Semi-empirical model potential study of Rydberg transitions in gold one-electron spectrum

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Semi-empirical model potential study of Rydberg transitions in gold one-electron spectrum J. Migdalek Institute of Computer Science, Pedagogical University of Cracow, Poland

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Article history: Received 19 December 2019 Accepted 21 January 2020 Available online xxxx

a b s t r a c t Semi-empirical model potential calculations of oscillator strengths for Rydberg transitions in principal, sharp and diffuse series of gold one-electron spectrum are presented. Relativistic effects are fully included through solving Dirac equations and core–valence electron correlation is represented in core polarization picture by appropriate part in model potential. The dipole-moment operator of transition is modified in accordance. Systematic trends are investigated in oscillator strengths along the spectral series showing hydrogen-like behavior for large n∗ effective quantum number of upper states of transitions. The so-called Cooper minimum in oscillator strengths is observed for principal series transitions. The influence of core–valence correlation (core polarization) on oscillator strengths along the spectral series is studied in detail demonstrating its importance even for transitions to rather highly excited Rydberg states with n ≈ 18. In addition, excitation energies for some levels which positions in the neutral gold spectrum is unknown so far, are also predicted. © 2020 Elsevier Inc. All rights reserved.

E-mail address: [email protected]. https://doi.org/10.1016/j.adt.2020.101324 0092-640X/© 2020 Elsevier Inc. All rights reserved.

Please cite this article in press as: J. Migdalek, Semi-empirical model potential study of Rydberg transitions in gold one-electron spectrum, Atomic Data and Nuclear Data Tables (2020) 101324, https://doi.org/10.1016/j.adt.2020.101324.

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Contents 1. 2. 3. 4.

Introduction......................................................................................................................................................................................................................... Calculations ......................................................................................................................................................................................................................... Discussion of results .......................................................................................................................................................................................................... Conclusions.......................................................................................................................................................................................................................... Declaration of competing interest.................................................................................................................................................................................... Acknowledgment ................................................................................................................................................................................................................ References ........................................................................................................................................................................................................................... Explanation of Tables ......................................................................................................................................................................................................... Table 1. Predicted excitation energies (vs. ground state) for some levels in gold one-electron spectrum which positions were unknown so far (in cm−1 ) ........................................................................................................................................................................ Table 2. Oscillator strengths for 6s − np transitions (principal series) .............................................................................................................. Table 3. Oscillator strengths for 6p − ns transitions (sharp series) ................................................................................................................... Table 4. Oscillator strengths for 6p − nd transitions (diffuse series) .................................................................................................................

1. Introduction Recent developments of theoretical and computational tools such as Multiconfigurational Dirac–Fock (MCDF) [1], relativistic many body perturbation theory (RMBPT) [2], relativistic random phase approximation (RRPA) [3], relativistic couple cluster method (RCC) [4] etc. allow to compute very accurate energies, oscillator strengths, lifetimes and other spectral characteristics of multi-electron atoms and ions. However, these very accurate methods are still tedious, complicated and time consuming and therefore mostly limited to few systems and limited number of transitions. On the other hand, the model potential techniques and particularly those of semi-empirical nature are very simple and allow to include relativistic and core–valence electron correlation effects (albeit in approximate core-polarization picture). They seem to be quite accurate, mostly due to adjustment of model potential to reproduce the experimental one-electron ionization energies (in accordance with Koopmans theorem [5]), and are very efficient in large-scale calculations of so-called one-electron spectra of atoms and ions. We had, in the past, proposed the whole family of such methods [6,7] that included relativistic effects (through direct solving of Dirac equations), core–valence electron correlation (in the core polarization picture, CP) and differed in representation of local core–valence electron exchange within the model potential scheme. To construct the model potential we employed the parent-ion-like core electron radial density obtained in both relativistic or nonrelativistic calculations. Some of these methods were later successfully used in numerous calculations of spectral characteristics of different atoms and ions. However, among those proposed earlier there was one form of model potential that was never used in practical calculations of energies and oscillator strengths. The idea of this approach was to treat in model potential the core–valence electron exchange as a reduction in the inter-electronic Coulomb repulsion caused by the Pauli exclusion principle (cf. next section). In this paper we intend to employ this simple method to study transitions to high Rydberg states in one-electron spectrum of gold, that had never been studied before and predict some energy levels that had not been determined neither theoretically nor experimentally so far. It is rather perplexing but there are only very scarce theoretical [2,6,8–10] as well as experimental data [11–15] on one-electron spectrum of gold and there are exclusively limited to lowest transitions of spectral series whereas higher transitions are completely unknown. However, the existing theoretical data demonstrate the importance of including in oscillator strength calculations of core– valence electron correlation. It had been widely believed in the past that this effect is important only for transitions to low lying

