text:
\section{Introduction}

A good starting point for the relativistic atomic calculations
is given by the Dirac-Coulomb-Breit (DCB) equation.
This equation can be solved with a high accuracy by
using either the configuration-interaction
Dirac-Fock (CI-DF) methods \cite{gra70,des75,bra77,ind90,koz01,gla04,tup05}
or the relativistic many-body perturbation theory (RMBPT) methods
\cite{dzu89,blu90,ind92,ynn94,saf99,por09,dzu12}.
In many cases
the precision of these calculations has reached
a level that requires evaluations of
quantum electrodynamics (QED) effects.
To date, the rigorous calculations of the QED effects
in middle- and high-Z systems
are fully restricted to
the 1/Z perturbation theory
 (see, e.g.,
\cite{yer99,art05,sha08,gla11,vol12} and references therein).
The perturbation theory
methods have been also extended to many-electron
ions and atoms employing an effective screening potential
instead of the Coulomb one
\cite{sap02,sap03,sha05a,sha05b,che06,art07,koz10,sap11}.
However, these methods are
too complicated to be directly included
into the DCB calculations.
 For this reason, numerous attempts
have been undertaken to propose simple methods
for incorporating
the QED corrections into
the CI-DF and RMBPT codes. These methods
(see, e.g., \cite{ind90,pyy03,dra03,fla05,thi10,pyy12,tup13,low13,rob13}
and references therein)
are generally based
on scaling the Lamb shift results for the Coulomb
potential to other atomic potentials
which include partially the screening effects.
Such a scaling can be done either directly by using the
Welton's idea \cite{wel48}
to express the main part of the self energy contribution
in terms of $\vec{\nabla}^2 V$ \cite{ind90,dra03,low13}
or by introducing an effective short-range potential
which fits the Lamb shifts for hydrogenlike ions
\cite{pyy03,fla05,thi10,pyy12,tup13,rob13}.

In Ref. \cite{sha93} it was shown that the QED corrections
can be systematically included into an effective Hamiltonian
 acting in the space of the Slater determinants made up of
one-electron positive-energy states whose total (many-electron)
energies are smaller
than the pair-creation threshold. To the lowest order,
this approach leads to a QED operator that, in principle,
can be added to DCB Hamiltonian. The main goal of this
paper is to represent this QED operator in a form that can
be easily included in any calculations based on the DCB
equation.

In the next section, we summarize the basic equations for the
effective Hamiltonian that includes the one-loop
QED corrections.   Then we approximate
the QED operator by a sum of  short-range local and nonlocal
potentials and calculate the model QED operator in a wide range
of $Z=10-120$. Finally, the model QED operator is applied
to calculations of the Lamb shifts in many-electron
atoms and ions, and the results obtained are compared
with other QED calculations.

Relativistic units ($\hbar=c=1$) are used in the paper.

\section{Effective Hamiltonian in the framework of QED}

The systematic method to derive a Schr\"odinger-like equation
for a relativistic many-electron atom from QED
can be formulated
within the two-time Green function (TTGF) method \cite{sha02}.
To determine such an equation, first of all one needs
to choose the active space in which the effective Hamiltonian acts.
Since rigorous calculations of the QED effects
in middle- and high-$Z$ systems
employ
the perturbation theory starting from
the Dirac equation with a local potential,
the active space is generally considered to be formed
either by a single or by (quasi)degenerate
unperturbed states. These states are given by the Slater
determinants made up of the solutions
of the Dirac equation with the local potential
considered. However, in Ref. \cite{sha93}
it was shown that the active space
can be extended
to  all unperturbed states made up of
one-electron positive-energy states whose total (many-electron)
energies are smaller
than the pair-creation threshold.
Moreover, if the consideration is restricted
to the lowest-order QED terms,
the active
space can be extended beyond the pair-creation
threshold. For simplicity, this extention, having
 no effect on the accuracy
of the calculations, is considered in the present
paper.
Using the derivations presented in Ref. \cite{sha93},
we summarize below the basic equations that are
obtained with this choice of the active space.

