# Data-Driven Energy Levels Calculation of Neutral Ytterbium (_Z_ =
70)††thanks: Project supported by the Fundamental Research Funds for the
Central Universities (Grant No. 10822041A2038).

Yushu Yu1,2, Chen Yang1,2, and Gang Jiang1,2
1Institute of Atomic and Molecular Physics, Sichuan University,
Chengdu 610065, China
2Key Laboratory of High Energy Density Physics and Technology,
Ministry of Education, Chengdu 610065, China Corresponding author. E-mail:
<EMAIL_ADDRESS>author. E-mail<EMAIL_ADDRESS>
###### Abstract

In view of the difficulty in calculating the atomic structure parameters of
high-$Z$ elements, the HFR (Hartree-Fock with relativistic corrections) theory
in combination with the ridge regression (RR) algorithm rather than the Cowan
code’s least squares fitting (LSF) method is proposed and applied. By
analyzing the energy level structure parameters of the HFR theory and using
the fitting experimental energy level extrapolation method, some excited state
energy levels of the Yb I ($Z=70$) atom including the $4f$ open shell are
calculated. The advantages of the ridge regression algorithm are demonstrated
by comparing it with Cowan’s least squares results. In addition, the results
obtained by the new method are compared with the experimental results and
other theoretical results to demonstrate the reliability and accuracy of our
approach.

Keywords: atomic data, ytterbium, energy levels, ridge regression algorithm

PACS: 31.15.xr, 32.30.-r, 95.30.Ky

##  1 Introduction

The atomic structure parameters of high-$Z$ atoms play fundamental and
critical roles in atomic physics. Accurate measurement or calculation of
atomic structure parameters is of great significance for understanding the
structure of atoms and their ionic states and in some technical fields, such
as biological imaging[1], medical diagnosis[2], atomic light clocks[3], etc.
have a vast range of applications.

In recent years, with the development of photoionization and detection
technology, substantial progress has been made in the spectral measurement of
complex atomic systems.[4, 5, 6, 7, 8, 9, 10, 11] However, from the NIST
Atomic Spectra Database, there is no systematic experimental data for high-$Z$
elements, especially lanthanides and actinides.[12] On the one hand, the high
cost of sample preparation and high radioactivity bring much inconvenience in
the experimental measurement. On the other hand, although the theoretical
calculation method has no cost and safety concerns, with the increase of
atomic number, complex electron correlation effects and relativistic effects
make the computational complexity of the _ab initio_ theoretical method
increase exponentially, which is a significant obstacle to using complex
atomic systems in fundamental research.

When dealing with a large amount of experimental data, it is appropriate to
use machine learning for data mining and modeling research instead of
expressing physical equations explicitly in the traditional scientific
paradigm. Therefore, data-driven methods are a good way to obtain more atomic
structure data, avoid expensive _ab initio_ calculations, and improve the
accuracy of high-$Z$ atomic data. Nowadays, widely used atomic structure
packages can be divided into two categories. The first category is non-
relativistic with relativistic corrections packages. For example, the Cowan
code[13] developed by Cowan in 1968 was rather primitive, which adopted the
single-configuration radial wave function Hartree-Fock (HF) method. There are
also ATSP2K[14] developed by Fischer _et al._ in 2007 using a non-relativistic
Multiconfiguration HF (MCHF) method, CIV3[15] developed by Hibbert, etc. The
second category of _ab initio_ package is based on the Dirac equation that
takes relativistic effects into account, including the GRASP2K[16] _ab initio_
package based on the multi-configuration Dirac-Hartree-Fock method published
by Grant _et al._ in 2013; the FAC[17] program with many-body perturbation
theory developed by Gu; the MDFGME[18, 19, 20] program based on the multi-
configuration Dirac-Fock matrix element method by Desclaux’s group, etc.

Multi-reference packages, e.g. ATSP2K, GRASP2K, etc. are preferable in
calculating the low-level spectra of lighter or specific elements. At the same
time, the Cowan code may yield results that are as accurate as or even more
accurate in the case of elements containing $nd$ and $nf$ electrons. A major
reason is the Cowan code performs the least squares fitting of the _ab initio_
calculation results and the existing experimental data on the parameters of
atomic structures. As a result, many effects such as the relativistic effect,
core polarization, and electron correlation are incorporated into the ab
initio calculation. The semi-empirical method has been widely used because of
its advantages, such as a relatively simple calculation model and absolute
convergence of the iterative process, which significantly improves the
calculation accuracy. In 1996, P. Uylings and co-workers[21] proposed applying
the orthogonal operator method in spectral calculations. The orthogonal
operator method can be considered an extension and improvement of the
traditional least squares fitting (LSF) method, introducing many small
parameters to account for specific interactions. The orthogonal operator has
many advantages. It uses the same non-relativistic approximation as the Cowan
code, with relativistic corrections as a perturbation method. In addition, it
takes into account small second-order interactions. However, the orthogonal
operator method is only applicable to the $d^{n},d^{n}l,d^{n}l^{n}l^{2}$, and
$f^{2}$ configurations so far; for other configurations that partially fill
the $f$ shell, the theory remains to be developed. Inspired by the Cowan code,
if the current state-of-the-art data-driven machine learning (ML) method is
integrated into the calculation of atomic structure parameters, the accuracy
of the current atomic structure calculation software can be improved.

Ytterbium ($Z=70$) is one of the typical lanthanide elements, and its neutral
atom ground state is $4f^{14}6s^{2}~{}^{1}\\!S_{0}$, and two energy level
systems are generated about the neutral Yb excited state: The
$4f^{14}nln^{\prime}l^{\prime}$ configuration excited by $6s$ shell electrons
and the $4f^{13}nln^{\prime}l^{\prime}n^{\prime\prime}l^{\prime\prime}$
configuration level system excited by $4f$ shell electrons. Yb I has been
recognized by some groups as an excellent optical clock candidate because of
its narrow $4f^{14}6s^{2}~{}^{1}\\!S_{0}\rightarrow
4f^{14}6s6p~{}^{3}\\!P_{0}$ transition.[22] Energy levels and Landé
$g$-factors, as the most basic features of atomic structure spectroscopy, have
been studied for a long time. King has measured the spectrum of neutral Yb
atoms using an electric furnace in 1931.[23] Meggers and his colleagues
studied atomic spectra of rare earth elements, reporting the wavelengths and
estimated intensities of 1791 lines of neutral ytterbium in the region from
2155 to 31308 $\mathrm{\SIUnitSymbolAngstrom}$.[24]Recently, using microwave
and radio frequency resonance methods, F. Niyaz _et al._ observed the
transition of ytterbium from $6s(n+3)d~{}^{1}\\!D_{2}\rightarrow 6snl$, $4\leq
l\leq 6$.[25] In addition to experimental observations, many research groups
have developed new theoretical methods to calculate the structural parameters
of Yb atoms. V. B. Ternovsky _et al._[26] applied the method of relativistic
many-body perturbation and the Dirac-Kohn-Sham zero approximation combined
with the generalized theoretical relativistic energy method to calculate the
energy and width of the self-ionizing resonance of $4f^{14}7s6p$,
$4f^{13}[^{2}\\!F_{7/2}]6s^{2}np[5/2]_{2}$,
$4f^{13}[^{7}\\!F_{7/2}]6s^{2}nf[5/2]_{2}$. V. A. Dzuba and V. V. Flambaum[27]
developed and used an efficient version of the multi-electron configuration
interaction method, the fast configuration-interaction method (FCI), to
calculate some low states of ytterbium energy levels.

