Radius of convergence and analytic behavior of the<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"><mml:mfrac><mml:mrow><mml:mn>1</mml:mn></mml:mrow><mml:mrow><mml:mi>Z</mml:mi></mml:mrow></mml:mfrac></mml:math>expansion

Baker, Jonathan D.; Freund, David; Hill, Robert Nyden; Morgan, John D.

doi:10.1103/physreva.41.1247

Cited by 234 publications

(200 citation statements)

References 67 publications

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“…The series of perturbation converges towards the bound-state energy and wave function provided 1/Z 1.09766, according to the detailed studies by Baker et al [136], and Ivanov [137]. This means that (∞ Z , m − , m − ) is bound for Z 0.911.…”

Section: H − -Like Statesmentioning

confidence: 99%

“…A rescaling is often used to set the attraction strength to unity and the repulsion strength to 1/Z, leading to the well-known "1/Z" expansion. See, for instance, [136,137]. When the repulsion among electrons is neglected, one can solve exactly the unperturbed Hamiltonian H 0 = H − 1/r 12 and obtain for the ground state an energy E 0 = −Z 2 and a wave function Ψ 0 ∝ exp(−Zr 1 − Zr 2 ).…”

Section: Appendix D: Elementary Three-and Four-body Calculations 1 Imentioning

confidence: 99%

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Stability of few-charge systems in quantum mechanics

2005

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We consider non-relativistic systems in quantum mechanics interacting through the Coulomb potential, and discuss the existence of bound states which are stable against spontaneous dissociation into smaller atoms or ions. We review the studies that have been made of specific mass configurations and also the properties of the domain of stability in the space of masses or inverse masses. These rigorous results are supplemented by numerical investigations using accurate variational methods. A section is devoted to systems of three arbitrary charges and another to molecules in a world with two space-dimensions.

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Section: H − -Like Statesmentioning

confidence: 99%

Section: Appendix D: Elementary Three-and Four-body Calculations 1 Imentioning

confidence: 99%

Stability of few-charge systems in quantum mechanics

2005

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“…For the case of the Coulomb interaction, an essentially exact curve has been calculated with a highly accurate 418-term Hylleraas-type wave function [73][74][75]. For comparison, we also show the results obtained with the single-determinant HF and RSH wave functions (with the cc-pV6Z basis) which just give horizontal lines since they do not depend on θ.…”

Section: A Convergence Of the Wave Functionmentioning

confidence: 99%

“…(measured from the nucleus) and the position of the second electron is varied on a circle at the same distance of the nucleus, r 2 = (0.5 cos θ, 0.5 sin θ, 0.). For the Coulomb interaction, we compare with the essentially exact curve obtained with a highly accurate 418-term Hylleraas-type wave function [73][74][75]. The curve of Ψ lr,µ (r 1 , r 2 ) as a function of θ reveals the angular correlation between the electrons [82].…”

Section: A Convergence Of the Wave Functionmentioning

confidence: 99%

Basis convergence of range-separated density-functional theory

Franck

Mussard

Luppi

et al. 2015

The Journal of Chemical Physics

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Range-separated density-functional theory is an alternative approach to Kohn-Sham densityfunctional theory. The strategy of range-separated density-functional theory consists in separating the Coulomb electron-electron interaction into long-range and short-range components, and treating the long-range part by an explicit many-body wave-function method and the short-range part by a density-functional approximation. Among the advantages of using many-body methods for the long-range part of the electron-electron interaction is that they are much less sensitive to the oneelectron atomic basis compared to the case of the standard Coulomb interaction. Here, we provide a detailed study of the basis convergence of range-separated density-functional theory. We study the convergence of the partial-wave expansion of the long-range wave function near the electron-electron coalescence. We show that the rate of convergence is exponential with respect to the maximal angular momentum L for the long-range wave function, whereas it is polynomial for the case of the Coulomb interaction. We also study the convergence of the long-range second-order Møller-Plesset correlation energy of four systems (He, Ne, N2, and H2O) with the cardinal number X of the Dunning basis sets cc-p(C)VXZ, and find that the error in the correlation energy is best fitted by an exponential in X. This leads us to propose a three-point complete-basis-set extrapolation scheme for range-separated density-functional theory based on an exponential formula.

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“…For the helium atom, the inclusion of r 12 terms [5] yields an expansion in partial waves converging asymtotically as l −6 or better, instead of a slow l −3 , where l is the maximum angular momentum in the expansion. Baker et al [14] also showed that using basis functions with the same analytic structure as the true wave function (depending on powers and logarithms of r 12 ) leads to a reduced expansion length. The success of methods explicitly including r 12 terms has been attributed to their correct description of the short-range part of the interaction.…”

Section: Introductionmentioning

confidence: 99%

Impact of electron–electron cusp on configuration interaction energies

Nolan

Filippi

Fahy

et al. 2001

The Journal of Chemical Physics

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The effect of the electron-electron cusp on the convergence of configuration interaction (CI) wave functions is examined. By analogy with the pseudopotential approach for electron-ion interactions, an effective electron-electron interaction is developed which closely reproduces the scattering of the Coulomb interaction but is smooth and finite at zero electron-electron separation. The exact many-electron wave function for this smooth effective interaction has no cusp at zero electronelectron separation. We perform CI and quantum Monte Carlo calculations for He and Be atoms, both with the Coulomb electron-electron interaction and with the smooth effective electron-electron interaction. We find that convergence of the CI expansion of the wave function for the smooth electron-electron interaction is not significantly improved compared with that for the divergent Coulomb interaction for energy differences on the order of 1 mHartree. This shows that, contrary to popular belief, description of the electron-electron cusp is not a limiting factor, to within chemical accuracy, for CI calculations.

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Radius of convergence and analytic behavior of the1Zexpansion

Cited by 234 publications

References 67 publications

Stability of few-charge systems in quantum mechanics

Stability of few-charge systems in quantum mechanics

Basis convergence of range-separated density-functional theory

Impact of electron–electron cusp on configuration interaction energies

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