Extensive and accurate relativistic calculations of atomic data on the 83Kr I spectrum

We provide an extensive, accurate and complete new spectroscopic data set for many levels belonging to the intermediate perturbing Rydberg series ns[K]J (5 ≤ n ≤ 8), nd[K]J (4 ≤ n ≤ 7) and the autoionizing Rydberg series np[K]J (5 ≤ n ≤ 7) and nf[K]J (4 ≤ n ≤ 5) relative to the ground state 4p6 1S0 for neutral krypton-83 isotope. The data set is composed of the energy levels, the oscillator strengths f i j , the radiative transition rates A i j , the Landé g-factors, the magnetic dipole and electric quadrupole hyperfine constants. The values of these spectroscopic parameters are calculated in the multiconfiguration Dirac-Hartree–Fock (MCDHF) framework. The calculations are carried out in the active space where the electron correlation effects, contributions from relativistic configuration interaction (RCI), the quantum electrodynamics (QED) effects with transverse photon Breit interaction and vacuum photon polarization, specific mass shift and self-energy corrections are included. In addition, we provide the oscillator strengths and transition rates in Coulomb and Babushkin gauges for the 4p5(2P)4d–4p5(2P)5p, 4p5(2P)5s–4p5(2P)5p, 4p5(2P)6s–4p5(2P)5p, 4p5(2P)4d–4p5(2P)44f, 4p5(2P)5d–4p5(2P)4f, 4p5(2P)5d–4p5(2P)6p, 4p5(2P)7s–4p5(2P)6p, 4p5(2P)4d–4p5(2P)5f, 4p5(2P)5d–4p5(2P)5f, 4p5(2P)6d–4p5(2P)5f, 4p5(2P)7p–4p5(2P)6d and 4p5(2P)8s–4p5(2P)7p transition arrays. Extensive comparisons with other experimental and theoretical data from the reference databases are carried out in order to judge the reliability of our data. These comparisons indicate that our results are accurate. The present spectroscopic data may be useful for line identification in observed spectra as well as modelling and diagnostics of astrophysical and fusion plasmas.

. Energy data cm 1 -( ) for 83 Kr I relative to the ground state 3p 6   We would also like to make some corrections to section 4 as some of the references were not cited in the correct place.
• Section 4.1, ' [83]' should be cited at the end of the second line. This line becomes: The relativistic Hamiltonian Ĥ describing the interaction of an atom with a uniform magnetic field B  has the form [83].
• [83] should be cited at the end of the line that after equation (4). This line becomes: where B m is the Bohr magneton [83].

