Three-body quantum Coulomb problem: Analytic continuation

Turbiner, A. V.; Vieyra, J. C. López; Pilón, Horacio Olivares

doi:10.1142/s021773231650156x

Cited by 9 publications

(17 citation statements)

References 32 publications

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“…In turn, at small Z, following the qualitative prediction by Stillinger and Stillinger [22] and further quantitative studies performed in [23], [24], there exists a certain value Z B > 0 for which the non-relativistic ground state energy with static nuclei is given by the Puiseux expansion in a certain fractional degrees…”

Section: Expansionssupporting

confidence: 60%

“…in the ground state energy for both systems. Surprisingly, this domain is described accurately by a 4th degree polynomial (without linear term) in variable λ = √ Z − Z B , where Z B is the 2nd critical charge [24]. This domain can also be fitted by the Majorana formula -the 2nd degree polynomial in Z (12) with two free parameters, while e 2 -the coefficient in front of Z 2 term -is kept fixed and equal to the sum of the (ground state) energies of 2(3)-Hydrogen atomswith similar accuracies of 4-3 s.d.!…”

Section: Discussionmentioning

confidence: 99%

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Few-electron atomic ions in non-relativistic QED: The ground state

2019

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Following detailed analysis of relativistic, QED and mass corrections for helium-like and lithiumlike ions with static nuclei for Z ≤ 20 the domain of applicability of Non-Relativistic QED (NRQED) is localized for ground state energy. It is demonstrated that for both helium-like and lithium-like ions with Z ≤ 20 the finite nuclear mass effects do not change 4-5 significant digits (s.d.) and the leading relativistic and QED effects leave unchanged 3-4 s.d. in the ground state energy. It is shown that the non-relativistic ground state energy can be interpolated with accuracy of not less than 6 decimal digits (d.d.) (or 7-8 s.d.) for Z ≤ 50 for helium-like and for Z ≤ 20 for lithium-like ions by a meromorphic function in, which is well inside the domain of applicability of NRQED. It is found that both the Majorana formula -a second degree polynomial in Z with two free parameters -and a fourth degree polynomial in λ (a generalization of the Majorana formula) reproduce the ground state energy of the helium-like and lithium-like ions for Z ≤ 20 in the domain of applicability of NRQED, thus, at least, 3 s.d. It is noted that 99.9% of the ground state energy is given by the variational energy for properly optimized trial function of the form of (anti)-symmetrized product of three (six) screened Coulomb orbitals for two-(three) electron system with 3 (7) free parameters for Z ≤ 20, respectively. It may imply that these trial functions are, in fact, exact wavefunctions in non-relativistic QED, thus, the NRQED effective potential can be derived. It is shown that the sum of relativistic and QED effects in leading approximation -3 s.d. -for both 2 and 3 electron systems is interpolated by 4th degree polynomial in Z

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Section: Expansionssupporting

confidence: 60%

Section: Discussionmentioning

confidence: 99%

Few-electron atomic ions in non-relativistic QED: The ground state

2019

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“…where Z B > 0 is a certain critical charge and E B = E(Z B ). This was confirmed quantitatively in [28]- [29]. Moreover, the expansion (20) was derived numerically in [39] using highly accurate values of ground state energy in close vicinity of Z > Z B obtained variationally.…”

Section: Z and Puiseux Expansionssupporting

confidence: 56%

“…cf. [29]. Now, we take the same optimal parameters α = 0.929044, β = −0.254746 obtained at Z = 2, and expanding the function…”

Section: Z and Puiseux Expansionsmentioning

confidence: 99%

Helium-like ions in $d$-dimensions: analyticity and generalized ground state Majorana solutions

Escobar-Ruiz,

Olivares-Pilón,

Aquino

et al. 2021

Preprint

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Non-relativistic Helium-like ions (−e, −e, Ze) with static nuclei in a d−dimensional space R d (d > 1) are considered. Assuming r −1 Coulomb interactions, a 2-parametric correlated Hylleraastype trial function is used to calculate the ground state energy of the system in the domain Z ≤ 10. For the lowest values d = 2, 3, 4, 5, a closed analytical expression for the corresponding expectation value of the energy is derived explicitly. For odd d = 3, 5, it is given by a rational algebraic function of the variational parameters whilst for even d = 2, 4 it is shown for the first time that it corresponds to a more complicated non-algebraic expression. These two types of analyticity will hold for any d. It allows us to construct accurate compact generalized Majorana analytical solutions for the ground state energy. They reproduce with good precision the first leading terms in the celebrated 1 Z expansion, and serve as generating functions for certain correlation-dependent properties. The (first) critical charge Z c vs d and the Shannon entropy S (d) r vs Z are also displayed. In the light of these results, for the physically important case d = 3 a more general 3-parametric correlated Hylleraas-type trial is used to compute the finite mass effects in the Majorana solution for a three-body Coulomb system with arbitrary charges and masses. It admits a straightforward generalization to any d. Concrete results for the systems e − e − e + , H + 2 and H − are indicated as well. Our variational analytical results are in excellent agreement with the exact numerical values reported in the literature.

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“…This function is characterized by four free parameters with one constraint (24), a = À α + β, due to boundary condition for (22), or equivalently, K = 0 in (26). Extra conditions D 0 = D = 0 from (26)-the absence of the terms r 12 , r 2 12 -are fulfilled automatically.…”

Section: Compact Trial Functions For Spin-triplet Statementioning

confidence: 99%

Ultra‐compact accurate wave functions for He‐like and Li‐like iso‐electronic sequences and variational calculus:II.Spin‐singlet (excited) and spin‐triplet (lowest) states of helium sequence

Turbiner

Vieyra

Valle

et al. 2022

Int J of Quantum Chemistry

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As a continuation of Part I (Int. J. Quantum Chem. 2021; 121: qua.26586), dedicated to the ground state of He-like and Li-like isoelectronic sequences for nuclear charges Z ≤ 20, a few ultra-compact wave functions in the form of generalized Hylleraas-Kinoshita functions are constructed, which describe the domain of applicability of the Quantum Mechanics of Coulomb Charges (QMCC) for energies (4-5 significant digits [s.d.]) of two excited states of He-like ions: the spin-singlet (first) excited state 2 1 S and lowest spin-triplet 1 3 S state. For both states, it provides absolute accuracy for energy $10 À3 a.u., exact values for cusp parameters and also for six expectation values the relative accuracy $10 À2 . Bressanini-Reynolds observation about the special form of nodal surface of 2 1 S state of helium is confirmed and extended to He-like ions with Z > 2. Critical charges Z = Z B , where ultra-compact trial functions lose their square-integrability, are estimated: Z B (1 1 S) ≈ Z B (2 1 S) $ 0.905 and Z B (1 3 S) $ 0.902. For both excited states, the Majorana formula-the energy as the second degree polynomial in Z-provides accurately the 4-5 significant digits for Z ≤ 20.

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Three-body quantum Coulomb problem: Analytic continuation

Cited by 9 publications

References 32 publications

Few-electron atomic ions in non-relativistic QED: The ground state

Few-electron atomic ions in non-relativistic QED: The ground state

Helium-like ions in $d$-dimensions: analyticity and generalized ground state Majorana solutions

Ultra‐compact accurate wave functions for He‐like and Li‐like iso‐electronic sequences and variational calculus:II.Spin‐singlet (excited) and spin‐triplet (lowest) states of helium sequence

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