To simplify the equations,
we assume that
the active space is formed by the Slater determinants
made up of the positive-energy solutions of
the Dirac equation with the Coulomb potential
$V_{\rm C}(r)=-\alpha Z_{\rm nuc}(r)/r$ (the effective
charge $Z_{\rm nuc}(r)$  accounts for the nuclear charge
distribution):
\begin{eqnarray} \label{Dirac}
h^{\rm D}\psi_n = \varepsilon_n\psi_n\,,
\end{eqnarray}
where
\begin{eqnarray} \label{Dirac1}
h^{\rm D}=\vec{\alpha} \cdot \vec{p} +  m\beta +  V_{\rm C}(r)
\end{eqnarray}
is the one-electron Dirac Hamiltonian.
We note, however, that all the equations can be easily
adopted to the theory with the active space
formed by the solutions in an effective potential
$V_{\rm eff}(r)$
(e.g., the Dirac-Hartree, the Kohn-Sham or a local
version of the Dirac-Fock potential) that includes
partly the screening effect. In this case,
the interaction with
the related counterterm ($V_{\rm C}(r)-V_{\rm eff}(r)$)
must be included in the total Hamiltonian.
To construct the desired Hamiltonian, one should first
consider the contribution from the
 one-photon exchange Feynman diagram (Fig. 1).
\begin{figure}
\begin{center}
\includegraphics[angle=0,height=0.20\textheight, width=0.40\textwidth]{figure1.eps}
\caption{\label{fig1}
One-photon exchange diagram.
}
\end{center}
\end{figure}
Evaluation of this diagram with the TTGF method leads
to the following symmetric
form of the electron-electron interaction
operator
(for details see Refs. \cite{sha93,sha02}):
\begin{eqnarray}\label{int}
h^{\rm int} &=&
\sum_{k\ne l,m \ne n}^{(\varepsilon_k,\varepsilon_l,\varepsilon_m,\varepsilon_n>0)}
|\psi_k \psi_l \rangle
\langle \psi_k\psi_l |\frac{1}{2}[I(\varepsilon_{k}-\varepsilon_{m})
\nonumber\\
&& + I(\varepsilon_{l}-\varepsilon_{n})]|\psi_m\psi_n\rangle
\langle \psi_m \psi_n|\,,
\end{eqnarray}
where the indeces $k$, $l$, $m$, $n$ enumerate the positive-energy one-electron
Dirac states,
$|\psi_k \psi_l  \rangle \equiv |\psi_k\rangle | \psi_l  \rangle$
is the direct product of the one-electron Dirac wave functions
$\psi_k(\vec{r_1})$ and $\psi_l(\vec{r_2})$,
\begin{eqnarray}
I(\omega)=e^2\alpha_1^{\rho}\alpha_2^{\sigma}D_{\rho \sigma}(\omega,r_{12})\,,
\end{eqnarray}
 $\alpha^{\rho}\equiv\gamma^0 \gamma^{\rho}=(1,{\mbox{\boldmath$\alpha$}})$ are the Dirac matrices,
$D_{\rho \sigma}(\omega,r_{12})$
is the photon propagator, and $r_{12} = |\vec{r_1}-\vec{r_2}|$
is the interelectronic distance.
It should be noted that
the symmetric form of the frequency-dependent
electron-electron interaction was first considered in Ref.
\cite{mit72}.
 The operator (\ref{int}) defines
 the interaction between two electrons only.
To get the total electron-electron interaction operator for
a many-electron atom, one has to sum Eq. (\ref{int})
over all pairs of atomic electrons:
\begin{eqnarray}\label{int2}
H^{\rm int} = \sum_{i<j} h_{ij}^{\rm int}\,,
\end{eqnarray}
where
$h_{ij}^{\rm int}$ is the two-electron operator  (\ref{int})
taken for electrons $i$ and $j$.