Introduction
Knowledge of reliable spectroscopic data in krypton spectra with which we are involved, is required in various fields of physics, such as astrophysics, determination of plasma composition, physics of lasers and photoionization processes [1][2][3]. These data include the energy levels, the oscillator strengths f , ij the radiative transition rates A , ij the Landé g-factors, the magnetic dipole and electric quadrupole hyperfine constants. It is well known that the singly ionized krypton atoms have a ground term consisting of two odd-parity states 4p 5 2 P 3/2 and 4p 5 2 P 1/2 . Furthermore, these atoms possess two adjacent ionization limits commonly called I 3/2 and I 1/2 , corresponding to 4p 5 2 P 3/2 and 4p 5 2 P 1/2 , respectively. The most accurate value of the first ionization limit I 3/2 =112 914.41±0.02 cm −1 was reported by Bounakhla and co-workers [4] which is in excellent agreement with the values that reported by Delsart and co-workers [5] (I 3/2 =112 914.47±0.03 cm −1 ) and Aymar and co-workers [6] (I 3/2 =112 914.49±0.03 cm −1 ) whereas it is in a satisfactory agreement with the value that reported by Yoshino and Tanaka [7] (I 3/2 =112 914.6±0.1 cm −1 ). On the other side, the new value of the second limit of ionization I 1/2 =118 284.5±0.1 cm −1 was also reported by Bounakhla [4] which is in fair agreement with the value of 118 284.6±0.2 cm −1 as reported by Yoshino and Tanaka [7]. All observed Rydberg Any further distribution of this work must maintain attribution to the author(s) and the title of the work, journal citation and DOI. series converge to these limits of ionization. The ones converging to the first limit are bound whereas some of those converging to the second limit are autoionized. Thereupon, the energy levels belonging to the Rydberg series that located below the first limit of ionization overlap and interact with each other, resulting in perturbations, while those belonging to the Rydberg series above the first limit are autoionizing into the adjacent continuum [8]. Therefore, their experimental data are difficult to be investigated and hard to be obtained. To that end, optical and laser spectroscopy with classical methods of detection is insufficient to detect the electronic transitions from the ground state to the states of these series. The bound levels are quenched by collisional ionization while the autoionized ones are quenched by spontaneous autoionization decay [9,10].
Racah coupling scheme [11] is commonly used to label the energy terms of the excited levels. These levels are labelled by 4p 5 nl [k] J and 4p 5 n′l′ [k] J depending on the state of the core 2 P 3/2 and 2 P 1/2 , respectively. The prime is employed to denote the levels that are built on the 4p 5 2 P 1/2 parent ion level.
Over the last decades, a large number of various experimental and theoretical works have been made in order to determine the position of the energy levels of the high-lying Rydberg states of krypton. The energy of the levels belonging to the odd parity autoionizing Rydberg series of krypton has been investigated using two-photon laser optogalvanic spectroscopy or a vacuum ultraviolet (VUV) optical spectroscopy with excitation from the 4p 6 1 S 0 ground state. Ahmed and co-workers [8] used two-photon laser optogalvanic spectroscopy in order to provide new energy data on the odd-parity bound and autoionized Rydberg series of krypton. They provided the energy data for the series , and 4p 5 ns [3/2] 2 (9n42). In addition to their experimental results, Ahmed and co-workers used multichannel quantum defect theory (MQDT) [12] in order to investigate the interchannel interaction between the overlapping Rydberg series. Klar and co-workers [13] recorded accurate values for the resonance position of the low-lying autoionizing Rydberg series 4p 5 nd′ [K] 2,3 (6n7) using two-step laser excitation of metastable 4p 5 5s [3/2] 2 levels via selected 4p 5 n 5p[K] 1,2 levels. The accuracy of their experimental results are judged by comparing them with those carried out by means of MQDT. The energies of 187 levels belonging to seven different series , and 4p 5 nd [7/2] 3 (24n53) with an accuracy better than 0.1 cm −1 have been reported by Delsart and co-workers [14] by means of a two-step optical excitation from metastable levels using field ionization detection method. Using the same experimental setup, Delsart and co-workers [5] reported the energy of the levels of the odd series 4p 5 nd [3/2] 2 (15n25), and 4p 5 nd [7/2] 3 (15n25) with an accuracy of 0.03 cm −1 . In 2002, Brandi and coworkers [15] measured five transitions from the 4p 6 [16]. Using excitation from the ground state, photoelectron angular distribution in threephoton ionization spectrum of 8s′ [1/2] 1 , 6d′ [3/2] 1 , and 5g′ [7/2] 3 has been studied by Dehmer and co-workers [17]. Autoionizing resonances of some ns′ [1/2] 1 (n=8-10, 12) and 6d′ [3/2] 1 have been investigated by Wu and co-workers [18] by using synchrotron-based photoelectron spectroscopy of ground state Kr atoms with photon resolution less than 0.23 cm −1 . Maeda and co-workers [19] reported, with resolution of 0.074 cm −1 , the energies of the series ns′ [1/2] 1 (8n14) and nd′ [1/2] 1 (6n12) relative to the ground state using high resolution VUV absorption spectra in parallel with a complete MQDT analysis of the resonance lineshapes. In 1994, Koeckhoven and co-workers [20] investigated autoionizing resonances ns′ [1/2] 1 (8n20), nd′ [3/2] 1 (6n24), nd′ [5/2] 3 (8n14) and ng′ [7/2] 3 (5n11) by using three-photon excitation from the ground state and derived quantum defect and width parameters.
On the other hand, the investigation of even-parity autoionizing Rydberg states of krypton is much less comprehensive and the spectral resolution is much poorer [21][22][23][24][25][26]. Recently, Li and co-workers [21]  (5n34) using DC discharge in a beam combined with one-photon laser excitation using the time of flight (TOF) ion detection in the photon energy range of 29 000-40 000 cm .
In addition to the previous experimental works quoted above, various theoretical works via different theoretical models have been devoted to the spectroscopy of high-lying autoionizing Rydberg states of neutral krypton atom [3,6,8,13,15,19,24,[27][28][29][30][31]. Among these works, L'Huillier and co-workers [27] employed MQDT to the autoionizing states, which were formerly revealed by Dehmer and co-workers [17], showing a satisfactory agreement between the theoretical and the experimental results. Petrov and co-workers [28] brought calculations of the reduced autoionization width for the 4p 5 nl′ [K] J (0l′5) series to close agreement with experiment via the single-electron Pauli-Fock approximation.
After all, all of the hitherto experimental and theoretical results of the energy levels for krypton atom seem to be restricted to levels belonging to high-lying Rydberg series situated near the first limit of ionization. However, the studies on the low-lying Rydberg states remain non-systematic and the published data are scarce.
Heretofore, all of the experimental techniques used to determine directly oscillator strength for rare gas resonance transitions appear to have disadvantages and restrictions; either experimentally or theoretically. Many data have been obtained by small angle electronic scattering [33-39, 42, 49]. In this technique, the practical relationship between the differential cross-section at small angle for excitation of the resonance level is implicated and the optical oscillator strength is deduced from it. However, we do not have an idea about the range of scattering angles at which this relationship is valid. The second experimental method is total absorption [43,47] for transition to low lying resonance states in krypton. However, this method is not widely applicable due to the obvious usage of the extreme edges of the absorption line, provided that the resolution of the obtained spectra is rather low. The third experimental method is the self-absorption [33,34,39,42,49]. In this method, the resonance lineshapes are very sensitive to the influence of many-electron effects [59][60][61].
Regardless of these different experimental techniques and theoretical procedures, considerable disparities were observed between experimental results, between theoretical results, and between theory and experiments [10,32].
The third spectroscopic parameter with which we are concerned is the transition rate or Einstein coefficient of spontaneous emission. This coefficient for rare gases is of great interest in many fields of physics such as lasers, plasmas, astrophysics and space physics research. The current reference databases contain a small amount of experimental and theoretical data for rare gas atoms [62][63][64] in general and krypton atom in particular [52]. The scarcity of experimental data on transition rates for krypton atoms make comparison with theory difficult. Moreover, agreement between theoretical data is unsatisfactory [10].
The forth spectroscopic parameter we consider is the Landé g-factor. This factor defines sensitivity of an energy level to a magnetic field. Precise values of this factor are of great interest for atomic spectroscopy physicists. These factors enable accurate investigation of level-crossing experiments [65][66][67][68] and make it possible to semi-empirical compute hyperfine structure constants [69]. Literature data on Landé g-factors for both low and high excited configuration of krypton are incomplete and inaccurate. Husson  Clearly, most of the extensive experimental measurements and theoretical calculations of the Landé gfactors are restricted to a few levels belonging to some low-lying Rydberg series. From the experimental point of view, it seems to be difficult to carry out a magnetic resonance experiment on all states of krypton atom since the relaxation of these states in the target cell is too fast to be observed and the time of interaction of atoms with the radiofrequency field is too short to be detected. From a theoretical point of view, the calculated results of the Landé g-factors for some levels belonging to certain series might be affected by perturbation due to configuration interaction as well as spin-orbit interaction [10].
The last spectroscopic parameters considered here are the magnetic dipole and electric quadrupole hyperfine constants. For most of the rare-gases atoms including krypton, few data on these parameters are found in reference databases. However, several experimental techniques have been employed over the last decades in order to measure experimentally these constants for the configurations np 5 n′p of rare gas atoms. Among the more fruitful of these experiments have been the traditional optical and radio-frequency spectroscopy and levelcrossing technique. As an example of these techniques, we quote the measurements of hyperfine constants for the levels 5p[3/2] 1 , 5p′[3/2] 1 and 5p′[1/2] 1 of Kr 83 reported by Husson and co-workers [66] using a CW-Oxazin 750 dye laser and a level-crossing technique. Abu Safia and Husson [73] measured the hyperfine constants for the level 5p[1/2] 1 of Kr 83 using the same experimental-setup of Husson and co-workers [66]. Actually, these levels possess three components and their structure extends over a narrower range than lines due to transitions between two levels owning hyperfine structure. For that reason, these levels are the most preferable ones for accurate measurements of hyperfine constants. On the other hand, Jackson [74]  The hyperfine structure constants for many other levels belonging to the bound or the autoionized Rydberg series have not been experimentally measured or theoretically calculated. Most probably, levels with J=3 are connected by radiative transitions to only one lower level. Under this state of affairs, it is difficult to detect the atomic radiation that matches the one used for excitation in level-crossing experiment. Consequently, their accurate measurements present challenging difficulties.