In this work, we further develop a more efficient data-driven method to
replace the Cowan code’s LSF procedure with the ridge regression (RR)
algorithm. Taking the relative energy level of the Yb atom as an example, we
compared the energy level results of the partially excited state of the $6s$
shell and the partially excited state of the $4f$ shell calculated by using
the LSF and RR models respectively, illustrating the advantages of the RR
model in extrapolative prediction, and exploring new methods. In addition, the
improved program of RR was used to calculate the energy level and Landé
$g$-factor of other parts of the excited state of the Yb I atom, and reliable
results were obtained.

##  2 Calculation methods

###  2.1 Ridge regression algorithm

Ridge regression[28] is an improved linear regression algorithm. Unlike the
unbiased estimation of linear regression, ridge regression uses $L_{2}$
regularization to obtain regression coefficients at the expense of losing some
information and reducing accuracy, which is more practical and reliable.

For least squares linear regression, its loss function is

$J\left(\theta\right)=\sum_{i=1}^{m}\left(h_{\theta}\left(x^{\left(i\right)}\right)-y^{\left(i\right)}\right)^{2}.$
(1)

Using $L_{2}$ regularization, the loss function should be modified as

$J\left(\theta\right)=\sum_{i=1}^{m}\left(h_{\theta}\left(x^{\left(i\right)}\right)-y^{\left(i\right)}\right)^{2}+\lambda\sum_{j=0}^{n}\theta_{j}^{2}.$
(2)

Using the $L_{2}$ regularized ridge regression model, taking into account the
coefficient $\theta_{j}$ in the prediction function
$h_{\theta}\left(x^{\left(i\right)}\right)$, it serves as a penalty mechanism
to prevent overfitting of the lines.

###  2.2 HFR methods

Cowan code’s HFR approach optimizes the fine structure parameters to minimize
the average deviation between the calculated energy level and the selected
experimental value, in order to obtain the required calculated energy level.
For an atomic system with nuclear charge number $Z_{0}$ and electron number
$N$, in atomic units, the Hamiltonian of non-relativistic with relativistic
correction is

$\boldsymbol{H}=-\sum_{i}\nabla_{i}^{2}-\sum_{i}\frac{2Z_{0}}{r_{i}}+\sum_{i>j}\frac{2}{r_{ij}}+\sum_{i}\zeta_{i}\left(r_{i}\right)\boldsymbol{l}_{i}\boldsymbol{s}_{i},$
(3)

where the first term is the kinetic energy term of the electron, the second
term is the potential energy term of the Coulomb interaction between the
nucleus and the electron, the third term is the Coulomb energy term between
the electron and the electron, and the fourth term is the spin-orbit
interaction term introduced by the relativistic correction.

In theory, solve the Schrödinger equation

$H\psi=E\psi,$ (4)

the atomic energy levels $E$ and the corresponding atomic wave functions
$\psi$ can be obtained. The fitting calculation method is based on the Slater-
Condon theory, which expands the unknown atomic wave function $\psi^{k}$ with
a set of known orthonormalized basis functions $\psi_{b}$ (calculated using
Hartree-Fock or several more approximate methods),

$\psi^{k}=\sum_{b}Y_{b}^{k}\psi_{b}.$ (5)

The atomic wave function is obtained by solving the matrix equation,

$\boldsymbol{H}\boldsymbol{Y}^{k}=\boldsymbol{E}^{k}\boldsymbol{Y}^{k}.$ (6)

According to the Slater-Condon theory[29] used by the HFR method, for the
single configuration with the electron occupation number $w_{i}$ of the
orbital
$n_{i}l_{i},\left(n_{1}l_{1}\right)^{w_{1}}\left(n_{2}l_{2}\right)^{w_{2}}\ldots\left(n_{q}l_{q}\right)^{w_{q}}$,
its Hamiltonian matrix element is

$\displaystyle\boldsymbol{H}_{ab}$
$\displaystyle=\delta_{ab}E_{\text{av}}+\sum_{j=1}\left[\sum_{k>0}\left(f_{k}\left(l_{j}l_{j}\right)\right)_{ab}F^{k}\left(l_{j}l_{j}\right)+\left(d_{j}\right)_{ab}\zeta_{j}\right]$
$\displaystyle+\sum_{i=1}^{q-1}\sum_{j=i+1}^{q}\left[\sum_{k>0}\left(f_{k}\left(l_{i}l_{j}\right)\right)_{ab}F^{k}\left(l_{i}l_{j}\right)\right.+\left.\sum_{k}\left(g_{k}\left(l_{i}l_{j}\right)\right)_{ab}G^{k}\left(l_{i}l_{j}\right)\right],$
(7)

where $E_{\text{av}}$ is the configuration average energy,
$F^{k}\left(l_{j}l_{j}\right)$ is the Slater direct integral between
equivalent electrons, $F^{k}\left(l_{i}l_{j}\right)$ is the Slater direct
integral between non-equivalent electrons, $G^{k}\left(l_{i}l_{j}\right)$ is
Slater exchange integral between non-identical electrons, $\zeta_{j}$ is the
radial integral related to the spin-orbit interaction.

In the HFR method, the one-electron radial wave function for each of the
specified electron configurations is first calculated using Hartree-Fock or
several more approximate methods, and the $E_{\text{av}}$ for each
configuration is obtained along with the Coulomb and spin-orbit interaction
_ab initio_ value of the integral. Then, an energy matrix is set up for each
possible value of $J$, and each matrix is diagonalized to obtain eigenvalues
(energy levels) and eigenvectors. Furthermore, relativistic corrections are
limited to calculations of mass velocity and Darwin corrections. The effects
not considered in other calculations are put into the ridge regression
algorithm by treating the integral term in Eq. (7) as a variable parameter,
and by minimizing its loss function,

$R\left(x_{l}\right)=\sum_{k}\left(E^{k}-T^{k}\right)^{2}+\lambda\sum_{l}x_{l}^{2},$
(8)

where $E^{k}$ and $T^{k}$ are the calculated values of HFR and experimental
energy level, respectively, $x_{l}^{2}$ is the radial integral value, and
$\lambda$ is the regularization coefficient. We implemented the ridge
regression model to optimize these radial integral parameters, and the average
deviation between the calculated energy level and the experimental value is
minimized. Therefore, the trained model can achieve a high-precision
calculation of the corresponding energy level.

