Atomic Structure Calculations for Neutral Oxygen

Energy levels and oscillator strengths for neutral oxygen have been calculated using the Cowan (CW), SUPERSTRUCTURE (SS), and AUTOSTRUCTURE (AS) atomic structure codes. The results obtained with these atomic codes have been compared with MCHF calculations and experimental values from the National Institute of Standards and Technology (NIST) database.


Introduction
Oxygen atom (O I) is the most abundant element after hydrogen and helium in the Universe.Its spectroscopic study is very important for the knowledge of the structure of stars, galaxies and in general the whole Universe.It is also important for studying the life on the earth and the possibility of life on other planets or exoplanets.The studies of earth's atmosphere and its radiative properties need these data.Industrial and technical applications need the characteristics of this element.
Pradhan and Saraph [1] calculated oscillator strengths for dipole transitions in O I using the SUPERSTRUCTURE (SS) code [2] with spectroscopic type orbitals for 1s, 2s, and 2p and correlation type orbitals for the 3d.Tayal and Henry [3] calculated oscillator strengths and electron collisional excitation cross sections for O I.They used the Hibbert CIV3 atomic structure code [4] with the eight orthogonal one-electron orbitals 1s, 2s, 2p, 3s, 3p, 3d, 4s, and 4p.Using the same CIV3 atomic structure code, Bell and Hibbert [5] calculated oscillator strengths for allowed transitions in O I with more single electron orbitals.Hibbert et al. (HBGV) [6] used the CIV3 code to calculate E1 transitions connecting the  = 3 and  = 4 energy levels in O I. Biémont et al. [7] calculate oscillator strengths of astrophysical interest for O I using the CIV3 configuration interaction code and the Hartree-Fock pseudorelativistic (HFR) suite of Cowan (CW) codes [8].Using the SS code, Biémont and Zeippen [9] calculated oscillator strengths for 2p 4 -3s and 3s-3p allowed or spin-forbidden transitions in O I. Zheng and Wang [10] used the Weakest Bound Electron Potential Model (WBEPM) theory to calculate radiative lifetime, transition probabilities, and oscillator strengths for atomic carbon and oxygen.Using the Multiconfiguration Hartree-Fock (MCHF) method [11], Tachiev and Froese Fischer (TFF) [12] calculated ab initio Breit-Pauli energy levels and transition rates for nitrogen-like and oxygen-like sequences.Froese Fischer and Tachiev (FFT) [13] calculated Breit-Pauli energy levels, lifetimes, and transition probabilities for the beryllium-like to neon-like sequences in the adjusted with experimental values.Fan et al. [14] used the WBEPMT theory to calculate energy levels of high states in O I. C ¸elik and Ates ¸ [15] employed the WBEPMT theory to calculate radial transition matrix elements and then atomic transition probabilities for O I.
Using CW or SS or AS codes, we did atomic structure calculations for several atoms and ions [16][17][18] that are needed for ab initio Stark broadening calculations [19,20] and for emission line ratio calculations [21], but we never compare results obtained by the three codes for the same element.
About O I atomic data in databases, we used the National Institute of Standards and Technology (NIST) data [22] for fine structure energy levels and oscillator strengths.There are energy levels and oscillator strengths of O I without fine structure in the Opacity Project TOPbase [23] and NORAD-Atomic-Data [24] atomic structure databases.TIPbase database [25] of the Opacity Project used NIST data for the fine structure energy levels and Galavis et al. [26] data for the oscillator strengths fine structure data.Galavis et al. [26] used the SS atomic structure code with spectroscopic type 2 International Journal of Spectroscopy orbitals for 1s, 2s, and 2p and correlation type orbitals for 3s, 3p, 3d, and 4f.
In the Chianti project [27], they used the NIST database for experimental energy levels and oscillator strengths.For theoretical energy levels they used the Zatsarinny and Tayal [28] and FFT [13] for the theoretical oscillator strengths.
In this work, we will calculate atomic data for transitions with fine structure in O I using CW and SS and AS codes.Comparison with other theoretical and experimental data available in the literature will be presented.

