Finite‐element calculations for the helium atom

Zheng, Weiying; Ying, Lung-An

doi:10.1002/qua.10770

Cited by 9 publications

(7 citation statements)

References 33 publications

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“…Finite difference methods (FDM) [45][46][47][48] and Finite element methods (FEM) [49][50][51][52][53][54] represent the solution and the differential equation on a discrete grid. The FDM grid is usually evenly spaced with derivatives calculated to some small (usually second) order.…”

Section: E Direct Solution Of Partial Differential Equation For Bound...mentioning

confidence: 99%

“…FEM uses subdomains, concentrating grid points where more accuracy is needed. Recent work achieves as many as seven decimal places in the energy of the ground state but produces surprisingly nonsmooth wave functions [53,54]. The rate of convergence of these methods is limited by the order of the representation of derivatives and is always algebraic with some small index dependent on the order used for derivatives.…”

Section: E Direct Solution Of Partial Differential Equation For Bound...mentioning

confidence: 99%

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Pseudospectral calculation of the wave function of helium and the negative hydrogen ion

Grabowski

Chernoff

2010

Phys. Rev. A

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We study the numerical solution of the nonrelativistic Schrödinger equation for two-electron atoms in ground and excited S states using pseudospectral (PS) methods of calculation. The calculation achieves convergence rates for the energy, Cauchy error in the wave function, and variance in local energy that are exponentially fast for all practical purposes. The method requires three separate subdomains to handle the wave function's cusplike behavior near the two-particle coalescences. The use of three subdomains is essential to maintaining exponential convergence and is more computationally efficient than a single subdomain. A comparison of several different treatments of the cusps suggests that the simplest prescription is sufficient. We investigate two alternate methods for handling the semi-infinite domain, one which involves a sequence of truncated versions of the domain and the other which employs an algebraic mapping of the semi-infinite domain to a finite one and imposes no explicit cutoffs on the wave function. The latter prescription proves superior. For many purposes it proves unnecessary to handle the three-particle coalescence in a special way. The presence of logarithmic terms in the exact solution is expected to limit the convergence to being nonexponential but the only clear evidence of that is the rate of convergence of derivatives near the three-particle coalescence point. Higher resolution than achieved in this work will ultimately be needed to see its limiting effect on other measures of error. As developed and applied here the PS method has many virtues: no explicit assumptions need be made about the asymptotic behavior of the wave function near cusps or at large distances, the local energy (Hψ/ψ) is exactly equal to the calculated global energy at all collocation points, local errors go down everywhere with increasing resolution, the effective basis using Chebyshev polynomials is complete and simple, and the method is easily extensible to other bound states. As the number of collocation points grows, the method achieves exponential convergence up to the resolution tested. This study serves as a proof-of-principle of the method for more general two-and possibly three-electron applications.

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Section: E Direct Solution Of Partial Differential Equation For Bound...mentioning

confidence: 99%

Section: E Direct Solution Of Partial Differential Equation For Bound...mentioning

confidence: 99%

Pseudospectral calculation of the wave function of helium and the negative hydrogen ion

Grabowski

Chernoff

2010

Phys. Rev. A

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“…For example, the Numerov and Runge‐Kutta methods, as in the latter case, are local approaches to the unknown function by a sequence of overlapping low‐order polynomials in a small subset of user‐defined grid points. [ 11,12 ] The GPS method and its variants in different forms are generally global approaches to the unknown function using global basis functions with a high degree, for example, the trigonometric functions or the orthogonal polynomials of a Sturm‐Liouville problem. [ 6 ] It has also been shown that the GPS method can produce much smoother solutions with incredible exponential convergence, which is significantly accurate compared to other DVR methods.…”

Section: Introductionmentioning

confidence: 99%

Revisiting the generalized pseudospectral method: Radial expectation values, fine structure, and hyperfine splitting of confined atom

Zhu

Jiao

et al. 2020

Int J of Quantum Chemistry

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This paper revisits the generalized pseudospectral (GPS) method on the calculation of various radial expectation values of atomic systems, especially on the spatially confined hydrogen atom and harmonic oscillator. As one of the collocation methods based on global functions, the powerfulness and robustness of the GPS method has been well established in solving the radial Schrödinger equation with high accuracy.However, in our recent work, it was found that the previous calculations based on the GPS method for the radial expectation values of confined systems show significant discrepancies with other theoretical methods. In this work we have tackled such a problem by tracing its source to the GPS method and found that the method itself may not be able to obtain the system wave function at the origin. Combined with an extrapolation method developed here, the GPS method can fully reproduce the radial quantities obtained by other theoretical methods, but with more flexibility, efficiency, and accuracy. We apply the GPS-extrapolation method to investigate the relatievistic fine structure and hyperfine splitting of confined hydrogen atom in s-wave states where the zero-point wave function dominates. Good agreement with previous predictions is obtained for confined hydrogen in low-lying states, and benchmark results are obtained for high-lying excited states. The perturbation treatment of the fine and hyperfine interactions is validated in the confining environment. K E Y W O R D Sconfined atom, fine structure, generalized pseudospectral method, hyperfine splitting, radial expectation value | INTRODUCTIONIn the past few years, the generalized pseudospectral (GPS) method, which is also known as the semispectral method or collocation method, has been successfully applied to investigate the structural properties of atomic and molecular systems, [1][2][3][4][5] the quantum scattering processes in nuclear and atomic physics, [6,7] and the laser-atom interactions. [8][9][10] Its usefulness has been continuously revealed in recent years in accurately and efficiently solving both the time-independent and time-dependent Schrödinger and Dirac equations. Being a type of discrete variable representation (DVR) method, the GPS method shows its special superiority over other generalizations of DVR, such as the finite difference and finite element methods. For example, the Numerov and Runge-Kutta methods, as in the latter case, are local approaches to the unknown function by a sequence of overlapping low-order polynomials in a small subset of user-defined grid points. [11,12] The GPS method and its variants in different forms are generally global approaches to the unknown function using global basis functions with a high degree, for example, the trigonometric

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“…Its adaptability to complex problems and unusual geometries has made FEM popular for problems in classical dynamics, and it has seen many recent applications to quantum electronic structure calculations, particularly of quantum dots [13][14][15]. Its application to vibrational quantum mechanics has been sporadic, although its success in solving the 2D methyl bending vibrational states of toluene dates from 1978 [16].…”

Section: Introductionmentioning

confidence: 98%

Solving the vibrational Schrödinger equation on an arbitrary multidimensional potential energy surface by the finite element method

Stare

Cooksy

2009

Computer Physics Communications

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Finite‐element calculations for the helium atom

Cited by 9 publications

References 33 publications

Pseudospectral calculation of the wave function of helium and the negative hydrogen ion

Pseudospectral calculation of the wave function of helium and the negative hydrogen ion

Revisiting the generalized pseudospectral method: Radial expectation values, fine structure, and hyperfine splitting of confined atom

Solving the vibrational Schrödinger equation on an arbitrary multidimensional potential energy surface by the finite element method

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