Precise nonvariational calculation of excited states of helium with the correlation-function hyperspherical-harmonic method

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

doi:10.1103/physreva.44.7158

Cited by 20 publications

(23 citation statements)

References 30 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…Thk optimization may include refinement cycles as well, if necessary. The results we obtained in this way are compared to the "exact" (i.e., the best calculation in the literature) values [18] in Table 2.1 One can thus get as accurate energies for the excited states as for the ground state.…”

Section: *On Leave From:institutefor Nuclear Research Of the Hungariamentioning

confidence: 86%

Recent applications of the stochastic variational method

Varga

Usukura

Suzuki

1999

Few-Body Problems in Physics ’98

View full text Add to dashboard Cite

show abstract

Section: *On Leave From:institutefor Nuclear Research Of the Hungariamentioning

confidence: 86%

Recent applications of the stochastic variational method

Varga

Usukura

Suzuki

1999

Few-Body Problems in Physics ’98

View full text Add to dashboard Cite

show abstract

“…They include the Hartree-Fock method [15], the finite difference method [19], [35], the correlation-function hyperspherical-harmonic method [18], [24], and various variational approximations. For the Hartree-Fock method, every electron is considered independently to be in a central electric field formed by the nucleus and other electrons.…”

Section: Introductionmentioning

confidence: 99%

Finite Element Approximations to the Discrete Spectrum of the Schrödinger Operator with the Coulomb Potential

Zheng¹,

Ying²

2004

SIAM J. Numer. Anal.

View full text Add to dashboard Cite

In the present paper, the authors consider the Schrödinger operator H with the Coulomb potential defined in R 3m , where m is a positive integer. Both bounded domain approximations to multielectron systems and finite element approximations to the helium system are analyzed. The spectrum of H becomes completely discrete when confined to bounded domains. The error estimate of the bounded domain approximation to the discrete spectrum of H is obtained. Since numerical solution is difficult for a higher-dimensional problem of dimension more than three, the finite element analyses in this paper are restricted to the S-state of the helium atom. The authors transform the six-dimensional Schrödinger equation of the helium S-state into a three-dimensional form. Optimal error estimates for the finite element approximation to the three-dimensional equation, for all eigenvalues and eigenfunctions of the three-dimensional equation, are obtained by means of local regularization. Numerical results are shown in the last section. 1. Introduction. The multielectron Coulomb problem in quantum mechanics cannot be solved in a finite form. Nevertheless it challenges and stimulates many mathematicians and physicists to devote themselves to developing efficient methods for solving the system. Several successful approximation techniques in quantum physics/chemistry have been developed for this problem. They include the Hartree-Fock method [15], the finite difference method [19], [35], the correlation-function hyperspherical-harmonic method [18], [24], and various variational approximations. For the Hartree-Fock method, every electron is considered independently to be in a central electric field formed by the nucleus and other electrons. The finite difference method needs a rectangular domain in R N and uniform grids. The double and triple basis set methods (which are variational methods indeed) are very powerful for the eigenvalue problem of the helium atom. Kono and Hattori (see [21], [22]) used two sets of basis functions r i 1 r j 2 r k 12 e −ξr1−ηr2 A ("ξ terms") and r i 1 r j 2 r k 12 e −ζ(r1+r2) A ("ζ terms") to calculate the energy levels for the S, P , and D states of the helium atom. (A is an appropriate angular factor.) The former set of functions is expected to describe the whole wave function roughly, while the latter is expected to describe the short-and middle-range correlation effects. Their calculations yield 9-10 significant digits for S states. Kleindienst, Lüchow, and Merckens [20] and Drake and Yan [12] applied the double basis set method to S-states of helium. Their basis functions are r i 1 r j 2 r k 12 e −ξ1r1−η1r2 and r i 1 r j 2 r k 12 e −ξ2r1−η2r2 . Drake and Yan employed truncations to ensure numerical stability and convergence. By complete optimization of the exponential scale factors ξ 1 ,

show abstract

“…Some approaches to the problem include various variational methods, the Hartree-Fock method [1], the finite-difference method [2], and the correlationfunction hyperspherical-harmonic method [3,4]. The variational method is the most powerful among them.…”

Section: Introductionmentioning

confidence: 99%

Finite‐element calculations for the helium atom

Zheng

Ying

2003

Int J of Quantum Chemistry

View full text Add to dashboard Cite

ABSTRACT:We give the explicit form of the Hylleraas-Breit transform in the current paper and apply it to fixed-nucleus problems of heliumlike ions. Employing the relation between the total angular momentum and the Hamilton, the six-dimensional Schrö dinger equation is transferred into three-dimensional systems of equations. We use the Lagrange finite-element method to obtain numerical solutions for some lowlying S, 1s2p 3 P, and 1s3d 3 D states of the helium atom.

show abstract

Precise nonvariational calculation of excited states of helium with the correlation-function hyperspherical-harmonic method

Cited by 20 publications

References 30 publications

Recent applications of the stochastic variational method

Recent applications of the stochastic variational method

Finite Element Approximations to the Discrete Spectrum of the Schrödinger Operator with the Coulomb Potential

Finite‐element calculations for the helium atom

Contact Info

Product

Resources

About