Extended Hylleraas three-electron integral

Pachucki, Krzysztof; Puchalski, Mariusz

doi:10.1103/physreva.71.032514

Cited by 54 publications

(93 citation statements)

References 28 publications

(39 reference statements)

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“…Recursion relations for f (−1, n 2 , n 3 , n 4 , n 5 , n 6 ) have been derived in [10]. Among them, the most numerically unstable is the one which increases the parameter n 4 , see Eq.…”

Section: Extended Hyleraas Integral Withmentioning

confidence: 99%

“…(36) of Ref. [10]. Namely, close to singularity points, x = ±1/2 and for small ω 1 , we use Taylor expansion to avoid numerical instabilities.…”

Section: Extended Hyleraas Integral Withmentioning

confidence: 99%

“…By using integration by parts identities [14], the following differential equations for the h function have been obtained in Ref. [10] ∂h…”

Section: Extended Hyleraas Integral Withmentioning

confidence: 99%

“…, and the solution of the inhomogeneous equation is obtained by Euler's method of variation of constants [10] H(x, y) = 1…”

Section: Extended Hyleraas Integral Withmentioning

confidence: 99%

“…in [8], and later by Yan and Drake in Refs. [9] and by us in [2,10]. The use of recursion relations allows for a significant increase in the size of the basis set and the accuracy of the obtained results [2].…”

Section: Introductionmentioning

confidence: 99%

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Singular Hylleraas three-electron integrals

Pachucki

Puchalski

2008

Phys. Rev. A

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Calculations of the leading quantum electrodynamics effects in few electron systems involve singular matrix elements of the inter-electronic distances of the form 1/r 3 i and 1/r 3 ij . Integrals that result when the nonrelativistic wave function is represented with a Hylleraas basis representation are studied. Recursion relations for various powers of the electron coordinates and the master integrals are derived in a form suited for high precision numerical evaluations.

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“…Recursion relations for f (−1, n 2 , n 3 , n 4 , n 5 , n 6 ) have been derived in [10]. Among them, the most numerically unstable is the one which increases the parameter n 4 , see Eq.…”

Section: Extended Hyleraas Integral Withmentioning

confidence: 99%

“…(36) of Ref. [10]. Namely, close to singularity points, x = ±1/2 and for small ω 1 , we use Taylor expansion to avoid numerical instabilities.…”

Section: Extended Hyleraas Integral Withmentioning

confidence: 99%

“…By using integration by parts identities [14], the following differential equations for the h function have been obtained in Ref. [10] ∂h…”

Section: Extended Hyleraas Integral Withmentioning

confidence: 99%

“…, and the solution of the inhomogeneous equation is obtained by Euler's method of variation of constants [10] H(x, y) = 1…”

Section: Extended Hyleraas Integral Withmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Singular Hylleraas three-electron integrals

Pachucki

Puchalski

2008

Phys. Rev. A

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Few‐parameter exponentially correlated wavefunctions for the ground state of lithium

Albert

Guevara

Sabin

et al. 2009

Int J of Quantum Chemistry

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ABSTRACT:Compact, but relatively accurate wavefunctions for the ground state of the Li atom were obtained through the use of a limited basis of exponentially correlated functions with optimized nonlinear parameters. In contrast to our earlier work, the basis contains pre-exponential factors that improve the rate of convergence of the basis-set expansion. The matrix elements needed in the present work were evaluated analytically using recursive methods reported recently by one of us; a check on the programming was provided by comparison with numerical evaluations carried out by Turbiner and Guevara. The rate of convergence of the expansion is compared with those of Hylleraas-basis computations, and a comparison is also made with exponentially correlated studies of He-like systems.

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Precision Calculations for Three-Body Molecular Bound States

Karr¹,

Haidar²,

Hilico³

et al. 2020

Recent Progress in Few-Body Physics

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Although they do not lend themselves to analytical resolution, three-body atomic or molecular systems are still simple enough to allow for very precise theoretical predictions of their energy levels, which makes them attractive candidates for fundamental tests and determination of fundamental physical constants. Focusing on the hydrogen molecular ions (H + 2 , HD + , D + 2 ), we outline the methods which have been used to improve the theoretical accuracy by several orders of magnitude over the last two decades. The three-body Schrödinger equation can be solved with extreme precision by variational methods with trial functions involving exponentials of interparticle distances. Quantum electrodynamics (QED) corrections are evaluated in the framework of nonrelativistic QED (NRQED). The current status of theory and possibilities of further improvement are briefly sketched.

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Extended Hylleraas three-electron integral

Cited by 54 publications

References 28 publications

Singular Hylleraas three-electron integrals

Singular Hylleraas three-electron integrals

Few‐parameter exponentially correlated wavefunctions for the ground state of lithium

Precision Calculations for Three-Body Molecular Bound States

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