Taking $D_{\rho \sigma}(\omega,r_{ij})$ in the Coulomb gauge  at zero energy
transfer  ($\omega=0$)
 leads to the Dirac-Coulomb-Breit Hamiltonian \cite{suc80}:
\begin{eqnarray}
H^{\rm DCB}=
\Lambda^{(+)}\Bigl[\sum_{i}h_i^{\rm D}  +\sum_{i<j}V_{ij}\Bigr]\Lambda^{(+)}\,,
\end{eqnarray}
where the indeces $i$ and $j$ enumerate the atomic electrons,
$\Lambda^{(+)}$ is the product of the one-electron
projectors on the positive-energy states
(which correspond to the potential $V_{\rm C}$),
$h_i^{\rm D}$ is the one-electron Dirac Hamiltonian
(\ref{Dirac1}) taken for electron $i$,
\begin{eqnarray}
V_{ij} &=& e^2\alpha_i^{\rho}\alpha_j^{\sigma}D_{\rho \sigma}(0,r_{ij})
= V^{\rm C}_{ij}+ V^{\rm B}_{ij}\nonumber \\
&=&\frac{\displaystyle \alpha}{\displaystyle r_{ij}}
-\alpha\Bigl[\frac{\displaystyle{ \vec{\alpha}_i\cdot \vec{\alpha}_j}}
{\displaystyle{ r_{ij}}}+\frac{\displaystyle 1}{\displaystyle 2}
(\vec{\nabla}_i\cdot \vec{\alpha}_i)(\vec{\nabla}_j\cdot \vec{\alpha}_j)
r_{ij}\Bigr]
\,
\end{eqnarray}
is the the sum of the Coulomb and Breit electron-electron interaction operators,
and $\alpha$ is the fine structure constant.
It is well known that the DCB Hamiltonian
 accounts for the nonrelativistic and lowest-order relativistic
contributions.
 In the Feynman gauge, to get the Hamiltonian to the same accuracy,
one has to account for the higher-order photon exchange diagrams (see Ref. \cite{sha93}
and references therein).

As the next step, one should consider the contributions from
the one-loop
self-energy (SE) and vacuum-polarization (VP) diagrams
presented in Figs. 2 and 3, respectively.
\begin{figure}
\begin{center}
\includegraphics[angle=0,height=0.12\textheight, width=0.30\textwidth]{figure2.eps}
\caption{\label{fig2}
Self-energy diagram.
}
\end{center}
\end{figure}
\begin{figure}
\begin{center}
\includegraphics[angle=0,height=0.158\textheight, width=0.17\textwidth]{figure3.eps}
\caption{\label{fig3}
Vacuum-polarization diagram.
}
\end{center}
\end{figure}
The direct calculation of these
contributions
within the TTGF method
leads to the following symmetric form
of the one-electron QED
 operator $h^{\rm QED}$ \cite{sha93}:
\begin{eqnarray}\label{oper1}
h^{\rm QED}&=&h^{\rm SE}+h^{\rm VP}=  \sum_{k,l}^{(\varepsilon_k,\varepsilon_l >0)}
|\psi_k\rangle
\langle \psi_k|
\nonumber \\ && \times
\bigl[\frac{1}{2}(\Sigma^{\rm SE}(\varepsilon_k)+\Sigma^{\rm SE}(\varepsilon_l))+V^{\rm VP}\bigr]|\psi_l\rangle
\langle\psi_l|
\,,  \nonumber \\
\end{eqnarray}
where $\Sigma^{\rm SE}(\varepsilon_k)$ and $ V^{\rm VP}$ are the renormalized SE and VP operators,
respectively, and the sums over  $k$ and $l$ go over all
 the positive-energy one-electron
Dirac states.
To get the total QED operator for a many-electron atom,
one has to sum Eq. (\ref{oper1}) over all
atomic electrons:
\begin{eqnarray}\label{oper1a}
H^{\rm QED} &=& H^{\rm SE}+ H^{\rm VP}\nonumber\\
&=&\sum_{i} h_i^{\rm QED}
= \sum_{i} (h_i^{\rm SE}+h_i^{\rm VP}) \,,
\end{eqnarray}
where $h_i^{\rm QED}$,
$h_i^{\rm SE}$, and  $h_i^{\rm VP}$ are the
one-electron operators (\ref{oper1}) taken for electron $i$.

