Full-scale analytical perturbation calculation of He-isoelectronic series with Hydrogenic bound states via Green's function expansion (monopole) of Coulomb interactions

Sharma, Shivalika; Aggarwal, Priyanka; Hazra, Ram Kuntal

doi:10.1080/00268976.2020.1770881

Cited by 5 publications

(7 citation statements)

References 21 publications

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“…. [36][37][38]47] The Coulomb Green function can be exploited for Z/r 1A , Z/r 2A , Z/r 1B , Z/r 2B and 1/r 12 in multipole expansion as: [48,49] Figure 1. 2c-2e diatomic molecule.…”

Section: Theoretical Developmentmentioning

confidence: 99%

“…Sharma et al and Kapil et al improvised analytical treatments to Coulomb and exchange interactions for 3D and 2D Heisoelectronic series and heavy-hole trions. [36][37][38][39] Kapil et al recently emphasized on analytical multiconfiguration calculation for Period-II elements with hydrogenic spin orbitals as basis set. [40] In this paper, we have formulated analytical diagonal Coulomb integrals (Js) over hydrogenic orbitals employing Green function expansion.…”

Section: Introductionmentioning

confidence: 99%

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Lah Number Mediated Analytical Two‐center Two‐electron Integrals of Coulomb Green Function of H‐like Orbitals for ¹Σ States

Kapil,

Sharma,

Aggarwal

et al. 2023

ChemistrySelect

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Energetics of two‐center two‐electron (2c–2e) systems carry challenges in theoretical understanding of Schrödinger equation (SE) for well‐known divergence of Coulomb interactions and nuclear separation (R) in modified H‐like AOs, Slater type orbitals (STOs), Gaussian type orbitals (GTOs), B‐spline, Sturmian function and etc. employed to VBT and MOT. Certain elegant computational and analytical techniques were developed for STO, GTO and other square integrable basis set within Born‐Oppenheimer (BO) approximation. STOs and GTOs have an essential limitation of absence of radial nodes. Thus, analytical treatment has become an urge for H‐like AOs. We have considered the diatomic molecules only for the sake of simplicity. Employing Sheffer identity in associated Laguerre polynomial/Whittaker‐M function forms of H‐like AOs and transforming integrals into elliptic coordinates with two nuclei on two foci furnishes exact, analytical and simple Coulomb integrals (Js) in terms of R. Lah number originated from for nuclear coordinates only due to Sheffer identity shows that energetics of diatomic molecules can be anticipated as extremum function of R. Therefore, the optimization of potential energy surface (PES) of electrons as gradient of R may lead to σ‐bond formation. In this paper, we have developed diagonal Js for bound states of H2 molecule.

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“…. [36][37][38]47] The Coulomb Green function can be exploited for Z/r 1A , Z/r 2A , Z/r 1B , Z/r 2B and 1/r 12 in multipole expansion as: [48,49] Figure 1. 2c-2e diatomic molecule.…”

Section: Theoretical Developmentmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Lah Number Mediated Analytical Two‐center Two‐electron Integrals of Coulomb Green Function of H‐like Orbitals for ¹Σ States

Kapil,

Sharma,

Aggarwal

et al. 2023

ChemistrySelect

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“…The wavefunction, ξ 0 (r, θ, ϕ)=u(r)Y m l (θ, ϕ), is product of radial and angular part respectively where m=0,±1,±2,...,±l. α 2 =−8E 0 , z=αr, u(r) =⇒ U (z), U (z)=z −1 M (z) and β=2Z/α for bound H-like orbitals (E 0 <0) transform the radial SE as [17,18]:…”

Section: Theoretical Developmentmentioning

confidence: 99%

“…and scaling factors for bra-space coordinates as α 1 =2Z/n 1 , α 3 =2Z/n 3 and for ket-space coordinates as α 2 =2Z/n 2 , α 4 =2Z/n 4 , subject to singly excited (n 2 =1, n 4 ̸ =1 ) or doubly excited states (n 2 , n 4 ̸ =1) for the radial integral (I M ) [17][18][19]. Evaluation of the angular part results in the cancellation of the factor 4π [eq.…”

