- Departamento de Física Atómica, Molecular y Nuclear, Facultad de Física, Universidad de Sevilla, Sevilla, Spain
- CN de Aceleradores (U. Sevilla, J. Andalucía, CSIC), Sevilla, Spain
| | - E. Orgaz
- Facultad de Química, Universidad Nacional Autónoma de México, Coyoacan, México, DF, México
| | - R. Lemus
- Instituto de Ciencias Nucleares, Universidad Nacional Autónoma de México, Coyoacan, México, DF, México
| |
Collapse
|
6
|
Abstract
Two algebraic approaches based on a discrete variable representation are introduced and applied to describe the Stark effect in the non-relativistic Hydrogen atom. One approach consists of considering an algebraic representation of a cutoff 3D harmonic oscillator where the matrix representation of the operators r2 and p2 are diagonalized to define useful bases to obtain the matrix representation of the Hamiltonian in a simple form in terms of diagonal matrices. The second approach is based on the U(4) dynamical algebra which consists of the addition of a scalar boson to the 3D harmonic oscillator space keeping constant the total number of bosons. This allows the kets associated with the different subgroup chains to be linked to energy, coordinate and momentum representations, whose involved branching rules define the discrete variable representation. Both methods, although originating from the harmonic oscillator basis, provide different convergence tests due to the fact that the associated discrete bases turn out to be different. These approaches provide powerful tools to obtain the matrix representation of 3D general Hamiltonians in a simple form. In particular, the Hydrogen atom interacting with a static electric field is described. To accomplish this task, the diagonalization of the exact matrix representation of the Hamiltonian is carried out. Particular attention is paid to the subspaces associated with the quantum numbers n=2,3 with m=0.
Collapse
|
7
|
A numerical analysis of motion in symmetric double-well harmonic potentials using pseudospectral methods. Chem Phys Lett 2020. [DOI: 10.1016/j.cplett.2019.136941] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/19/2022]
|
8
|
Zhan Y, Shizgal BD. Diffusion in a bistable system: The eigenvalue spectrum of the Fokker-Planck operator and Kramers' reaction rate theory. Phys Rev E 2019; 99:042101. [PMID: 31108642 DOI: 10.1103/physreve.99.042101] [Citation(s) in RCA: 3] [Impact Index Per Article: 0.6] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 12/10/2018] [Indexed: 11/07/2022]
Abstract
The time-dependent solution of the Fokker-Planck equation with bistable potentials is considered in terms of the eigenfunctions and eigenvalues of the linear Fokker-Planck operator. The Fokker-Planck equation is the high friction limit of the corresponding Kramers' equation. Two different potentials are considered defined with a constant diffusion coefficient, ε, and position-dependent drift coefficients. The smallest nonzero eigenvalue of the Fokker-Planck operator, λ_{1}, provides the long-time rate coefficient for the transformation of the different species in the two stable states. A novel pseudospectral method with nonclassical polynomials is applied to this class of systems. The convergence of the eigenvalues and eigenfunctions of the Fokker-Planck operator versus the number of basis functions is studied and compared with previous results. The results are consistent with Kramers' theory, and a linear relationship between lnλ_{1} and 1/ε for sufficiently small ε values is verified. A comparison with analytic approximations to λ_{1} is provided.
Collapse
Affiliation(s)
- Yahui Zhan
- Department of Mathematics, University of British Columbia Vancouver, British Columbia, V6T1Z1 Canada
| | - Bernie D Shizgal
- Department of Chemistry, University of British Columbia Vancouver, British Columbia, V6T1Z1 Canada
| |
Collapse
|
9
|
Bao J, Shizgal BD. Pseudospectral method of solution of the Schrödinger equation for the Kratzer and pseudoharmonic potentials with nonclassical polynomials and applications to realistic diatom potentials. COMPUT THEOR CHEM 2019. [DOI: 10.1016/j.comptc.2019.01.001] [Citation(s) in RCA: 6] [Impact Index Per Article: 1.2] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 10/27/2022]
|
10
|
Shizgal BD. A comparison of pseudospectral methods for the solution of the Schrödinger equation; the Lennard-Jones ( n , 6) potential. COMPUT THEOR CHEM 2017. [DOI: 10.1016/j.comptc.2017.05.009] [Citation(s) in RCA: 2] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 10/19/2022]
|
11
|
Shizgal BD. Pseudospectral method of solution of the Schrödinger equation with non classical polynomials; the Morse and Pöschl–Teller (SUSY) potentials. COMPUT THEOR CHEM 2016. [DOI: 10.1016/j.comptc.2016.03.002] [Citation(s) in RCA: 8] [Impact Index Per Article: 1.0] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 10/22/2022]
|
12
|
Szalay V, Ádám P. Variational properties of the discrete variable representation: discrete variable representation via effective operators. J Chem Phys 2012; 137:064118. [PMID: 22897266 DOI: 10.1063/1.4740486] [Citation(s) in RCA: 4] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/15/2022] Open
Abstract
A variational finite basis representation/discrete variable representation (FBR/DVR) Hamiltonian operator has been introduced. By calculating its matrix elements exactly one obtains, depending on the choice of the basis set, either a variational FBR or a variational DVR. The domain of grid points on which the FBR/DVR is variational has been shown to consist of the subsets of the set of grid points one obtains by diagonalizing commuting variational basis representations of the coordinate operators. The variational property implies that the optimal of the subsets of a fixed number of points, i.e., the subset which gives the possible highest accuracy eigenpairs, gives the DVR of the smallest trace. The symmetry properties of the variational FBR/DVR Hamiltonian operator are analyzed and methods to incorporate symmetry into FBR/DVR calculations are discussed. It is shown how the Fourier-basis FBR/DVR suitable to solving periodic systems arise within the theory presented. Numerical examples are given to illustrate the theoretical results. The use of variational effective Hamiltonian and coordinate operators has been instrumental in this study. They have been introduced in a novel way by exploiting quasi-Hermiticity.