##  3 Results and discussion

In this work, we calculated the relative energy levels and the Landé
$g$-factor values of some excited states of Yb I atom by using the HFR method
improved by the ridge regression model and the least square method used by the
Cowan code, respectively, and compared them with the experimental data
provided by NIST. In the calculation, the energy level values are relative to
the ground state energy level [Xe]$4f^{14}6s^{2}$, and sorted in order from
low to high energy levels. Taking the odd-parity configuration
$4f^{13}5d6s^{2}$ excited by the $4f$ subshell as an example, we analyzed the
advantages of applying the ridge regression algorithm, obtained the connection
between the parameters and energy levels, and then applied this law to other
highly excited states.

In Table 1, we list the different methods to calculate the energy level values
of all possible spectral items of the odd-parity configuration
$4f^{13}5d6s^{2}$, and compare them with the experimental values from NIST
respectively, and calculate the mean absolute error (MAE)
$R=\sqrt{\sum_{i}^{N}{\frac{(E^{i}-T^{i})^{2}}{N}}}$ for each symmetric block
(a symmetric block is a state whose state function differs only by the total
angular momentum $J$). The _ab initio_ results come from the non-relativistic
Hartree-Fock method, with only some of the relativistic effects accounted for
as perturbations. The data with asterisks in Table 1 are the data fitted with
the experimental values, Fit I–III are the results of fitting with different
numbers of experimental values, and the number of experimental values added
gradually increases from I–III. It can be clearly seen that adding the
experimental energy level effectively improves the accuracy of the energy
level calculation, and with the increase of the fitting experimental energy
level, the accuracy is higher. In addition, adding experimental energy levels
to a certain symmetric block also improves the accuracy of other symmetric
blocks. It can be seen from Fit I that when we only add two experimental
energy levels to the spectral term (7/2, 1/2), although the Cowan least
squares method achieves zero MAE of the symmetric block, however, it affects
the results of other symmetric blocks and loses the physical significance.
When enough experimental data is added, accurate results can be obtained, and
the error with the experimental value is less than 50
$\mathrm{c}\mathrm{m}^{-1}$. In the case of adding the same less experimental
data, comparing the calculation results after adding the ridge regression
model program and Cowan code’s calculation results, e.g., Fit I and II
demonstrate that RR provides more accurate results than Cowan code’s LSF
procedure, which indicates that the RR algorithm has stronger generalization
ability. With less data, it can capture the characteristics of the data more
accurately and deliver more accurate results.

Table 1: Yb I $4f^{13}6s^{2}6p$ fitting calculation of energy levels (in
$\mathrm{c}\mathrm{m}^{-1}$). The data with asterisks (*) are the data fitted
with the experimental values.

|  |  |  |  | Fit I | Fit II | Fit III
---|---|---|---|---|---|---|---
Configuration | Term | $J$ | Expt. | _Ab initio_ | Cowan code | This work | Cowan code | This work | Cowan code | This work
$4f^{13}6s^{2}6p$ | (7/2, 1/2) | 3 | $32\,065.3$ | $32\,277.2$ |  $32\,065.3$* |  $32\,122.1$* |  $31\,654.0$* |  $32\,065.3$* |  $32\,069.0$* |  $32\,069.0$*
|  | 4 | $32\,273.6$ | $32\,371.3$ |  $32\,273.6$* |  $32\,216.6$* |  $33\,456.0$* |  $32\,273.6$* |  $32\,280.3$* |  $32\,280.3$*
$R_{1}$a |  |  |  | 165.0 | 0.0 | 56.9 | 885.2 | 0.0 | 5.4 | 5.4
| (7/2, 3/2) | 5 | $35\,178.8$ | $34\,455.3$ | $32\,969.5$ | $34\,355.9$ |  $34\,772.2$* |  $35\,165.5$* |  $35\,134.8$* |  $35\,134.8$*
|  | 2 | $35\,197.0$ | $34\,500.5$ | $35\,023.8$ | $34\,406.9$ |  $34\,839.4$* |  $35\,210.3$* |  $35\,205.4$* |  $35\,205.4$*
|  | 3 | $35\,807.5$ | $34\,790.8$ | $33\,314.0$ | $34\,690.0$ | $39\,778.9$ | $35\,482.9$ |  $35\,789.2$* |  $35\,789.2$*
|  | 4 | $36\,061.0$ | $34\,967.0$ | $33\,471.0$ | $34\,867.9$ | $40\,314.2$ | $35\,747.4$ |  $36\,104.5$* |  $36\,104.5$*
$R_{2}$ |  |  |  | 899.9 | 2111.7 | 996.7 | 2922.1 | 225.9 | 32.5 | 32.5
| (5/2, 1/2) | 3 |  | $43\,213.3$ | $46\,988.0$ | $43\,331.2$ | $42\,537.0$ | $42\,416.3$ | $42\,343.4$ | $42\,343.4$
|  | 2 | $42\,531.9$ | $43\,321.1$ | $49\,249.4$ | $43\,443.7$ |  $42\,196.1$* |  $42\,531.9$* |  $42\,521.9$* |  $42\,521.9$*
$R_{3}$ |  |  |  | 789.2 | 6717.5 | 911.8 | 335.8 | 0.0 | 10.0 | 10.0
| (5/2, 3/2) | 3 |  | $45\,833.1$ | $48\,265.1$ | $46\,007.1$ | $49\,006.2$ | $45\,962.9$ | $46\,218.5$ | $46\,218.5$
|  | 1 | $44\,834.6$ | $45\,122.9$ | $47\,625.7$ | $45\,297.5$ | $40\,650.9$ | $45\,258.9$ |  $44\,877.6$* |  $44\,877.6$*
|  | 4 | $45\,497.6$ | $45\,434.4$ | $47\,998.0$ | $45\,610.6$ | $43\,712.0$ | $45\,728.3$ |  $45\,473.3$* |  $45\,473.3$*
|  | 2 | $45\,931.9$ | $45\,654.2$ | $48\,044.8$ | $45\,828.1$ | $48\,632.9$ | $45\,776.1$ |  $45\,905.1$* |  $45\,905.1$*
$R_{4}$ |  |  |  | 234.0 | 2483.7 | 281.6 | 3054.4 | 293.0 | 32.4 | 32.4

  * a

$R_{n}$ is the mean absolute error (MAE)
$R=\sqrt{\sum_{i}^{N}{\frac{(E^{i}-T^{i})^{2}}{N}}}$ for each symmetric block
(a symmetric block is a state whose state function differs only by the total
angular momentum $J$).