2 2 3 4 5 5 5 6 6 6 6 6

states in neutral or lowly ionized isoelectronic systems. Recently, we had demonstrated [16] that this effect is important also for transitions in highly ionized members of gold isoelectronic system. Now it is our aim to study transitions to high lying Rydberg states in one-electron spectrum of neutral gold as well as to evaluate the influence of core–valence electron correlation (in core polarization picture) on oscillator strengths for these transitions. 2. Calculations As mentioned in previous section we used in this paper the relativistic model potential method proposed by us earlier [7]. In this approach the appropriate Dirac equations are solved with the effective potential V (r) = −Z /r + Vcx (r) + Vcp (r)

(1)

where Vcx (r) represents direct Coulomb interaction (repulsion) between the core and valence electron combined with local valence–core electron exchange. The combined inter-electronic Coulomb and exchange operator for a closed-shell core operating on a valence electron at r is deduced from the assumption that the exchange interaction arises from a reduction caused by the Pauli exclusion principle in the inter-electronic Coulomb repulsion and can be written as Vcx =

∑

⟨φc (rc )|r − rc |−1 [2 − exp(i/h(pc − p) · (rc − r))]|φc (rc )⟩ (2)

c

where φc (rc ) is a doubly occupied core orbital, pc and p are momentum operators for core and valence electrons, respectively, and the exponential operator exchanges the valence electron with a core electron with the same spin orientation. Eq. (2) can be approximated by expression Vcx (r) ≈ Vc (r)[1 − (aZc−2/3 )e−r/µ ] = Vc (r) + Vx (r)

(3)

2/3

where a = 5/3(2/2π ) ≈ 1.018, Zc is the number of core electrons and Vc and Vx are direct Coulomb inter-electronic repulsion and local core–valence electron exchange, respectively. −2/3 The coefficient aZc was determined by comparing Coulomb and exchange energies of an electron gas of constant density in a spherical container and µ is an adjustable parameter. The direct Coulomb interaction between the core and valence electron Vc (r) can be expressed in terms of electron radial density of the core ρc (r) (the parent ion of the system studied is considered as a core) Vc (r) =

1 r

r

∫

ρc (r ′ )dr ′ + 0

∞

∫ r

ρc (r ′ ) r′

dr ′

(4)

In order to make the approach as simple as possible and independent from the need of any other additional calculations the

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J. Migdalek / Atomic Data and Nuclear Data Tables xxx (xxxx) xxx

core electron radial density ρc (r) was determined from tabulated analytical simplified self-consistent-field calculation of Gombas and Szondy [17]. On the other hand, one can also adopt radial density computed with analytical or numerical Dirac–Fock method. The high accuracy of densities used is not crucial here since the potential seen by the valence electron is modified by adjustment of parameter µ in Eq. (3) to make the one-electron energy eigenvalue in Dirac equations equal to experimental ionization energy for given state (Koopmans [5]). We will discuss the influence of the accuracy of radial core density used later. The Vcp (r) in Eq. (1) is the core polarization potential representing valence–core electron correlation in the semi-classical picture 1 Vcp (r) = − α r 2 (r 2 + r02 )−3 (5) 2 where α is the dipole polarizability of the parent-ion-like core and r0 is the cut-off radius which removes singularity of the potential for r = 0 and is considered to be of the size of the core. The values of α = 13.38 au and r0 = 1.210 au were taken from Fraga et al. [18] for Au+ which is treated as a core for neutral gold (the value of r0 corresponds to maximum of outermost spinorbital of Au+ ). Since the perturbation theory demonstrates that core polarization corrections in model potential and transition matrix element are of the same order, therefore in oscillator strength calculations the dipole-moment operator of transition d = −r has to be replaced by d + dc where dc = α r3 (r 2 + r02 )−3

(6)

is the additional dipole moment induced by the valence electron. One has to note that the choice of cut-off function and cut-off radius r0 in Eqs. (5) and (6) (important for small r region only) is arbitrary and is matter of experience since theory predicts only the asymptotic behavior of core polarization corrections for large r region. Calculations which include both core polarization corrections are named RMP+CP II (Relativistic Model Potential with Core Polarization). To separate the influence of CP potential (Eq. (5)) from that due to CP correction in dipole moment of transition (Eq. (6)), the additional oscillator strength calculations were performed which include the CP potential only (RMP+CP I). In order to evaluate the overall contribution of core polarization to oscillator strengths we performed also the RMP calculations where core polarization was entirely omitted. To estimate the sensitivity of obtained oscillator strengths on accuracy of core density used in model potential the additional RMPD+CP computations were also performed for some transitions with radial core density ρc (r) computed with accurate Dirac–Fock method [19] instead of simplified non-relativistic selfconsistent-field densities of Gombas and Szondy [17]. As can be seen from following section the differences between these two sets of data are not meaningful. In all types of calculations presented in this study the µ parameter in the model potential (Eq. (3)) was adjusted to fit the eigenenergy of Dirac equations to the experimental ionization energy for each |nlj⟩ state for which the experimental energy was available. The experimental ionization energies were calculated from excitation energies tabulated by Moore [20]. For higher levels where the experimental ionization energies were not available, the eigenenergies were computed by freezing the value of corresponding µ parameter obtained for the highest |nlj⟩ state for which the experimental ionization energy was known, namely: 14s1/2 , 8p1/2,3/2 and 14d3/2,5/2 for principal, sharp and diffuse series, respectively. The computed eigenenergies are, according to Koopmans theorem, the predicted values of ionization energies