Wavefunctions and Hamiltonian
Accurate calculations of spectroscopic parameters of a many-electrons atom cannot be done without the involvement of relativistic corrections. To take into account these corrections, we have to construct the N-electron wavefunction, where N is the number of electrons. These wavefunctions are central-field oneelectron orbitals. The one-electron wavefunction of both the core-and valence-excited configurations can be written in terms of the Dirac four components spinors r iP r Q r , ,  The Pauli spherical spinors are simultaneous eigenfunctions of the parity operator Π, and the total angular momentum operators J and J . 2 The quantum numbers j, k and l are related via The radial functions are the solutions of the coupled Dirac equation for local central field V(r) and a given orbital nk [80]: Where V r nk ( ) is the sum of the nuclear potential and the direct Coulomb potential, P C and Q C include all the two electrons interactions expected for the direct Coulomb instantaneous repulsion, and nk  are Lagrange parameters employed in order to put into effect the orthonormality restrictions P r P r Q r Q r dr 6 n k nk n k nk n n k k 0 On the other hand, the radial functions P r nk ( ) and Q r nk ( ) must satisfy the boundary conditions: The Dirac four components spinors given by equation ( where the first term H i , D ( ) which incorporates the major relativistic effects upon the electrons distribution from the outer shells, represents the Hermitian single-electron Dirac Hamiltonian due to the potential of the nuclear charge, and the second term is the Breit operator for electron-electron interaction where c=137 in atomic units, V r i ( ) is the electric scalar potential due to the interaction of the electron with the nucleus, A r   ( ) is the magnetic vector potential, P i = - is the electron momentum operator, 0 0 a s s = ⎛ ⎝ ⎜ ⎞ ⎠ ⎟ˆis a matrix known as Dirac operator and ŝ represents the Pauli matrices, ⎟ˆˆis a matrix known as the Dirac basis where Iˆis the unit matrix, r ij represents the distance between the electrons and ij w is the exchange photon frequency. In central field, the Dirac Hamiltonian H D given in equation (11) becomes where V r N ( ) is the potential due to the nuclear charge, given by where Z is the atomic number and R A 2.022 10 n 5 1 3 =´-/ denotes an empirically fixed nucleus radius, while A is the nucleus mass number.
Obviously, these Hamiltonians contains the instantaneous Coulomb repulsion, the magnetic interaction and the electron-electron interaction due to the finite value of the speed of light. At this stage, it is worthwhile to reveal that the Breit interaction is further included in sequence in the calculation of relativistic configuration interaction (RCI).