For the fitting of configuration $4f^{13}5d6s^{2}$ using different methods, we
also analyzed the variation trend of the radial integral parameters of its
Hamiltonian matrix elements (Table 2). Different interactions between
electrons are expressed for the six parameters included in its Hamiltonian.
From the results given by the Cowan code in Fit I and II, we observed some
unreasonable parameter values after fitting that are too large or equal to 0,
which is the result of overfitting by the LSF. While the LSF modifies the _ab
initio_ results to a certain extent, the physically meaningful Slater
parameters are trimmed excessively. Compared with the radial integral
parameters of the Hamiltonian matrix elements calculated by the Cowan code,
the parameter results we give are more stable, avoiding the result that the
integral term is equal to 0 or too large, and the deviation from the _ab
initio_ results is not large. Parameter values are more physically meaningful.

Table 2: Yb I $4f^{13}6s^{2}6p$ fitting calculation parameter values

|  | Fit I | Fit II | Fit III
---|---|---|---|---
Parameters | _Ab initio_ | Cowan code | This work | Cowan code | This work | Cowan code | This work
${E}_{{\text{av}}}$ | $38\,565.6$ | $39\,467.2$ | $38\,565.6$ | $39\,659.1$ | $38\,755.0$ | $38\,796.9$ | $38\,796.9$
${\zeta}_{{f}}$ | $3111.8$ | $4243.5$ | $3190.1$ | $2320.3$ | $2947.7$ | $2903.0$ | $2903.0$
${\zeta}_{{p}}$ | $1514.7$ | $625.1$ | $1552.8$ | $1170.4$ | $2141.1$ | $2136.2$ | $2136.2$
${F}^{{2}}\left({\text{fp}}\right)$ | $2396.6$ | $2102.2$ | $2396.6$ | $24\,744.7$ | $2396.6$ | $4474.6$ | $4474.6$
${G}^{{2}}\left({\text{fp}}\right)$ | $566.1$ | $6214.3$ | $580.3$ | $6412.3$ | $566.1$ | $1046.8$ | $1046.8$
${G}^{{4}}\left({\text{fp}}\right)$ | $500.8$ | $1211.0$ | $513.4$ | $0.0$ | $1715.6$ | $1027.2$ | $1027.2$

In addition, we also compared some configurations of single-electron
excitations in the $6s$ shell under different methods. From Fit I, the results
obtained by the ridge regression model and the least squares method used by
Cowan can fully demonstrate the advantages of the ridge regression model in
the case of fewer data as well. Adding the same amount of experimental data,
the MAE of our improved results and NIST experimental values are an order of
magnitude higher than Cowan code’s accuracy. Importantly, more physically
meaningful results have been obtained. For configuration $4f^{14}6s6d$, in Fit
I, we add two experimental energy levels to fit the calculation results.
According to the results calculated by the Cowan code, the two spectral terms
${}^{1}\\!D_{2}$ and ${}^{3}\\!D_{3}$ without experimental values are 61154.1
and 61110.0 $\mathrm{c}\mathrm{m}^{-1}$, respectively, which deviates from the
experimental and _ab initio_ results by 2000 $\mathrm{c}\mathrm{m}^{-1}$. The
reason is the over-fitting phenomenon caused by the least square method. In
order to keep the fitted function more consistent with the results of the
added experimental data, there is a large error in the extension. However,
when the same data is added to the ridge regression model, the above situation
does not occur, and the extrapolated results are more realistic, with a
difference of less than 300 $\mathrm{c}\mathrm{m}^{-1}$ from the experimental
energy level. There is no doubt that the accuracy can also be greatly improved
if more experimental data is included. It is only 0.9
$\mathrm{c}\mathrm{m}^{-1}$ different from the experimental data for the same
symmetric block.

Table 3: Yb I $4f^{14}6snl$ fitting calculated energy level values (in
$\mathrm{c}\mathrm{m}^{-1}$). The data with asterisks (*) are the data fitted
with the experimental values.

|  |  |  |  | Fit I | Fit II
---|---|---|---|---|---|---
Configuration | Term | $J$ | Expt. | _Ab initio_ | Cowan code | This work | Cowan code | This work
$4f^{14}6s^{2}$ | ${}^{1}\\!S$ | 0 | $0.0$ | $0.0$ |  $0.0$* |  $0.0$* | $0.0$ | $0.0$
$4f^{14}6s6p$ | ${}^{3}\\!P$ | 0 | $17\,288.4$ | $11\,235.3$ |  $17\,269.9$* |  $17\,288.4$* |  $17\,288.4$* |  $17\,288.4$*
|  | 1 | $17\,992.0$ | $11\,668.9$ | $17\,930.3$ | $18\,009.0$ | $17\,908.8$ | $17\,908.8$
|  | 2 | $19\,710.4$ | $12\,626.1$ | $19\,924.4$ | $20\,421.9$ |  $19\,710.4$* |  $19\,710.4$*
| ${}^{1}\\!P$ | 1 | $25\,068.2$ | $26\,035.3$ |  $25\,132.0$* |  $25\,068.5$* |  $25\,068.2$* |  $25\,068.2$*
$R_{1}$a |  |  |  | 5651.2 | 116.2 | 355.8 | 32.2 | 32.2
$4f^{14}5d6s$ | ${}^{3}\\!D$ | 1 | $24\,489.1$ | $20\,789.3$ |  $24\,489.1$* |  $24\,489.1$* |  $24\,489.1$* |  $24\,489.1$*
|  | 2 | $24\,751.9$ | $21\,257.1$ | $24\,780.3$ | $24\,834.2$ |  $24\,751.9$* |  $24\,751.9$*
|  | 3 | $25\,270.9$ | $22\,128.8$ | $27\,069.3$ | $26\,865.7$ | $25\,275.2$ | $25\,275.2$
| ${}^{1}\\!D$ | 2 | $27\,677.6$ | $27\,658.7$ |  $27\,677.7$* |  $27\,678.5$* |  $27\,677.7$* |  $27\,677.7$*
$R_{2}$ |  |  |  | 2990.7 | 899.3 | 798.4 | 2.2 | 2.2
$4f^{14}6s7p$ | ${}^{3}\\!P$ | 0 | $38\,090.7$ | $33\,330.9$ |  $38\,090.7$* |  $38\,090.7$* |  $38\,067.6$* |  $38\,067.6$*
|  | 1 | $38\,174.2$ | $33\,421.3$ |  $38\,174.2$* |  $38\,174.2$* |  $38\,204.9$* |  $38\,204.9$*
|  | 2 | $38\,552.0$ | $33\,639.9$ | $38\,350.2$ | $38\,373.0$ |  $38\,544.6$* |  $38\,544.6$*
| ${}^{1}\\!P$ | 1 | $40\,564.0$ | $35\,109.1$ | $43\,119.1$ | $39\,849.0$ |  $40\,563.7$* |  $40\,563.7$*
$R_{3}$ |  |  |  | 4978.2 | 1281.5 | 368.5 | 19.6 | 19.6
$4f^{14}6s6d$ | ${}^{3}\\!D$ | 1 | $39\,808.7$ | $35\,330.3$ |  $39\,808.7$* |  $39\,808.7$* |  $39\,807.6$* |  $39\,807.6$*
|  | 2 | $39\,838.0$ | $35\,389.7$ |  $39\,838.0$* |  $39\,838.0$* |  $39\,839.3$* |  $39\,839.3$*
|  | 3 | $39\,966.1$ | $35\,526.8$ | $61\,110.0$ | $39\,887.1$ |  $39\,966.2$* |  $39\,966.2$*
| ${}^{1}\\!D$ | 2 | $40\,061.5$ | $35\,893.0$ | $61\,154.1$ | $40\,571.3$ |  $40\,061.3$* |  $40\,061.3$*
$R_{4}$ |  |  |  | 4385.4 | 14932.9 | 257.9 | 0.9 | 0.9