Collapse
Affiliation(s)
- Viktor Szalay
- Institute for Solid State Physics and Optics, Wigner Research Centre for Physics, Hungarian Academy of Sciences, P. O. Box 49, H-1525 Budapest, Hungary.
| | | |
Collapse
|
13
|
Lo J, Shizgal BD. Spectral convergence of the quadrature discretization method in the solution of the Schrödinger and Fokker-Planck equations: Comparison with sinc methods. J Chem Phys 2006; 125:194108. [PMID: 17129090 DOI: 10.1063/1.2378622] [Citation(s) in RCA: 4] [Impact Index Per Article: 0.2] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 01/04/2023] Open
Abstract
Spectral methods based on nonclassical polynomials and Fourier basis functions or sinc interpolation techniques are compared for several eigenvalue problems for the Fokker-Planck and Schrodinger equations. A very rapid spectral convergence of the eigenvalues versus the number of quadrature points is obtained with the quadrature discretization method (QDM) and the appropriate choice of the weight function. The QDM is a pseudospectral method and the rate of convergence is compared with the sinc method reported by Wei [J. Chem. Phys., 110, 8930 (1999)]. In general, sinc methods based on Fourier basis functions with a uniform grid provide a much slower convergence. The paper considers Fokker-Planck equations (and analogous Schrodinger equations) for the thermalization of electrons in atomic moderators and for a quartic potential employed to model chemical reactions. The solution of the Schrodinger equation for the vibrational states of I2 with a Morse potential is also considered.
Collapse
Affiliation(s)
- Joseph Lo
- Institute of Applied Mathematics, University of British Columbia, Vancouver, British Columbia V6T 1Z1, Canada.
| | | |
Collapse
|
14
|
Peng LY, Starace AF. Application of Coulomb wave function discrete variable representation to atomic systems in strong laser fields. J Chem Phys 2006; 125:154311. [PMID: 17059259 DOI: 10.1063/1.2358351] [Citation(s) in RCA: 16] [Impact Index Per Article: 0.9] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 01/15/2023] Open
Abstract
We present an efficient and accurate grid method for solving the time-dependent Schrodinger equation for an atomic system interacting with an intense laser pulse. Instead of the usual finite difference (FD) method, the radial coordinate is discretized using the discrete variable representation (DVR) constructed from Coulomb wave functions. For an accurate description of the ionization dynamics of atomic systems, the Coulomb wave function discrete variable representation (CWDVR) method needs three to ten times fewer grid points than the FD method. The resultant grid points of the CWDVR are distributed unevenly so that one has a finer grid near the origin and a coarser one at larger distances. The other important advantage of the CWDVR method is that it treats the Coulomb singularity accurately and gives a good representation of continuum wave functions. The time propagation of the wave function is implemented using the well-known Arnoldi method. As examples, the present method is applied to multiphoton ionization of both the H atom and the H(-) ion in intense laser fields. The short-time excitation and ionization dynamics of H by an abruptly introduced static electric field is also investigated. For a wide range of field parameters, ionization rates calculated using the present method are in excellent agreement with those from other accurate theoretical calculations.