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for levels unknown so far. They are later converted to energy levels and are presented in Table 1. Since the choice of cut-off function and cut-off radius r0 in core polarization corrections (Eqs. (5) and (6)) is, as mentioned above, an arbitrary one, we also tried an alternative approach where r0 instead of µ parameter is fitted to experimental ionization energies (µ is set to unity) and adjustment of model potential is due entirely to core polarization potential (Eq. (5)). Nevertheless, the results of these two approaches for transitions for which experimental data are available are quite similar as oscillator strengths computed with adjustment of µ for 6s1/2 − 6p1/2,3/2 transitions are 0.181, 0.407 and those resulting from adjustment of r0 are 0.189, 0.415, respectively. For 6p1/2,3/2 6d3/2,5/2 transitions the obtained oscillator strengths are 0.419, 0.0532, 0.468 and 0.425, 0.0535, 0.469, respectively. These both sets of data agree with experiment well within the range of differences between experimental results. This good agreement seems to support the belief that adjustment of r0 or µ parameter in core polarization potential (5) can compensate the differences in radial density used. The oscillator strength calculations were performed for relativistic components of 6s − np transitions with n = 6 − 16 (principal series), 6p − ns transitions with n = 7 − 18 (sharp series) and for 6p − nd transitions with n = 6 − 18 (diffuse series). The obtained values are presented and compared with other available theoretical and experimental data in Tables 2–4, respectively. 3. Discussion of results As can be seen from Tables 2–4 oscillator strengths computed in this study agree very favorably with available experimental data as well as with other theoretical values published so far which include core polarization effects. Unfortunately these values are rather scarce and limited mostly to lowest transitions in spectral series. One should stress a very good agreement of our values with more accurate results of RMBPT method [2] which treats core–valence electron correlation in completely different way. Our earlier theoretical data RHF+CP [9] represent this effect in core polarization picture similarly to our present calculations but differ in the way the valence–core exchange is treated. In our present approach the approximate local exchange is represented in model-potential way whereas in the RHF the accurate nonlocal exchange is fully accounted for in Dirac–Fock method. One should note, however, that our previous RHF+CP method is, in fact, also of semi-empirical nature as the cut-off radius in corepolarization potential is there treated as adjustable parameter and fitted to experimental ionization energies. The difference between our former RHF+CP and DF+CP data of Ref. [10] where the cut-off r0 parameter is not adjustable, is entirely due to the fit of cut-off parameter in RHF+CP method to experimental ionization energies since the core polarization corrections are exactly the same. As can be seen from Tables 2–4 replacement of simplified nonrelativistic self-consistent-field densities of Gombas and Szondy [17] by accurate Dirac–Fock [19] densities in RMPD+CP calculations leads for sharp and diffuse series to only marginal changes in computed oscillator strengths. More visible differences are observed only for results of principal series because their great sensitivity to any changes of parameters in vicinity of Cooper minimum [21], however even there differences are rather small. This is mostly due to the adjustment of model potential to reproduce the experimental ionization energies where the proper adjustment of parameter µ compensates differences in coreelectron density used. Figs. 1–3 represent behavior of (n∗)3 fik values vs. n∗ (n∗ being the effective quantum number for upper state of transition)

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Fig. 1. The dependence of (n∗)3 fJ −J ′ on effective quantum number n∗ for Rydberg transitions in the gold principal (6s − np, n = 6 − 16) series.

Fig. 2. The dependence of (n∗)3 fJ −J ′ on effective quantum number n∗ for Rydberg transitions in the gold sharp (6p − ns, n = 7 − 18) series.

for principal, sharp and diffuse series, respectively. Three observations can be deduced from these figures. First, oscillator strengths for all three series exhibit the well-known hydrogenlike ’constant (n∗)−3 ’ behavior for high n∗ values observed earlier for alkalis [22]. Second, differences between relativistic f1/2−1/2 and f1/2−3/2 components of sharp series and f1/2−3/2 , f3/2−3/2 and f3/2−5/2 components of diffuse series increase along the series which is also reflected in deviations found in computed relativistic line strength ratios from the values predicted for these series with neglect of spin–orbit interaction. Third, situation for principal series is more complicated as behavior of oscillator strengths along the series exhibits so-called Cooper minimum [21]. This effect is due to mutual cancellation of positive and negative contributions in the transition integral which changes sign at some value of n∗ along the spectral series resulting in minimum in oscillator strengths. As can be seen from Fig. 3 this minimum occurs separately for f1/2−1/2 and f1/2−3/2 components for different members of the series. In vicinity of the Cooper minimum oscillator strengths are not only anomalously small but they are extremely sensitive to core–valence electron effects which is responsible for large difference between our results for 6s − 7p transition and those computed with RMBPT approach [2] (see Table 2). It was also our aim to investigate the influence of core polarization on oscillator strengths along the principal, sharp and diffuse series in neutral gold. As can be seen from Table 3 this contribution is negative and rather weak for sharp series. It amounts to below 10% , is slightly higher for f3/2−1/2 than for