  * a

$R_{n}$ is the mean absolute error (MAE)
$R=\sqrt{\sum_{i}^{N}{\frac{(E^{i}-T^{i})^{2}}{N}}}$ for each symmetric block
(a symmetric block is a state whose state function differs only by the total
angular momentum $J$).

Based on the ridge regression model, in the case of few data, the _ab initio_
results can be modified to a certain extent for different excited states to
optimize the Hamiltonian. We use this method to perform fitting calculations
for other excited states, which are summarized in Table 4 and Table 5 (odd and
even parity, respectively).

Table 4: Yb I odd-parity excited state fitting calculated energy levels (in
$\mathrm{c}\mathrm{m}^{-1}$), Landé $g$-factor values.

|  |  | Level |  | Landé $g$ |
---|---|---|---|---|---|---
Configuration | Term | $J$ | Expt. | This work | $\Delta E$ | Expt. | This work | $\Delta g$
$4f^{14}6s6p$ | ${}^{3}\\!P$ | 0 | $17\,288.4$ | $17\,288.4$ | $0.0$ |  | $0.000$ |
|  | 1 | $17\,992.0$ | $17\,908.8$ | $83.2$ | $1.492\,82$ | $1.488$ | $0.004\,82$
|  | 2 | $19\,710.4$ | $19\,710.4$ | $0.0$ | $1.5$ | $1.501$ | $0.001$
| ${}^{1}\\!P$ | 1 | $25\,068.2$ | $25\,068.2$ | $0.0$ | $1.035$ | $1.013$ | $0.022$
$4f^{13}5d6s^{2}$ | (7/2, 3/2) | 2 | $23\,188.5$ | $23\,339.9$ | $151.4$ | $1.45$ | $1.461$ | $0.011$
|  | 5 | $25\,859.6$ | $25\,852.5$ | $7.1$ | $1.04$ | $1.023$ | $0.017$
|  | 3 | $27\,445.6$ | $27\,309.2$ | $136.4$ | $1.22$ | $1.223$ | $0.003$
|  | 4 | $28\,184.5$ | $28\,150.4$ | $34.1$ | $1.14$ | $1.137$ | $0.003$
| (7/2, 5/2) | 6 | $27\,314.9$ | $27\,035.7$ | $279.2$ | $1.16$ | $1.167$ | $0.007$
|  | 2 | $28\,196.0$ | $28\,358.0$ | $162.0$ | $1.02$ | $1.023$ | $0.003$
|  | 1 | $28\,857.0$ | $28\,857.8$ | $0.8$ | $1.2635$ | $1.272$ | $0.0085$
|  | 4 | $29\,775.0$ | $29\,885.2$ | $110.2$ | $1.09$ | $1.091$ | $0.001$
|  | 3 | $30\,207.4$ | $30\,215.0$ | $7.6$ | $1.08$ | $1.091$ | $0.011$
|  | 5 | $30\,524.7$ | $30\,556.1$ | $31.4$ | $1.18$ | $1.176$ | $0.004$
| (5/2, 5/2) | 0 |  | $36\,047.2$ |  |  | $0.000$ |
|  | 1 |  | $39\,156.9$ |  |  | $0.768$ |
|  | 5 |  | $39\,660.1$ |  |  | $1.035$ |
|  | 2 |  | $40\,499.2$ |  |  | $0.918$ |
|  | 3 |  | $41\,784.5$ |  |  | $0.979$ |
|  | 4 |  | $42\,349.4$ |  |  | $1.045$ |
| (5/2, 3/2) | 4 |  | $37\,291.3$ |  |  | $0.827$ |
|  | 2 |  | $38\,593.0$ |  |  | $0.933$ |
|  | 1 |  | $41\,961.5$ |  |  | $0.96$ |
|  | 3 |  | $40\,740.8$ |  |  | $0.874$ |
$4f^{14}6s7p$ | ${}^{3}\\!P$ | 0 | $38\,090.7$ | $38\,067.6$ | $23.1$ |  | $0.000$ |
|  | 1 | $38\,174.2$ | $38\,204.9$ | $30.7$ | $1.14$ | $1.497$ | $0.357$
|  | 2 | $38\,552.0$ | $38\,544.6$ | $7.4$ | $1.5$ | $1.501$ | $0.001$
| ${}^{1}\\!P$ | 1 | $40\,564.0$ | $40\,563.7$ | $0.3$ | $1.01$ | $1.005$ | $0.005$
$4f^{14}6s5f$ | ${}^{1}\\!F$ | 3 |  | $42\,680.6$ |  |  | $1.045$ |
| ${}^{3}\\!F$ | 4 |  | $42\,652.3$ |  |  | $1.251$ |
|  | 3 |  | $43\,456.6$ |  |  | $1.039$ |
|  | 2 | $43\,433.9$ | $43\,433.8$ | $0.0$ | $0.68$ | $0.666$ | $0.014$
$4f^{14}6s8p$ | ${}^{3}\\!P$ | 0 | $43\,614.3$ | $43\,616.9$ | $2.6$ |  | $0.000$ |
|  | 1 | $43\,659.4$ | $43\,656.4$ | $3.0$ | $1.48$ | $1.469$ | $0.011$
|  | 2 | $43\,805.7$ | $43\,805.9$ | $0.2$ | $1.49$ | $1.501$ | $0.011$
| ${}^{1}\\!P$ | 1 | $44\,017.6$ | $44\,017.8$ | $0.2$ | $1$ | $1.033$ | $0.033$
$4f^{14}6s11p$ | (1/2, 1/2) | 0 |  | $48\,192.7$ |  |  | $0.000$ |
|  | 1 | $48\,212.1$ | $48\,198.6$ | $13.5$ |  | $1.49$ |
| (1/2, 3/2) | 2 |  | $48\,215.5$ |  |  | $1.501$ |
|  | 1 | $48\,258.5$ | $48\,271.9$ | $13.4$ |  | $1.011$ |
$4f^{14}6s6f$ | ${}^{1}\\!F$ | 3 |  | $45\,143.7$ |  |  | $1.000$ |
| ${}^{3}\\!F$ | 2 | $45\,956.3$ | $45\,956.3$ | $0.0$ | $0.72$ | $0.666$ | $0.054$
|  | 3 |  | $46\,169.9$ |  |  | $1.084$ |
|  | 4 |  | $46\,360.1$ |  |  | $1.251$ |
$4f^{14}6s9p$ | ${}^{3}\\!P$ | 0 | $46\,082.2$ | $46\,068.5$ | $13.7$ |  | $0.000$ |
|  | 1 | $46\,078.9$ | $46\,095.4$ | $16.5$ | $1.34$ | $1.481$ | $0.141$
|  | 2 | $46\,184.2$ | $46\,182.0$ | $2.2$ | $1.5$ | $1.501$ | $0.001$
| ${}^{1}\\!P$ | 1 | $46\,370.3$ | $46\,369.6$ | $0.7$ | $1.07$ | $1.02$ | $0.05$

