Installment 6 of SQM vs GUTCP Helium Atom Model

https://www.researchgate.net/publication/342343819_Accurate_Ground_State_Energies_of_Helium-like_Ions_Using_a_Simple_Parameter-free_Matrix_Method

Accurate Ground State Energies of Helium-like Ions Using a Simple Parameter-free Matrix Method

June 2020

DOI: 10.13140/RG.2.2.15166.72008

Project: Approximate Ground and First Excited State Energies of He-like Ions using Matrix Method based on Hydrogenic Orbitals

[...]

Theoretical calculations of energies of two-electron systems such as helium and helium-like ions have been attractive since the discovery of quantum mechanics [1] because these systems are the most simple many-body systems and therefore, traditionally used as a testing ground for various methods in theoretical quantum calculations [2].

A large number of theoretical studies has long been conducted to calculate energies of helium-like ions accurately. The methods used in the studies ranged from the most sophisticated to the simplest ones. Some relatively new advanced calculations were quantum electrodynamics calculations [2,3], iterative complement interaction (ICI) method [4,5], and discrete variational-perturbation approach based on explicitly correlated wave functions [6-8]. Older sophisticated calculations include iteration method with the use of perimetric coordinates in the wave function expansion [9-13], numerical Hartree-Fock methods [14], variational calculations with 230-term wave functions [15], variational calculations including relativistic and mass polarization corrections [16], 401-order perturbation calculations [17], and improved Roothaan-Hartree-Fock methods [18]. In general, up to thousands of variational parameters were used in most of such sophisticated calculations to produce highly accurate energies of He-like ions [19]. However, these methods mostly involved tedious analytical calculations and or expensive numerical calculations, mainly due to the use of many parameters in their calculations. Wavefunctions obtained from such methods are often computationally prohibitive to be used in calculating cross-sections of many physical processes such as double ionization of atoms by ion [19]. Therefore, to find more simple wavefunctions with a low number of parameters is essential for the study of those physical processes. [...] With this parameter-free and straightforward method, the aim of this work is not to better highly advanced variational calculation methods but instead to produce more accurate ground state energies of helium-like ions compared to other simple approaches in the literature such as the standard hydrogenic perturbation theory, the uncertainty principle-variational approach and the geometrical model, with much less algebra and minimum computational time. Hartree atomic units were used throughout this work.

[...]
The exact energies for He-like ions with 3 ≤ Z ≤ 10 are -7.2799 a.u, 13.6556 a.u, -22.0310 a.u, -32.4062 a.u, -44.7814 a.u, -59.1566 a.u, -75.5317 a.u, and -93.9068 a.u, respectively [9]. Errors in Table 1 were calculated based on these exact values. Table 1 Ground state energies of light he-like ions (with their % errors) from our analytical calculation (AC), our numerical calculation (NC), the geometrical model (GM) [24], standard perturbation theory (SPT) [25], and the uncertainty principle-variational approach (UPV) [28].**

Ion AC (%error) NC (% error) SPT (% error) GM(% error) UPV (% error)

Li+-7.1886(1.25%) -7.2053(1.02%) -7.1250(2.16%) -7.3249(0.62%) -7.1880(1.26%)

Be2+-13.5570(0.72%) -13.5740(0.60%) -13.5000(1.18%) -13.7373(0.60%) -13.5486(0.78%)

B3+-21.9285(0.47%) -21.9455(0.39%) -21.8750(0.75%) -22.1477(0.53%) -21.9083(0.56%)

C4+-32.3014(0.32%) -32.3183(0.27%) -32.2500(0.52%) -32.5573(0.47%) -32.2672(0.43%)

N5+-44.6749(0.24%) -44.6918(0.20%) -44.6250(0.39%) -44.9661(0.41%) -44.6252(0.35%)

O6+-59.0489(0.18%) -59.0657(0.15%) -59.0000(0.31%) -59.3751(0.37%) -58.9824(0.29%)

F7+-75.4231(0.14%) -75.4398(0.12%) -75.3750(0.25%) -75.7833(0.33%) -75.3388(0.26%)

Ne8+-93.7975(0.12%) -93.8142(0.10%) -93.7500(0.21%) -(-) -93.6943(0.23%)

Notice how the "exact" "ground state" energies of Helium-like ions:

-7.2799 a.u, 13.6556 a.u, -22.0310 a.u, -32.4062 a.u, -44.7814 a.u, -59.1566 a.u, -75.5317 a.u, and -93.9068 a.u ...don't correspond to the experimental first-ionization energies. How do you suppose these "exact" "ground state" energies are derived? Without parameters? How are these "exact" "ground state" energies observed? I.e. what do they mean for the experimenter?

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