Fig. 3. The dependence of (n∗)3 fJ −J ′ on effective quantum number n∗ for Rydberg transitions in the gold diffuse (6p − nd, n = 6 − 18) series.

f1/2−1/2 and increases slowly along the series. The core polarization influence is also negative but much stronger for diffuse series (Table 4), ranges between 20%–30%, is largest for f1/2−3/2 component and again rises slowly along the series. For principal series the precise evaluation of core polarization contribution is difficult because of the presence of Cooper minimum and consequent extreme sensitivity of oscillator strengths to this effect. It amounts to negative 80%–90% contribution for 6s − 6p transition which is confirmed by available reliable experimental data and to almost −1750% for f1/2−3/2 component of 6s − 7p transition close to Cooper minimum whereas for f1/2−1/2 component of this transition it is only −23%. For higher transitions more distant from Cooper minimum contribution of CP diminishes but even for 6s − 16p transition it is close to 100% and is positive for f1/2−1/2 component and negative for f1/2−3/2 . A more detailed analysis of separate relative contributions of CP potential (∆CP I) and CP correction to transition dipole moment (∆CP II) (cf. Tables 2–4) demonstrates that the latter dominates clearly for diffuse series whereas such domination, albeit on much smaller scale, is visible also for f3/2−1/2 components for sharp series transitions (with exception of 6p − 7s). For f 1/2 − 1/2 transition of this series the influence of CP potential is more important. For principal series the influence of CP correction to dipole moment of transition decisively prevails for all transitions (particularly for the f1/2−3/2 component) and its magnitude strongly depends on a distance from Cooper minimum. Finally, it may be observed that combined relative contribution of core polarization to oscillator strengths is rather stable along the series (except for vicinity of Cooper minimum observed for principal series) which stems from the fact that absolute core polarization contributions diminish along the series with increasing principal quantum number n of upper level in more or less similar pace as oscillator strengths themselves. 4. Conclusions Relativistic semi-empirical model potential calculations of oscillator strengths for high Rydberg transitions in principal, sharp and diffuse series of gold one-electron spectrum were performed. Good agreement was obtained with the available experimental and other theoretical data. Systematic trends were investigated in oscillator strengths along the spectral series showing hydrogenlike behavior for high effective quantum number n∗ of upper states of transition. The influence of core–valence correlation (core polarization) on oscillator strengths was closely monitored along the spectral series demonstrating its importance even for transitions to rather highly excited Rydberg states with n ≈ 18. Great sensitivity of oscillator strengths to core polarization was

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observed near vicinity of Cooper minimum in principal series. In addition, some excitation energies for levels which positions in the gold spectrum was unknown so far, were also predicted. The proposed very simple semi-empirical model potential approach was found a very useful tool for large-scale computations of oscillator strengths as well as for investigation of their behavior along spectral series. Declaration of competing interest The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper. Acknowledgment This study was supported by a Pedagogical University of Cracow Statutory Research Fund. References [1] I.P. Grant, Relativistic Quantum Theory of Atoms and Molecules: Theory of Computations, Springer, New York, 2006.

