A fast convergent hyperspherical expansion for the helium ground state

Haftel, M. I.; Mandelzweig, V. B.

doi:10.1016/0375-9601(87)90215-5

Cited by 61 publications

(34 citation statements)

References 28 publications

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“…This method has been applied by a number of authors to H − and He systems [31,32,33,34], but the rate of convergence is rather slow. To improve the rate of convergence, Haftel and Mandelzweig [35,36,37,38,39] introduced an exponential factor in the expansion, ψ = χΦ, where χ is chosen to be of the form χ = exp{−a(r 1 + r 2 ) + br 12 }, with a and b to be obtained variationally or by some ansatz. If a and b are appropriately chosen, the singularity in the Coulomb potential can be removed, and it is then possible to expand Φ in terms of hyperspherical harmonics with rapid convergence.…”

Section: Hyperspherical Coordinates Methodsmentioning

confidence: 99%

Kinetic energy operator approach to the quantum three-body problem with Coulomb interactions

Chi

Fang

Hsiang

et al. 2007

Solid State Communications

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Abstract. We present a review of the quantum three-body problem, with emphasis on the different methodologies, different three-body atomic systems and their historical interest. With the review as the background, a more recently proposed non-variational, kinetic energy operator approach to the solution of quantum three-body problem is presented, based on the utilization of symmetries intrinsic to the kinetic energy operator, i.e., the three-body Laplacian operator with the respective masses. Through a four-step reduction process, the nine dimensional problem is reduced to a one dimensional coupled system of ordinary differential equations, amenable to accurate numerical solution as an infinite-dimensional algebraic eigenvalue problem. A key observation in this reduction process is that in the functional subspace of the kinetic energy operator where all the rotational degrees of freedom have been projected out, there is an intrinsic symmetry which can be made explicit through the introduction of Jacobi-spherical coordinates. A numerical scheme is presented whereby the Coulomb matrix elements are calculated to a high degree of accuracy with minimal effort, and the truncation of the linear equations is carried out through a systematic procedure. The resulting matrix equations are solved through an iteration process. Numerical results are presented for (1) the negative hydrogen ion H − , (2) the helium and heliumlike ions (Z = 3 ∼ 6), (3) the hydrogen molecular ion H + 2 , and (4) the positronium negative ion Ps − . Up to thirteen-significant-figure accuracy is achieved for the ground state eigenvalues when double precision programming is used. Comparison with the variational and other approaches shows our ground state eigenvalues to be comparable, generally with less decimal digits than the variational results, but can yield highly accurate wavefunctions as by-products. Results on low-lying excited states and their wavefunctions are obtained simultaneously with the ground state properties, some at accuracies not achieved by other methods. In particular, for the doubly excited state 3 P e of H − and the 1,3 P e states of helium, some results are obtained for the first time. Also, we have calculated fourteen H + 2 excited states, up to its dissociation level. Analysis of the wavefunction characteristics, especially in relation to the electronelectron correlation effects, are presented. A significant advantage of the kinetic energy operator approach is its general applicability to different three-body systems, with only the charges, masses, and the symmetry of the desired state as the required inputs. Potential applications of the present approach to scattering and other problems are noted.

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Section: Hyperspherical Coordinates Methodsmentioning

confidence: 99%

Kinetic energy operator approach to the quantum three-body problem with Coulomb interactions

Chi

Fang

Hsiang

et al. 2007

Solid State Communications

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“…Although their method was incomplete, it contained a good idea for simplifying the quantum three-body problem. Developing their idea, we suggest an improved hyperspherical harmonic method to separate three rotational degrees of freedom completely from the internal ones, and the decomposition of the function for fastening the convergence of the series in the hyperspherical harmonic method [15][16][17] is still effective in the improved method.…”

Section: Introductionmentioning

confidence: 99%

“…In the hyperspherical harmonic method [15][16][17], the six Jacobi coordinates, after separation of the center-of-mass motion, are separated to one hyperradial variable ρ and five hyperangular variables Ω. The wave function is presented as a sum of products of hyperradial function and the hyperspherical harmonic function, depending on Ω.…”

Section: Introductionmentioning

confidence: 99%

“…Another difficulty in the practical calculations by the hyperspherical harmonic method is the slow convergence of the series. The convergence was fastened by decomposing the wavefunction ψ = χφ, where χ is chosen to take into account the singularities of the potential and the clustering properties of the wavefunction, and φ is the part to be expanded in hypersphericals [15].…”

Section: Introductionmentioning

confidence: 99%

“…There are many reasons to prefer the direct solutions of the three-body Schrödinger equation over the variational one, for example, the analytic structure of the variational wavefunction is chosen arbitrarily. The accurate direct solution of the three-body Schrödinger equation with the separated center-of-mass motion has been sought based on different numerical methods, such as the finite difference [11], finite element [12], complex coordinate rotation [13], hyperspherical coordinate [14], and hyperspherical harmonic [15][16][17] methods.…”

Section: Introductionmentioning

confidence: 99%

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Quantum three-body problems

2000

Sci. China Ser. A-Math.

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An improved hyperspherical harmonic method for the quantum threebody problem is presented to separate three rotational degrees of freedom completely from the internal ones. In this method, the Schrödinger equation of three-body problem is reduced to a system of linear algebraic equations in terms of the orthogonal bases of functions. As an important example in quantum mechanics, the energies and the eigenfunctions of some states of the helium atom and the helium-like ions are calculated. PACS number(s): 03.65. Ge, and 02.30.Gp

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