Thus, within the lowest-order QED approximation
 the total effective Hamiltonian can be presented as
\begin{eqnarray} \label{Htot}
H=\Lambda^{(+)}\Bigl[\sum_{i} (h_i^{\rm D}+ h_i^{\rm QED})
+\sum_{i<j} h_{ij}^{\rm int}
\Bigr]  \Lambda^{(+)},
\end{eqnarray}
where  the sums go over all atomic electrons
and  $h_{ij}^{\rm int}$, which
 is the two-electron operator  (\ref{int})
considered for electrons $i$ and $j$,
 must be taken in the Coulomb gauge
(if it is taken in the Feynman gauge,
 an additional term must be
added to keep the same accuracy level \cite{sha93}).
In Eq. (\ref{Htot}), all the operators are
 defined for the Coulomb potential $V_{\rm C}$.
To get the corresponding Hamiltonian
in the active space formed by the solutions in
an effective potential  $V_{\rm eff}$, one should do
the evident replacements in all the operators, including
the projector, and add the term
$\Lambda^{(+)}\sum_{i} (V_{\rm C}(r_i)-V_{\rm eff}(r_i))\Lambda^{(+)}$.

\section{Model QED operator}

Since the one-electron VP operator is given by the sum of
the local Uehling and Wichmann-Kroll potentials  \cite{moh98},
$V^{\rm VP} =  V_{\rm Uehl} +  V_{\rm WK}$, it can be easily
included into the DCB equations.
 As is known,
the dominant part
of the VP contribution is represented  by the Uehling
potential:
\begin {eqnarray} \label{uehlexpr}
V_{\rm Uehl}(r)&=&-\alpha Z
\frac{2\alpha}{3\pi}
\int\limits_0^\infty dr'\; 4\pi r'\rho
(r')
\nonumber \\ &&\times
\int\limits_1^\infty dt \;
(1 +\frac{1}{2t^2})
\frac{\sqrt{t^2-1}}{t^{2}}\nonumber \\
&&\times \frac{e^{(-2m|r-r'|t)}-e^{(-2m(r+r')t)}}
{4mrt} \,,\nonumber \\
\end{eqnarray}
where
 $Z\rho(r)$ is the density of
 the nuclear charge distribution
($\int \rho(r) d{\bf r}=1$).
The Uehling potential can easily be evaluated employing the approximate
formulas from Ref. \cite{ful76}. Including the screening effect  into this
potential causes no problem but hardly affects the result (see, e.g.,
Ref. \cite{sha05b}).
Evaluation of the
Wichmann-Kroll potential is a much more difficult problem \cite{sof88,man89,per93}.
However, to a good accuracy, it can be calculated with the help of the approximate
formulas derived in Ref. \cite{fai91}. Therefore,
in what follows, we restrict our consideration to the SE contribution only.

In principle, the operator $H^{\rm SE}$ defined by Eqs. (\ref{oper1})-(\ref{oper1a})
can be added to the DCB Hamiltonian
to account for the lowest-order SE corrections.
In practice, however, due to absence of rather simple
algorithms for evaluation of the SE contributions for
arbitrary states,
we have to restrict
 $h^{\rm SE}$ to its matrix elements
between a finite number of low-lying one-electron
Dirac-Coulomb (or other effective potential)
wave functions. This restriction
strongly enlarges
the interaction range of the SE operator
and
excludes highly-excited bound and continuum spectrum components
from the active space.
As a result, such a potential
may lead to the SE corrections which strongly deviate from the correct ones.
For instance, in case of H-like ions
 it gives zero results for the states with the principal quantum number
 $n\ge 4$, provided the sums
in (\ref{oper1}) are limited by $n_k,n_l\le 3$.
 To solve these problems, we modify
the operator   $h^{\rm SE}$ in two steps.