Methods for Calculation
2.1.Hartree-Fock Pseudorelativistic (HFR) Method.In this method a set of orbitals are obtained for each electron configuration by solving the Hartree-Fock equations [8].A totally antisymmetric wave-function is a combination of single electron solution of the hydrogen atom Ψ (Slater determinant): () means that the th electron's space and spin are in the one-electron state   .This will automatically satisfy the Pauli principle, because a determinant vanishes if two columns are the same.Relativistic corrections are introduced by a Breit-Pauli Hamiltonian and treated by the perturbation theory.The relativistic corrections include the Blume-Watson spin-orbit, mass-variation, and one-body Darwin terms.The Blume-Watson spin-orbit term contains the part of the Breit interaction that can be reduced to a one-body operator.
The Cowan (CW) atomic structure suite of codes (RCN, RNC2, RCG, and RCE) uses this HFR method.The three first codes are for ab initio atomic structure calculations and the fourth one (RCE) is used to make least-squares fit calculations using an iterative procedure.

2.2.
Thomas-Fermi-Dirac-Amaldi (TFDA) Method.In this method and to have atomic parameters of an atom or ion, a statistical TFDA potential is used.For an atom or ion having  protons and  electrons, this potential is in the following form [29]: where with  the constant: and  ℓ are the orbital scaling parameters.
The function  verifies the following equation: with the boundary conditions: The SUPERSTRUCTURE (SS) and AUTOSTRUCTURE (AS) atomic structure codes use this method.Relativistic corrections are also done by a perturbation method using the Breit-Pauli Hamiltonian.The SS atomic structure code used in this work [30] is an updated version of the original one of 1974 [2].Some relativistic corrections are introduced in this version and orbital scaling parameters  ℓ are dependent on  and ℓ [31] and not like the original SS version of 1974, where scaling parameters were depending only on ℓ ( ℓ ).The AS code [32,33] is an extension of the SS code incorporating various improvements and new capabilities like two-body nonfine-structure operators of the Breit-Pauli Hamiltonian and polarization model potentials.Comparing the two atomic structure codes SS and AS we can see that even they used the same techniques in general; they gave different results mainly because they incorporated different relativistic corrections of the Hamiltonian.For the comparison between the two codes, we can refer to the work of Elabidi and Sahal-Bréchot [34] where they studied excitation cross section by electron impact for O V and O VI levels.They showed that the incorporation of the two-body non-fine-structure operators (contact spinspin, two-body Darwin, and orbit-orbit) in AS and not in the initial SS code is the main reason of the different results obtained by the two codes.

Results and Discussion
3.1.Energy Levels.We performed ab initio calculations of energy levels for O I using the three atomic structure codes CW, SS, and AS with the 5 configurations expansion 2p 4 , 2p 3 3s, 2p 3 3p, 2p 3 3d, and 2p 3 4s.This same set of configurations expansion was used by Tachiev and Froese Fischer (TFF) in the ab initio calculations [12] and by Froese Fischer and Tachiev in the adjusted with experimental values calculations [13].For the SS and AS atomic codes, the scaling parameters are determined variationally by minimizing the sum of all the nonrelativistic term energies (Table 1).
In Tables 2-6, calculated fine structure energy levels are presented.The obtained values are compared with the NIST atomic database [22] and with Tachiev and Froese Fischer ab initio calculations [12] using the Multiconfiguration Hartree-Fock (MCHF) method [11].

Oscillator Strengths.
We have also computed oscillator strengths for three multiplets of the transition 2p 4 -2p 3 3s, four multiplets of the transition 2p 3 3s-2p 3 3p, one multiplet of the transition 2p 4 -3d, and four multiplets of the transition 2p 3 3p-2p 3 3d using the atomic structure codes CW, SS, and AS (see Tables 7-10).They are compared with those of FFT [13] and HBGV [6] and tabulated in NIST [22].Our calculations with CW code give an agreement of about 2% with NIST values while FFT gives 8% with the NIST ones.The HBGV values have an agreement of 2% with the NIST ones but many data are missing.

Conclusions
The comparison between the energy levels calculated by the different atomic codes indicates that the agreement with NIST data is generally less than 20% except for the  We obtained, respectively, six and four energy levels corresponding to the configurations 2p 3 3p and 2p 3 3d which are not in the NIST atomic database.
TFF energy levels values are nearly the same compared to NIST ones with less than 1% difference but many energy levels are missed in these calculations.
When comparing oscillator strengths calculated by AS code, we obtain good agreement with NIST.
We can say that there is no best atomic code to use and we have to calculate atomic structure data with more than one code to compare results between them and with other theoretical and experimental sources.

Table 1 :
The scaling parameters used after minimization for the SS and AS atomic structure codes.

Table 4 :
The same as Table2for configuration 2s 2 2p 3 3p except that we did not put the AS data because they are too far from the other results.
* Wavelengths are from FFT.