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[2] U.I. Safronova, W.R. Johnson, Phys. Rev. A 69 (2004) 052511. [3] W.R. Johnson, D. Kolb, K.-N. Huang, At. Data Nucl. Data Tables 28 (1983) 333. [4] S.A. Blundell, W.R. Johnson, J. Sapirstein, Phys. Rev. A 43 (1991) 3407. [5] T. Koopmans, Physica 1 (1933) 104. [6] J. Migdalek, J. Quant. Spectrosc. Radiat. Transfer 20 (1978) 81. [7] J. Migdalek, W.E. Baylis, Phys. Rev. A 22 (1980) 22. [8] J. Migdalek, W.E. Baylis, J. Quant. Spectrosc. Radiat. Transfer 22 (1979) 113. [9] H.-S. Chou, W.R. Johnson, Phys. Rev. A 56 (1997) 2424. [10] J. Migdalek, M. Garmulewicz, J. Phys. B: At. Mol. Phys. 33 (2000) 1735. [11] N.P. Penkin, I.Y. Slavenas, Opt. Spectrosc. 15 (1963) 3. [12] P. Hannaford, P.L. Larkins, R.M. Lowe, J. Phys. B: At. Mol. Phys. 14 (1981) 2321. [13] D. Einfeld, J. Ney, J. Wilson, Z. Naturf. 26 (1971) 668. [14] G.L. Plekhotkina, Opt. Spectrosc. 51 (1981) 207. [15] M. Zhang, et al., J. Phys. B: At. Mol. Opt. Phys. 51 (2018) 205001. [16] J. Migdalek, Can. J. Phys. 96 (2018) 610. [17] P. Gombas, T. Szondy, Acta Phys. Hung. 25 (1968) 345. [18] S. Fraga, K.M.S. Saxena, J. Karwowski, Handbook of Atomic Data, Elsevier, Amsterdam, 1976. [19] J.P. Desclaux, Comput. Phys. Comm. 9 (1975) 31. [20] C.E. Moore, Atomic Energy Levels. NBS Circular 467, vol. III, U.S. Government Printing Office, Washington, 1970. [21] U. Fano, J.W. Cooper, Rev. Modern Phys. 40 (1968) 441. [22] W.L. Wiese, A.W. Weiss, Phys. Rev. 175 (1968) 50.

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Explanation of Tables Table 1.

Predicted excitation energies unknown so far (in cm−1 ) Level J Energy

(vs. ground state) for some levels in gold one-electron spectrum which positions were Designation of level configuration (principal and orbital quantum numbers) Fine structure level total quantum number Excitation energy (vs. the ground state) in cm−1

Table 2.

Oscillator strengths for 6s − np transitions (principal series) Transition Designation of lower and upper level, respectively J − J′ Fine structure level total quantum number of lower and upper level, respectively RMP Present study: Oscillator strengths computed with core polarization neglected RMP I Present study: Oscillator strengths computed with core polarization included in model potential only RMP II Present study: Oscillator strengths computed with core polarization included in both model potential and dipole-transition moment RMPD II Present study: Additional calculations of oscillator strengths with core polarization in both model potential and dipole-transition moment but with core density computed with Dirac–Fock method instead of simplified self-consistent-field calculation of Gombas and Szondy (Ref. [17]) as used in RMP, RMP I and RMP II computations. RHF+CP Other theory: Relativistic Hartree–Fock calculations of Ref. [8] with adjusted core polarization corrections. DF+CP Other theory: Dirac–Fock calculations of Ref. [10] with not adjusted core polarization corrections Experiment Experimental data of a Ref. [8], b Ref. [10], c Ref. [2], d Ref. [11], e Ref. [12], f Ref. [15], g Ref. [13] Relative core polarization (CP) contribution to RMP+CP oscillator strengths in [%] ∆CPI Core polarization contribution from CP potential ∆CPII Core polarization contribution from CP correction to dipole moment transition operator ∆CPI+II Core polarization contribution from both CP potential and CP correction to dipole moment transition operator

Table 3.

Oscillator strengths for 6p − ns transitions (sharp series) Transition Designation of lower and upper level, respectively J − J′ Fine structure level total quantum number of lower and upper level, respectively RMP Present study: Oscillator strengths computed with core polarization neglected RMP I Present study: Oscillator strengths computed with core polarization included in model potential only RMP II Present study: Oscillator strengths computed with core polarization included in both model potential and dipole-transition moment RMPD II Present study: Additional calculations of oscillator strengths with core polarization in both model potential and dipole-transition moment but with core density computed with Dirac–Fock method DF+CP Other theory: Dirac–Fock calculations with not adjusted core polarization corrections of Ref. [10] RMBPT Other theory: Relativistic Many-Body Perturbation method of Ref. [2]. Experiment No experimental data available Relative core polarization (CP) contribution to RMP+CP oscillator strengths in [%] ∆CPI Core polarization contribution from CP potential ∆CPII Core polarization contribution from CP correction to dipole moment transition operator ∆CPI+II Core polarization contribution from both CP potential and CP correction to dipole moment transition operator

Table 4.

Oscillator strengths for 6p − nd transitions (diffuse series) Transition Designation of lower and upper level, respectively J − J′ Fine structure level total quantum number of lower and upper level, respectively RMP Present study: Oscillator strengths computed with core polarization neglected RMP I Present study: Oscillator strengths computed with core polarization included in model potential only RMP II Present study: Oscillator strengths computed with core polarization included in both model potential and dipole-transition moment RHF+CP Other theory: Relativistic Hartree–Fock calculations with adjusted core polarization potential of Ref. [8] DF+CP Other theory: Dirac–Fock calculations with not adjusted core polarization corrections of Ref. [10] RMBPT Other theory: Relativistic Many-Body Perturbation method of Ref. [2] Experiment Experimental data of d Ref. [14] Relative core polarization (CP) contribution to RMP+CP oscillator strengths in [%] ∆CPI Core polarization contribution from CP potential ∆CPII Core polarization contribution from CP correction to dipole moment transition operator ∆CPI+II Core polarization contribution from both CP potential and CP correction to dipole moment transition operator