Table 5: Yb I even-parity excited state fitting calculated energy levels (in
$\mathrm{c}\mathrm{m}^{-1}$), Landé $g$-factor values

|  |  | Level |  | Landé $g$ |
---|---|---|---|---|---|---
Configuration | Term | $J$ | Expt. | This work | $\Delta E$ | Expt. | This work | $\Delta g$
$4f^{14}6s^{2}$ | ${}^{1}\\!S$ | 0 | $0.0$ | $0.0$ | $0.0$ |  | $0.000$ |
$4f^{14}5d6s$ | ${}^{3}\\!D$ | 1 | $24\,489.1$ | $24\,489.1$ | $0.0$ | $0.5$ | $0.499$ | $0.001$
|  | 2 | $24\,751.9$ | $24\,751.9$ | $0.0$ | $1.16$ | $1.164$ | $0.004$
|  | 3 | $25\,270.9$ | $25\,275.2$ | $4.3$ | $1.34$ | $1.334$ | $0.006$
| ${}^{1}\\!D$ | 2 | $27\,677.6$ | $27\,677.7$ | $0.1$ | $1.01$ | $1.003$ | $0.007$
$4f^{14}6s7s$ | ${}^{3}\\!S$ | 1 | $32\,694.7$ | $32\,694.7$ | $0.0$ | $2.01$ | $2.002$ | $0.008$
| ${}^{1}\\!S$ | 0 | $34\,350.6$ | $34\,350.6$ | $0.0$ |  | $0.000$ |
$4f^{14}6s6d$ | ${}^{3}\\!D$ | 1 | $39\,808.7$ | $39\,807.6$ | $1.1$ | $0.5$ | $0.499$ | $0.001$
|  | 2 | $39\,838.0$ | $39\,839.3$ | $1.3$ | $1.16$ | $1.143$ | $0.017$
|  | 3 | $39\,966.1$ | $39\,966.2$ | $0.1$ | $1.33$ | $1.334$ | $0.004$
| ${}^{1}\\!D$ | 2 | $40\,061.5$ | $40\,061.3$ | $0.2$ | $1.03$ | $1.024$ | $0.006$
$4f^{13}6s^{2}6p$ | (7/2, 1/2) | 3 | $32\,065.3$ | $32\,069.0$ | $3.7$ | $1.23$ | $1.259$ | $0.029$
|  | 4 | $32\,273.6$ | $32\,280.3$ | $6.7$ |  | $1.063$ |
| (7/2, 3/2) | 5 | $35\,178.8$ | $35\,134.8$ | $44.0$ |  | $1.200$ |
|  | 2 | $35\,197.0$ | $35\,205.4$ | $8.4$ | $1.05$ | $1.057$ | $0.007$
|  | 3 | $35\,807.5$ | $35\,789.2$ | $18.3$ | $1.08$ | $1.088$ | $0.008$
|  | 4 | $36\,061.0$ | $36\,104.5$ | $43.5$ |  | $1.200$ |
| (5/2, 1/2) | 3 |  | $42\,343.4$ |  |  | $0.799$ |
|  | 2 | $42\,531.9$ | $42\,521.9$ | $10.0$ |  | $0.965$ |
| (5/2, 3/2) | 3 |  | $46\,218.5$ |  |  | $1.021$ |
|  | 1 | $44\,834.6$ | $44\,877.6$ | $43.0$ | $0.66$ | $0.499$ | $0.161$
|  | 4 | $45\,497.6$ | $45\,473.3$ | $24.3$ |  | $1.037$ |
|  | 2 | $45\,931.9$ | $45\,905.1$ | $26.8$ |  | $0.811$ |
$4f^{14}6s8s$ | ${}^{3}\\!S$ | 1 | $41\,615.0$ | $41\,615.0$ | $0.0$ | $2.02$ | $2.002$ | $0.018$
| ${}^{1}\\!S$ | 0 | $41\,939.9$ | $41\,939.9$ | $0.0$ |  | $0.000$ |
$4f^{14}6s7d$ | ${}^{3}\\!D$ | 1 | $44\,311.4$ | $44\,302.3$ | $9.1$ |  | $0.499$ |
|  | 2 | $44\,313.1$ | $44\,314.1$ | $1.0$ |  | $1.157$ |
|  | 3 | $44\,357.6$ | $44\,345.3$ | $12.3$ | $1.32$ | $1.334$ | $0.014$
| ${}^{1}\\!D$ | 2 | $44\,380.8$ | $44\,401.1$ | $20.3$ | $1.1$ | $1.010$ | $0.09$
$4f^{14}6p^{2}$ | ${}^{3}\\!P$ | 0 | $42\,436.9$ | $42\,447.5$ | $10.6$ |  | $0.000$ |
|  | 1 | $43\,805.4$ | $43\,763.7$ | $41.7$ | $1.47$ | $1.501$ | $0.031$
|  | 2 | $44\,760.4$ | $44\,806.1$ | $45.7$ | $1.34$ | $1.436$ | $0.096$
| ${}^{1}\\!D$ | 2 | $47\,821.8$ | $47\,819.8$ | $2.0$ | $1.04$ | $1.065$ | $0.025$
| ${}^{1}\\!S$ | 0 |  | $49\,935.2$ |  |  | $0.000$ |
$4f^{14}5d^{2}$ | ${}^{3}\\!F$ | 2 | $47\,634.4$ | $47\,634.4$ | $0.0$ |  | $0.668$ |
|  | 3 | $47\,860.2$ | $47\,860.3$ | $0.1$ | $1.02$ | $1.084$ | $0.064$
|  | 4 |  | $48\,130.4$ |  |  | $1.25$ |
| ${}^{1}\\!D$ | 2 |  | $50\,036.9$ |  |  | $1.048$ |
| ${}^{3}\\!P$ | 0 |  | $50\,404.0$ |  |  | $0.000$ |
|  | 1 |  | $50\,492.5$ |  |  | $1.501$ |
|  | 2 |  | $50\,697.6$ |  |  | $1.45$ |
| ${}^{1}\\!G$ | 4 |  | $51\,319.5$ |  |  | $1.000$ |
| ${}^{4}\\!S$ | 0 |  | $56\,283.8$ |  |  | $0.000$ |
$4f^{14}6s8d$ | ${}^{3}\\!D$ | 1 | $46\,445.0$ | $46\,451.6$ | $6.6$ | $0.49$ | $0.499$ | $0.009$
|  | 2 | $46\,467.7$ | $46\,459.7$ | $8.0$ | $1.12$ | $1.156$ | $0.036$
|  | 3 | $46\,480.7$ | $46\,482.0$ | $1.3$ | $1.35$ | $1.334$ | $0.016$
| ${}^{1}\\!D$ | 2 |  | $46\,519.5$ |  |  | $1.011$ |