Reduction to radial integrals
The expectation value of the total Hamiltonian will include corrections from the one particle Dirac Hamiltonian, the Coulomb repulsion and the Breit interaction. Using equation ( In equation (17), the first integral includes the contribution of subtracting the rest mass energy, the second integral includes the kinematics operator P i a ⋅ˆand the third integral includes the electron-nucleus interaction.
For the two-electrons operator given by equation (12) Here the smaller and the greater of the vectors r i  and r j  are denoted by r < and r , > P k are the Legendre polynomials and δ is the angle between the vectors r i  and r ; j  more precisely, between the directions 1 1 q j and . 2 2 q j By using the theorem of addition of spherical harmonics, equation (18) can be expressed in terms of tensor product as follows By means of equation (19), the two-electrons antisymmetrised matrix elements can be written as where a denotes the shell with quantum numbers n k a b and the same things for b, c, and d, and the radial part R k is the relativistic Slater integral given as R abcd r and the angular part can be expressed in terms of 3-j symbols as follows C k m sljm C sl j m j j j k j j k j m q m km 1 2 1 2 1 by assuming that the radial functions are the same within certain shells a, b, c, and d. Here ( ) represents the 3jsymbol. However, the following triangular condition must be satisfied In addition to that condition, l l k a c + + and l l k b d + + must both be even.
On the other hand, the reduction to radial integrals of the second term of equation (12) becomes Here j k and k h are the regular and irregular spherical Bessel functions, respectively. The following triangular conditions must be satisfied with the same triangular condition and parity given by equation (27).
The relativistic radiative transition rates A fi and the absorption oscillator strength f if

The relativistic radiative transition rates A fi
The relativistic radiative transition rates A if per second corresponding to a transition from the initial state i to the final state f for an electron outside a closed shell [81,82] is where M if is the matrix element for a relativistic radiative transition of a single electron multipole operator of order L [82]. This matrix element is either the integral for electric multipole transitions or the integral for the magnetic multipole transitions [81].
where j l is the spherical Bessel function and G in equation (34) is the gauge parameter. It takes the value 0 in Coulomb gauge and L L 1 + in Babushkin gauge. In the non-relativistic limit, G=0 gives the velocity form of radiation matrix elements, while G L L 1 = + gives the length form.

The absorption oscillator strength f if
The dimensionless absorption oscillator strength f if for a transition from the initial level i to the final level f are calculated in the single multipole approximation via transition probability for each multipole type. In this approximation, the transition rates of different multipole type will be summed up to obtain the total radiative rate. Therefore, we will ignore any interference effects that could hypothetically exist due to different phases (signs) of various contributions to the transition matrix element. Under these conditions, the oscillator strength f if for the transition from initial atomic state function i Y to final state function f Y induced by a multipole The last equation can be written in terms of the reduced matrix elements as