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Table 1 Predicted energies (vs. ground state) for some levels in neutral gold which positions were unknown so far (in cm−1 ). Level

J

Energy

Level

J

Energy

Level

J

Energy

15s

1/2

73391

9p

1/2 3/2

69567.1 69716.5

15d

3/2 5/2

73645.8 73647.5

16s

1/2

73562.4

10p

1/2 3/2

71108.4 71192.9

16d

3/2 5/2

73759.0 73760.4

17s

1/2

73693.9

11p

1/2 3/2

72014.4 72066.7

17d

3/2 5/2

73848.8 73849.9

18s

1/2

73796.8

12p

1/2 3/2

72592.3 72627.0

18d

3/2 5/2

73921.2 73922.1

13p

1/2 3/2

72983.6 73007.8

14p

1/2 3/2

73260.8 73278.4

15p

1/2 3/2

73464.5 73477.5

16p

1/2 3/2

73618.4 73628.2

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Transition

6s-6p 6s-7p 6s-8p 6s-9p 6s-10p 6s-11p 6s-12p 6s-13p 6s-14p 6s-15p 6s-16p a

Ref. [8]. Ref. [10]. c Ref. [2]. d Ref. [11]. e Ref. [12]. f Ref. [15]. g Ref. [13]. b

J-J’

1/2-1/2 1/2-3/2 1/2-1/2 1/2-3/2 1/2-1/2 1/2-3/2 1/2-1/2 1/2-3/2 1/2-1/2 1/2-3/2 1/2-1/2 1/2-3/2 1/2-1/2 1/2-3/2 1/2-1/2 1/2-3/2 1/2-1/2 1/2-3/2 1/2-1/2 1/2-3/2 1/2-1/2 1/2-3/2

Present study

Other theory

Experiment

RMP

RMP +CP I

RMP +CP II

RMPD CPII

RHF +CPa

DF +CPb

RMBPTc

3.47E−01 7.27E−01 8.98E−04 1.16E−02 5.23E−06 1.50E−03 8.25E−06 5.90E−04 6.70E−06 2.98E−04 4.94E−06 1.73E−04 3.61E−06 1.10E−04 2.68E−06 7.44E−05 2.02E−06 5.28E−05 1.55E−06 3.89E−05 1.22E−06 2.95E−05

3.37E−01 7.07E−01 9.51E−04 1.19E−02 3.11E−05 1.29E−03 2.74E−05 4.89E−04 1.92E−05 2.42E−04 1.32E−05 1.39E−04 9.33E−06 8.75E−05 6.76E−06 5.89E−05 5.03E−06 4.17E−05 3.83E−06 3.06E−05 2.98E−06 2.32E−05

1.81E−01 4.07E−01 7.28E−04 6.33E−04 1.43E−03 1.81E−04 7.55E−04 1.49E−04 4.39E−04 1.04E−04 2.76E−04 7.15E−05 1.84E−04 5.05E−05 1.29E−04 3.66E−05 9.38E−05 2.73E−05 7.03E−05 2.08E−05 5.40E−05 1.62E−05

1.91E−01 4.26E−01 8.58E−04 5.57E−04 1.55E−03 2.12E−04

1.58E−01 3.60E−01

1.83E−01 4.18E+00

1.88E−01 4.08E−01 1.00E−04 8.20E−03

1.90E−01d 4.10E−01

Relative CP contribution in [%]

1.76E−01e 3.51E−01

1.57E−01f 3.90E−01g

∆ CPI

∆CPII

∆CPI+II

−5.52 −4.91

−86.19 −73.71 −30.63 −1779.94

−91.71 −78.62 −23.35 −1732.54

7.28 47.39 1.81 −116.02 2.54 −67.79 2.85 −53.85 2.99 −47.55 3.11 −44.55 3.16 −42.35 3.21 −40.66 3.24 −39.90 3.26 −38.89

97.83

99.63

−612.71

−728.73

96.37

98.91

−228.19

−295.97

95.63

98.47

−132.69

−186.54

95.22

98.21

−94.41

−141.96

94.93

98.04

−73.27

−117.82

94.76

97.92

−60.93

−103.28

94.64

97.85

−52.75

−93.41

94.55

97.8

−47.12

−87.02

94.48

97.74

−43.21

−82.10

J. Migdalek / Atomic Data and Nuclear Data Tables xxx (xxxx) xxx

Please cite this article in press as: J. Migdalek, Semi-empirical model potential study of Rydberg transitions in gold one-electron spectrum, Atomic Data and Nuclear Data Tables (2020) 101324, https://doi.org/10.1016/j.adt.2020.101324.