##  4 Conclusion

For the Yb I atom, we propose and use the data-driven method of ridge
regression model fitting to calculate a total of 94 energy levels for the
even-parity configuration $4f^{14}6s^{2}$ \+ $4f^{14}5d6s$ \+ $4f^{14}6s7s$ \+
$\ldots{}$ and the odd-parity group $4f^{14}6s6p$ \+ $4f^{13}5d6s^{2}$ \+
$4f^{14}6s7p$ \+ $\ldots{}$, where there are 66 experimental energy levels
added, and these experimental energy levels fully consider the electron
correlation. The $4f$ shell is full of electrons, and the absolute error
between the calculated energy level and the NIST experimental value is less
than 50 $\mathrm{c}\mathrm{m}^{-1}$, and the relative is less than 0.1%. Among
them, the $4f^{13}5d6s^{2}$ configuration has an average absolute error of 70
$\mathrm{c}\mathrm{m}^{-1}$ and a relative error of less than 0.3%. Moreover,
reliable predicted data are made for energy levels that are experimentally
unattainable. This work demonstrates that the ridge regression model has a
stronger generalization ability when the amount of data is insufficient, and
the predicted data is more reliable. It is expected that a more accurate
machine learning model will be applied to the atomic structure calculation.

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interpreted the use of public keys as the maximum minimisation of information
and argued that they could fulfil the data minimisation requirements of the
GDPR [74, 51].

Finally, territorial scope is another major theme identified in the literature
in the sense of the GDPR compliance issues of public blockchains. The
constraints regarding territorial scope defined by the GDPR was found
difficult to implement by several researchers due to the lack of geographical
border of public decentralised systems. We believe that similar to the
definitions made for data processors and controllers by the GDPR, this is
another area that should be handled in a clearer way in the regulations
considering the inevitable increase in globalisation of information-centric
businesses.

###  5.2 Solutions Proposed for GDPR Compliance (RQ2)

As a response to our second research question (RQ2), our results revealed
three major research areas in the literature: techniques proposed to overcome
problems regarding deletion of personal data; management of consent and
privacy preferences; and techniques to secure data and ensure data privacy.

Regarding the first research area, hashing out is far most the most common
technique proposed in the literature with the aim of implementing the RTBF. In
this approach, personal data is stored in off-chain repositories and only the
hash data pointing to offline storage is stored in the distributed ledgers.
This allows off-chain data to be deleted once a request to erasure is
received. In the second approach, a private key is transmitted to the data
subject which is used to link the blockchain hash pointer to the data located
in distributed off-chain repositories and erasure is accomplished via key
destruction. Even though both of the techniques were claimed to be effective
in means of assuring RTBF in many studies, they fall short in maintaining the
decentralisation principle of blockchains and reintroduce several security
problems such as trusted third parties.

In Figure 4, we summarised major solutions proposed in the literature for
addressing the RTBF problem, where red hexagons represent limitations argued
for each solution. Here, it is important to underline that none of those
solutions can be used to delete metadata in blockchains even though it is
recognised as personal data by majority of the studies, and therefore, fall in
the scope of the GDPR.

Figure 4: Solutions proposed for addressing the RTBF

One interesting detail for this figure is perhaps the case where data subject
is data controller. Once this assumption is made, many of the data processing
requirements including the RTBF become unnecessary, and therefore, this is
added as a solution in Figure 4.

In the second research area, smart contracts were heavily discussed as a
solution to managing consent and controlling privacy preferences. They mainly
enable to translate privacy preferences into automated rules which can check
the validity of a data access request by a third party and handle it
transparently for all parties involved. This approach has two main use cases:
preventing unauthorised data access and providing proofs for privacy
violations. As this process is fully automated, it allows detection of consent
violations in a publicly verifiable way and can demonstrate legally valid
processing of personal information. The immutable nature of blockchains also
enables to assure the consent integrity, non-repudiation and versioning in a
publicly verifiable manner. However, it was also noted in the literature that
in an ideal sense, data subjects should be able to obtain human interventions
and challenge the decision after the smart contract has been executed, which
is not possible via smart contracts [86]. We identified one study in the
literature that recognised this problem and proposed to use smart contracts
through a securely hosted decryption function that enables the data subject
who is the single source with the key used for decryption to be notified
whenever their personal data is accessed [128]. Therefore, designing
blockchain solutions that can support human interventions could be reported as
a potential research area.

Regarding techniques to assure data security and privacy, there is a variety
of techniques proposed with two main goals: assuring data integrity, and
assuring data privacy or data confidentiality. ZKP is the most commonly cited
technique that has been proposed to assure data confidentiality. Even though
not received as much attention as ZKP, Merkle trees have also been proposed in
the literature. Ring signatures, SMPC and tumbling techniques are the emerging
themes identified, all of which can be used to assure data integrity or
confidentiality of personal data on the chain. A full list of solutions and
their consequences reported in the literature can be seen in Figure 5.

Figure 5: Solutions proposed to assure data security and privacy

Similar to the limitations in approaches regarding implementing the RTBF, one
research gap in the literature regarding security and privacy of data on
blockchains is the low number of solutions that focus on public keys. A
majority of the solutions were proposed to assure the GDPR compliance of
personal data in transactions and they tend to overlook public keys. In Figure
5, we have demonstrated the solutions proposed, where yellow boxes represent
the solutions for transaction data and green ones represent the techniques
that considers the privacy of both transaction data and public keys. Similar
to Figure 4, red hexagons indicate problems raised in the literature for the
proposed solutions.