The conventional hyperfine Hamiltonian
The relativistic Hamiltonian H describing the interaction of an atom with a uniform magnetic field B  has the form where the summation is taken over all atomic i th electron, e is the electron charge and 0 0 a s s = ⎛ ⎝ ⎜ ⎞ ⎠ ⎟ˆis a four matrix known as Dirac operator and ŝ represents the Pauli matrices. For simplicity, we shall drop the index i in the following equations. In terms of the scalar product of magnetic dipole moment spherical tensor of rank 1, , represent the spherical harmonic vector. The Landé g-factor g J is defined by the magnetic dipole moment operator m of an atom in the state JM J ñ | as m is the Bohr magneton. Since matrix elements of the Hamiltonian given by equation (46) can be calculated in the state JM , J ñ | the matrix elements of the Hamiltonian Ĥ are On the other hand, the renowned spin factor g s [84] can be represented as denoting the fine structure constant.
The correction to the electron g s factor leads to a correction to the relativistic interaction Hamiltonian by an amount where J G and J g represent the configuration and any other quantum number required to specify a CSF, and C r are the expansion coefficients. On the other hand, the CSFs are sums of product of four components spin-orbital function named Dirac wavefunctions. The spin spherical harmonics km c defined in equation (2) can be written in the LSJcoupling as Moreover, an angular recoupling computer program [85] is used in order to reduce the matrix elements to terms involving single-particle orbitals only: and the {} denotes the 6j-symbol. However, the single-particle matrix elements can be reduced into angular factors and radial integral:

The traditional hyperfine interaction
In the relativistic framework, the hyperfine Hamiltonian is where T n k ( ) and T e k ( ) are the spherical tensor operator of rank k, representing the nuclear and electron angular momentum, respectively, j and i designate the various protons and electrons, respectively.
The mean value of H hfŝ in a fine structure state J I F M F ñ | is given to first order as where I is the nuclear spin and F=J+I is the total hyperfine angular momentum quantum number.
For the magnetic dipole case, k=1, while in the electric quadrupole case k=2.
For sake of simplicity, we express the nuclear dipole moment in nuclear magneton n m units, the nuclear dipole moment I m can be written as  I M   I T I M  I  I  k  I  I  I  I T  I  1  0 64 On the other hand, the operator T q 1 ( ) is given by [86,87] T t ie r Y r 8 3 where e is the absolute value of the electron charge and j denotes the jth electron in the atom

Calculation of the magnetic dipole hyperfine constant A MD
The hyperfine constant A MD is related to to the nuclear magnetic moment by By means of equation (63), the hyperfine energy splitting E MD due to the magnetic dipole is

Calculation of the electric quadrupole constant B EQ
The electric quadrupole constant B EQ is related to the nuclear quadrupole moment Q. The latter is given by Since e is the charge and Q is the operator for the proton, then e Q represents the charge distribution in the nucleus.
On the other hand, the operator T q 2 ( ) is given by [88] as By means of equation (70), the hyperfine energy splitting E EQ due to the electric quadrupole moment of the nucleus

Evaluation of uncertainty in theoretical results
The evaluation of uncertainties associated with the theoretical results is mandatory for comparison with the experimental results. Uncertainty is shown to act as a judge of acceptability of theoretical predictions as compared with experimental measurements. The uncertainty of the calculated transition energy is given by [89,90] as Similarly, the uncertainty in the theoretical values of the Landé g-factor g can be estimated as an root mean square of errors for those levels whose Landé g-factors are experimentally known