Collapse
Affiliation(s)
- Liang-You Peng
- Department of Physics and Astronomy, The University of Nebraska-Lincoln, Nebraska 68588-0111, USA.
| | | |
Collapse
|
15
|
Szalay V. Optimal grids for generalized finite basis and discrete variable representations: definition and method of calculation. J Chem Phys 2006; 125:154115. [PMID: 17059247 DOI: 10.1063/1.2358979] [Citation(s) in RCA: 9] [Impact Index Per Article: 0.5] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/14/2022] Open
Abstract
The method of optimal generalized finite basis and discrete variable representations (FBR and DVR) generalizes the standard, Gaussian quadrature grid-classical orthonormal polynomial basis-based FBR/DVR method to general sets of grid points and to general, nondirect product, and/or nonpolynomial bases. Here, it is shown how an optimal set of grid points can be obtained for an optimal generalized FBR/DVR calculation with a given truncated basis. Basis set optimized and potential optimized grids are defined. The optimized grids are shown to minimize a function of grid points derived by relating the optimal generalized FBR of a Hamiltonian operator to a non-Hermitian effective Hamiltonian matrix. Locating the global minimum of this function can be reduced to finding the zeros of a function in the case of one dimensional problems and to solving a system of D nonlinear equations repeatedly in the case of D>1 dimensional problems when there is an equal number of grid points and basis functions. Gaussian quadrature grids are shown to be basis optimized grids. It is demonstrated by a numerical example that an optimal generalized FBR/DVR calculation of the eigenvalues of a Hamiltonian operator with potential optimized grids can have orders of magnitude higher accuracy than a variational calculation employing the same truncated basis. Nevertheless, for numerical integration with the optimal generalized FBR quadrature rule basis optimized grids are the best among grids of the same number of points. The notions of Gaussian quadrature and Gaussian quadrature accuracy are extended to general, multivariable basis functions.
Collapse
Affiliation(s)
- Viktor Szalay
- Crystal Physics Laboratory, Research Institute for Solid State Physics and Optics, Hungarian Academy of Sciences, P.O. Box 49, H-1525 Budapest, Hungary.
| |
Collapse
|
16
|
Hu XG, Ho TS, Rabitz H. Solving the bound-state Schrodinger equation by reproducing kernel interpolation. PHYSICAL REVIEW. E, STATISTICAL PHYSICS, PLASMAS, FLUIDS, AND RELATED INTERDISCIPLINARY TOPICS 2000; 61:2074-2085. [PMID: 11046499 DOI: 10.1103/physreve.61.2074] [Citation(s) in RCA: 4] [Impact Index Per Article: 0.2] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 11/16/1998] [Indexed: 05/23/2023]
Abstract
Based on reproducing kernel Hilbert space theory and radial basis approximation theory, a grid method is developed for numerically solving the N-dimensional bound-state Schrodinger equation. Central to the method is the construction of an appropriate bounded reproducing kernel (RK) Lambda(alpha)( ||r ||) from the linear operator -nabla(2)(r)+lambda(2) where nabla(2)(r) is the N-dimensional Laplacian, lambda>0 is a parameter related to the binding energy of the system under study, and the real number alpha>N. The proposed (Sobolev) RK Lambda(alpha)(r,r(')) is shown to be a positive-definite radial basis function, and it matches the asymptotic solutions of the bound-state Schrodinger equation. Numerical tests for the one-dimensional (1D) Morse potential and 2D Henon-Heiles potential reveal that the method can accurately and efficiently yield all the energy levels up to the dissociation limit. Comparisons are also made with the results based on the distributed Gaussian basis method in the 1D case as well as the distributed approximating functional method in both 1D and 2D cases.
Collapse
Affiliation(s)
- XG Hu
- Department of Chemistry, Princeton University, Princeton, New Jersey 08544-1009, USA
| | | | | |
Collapse
|
17
|
Wei GW. Discrete singular convolution for the solution of the Fokker–Planck equation. J Chem Phys 1999. [DOI: 10.1063/1.478812] [Citation(s) in RCA: 226] [Impact Index Per Article: 9.0] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/14/2022] Open
|
18
|
Drozdov AN, Hayashi S. Improved power series expansion for the time evolution operator: Application to two-dimensional systems. J Chem Phys 1999. [DOI: 10.1063/1.477855] [Citation(s) in RCA: 2] [Impact Index Per Article: 0.1] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/14/2022] Open
|
19
|
Hu XG, Ho TS, Rabitz H. Variational reproducing kernel Hilbert space (RKHS) grid method for quantum mechanical bound-state problems. Chem Phys Lett 1998. [DOI: 10.1016/s0009-2614(98)00341-8] [Citation(s) in RCA: 7] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/26/2022]
|
20
|
Zhang D, Wei G, Kouri D, Hoffman D. Lagrange distributed approximating functional method for the solution of the Schrödinger equation. Chem Phys Lett 1998. [DOI: 10.1016/s0009-2614(97)01360-2] [Citation(s) in RCA: 6] [Impact Index Per Article: 0.2] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 10/18/2022]
|
21
|
Zhang DS, Wei GW, Kouri DJ, Hoffman DK. Distributed approximating functional approach to the Fokker–Planck equation: Eigenfunction expansion. J Chem Phys 1997. [DOI: 10.1063/1.473520] [Citation(s) in RCA: 24] [Impact Index Per Article: 0.9] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/14/2022] Open
|