We consider the categorisation presented here a major contribution of this
study, as it differentiates the techniques based on their focus (transactional
data, meta data or both) and demonstrates both the strength and limitations of
the current progress in this research area.

###  5.3 Legal Roles and Responsibilities (RQ3)

The third research question (RQ3) aims to investigate how legal roles and
responsibilities of different stakeholders of public blockchain networks have
been considered in the literature, e.g., who are considered data controllers
and processors. This is one of the most common debates in the literature in
the context of GDPR compliance of public blockchains. However, it is not
possible to report a consensus among the researchers in the sense of
identification of data controllers. The ambiguities in this context even yield
some researchers to claim that there is no controller in public blockchains,
and therefore, the privacy and data protection law is not applicable [44].

However, a majority of the studies covered in our SLR discussed several
different entities and their roles and responsibilities. In total, three main
categories are discussed as controllers in the literature: nodes (lightweight
nodes, full nodes, miners, all nodes as joint controller); developers; and
users. However, none of the categories dominate the discussions. On the other
hand, regarding processors, users and nodes on the blockchain are argued to be
processors in almost all of the studies.

It is important to highlight here that a vast amount of past research made a
over-simplified assumption: personal data is added to the system by their
owners (see Use Case 1 in Figure 6). Past research generally overlooked cases
where personal data is added to the system by others on behalf of the data
subject, as shown in Use Case 2 in Figure 6. This largely neglected use case
impacts on several aspects including the identification of data controller and
the ability of data subjects to exercise their rights given by the GDPR. CNIL
slightly covered this use case in one of their reports [24] and explained that
a natural person who sells or buys Bitcoin on their own account is not
considered as a data controller since s/he is entering personal data outside a
professional or commercial activity (in accordance with the principle of
domestic exception provided for in Article 2 of the GDPR). On the other hand,
if they carry out these transactions on behalf of other natural persons inside
their professional or commercial activity, the CNIL report states that they
may be considered controllers. This definition is unhelpful to understand the
use case where a natural person enters information of others in the course of
a purely personal or household activity, with no connection to a professional
or commercial activity. The second use case helps indicate that researchers
should be more comprehensive when considering the GDPR compliance issue of
public bloclchains since they can change many aspects drastically. In addition
to the above two use cases, more could be constructed, especially for hybrid
blockchain systems where multiple blockchain systems and non-blockchain
systems intermingle, leading to even more complicated relationships and
behaviours of different entities in the larger ecosystem.

Figure 6: Two different use cases of blockchains in terms of roles and
responsibilities in the GDPR context

Ibáñez et al. put forward the closest approach to cover this variation and
discussed GDPR compliance of blockchains in three most common scenarios of how
a data subject interacts with a blockchain [62]. They evaluated the role
assignment, and enumerated applicable strategies for data minimisation and the
RBTF and amendments for each scenario individually. Their scenarios are given
as follows: an individual interacts directly with a permissionless blockchain;
applications that use permissionless blockchains as the backend; and
permissioned blockchains. We do not cover the third scenario in this SLR since
permissioned blockchains are out of the scope of our work. For the first
scenario, Ibáñez et al. argued that it is not possible to identify a
controller and the onus of compliance might need to be put on users
themselves. For the second scenario, the owners of the intermediary
application were argued as data controllers. While this separation is
valuable, it falls short of addressing the situation where people put data of
others into the blockchain system. This use case reflects a process that
completely predates the person being able to give or withdraw consent and
exercise other rights. Considering the immutable nature of blockchains, such a
use case raises significant privacy concerns. More interestingly, we can also
speculate that if a person can or should make irrevocable decisions on their
future self’s behalf, echoing the importance of the RTBF if the initial
decision turns out to be wrong in future. We believe that impossibility of
exercising the right to withdraw consent or the RTBF leads to major technical,
philosophical, legal, ethical, moral and societal concerns and challenges.
Limiting the types of data (e.g., disallowing sensitive personal data or
illegal materials) or developing solutions that are not harmful for kids,
younger people and other vulnerable people could be argued as responsibilities
of service providers, who then should assume the role of data controllers.

##  6 Conclusion

We conducted an SLR of 114 research papers to produce a more in-depth and up-
to-date understanding of the current research literature on GDPR compatibility
of public blockchain systems. Due to a variety of challenges discussed and
solutions proposed for such challenges, as well as the different
interpretations regarding complying issues and overlapping concepts, it is
challenging to categorise discussions in the literature in a systematic way.
However, our study recommends six broad categories to consider while
evaluating the GDPR compatibility of public blockchain systems: categories of
personal data in blockchains; roles and responsibilities in blockchains;
technical solutions proposed for protection of personal data; discussions
regarding how to exercise data subject rights guaranteed by the GDPR;
discussions regarding lawfulness, fairness and transparency; and discussions
regarding GDPR principles and other related elements.

Our contributions are four-fold. Firstly, the variety of personal data on a
blockchain has been recognised in our study, which is generally overlooked in
similar past studies and we identified the challenges and proposed solutions
for each category accordingly. Secondly, we identified and reported
limitations and consequences of solutions proposed in the literature as well
as contradicting opinions, allowing to get a better idea about the current
status of the research in this area. Thirdly, we provided perspectives from
both the network and application domain perspectives and categorised
discussions accordingly, which have not been covered comprehensively in past
studies. Finally, we include a broad scope of GDPR elements in our study and
provided a much deeper and concise representation of the literature,
identifying not only the conflicts (negative aspects) but also the compliance
(positive) aspects between public blockchains and the GDPR.

While our study was comprehensive in means of identifying studies regarding
GDPR compatibility aspects of public blockchain systems, we would like to
acknowledge some limitations. The first limitation is about the databases we
used. Due to limited search facilities, Google Scholar can provide the high
number of candidate data items if searched into fulltext, however due to the
large number of items and the low hit rate we decided to search into titles
only, which might have led us to miss some relevant studies. Despite this
limitation, as we also used two additional scientific databases (Scopus and
WoS), two most widely used databases for SLRs with a very comprehensive
coverage of research papers, we believe that the research papers we covered
still represent a very good representation of the current research. Secondly,
as stated before, we set our scope to be limited to public blockchains only
because it is relatively straightforward to comply with the GDPR for private
and consortium blockchains. Therefore, our SLR does not cover solutions
proposed for GDPR compatibility that are based on or for private and
consortium blockchains only. To address the second limitation, a new SLR can
be conducted to focus on private and consortium blockchains, and compare the
results with what are reported in this paper for public blockchains.

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