Calculation proceeding
All the above equations are implanted in the General Relativistic Structure Package code called GRASP2K [91]. This code utilizes a fully relativistic treatment that could be applied efficiently for an arbitrary atomic and ionic system. We first start our calculations by generating the relativistic configuration state wavefunctions, by combining two different methods; the relativistic Multiconfiguration Dirac-Hartree-Fock (MCDHF) method, for generating one-electron wavefunctions, and the relativistic configuration interaction (RCI) method to include further electron correlation effects as well as the Breit and quantum electrodynamics (QED) correction and the effect of nuclear charge distribution. Whereby we consider a single excitation from the [Ar] 3d 10 4s 2 4p 6 reference configuration to a set of spectroscopic levels with angular momenta up to the g-shell, which are used to generate the multiconfiguration basis. Core-valence correlation was accounted for by performing a single excitation from the 4p-shell orbital to one of the valence shell orbitals. Moreover, we consider all the n=4 core subshell orbitals as active orbitals; in which single and double excitations among those spectroscopic orbitals is allowed, this effectively describes the core polarizations effects. In these excitations, the set of orbitals {1s, 2s, 2p, 3s, 3p, 3d} is designated as n=3 set, while the sets of orbitals {1s, 2s, 2p, 3s, 3p, 3d, 4s, 4p, 4d, 4f} and {1s, 2s, 2p, 3s, 3p, 3d, 4s, 4p, 4d, 4f, 5s, 5p, 5d, 5f, 5g} are designated as n=4, 5 sets, respectively and the set of orbitals {1s, 2s, 2p, 3s, 3p, 3d, 4s, 4p, 4d, 4f, 5s, 5p, 5d, 5f, 5g, 6s, 6p, 6d, 6f, 6g} is designated as n=6 set.
At the second stage of the calculations, with a well-optimized relativistic orbital basis at hand, additional effects from the Breit interaction and the radiative QED corrections in the form of the electron self-energy and vacuum-polarization are included in the wavefunctions through a subsequent RCI calculation. Once the wavefunctions have been obtained, we calculated all other physical observables of interest, such as excitation energies, radiative transition rates, oscillator strengths, hyperfine parameters, and Landé g-factors.
The radial functions and expansion coefficients were improved to self-consistency using the multiconfiguration self-consistent field (MCSCF) procedure [92]. The corresponding eigenvalues and eigenvectors are determined by means of the iterative Davidson method [93]. Hereafter, we used these eigenvectors in order to calculate the required allowed transition data between levels of odd and even parities. Selection rules for electric-dipole transitions in the jk coupling scheme are: Δ J=0, ±1, Δ J=0, Δ k=0, ±1 and Δ l=±1. However, these rules are not strictly respected, since transitions with change of the ionic core as well as transitions with Δ l=3 are observed (due to configuration-interaction effects) [94].
Moreover, the electric dipole transition rule requires that the parity of initial and final states must be different for an allowed transition.
The electron correlation effects are taken into account systematically with the active space approach. The building of the configuration space is tuned not only to capture the electron correlation effect efficiently, but also for circumventing the convergence problem that one frequently encounters in SCF calculations. The correlation between the valence electron and the core electrons as well as the core-core correlation are taken into account by single excitation of electrons from the external orbital 4p 6 of krypton to n′l′-valance orbital and keeping the core-orbitals active ([Ar]3d(10, * ) 4s(2, * ) 4p(5, * ) n′l′). In this stage, the ASFs are expanded over even 4s 2 4p 5 n′l′ configuration states and one set of orbitals is optimized. For the levels under consideration, only 5s, 6s, 7s, 8s, 5p, 6p, 7p, 4d, 5d, 6d, 7d, 4f and 5f orbitals were treated as valence and varied in the calculations that were performed for 4n8. For n=8, only 8s orbitals were considered. The other orbitals were treated as core ones. To first order correction by perturbation theory, the first order correction of ASFs (1) is written as a linear combination of all CSFs. These CSFs interact with the zero-order ASFs (0) , and thusly can be written as a linear combination of CSFs that are obtained via single excitation from occupied orbital of the reference configuration to the virtual orbitals. The virtual orbitals were added layer by layer where we generated four layers containing the orbitals s, p, d and f. In order to systematically emphasize the effects of single excitation, we start by involving the CSF's that arise from the excitation of one electron to account for the valence-core interactions. The changes in the transition energies and rates comprise the core-valence contribution.
Furthermore, the virtual orbitals were produced in a restrained configuration space. In this space, only a certain number of electron-pair correlation were counted in the MCDHF. Afterward, a series of CI calculations including different electron correlation effects are carried out in order to select the significant ones.
The hyperfine structure constants and Landé g-factors are calculated to first order hyperfine interaction in frame of the MCDHF method. Hence, the distortions of electron shell by nuclear moments are not taken into account in the calculations [95,96]. The selected wave functions J M , J ñ | used in MCDHF consist of all possible jj-coupled states that come up from the 4p 5 nl electronic configurations. By means of these wave functions, we built the eigenstates I J F M , , , F ñ | of the hyperfine Hamiltonian. Here I and J being the total angular momentum of the nucleus and the electron state, respectively, and F I J.
More details of this computational approach can be found in [97][98][99]. 83  between the present and NIST values may be due to several factors. The first factor may be attributed to the errors in the calculation of the correlation effects on the ground state and on the excited states. These errors on the excited states largely (but not totally) cancel out in the excitation energies. So, it is the difference in the amount of included correlation effects in the two levels that may influence the accuracy of our calculated values. The second one might due to the nucleus mass and spin which vary from isotope to another. Both nucleus mass in atomic unit and spin should be taken into account during the calculations.