Table 2 Oscillator strengths for 6s-np transitions (principal series). For description of computational methods see Explanation of Tables.

J. Migdalek / Atomic Data and Nuclear Data Tables xxx (xxxx) xxx

9

Table 3 Oscillator strengths for 6p − ns transitions (sharp series). For description of computational methods see Explanation of Tables. Transition

6p-7s 6p-8s 6p-9s 6p-10s 6p-11s 6p-12s 6p-13s 6p 14s 6p-15s 6p-16s 6p-17s 6p-18s a b

J-J’

1/2-1/2 3/2-1/2 1/2-1/2 3/2-1/2 1/2-1/2 3/2-1/2 1/2-1/2 3/2-1/2 1/2-1/2 3/2-1/2 1/2-1/2 3/2-1/2 1/2-1/2 3/2-1/2 1/2-1/2 3/2-1/2 1/2-1/2 3/2-1/2 1/2-1/2 3/2-1/2 1/2-1/2 3/2-1/2 1/2-1/2 3/2-1/2

Present study

Other theory

Exp.

Relative CP contribution in [%]

RMP

RMP +CP I

RMP +CP II

RMPD CPII

DF +CPa

RMBPTb

∆ CPI

1.42E−01 1.75E−01 1.67E−02 1.57E−02 5.73E−03 5.15E−03 2.72E−03 2.40E−03 1.52E−03 1.33E−03 9.44E−04 8.23E−04 6.25E−04 5.41E−04 4.33E−04 3.72E−04 3.16E−04 2.70E−04 2.37E−04 2.03E−04 1.83E−04 1.56E−04 1.44E−04 1.23E−04

1.35E−01 1.69E−01 1.60E−02 1.53E−02 5.52E−03 5.03E−03 2.62E−03 2.35E−03 1.47E−03 1.30E−03 9.13E−04 8.07E−04 6.06E−04 5.32E−04 4.23E−04 3.68E−04 3.08E−04 2.68E−04 2.31E−04 2.01E−04 1.78E−04 1.55E−04 1.40E−04 1.22E−04

1.37E−01 1.66E−01 1.63E−02 1.47E−02 5.63E−03 4.79E−03 2.68E−03 2.22E−03 1.50E−03 1.23E−03 9.32E−04 7.59E−04 6.18E−04 4.99E−04 4.29E−04 3.43E−04 3.12E−04 2.49E−04 2.35E−04 1.87E−04 1.81E−04 1.44E−04 1.42E−04 1.13E−04

1.40E−01 1.70E−01 1.67E−02 1.50E−02 5.76E−03 4.89E−03 2.74E−03 2.27E−03

1.52E−01 1.81E−01

1.39E−01 1.67E−01 1.71E−02 1.42E−02

−5.11 −3.61 −4.29 −2.72 −3.73 −2.51 −3.73 −2.25 −3.33 −2.44 −3.33 −2.11 −3.07 −1.8 −2.33 −1.17 −2.56 −0.8 −2.55 −1.07 −2.76 −0.69 −2.82 −0.88

∆CPII

∆CPI+II

1.46

−3.65 −5.42 −2.45 −6.8 −1.78 −7.52 −1.49 −8.11 −1.33 −8.13 −1.29 −8.43 −1.13 −8.42 −0.93 −8.45 −1.28 −8.43 −0.85 −8.56 −1.10 −8.33 −1.41 −8.85

−1.81 1.84

−4.08 1.95

−5.01 2.24

−5.86 2.00

−5.69 2.04

−6.32 1.94

−6.61 1.4

−7.29 1.28

−7.63 1.70

−7.49 1.66

−7.64 1.41

−7.96

Ref. [10]. Ref. [2].

Please cite this article in press as: J. Migdalek, Semi-empirical model potential study of Rydberg transitions in gold one-electron spectrum, Atomic Data and Nuclear Data Tables (2020) 101324, https://doi.org/10.1016/j.adt.2020.101324.

10

J. Migdalek / Atomic Data and Nuclear Data Tables xxx (xxxx) xxx

Table 4 Oscillator strengths for 6p-nd transitions (diffuse series). For description of theoretical methods see Explanation of Tables. Transition

6p-6d

6p-7d

6p-8d

6p-9d

6p-10d

6p-11d

6p-12d

6p-13d

6p-14d

6p-15d

6p-16d

6p-17d

6p-18d

a

Ref. Ref. c Ref. d Ref. b

J-J’

1/2-3/2 3/2-3/2 3/2-5/2 1/2-3/2 3/2-3/2 3/2-5/2 1/2-3/2 3/2-3/2 3/2-5/2 1/2-3/2 3/2-3/2 3/2-5/2 1/2-3/2 3/2-3/2 3/2-5/2 1/2-3/2 3/2-3/2 3/2-5/2 1/2-3/2 3/2-3/2 3/2-5/2 1/2-3/2 3/2-3/2 3/2-5/2 1/2-3/2 3/2-3/2 3/2-5/2 1/2-3/2 3/2-3/2 3/2-5/2 1/2-3/2 3/2-3/2 3/2-5/2 1/2-3/2 3/2-3/2 3/2-5/2 1/2-3/2 3/2-3/2 3/2-5/2

Present study

Other theory

Exp.