Section: Ii3 Monopole Factor (L=0)mentioning

confidence: 99%

“…Therefore, H-like orbitals can only address phenomena at atomic scale. Recently, Sharma et al expounded calculations on He-isoelectronic ions as yardsticks up to third order perturbation [17][18][19]. Moreover, Hartree-Fock SCF has one of the well-known limitations of inability to reproduce dissociation energy of 2c-ne molecular systems [34].…”

Section: Introductionmentioning

confidence: 99%

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Analytical multiconfiguration treatment to one-center many-electron He-isoelectronic ions and Period-II elements with H-like bound-states

Kapil,

Sharma,

Aggarwal

et al. 2023

Theor Chem Acc

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Employing H-like spin-orbitals (SOs) in electronic structure theory is a long-awaited quantum problem as the analytical integral of Coulomb interaction is very difficult to solve for one-center manyelectron (1c-ne) system. He-isoelectronic ions become a benchmark. Complexity grows fast for Period-II s-and p-block elements with increasing number of electrons. Moreover, Hartree-Fock Self-Consistent Field (SCF) and post Hartree-Fock SCF theories generally make use of closed-shell, restricted and unrestricted open-shell single configurations (SCs) but actual electronic bound states urge for multiconfigurations (MCs). After Born-Oppenheimer (BO) approximation, utilization of associated Laguerre polynomial/Whittaker-M function basis sets of H-like SOs for the Coulomb Green ′ s function among electrons furnishes analytical, terminable, simple and finitely summed integrals in terms of Lauricella functions. MCs complying with so-called ground, singly and multiply excited states incurring s-and p-SOs are constructed to capture monopole and dipole factors only. However, we believe that quadrupole and higher order poles can be achieved as a product of angular integrals using Wigner 3-j symbols and closed forms of radial integrals. We have observed good agreement among literature and exact ground state energies (GSEs) of He-isoelectronic ions and Period-II elements with both their so-called ground electronic configurations as well as MCs. For certain elements, we have found satisfactory results for ionization energies (IEs).

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Analytical two-center one-electron overlap and exchange integrals for $$^{1}\Sigma$$ states: Lah number guided Coulomb Green function of H-like s-orbitals

Kapil

Hazra

2023

J Chem Sci

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Full-scale analytical perturbation calculation of He-isoelectronic series with Hydrogenic bound states via Green's function expansion (monopole) of Coulomb interactions

Cited by 5 publications

References 21 publications

Lah Number Mediated Analytical Two‐center Two‐electron Integrals of Coulomb Green Function of H‐like Orbitals for ¹Σ States

Lah Number Mediated Analytical Two‐center Two‐electron Integrals of Coulomb Green Function of H‐like Orbitals for ¹Σ States

Analytical multiconfiguration treatment to one-center many-electron He-isoelectronic ions and Period-II elements with H-like bound-states

Analytical two-center one-electron overlap and exchange integrals for $$^{1}\Sigma$$ states: Lah number guided Coulomb Green function of H-like s-orbitals

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Full-scale analytical perturbation calculation of He-isoelectronic series with Hydrogenic bound states via Green's function expansion (monopole) of Coulomb interactions

Cited by 5 publications

References 21 publications

Lah Number Mediated Analytical Two‐center Two‐electron Integrals of Coulomb Green Function of H‐like Orbitals for 1Σ States

Lah Number Mediated Analytical Two‐center Two‐electron Integrals of Coulomb Green Function of H‐like Orbitals for 1Σ States

Analytical multiconfiguration treatment to one-center many-electron He-isoelectronic ions and Period-II elements with H-like bound-states

Analytical two-center one-electron overlap and exchange integrals for $$^{1}\Sigma$$ states: Lah number guided Coulomb Green function of H-like s-orbitals

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Product

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Lah Number Mediated Analytical Two‐center Two‐electron Integrals of Coulomb Green Function of H‐like Orbitals for ¹Σ States

Lah Number Mediated Analytical Two‐center Two‐electron Integrals of Coulomb Green Function of H‐like Orbitals for ¹Σ States