Calculation of energy levels in
The third factor might due to perturbing levels of the strong and weak series. The series ns is assigned to 9d[1/2] 1 by Kaufman and Humphreys [101] whereas it was assigned to 7s′ [1/2] 1 by Moore [102] and Yoshino and Tanaka [7]. Moreover, the level at 111 072.5 cm −1 is assigned to 9d[1/2] 1 by [7]. Due to these perturbing series, many levels were left without measuring by Kaufman and Humphreys [101].
The last factor might be due to the spin-orbit interaction and the perturbation between the levels belonging to the same series. For the perturbing 5s, 6s, 7s, and 8s-series, the levels 3 P 1 is perturbed by the level 1 P 1 . By the same token, the spin-orbit interaction couples the level 3 F 3 with the level 1 F 3 while the levels 1 F 3 and 3 D 3 perturbed each other. The level 3 F 2 is perturbed by the level 1 D 2 in the 4d-series. The spin-orbit coupling and perturbation are formidably remarkable in the 5d, 6d and 7d-series. The spin-orbit interaction couples more than two levels. In the 5d-series, the levels 3 P 2 , 3 D 1 and 3 F 3 couple with 3 D 2 , 1 P 1 and 1 F 3 , respectively, while the level 3 D 3 is perturbed by the level 1 F 3 and the levels 3 F 2 and 1 D 2 perturb each other. In the 6d-series, the levels 3 P 2 and 3 F 3 couple via spin-orbit interaction, with the levels 3 D 2 and 1 F 3 , respectively. Moreover, the level 1 F 3 is perturbed by the level 3 D 3 . Similarly, the levels 3 D 1 and 3 F 2 are perturbed by the levels 1 P 1 and 1 D 2 , respectively. For the levels belonging to the 7d-series, the spin-orbit interaction couples the levels 1 D 2 and 3 D 2 as well as 3 F 3 with 1 F 3 . While the perturbation takes place between the levels 3 D 1 and 1 P 1 , 3 D 3 and 1 F 3 as well as between the levels 3 F 2 and 1 D 2 .
In the 5p and 6p-seris, the levels 3 D 2 couples with 1 D 2 via spin-orbit interaction, while the level 3 P 0 is perturbed by the level 1 S 0 and the level 3 D 1 is perturbed by the level 1 P 1. Whereas, in the 7p-series, only one kind of mixing takes place. In this series the level 3 D 2 is perturbed by the level 1 D 2 and the level 3 D 1 is perturbed by the level 1 P 1 .
Interpretation of the obtained theoretical results for the levels belonging to the 4f and 5f-series was a challenging task. On one hand, the energy positions of the levels belonging to these series are close to each other than in the other series. On the other hand, the level 3 D 3 couples with the level 3 F 3 , and 3 F 2 couples with 1 D 2 via spin-orbit interaction. After all, by virtue of these couplings and perturbation effects, assigning of the theoretically obtained energies to the corresponding experimental levels was an arduous mission.
Comparison of results obtained in the different gauges provides a means of estimating the uncertainties of the calculation. If the wavefunctions were analytically exact solutions of the equations solved, the results should be gauge-invariant. However, our solutions are found by an approximate numerical method, and the degree of departure from the gauge-invariance can serve as an indicator of numerical convergence.
The discrepancy between the values of the oscillator strengths in Babushkin and Coulomb gauges was also reported by Froese Fischer and Rubin [103], Bieron and co-workers [104,105], Irimia and Froese Fischer [106] and Zhou and co-workers [107]. This disparity may be ascribed to the fact that the amplitudes of the electric dipole transition in both Babushkin and Coulomb gauges are susceptible to diverse radial part of the wavefunctions. Thence, the difference between the values of the oscillator strengths in the two gauges cannot be considered as an error bar of itself. In contrast, the difference between the Babushkin and Coulomb gauge values clarify the fact that the core-polarization model is unsuitable to accurately compute the energy differences between the initial and final levels of all transitions as interpreted by Irimia and Froese Fischer [106]. Zhou and co-workers [107] noticed that the inconsistency in the oscillator strengths between Babushkin and Coulomb gauges is very large. In their work, the values of the oscillator strengths in the Coulomb gauge are much larger than those in the Babushkin gauge. Zhou and co-workers ascribed the gauge differences to the fact that the amplitudes of the electric dipole transition in both gauges are sensitive to different radial region of the wave functions. Bieron and co-workers [104] noticed a 20% difference between the values of the oscillator strengths calculated in the two gauges. They reported that the Babushkin gauge values fell within the experimental error limits. Also, they noted that the gauge difference was somewhat larger than 35% for the calculated values of the transition rates. They concluded that the agreement between the values of the oscillator strengths in Babushkin and Coulomb gauges is an indication (but not a proof) of convergence of the results [104], and it is a useful indicator of the degree of saturation of correlation effects in the partially saturated multiconfiguration calculations of transition rates [105].
From our point of view, the two gauges display different energy dependence. The Coulomb gauge is strongly energy dependent, whereas the Babushkin gauge is less dependent on the calculated values of transition energy. The transition rate in the Coulomb gauge is sensitive to the wavefunction accuracy at large distance from the nucleus, which is usually worse than near the nucleus. Moreover, some contributions to the transition rates in the Coulomb gauge cannot be computed with GRASP2K codes. For that reason, the values of the oscillator strengths and transitions rates obtained in Babushkin gauge usually adopted by researchers in order to compare these values with the experimental ones. On the other hand, the calculations of energy level differences require well balanced orbital sets and typically necessitate highly extensive multiconfiguration expansions. If a common set of orbitals is used for both states, the results in both gauges converge and will be in fair agreement with the experimental ones [105].
In order to obtain an idea about the accuracy of our computed results, we compare them with the experimental and theoretical results published over the last decades by other researchers. Tables 3 and 4 and the references there display this comparison for the 4p 6 1 S 0 -3p 5 5s 3 P 1 and 4p 6 1 S 0 -3p 5 5s 1 P 1 transition arrays,                        respectively. These tables exhibit discrepancies between different experimental and theoretical results. Our results appear to be the best ones that match most of the theoretical and experimental results. For the transition 4p 6 1 S 0 -3p 5 5s 3 P 1 , we perceive that our theoretical results in Babushkin and Coulomb gauges are in 90% agreement with the relativistic many-body calculations of Euripides and co-workers [50] and the parametrized potential calculations of Aymar [108] and Aymar and co-workers [52,55] and in 15% deviation from the other results published by different researcher. This agreement along with the consistency between the two gauges in different frameworks of calculations further warrants the accuracy of our computational procedure and implies dependable atomic wavefunctions.  D 1 -3 P 2 , 3 D 1 -3 P 1 , 3 D 1 -3 P 0 , 3 D 1 -1 P 1 , 3 D 2 -3 P 2 , 3 D 2 -3 P 1 , 3 D 2 -1 P 1 , 3 S 1 -3 P 2 , 3 S 1 -3 P 0 and 3 S 1 -1 P 1 ), our theoretical values of the transitions rates for seven hundred fifty seven allowed transitions are reported here for the first time.
Owning to the lack of available experimental and theoretical data of transition rates for the levels under consideration, we were unable to do a comparison in order to infer reliably the accuracy of our calculated values of the transition rates in both Babushkin and Coulomb gauges. Since the values of the transitions rates are related to the values of the oscillator strengths, then the accuracy of our data of the oscillator strengths assures the accuracy of our data of the transition rates.