As the first step, to minimize the deviation of
the representation  (\ref{oper1}), restricted to
a finite number of states, from the exact one,
 we separate out a local (with respect to $r$) potential $V^{\rm SE}_{\rm loc}$
from the SE operator and
employ formula (\ref{oper1}) for the remaining SE part only.
Such a separation can be justified by the fact that
a dominant part of the nonrelativistic SE operator can be
represented by a local short-range potential \cite{wel48}.
Due to the conservation
of the angular quantum numbers by the one-electron
SE operator, we can choose the local
part to be different for different $\kappa= (-1)^{j+l+1/2}(j+1/2)$.
With this in mind,
we introduce the projector $P_{\kappa}$ which acts only on the angular variables
and is defined by its kernel as
\begin{eqnarray}\label{projector}
&&P_{\kappa}(\bf n, \bf n^{\prime})
=
\nonumber \\
&&\Dmatrix{\sum_{m}\Omega_{\kappa\, m}({\bf n}) \Omega^{\dagger}_{\kappa\, m}({\bf n}^{\prime})}
{\;\;\displaystyle{0}}
{\displaystyle{0}}{\;\; \sum_{m}\Omega_{-\kappa\, m}({\bf n})\Omega^{\dagger}_{-\kappa\, m}({\bf n}^{\prime}) }\,,
\nonumber \\
\end{eqnarray}
where $ \Omega_{\kappa\, m}({\bf n})$ is a spherical spinor.
Then, the local potential is given by
\begin{eqnarray}\label{local1}
V^{\rm SE}_{\rm loc}=\sum_{\kappa}V^{\rm SE}_{{\rm loc},\kappa}P_{\kappa}\,,
\end{eqnarray}
where $V^{\rm SE}_{{\rm loc},\kappa}(r)$ is a short-range radial potential which
can be chosen differently for different values of $\kappa$.
Strictly speaking,
 the potential $V^{\rm SE}_{\rm loc}$ is semilocal. However, here and in what follows
we label it as "local",  keeping in mind that it is local only at a given
value of $\kappa$.
We put
\begin{eqnarray}\label{local}
 V^{\rm SE}_{{\rm loc},\kappa}(r) =  A_{\kappa} \exp{(-r/\lambdabar_C)}\,,
\end{eqnarray}
where the constant $A_{\kappa}$ is chosen to reproduce
the  SE shift for the lowest energy level at the given $\kappa$
in the corresponding H-like ion, and $\lambdabar_C=\hbar/(mc)$.

As the second step, we restrict the active space of
 the remaining SE operator, $h^{\rm SE}-V^{\rm SE}_{\rm loc}$,
to the basis functions $\{\phi_i({\bf r})\}_{i=1}^{n}$  which, having the same
angular parts as the
one-electron Dirac-Coulomb
functions $\{\psi_i({\bf r})\}_{i=1}^{n}$,
are localized at smaller distances compared to the
Dirac-Coulomb ones.
The specific choice of the  functions $\{\phi_i({\bf r})\}_{i=1}^{n}$
will be given below.
With these  functions, we approximate the
one-electron SE operator  as follows
\begin{eqnarray}\label{oper2}
h^{\rm SE} = V^{\rm SE}_{\rm loc} +
\sum_{i,k=1}^{n} |\phi_i\rangle B_{ik}\langle \phi_k|\,,
\end{eqnarray}
where the matrix $B_{ik}$ has to be determined in such a way
that the matrix elements of the model SE operator (\ref{oper2})
with the H-like wave functions, corresponding to the
space under consideration, coincide with the exact ones.
This leads to the equations
\begin{eqnarray}\label{oper2a}
\sum_{j,l=1}^{n} \langle \psi_i
|\phi_j\rangle B_{jl}\langle \phi_l|\psi_k\rangle
\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;
\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;
 \nonumber \\
=\langle \psi_i|\bigl[\frac{1}{2}(\Sigma(\varepsilon_i)+ \Sigma(\varepsilon_k))
-V^{\rm SE}_{\rm loc}\bigr]|\psi_k\rangle \,.
\end{eqnarray}
Introducing the matrix  $D_{ik} = \langle \phi_i|\psi_k\rangle $,
 we get
\begin{eqnarray}
B_{ik}&=&  \sum_{j,l=1}^{n} ((D^t)^{-1})_{ij}
\langle \psi_j|\bigl[\frac{1}{2}(\Sigma(\varepsilon_j)+ \Sigma(\varepsilon_l))
\nonumber\\
&& -V^{\rm SE}_{\rm loc}\bigr]|\psi_l\rangle
(D^{-1})_{lk}
\,.
\end{eqnarray}
Therefore, the model one-electron SE operator can be written as
\begin{eqnarray}\label{oper3}
h^{\rm SE}&=&  V^{\rm SE}_{\rm loc}+
\sum_{i,k=1}^n \sum_{j,l=1}^n
|\phi_i\rangle
((D^t)^{-1})_{ij}
 \nonumber \\ && \times
\langle \psi_j|\bigl[\frac{1}{2}(\Sigma(\varepsilon_j)+\Sigma(\varepsilon_l))-V^{\rm SE}_{\rm loc}
\bigr]|\psi_l\rangle]\nonumber\\
&&\times (D^{-1})_{lk}
\langle \phi_k|\,.
\nonumber \\
\end{eqnarray}