RMP

RMP +CP I

RMP +CP II

RMPD CPII

RHF +CPa

DF +CPb

RMBPTc

5.06E−01 6.24E−02 5.47E−01 1.32E−01 1.41E−02 1.26E−01 5.53E−02 5.54E−03 5.06E−02 2.94E−02 2.93E−03 2.69E−02 1.74E−02 1.70E−03 1.54E−02 1.12E−02 1.08E−03 9.80E−03 7.66E−03 7.33E−04 6.67E−03 5.42E−03 5.13E−04 4.77E−03 4.05E−03 3.84E−04 3.50E−03 3.09E−03 2.92E−04 2.66E−03 2.41E−03 2.27E−04 2.07E−03 1.92E−03 1.80E−04 1.65E−03 1.55E−03 1.46E−04 1.33E−03

4.77E−01 5.96E−02 5.21E−01 1.25E−01 1.35E−02 1.20E−01 5.31E−02 5.38E−03 4.86E−02 2.76E−02 2.78E−03 2.51E−02 1.65E−02 1.63E−03 1.47E−02 1.07E−02 1.04E−03 9.42E−03 7.30E−03 7.06E−04 6.42E−03 5.22E−03 4.98E−04 4.59E−03 3.88E−03 3.71E−04 3.38E−03 2.96E−03 2.82E−04 2.57E−03 2.31E−03 2.20E−04 2.00E−03 1.83E−03 1.74E−04 1.59E−03 1.48E−03 1.41E−04 1.29E−03

4.19E−01 5.32E−02 4.68E−01 1.05E−01 1.14E−02 1.02E−01 4.30E−02 4.37E−03 4.02E−02 2.24E−02 2.26E−03 2.09E−02 1.32E−02 1.30E−03 1.19E−02 8.47E−03 8.26E−04 7.57E−03 5.77E−03 5.57E−04 5.14E−03 4.08E−03 3.89E−04 3.66E−03 3.04E−03 2.90E−04 2.68E−03 2.31E−03 2.20E−04 2.04E−03 1.80E−03 1.71E−04 1.59E−03 1.43E−03 1.36E−04 1.26E−03 1.16E−03 1.10E−04 1.02E−03

4.22E−01 5.36E−02 4.71E−01 1.05E−01 1.14E−02 1.02E−01 4.28E−02 4.35E−03 4.01E−02 2.24E−02 2.26E−03 2.10E−02 1.32E−02 1.30E−03 1.19E−02

4.21E−01 5.29E−02 4.67E−01

4.59E−01 5.30E−02 5.24E−01

4.20E−01 5.30E−02 4.77E−01 1.01E−01 1.03E−01

4,20E−01d 4,60E−01

Relative CP contribution in [%]

∆ CPI

∆CPII

∆CPI+II

−6.92 −5.26 −5.56 −6.67 −5.26 −5.88 −5.12 −3.66 −4.98 −8.04 −6.64 −8.61 −6.82 −5.38 −5.88 −5.90 −4.84 −5.02 −6.24 −4.85 −4.86 −4.90 −3.86 −4.92 −5.59 −4.48 −4.48 −5.63 −4.55 −4.41 −5.56 −4.09 −4.40 −6.29 −4.41 −4.76 −6.03 −4.55 −3.92

−13.84 −12.03 −11.32 −19.05 −18.42 −17.65 −23.49 −23.11 −20.90 −23.21 −23.01 −20.10 −25.00 −25.38 −23.53 −26.33 −25.91 −24.44 −26.52 −26.75 −24.90 −27.94 −28.02 −25.41 −27.63 −27.93 −26.12 −28.14 −28.18 −25.98 −28.33 −28.65 −25.79 −27.97 −27.94 −26.19 −27.59 −28.18 −26.47

−20.76 −17.29 −16.88 −25.71 −23.68 −23.53 −28.60 −26.77 −25.87 −31.25 −29.65 −28.71 −31.82 −30.77 −29.41 −32.23 −30.75 −29.46 −32.76 −31.60 −29.77 −32.84 −31.88 −30.33 −33.22 −32.41 −30.60 −33.77 −32.73 −30.39 −33.89 −32.75 −30.19 −34.27 −32.35 −30.95 −33.62 −32.73 −30.39

[6]. [10]. [2]. [14].

Please cite this article in press as: J. Migdalek, Semi-empirical model potential study of Rydberg transitions in gold one-electron spectrum, Atomic Data and Nuclear Data Tables (2020) 101324, https://doi.org/10.1016/j.adt.2020.101324.