Calculation of the Hyperfine parameters and the Landé g-factor of levels of 83 Kr
The magnetic moment splitting factor and the quadrupole moment coupling factor hyper fine structure constants A MD and B , EQ respectively and the Lande g J -factor of the levels belonging to the intermediate perturbing series ns[K] J (5n8), nd[K] J (4n7) and the autoionizing Rydberg series np[K] J (5n7) and nf[K] J (4n5) have been calculated from the relativistic hyperfine structures of the levels. The obtained results are given in table 6. The scarcity of experimental and theoretical data of hyperfine parameters of 83 Kr indicates that many of these data displayed in table 6 are reported here for the first time.
A test of the accuracy of our results concerning the Lande g J -factor can be made by comparing them with the experimentally measured as well as theoretically calculated ones. Table 7 exhibits this comparison and shows that our data are in very good agreement with the experimental as well as the theoretical data published by other researchers. Such agreement testifies to reliability and the fairly accuracy of our theoretical results.
The accuracy of our theoretical values of the hyperfine constants is analyzed by comparing them with the corresponding experimental and theoretical data available in the literature. Table 8 displays this comparison. In spite of this comparison, there are a considerable gap in the available data of the hyperfine constants of 83 Kr. Clearly, most of the experimental data of the hyperfine constants available in the literature are those of the 4p 5 5p configuration. The hyperfine structure of this configuration is remarkable, due to the fact that eight of its ten states reveal interrelated hyperfine effects. The agreement between our theoretical results and the other experimental and theoretical results of the hyperfine constants A MD and B EQ of the levels 5s [ [126], whose large error bars encompass all the results, the disparity between our results and Jackson's [74] and Table 4. Comparison between experimental and theoretical values of the oscillator strength for the 4p 6 1 S 0 -4p 5 5s 1 P 1 transition in Kr I.
A test of accuracy of our results was made by comparing them with the experimentally measured as well as theoretically calculated ones. This comparison had two objectives. On one hand, it showed that our data are in good agreement with the experimental as well as the theoretical data published by other researchers. Such agreement testifies to reliability and the fair accuracy of our theoretical results. On the other hand, it showed the scarcity of available experimental and theoretical atomic data of neutral krypton-83 isotope. Consequently, the present work provides a data set to fill in the gaps in the spectroscopic data of 83 Kr.
Finally, it is hoped that this theoretical work will soon make contact with experiment in order to stimulate further accurate, and reliable experimental works on 83 Kr.