Now let us consider the choice of  the functions
 $\{\phi_i({\bf r})\}_{i=1}^{n}$. From one side,
since we use the SE matrix elements calculated with
the hydrogenlike wave functions, these functions
should be chosen rather close to the H-like ones.
From the other side, because of a short interaction range of the
SE operator, they should vanish at smaller distances
compared to the H-like wave functions.
With this in mind, we construct them using the H-like
wave functions multiplied with the factor
\begin{eqnarray}
\rho_l(r)=\exp{(-2\alpha Z (r/\lambdabar_C)/(1+l))},
\end{eqnarray}
where $l=|\kappa+1/2|-1/2$ is the orbital angular momentum
of the state under consideration.
The simple choice  $\phi_i({\bf r}) = \rho_{l_i}(r)\psi_i({\bf r})$
fits the goal but, due to a rather similar behaviour of the
 wave functions for different values of the principal quantum number
  at small $r$,
gives a matrix $D$ close to degenerate one and, therefore, leads to a rather
singular matrix $D^{-1}$. For this reason,
we consider
a slightly different choice. In what follows,
we restrict the basis functions by
 $ns$, $np_{1/2}$,  $np_{3/2}$, $nd_{3/2}$,
and   $nd_{5/2}$ states with the
principal quantum number  $n\le 3$
for the $s$ states and $n\le 4$
for the $p$ and $d$ states,
and put
\begin{eqnarray}
\phi_i({\bf r}) = \frac{1}{2}(I-(-1)^{s_i}\beta)
\rho(r)\psi_i({\bf r})\,,
\end{eqnarray}
where $I$ is the identity matrix,
$\beta$ is the standard Dirac matrix, the index $s_i=n_i-l_i$
enumerates the positive energy states at the given $\kappa$,
and $n_i$ is the principal quantum number.
With this choice,
one easily finds
\begin{widetext}
\begin{eqnarray}
&D_{11} =\int_0^\infty dr\, r^2 g_1^2(r)\,\rho_l(r)\,,\;\;\;\;
D_{12} =\int_0^\infty dr\, r^2 g_1(r)\,g_2(r)\,\rho_l(r)\,,\nonumber\\
&D_{13} = D_{31}=\int_0^\infty dr\, r^2 g_1(r)\,g_3(r)\,\rho_l(r)\,,\;\;\;\;
D_{21} =\int_0^\infty dr\, r^2 f_1(r)\,f_2(r)\,\rho_l(r)\,,\nonumber\\
&D_{22} =\int_0^\infty dr\, r^2 f_2^2(r)\,\rho_l(r)\,,\;\;\;\;
D_{23} =\int_0^\infty dr\, r^2 f_2(r)\,f_3(r)\,\rho_l(r)\,,\nonumber\\
& D_{32} =\int_0^\infty dr\, r^2 g_2(r)\,g_3(r)\,\rho_l(r)\,,\;\;\;\;
D_{33} =\int_0^\infty dr\, r^2 g_3^2(r)\,\rho_l(r)\,,
\end{eqnarray}
\end{widetext}
where  $g_i(r)$ and $f_i(r)$ are the upper and lower radial components of the
hydrogenlike wave functions and the index $i$ enumerates
the positive-energy states at the given $\kappa$.
The explicit formulas for the calculation of the inverse
matrix  can be found in the standard textbooks.

Thus, in what follows we use the
model one-electron SE operator given by
\begin{eqnarray} \label{hse_fin}
h^{\rm SE}&=& V^{\rm SE}_{\rm loc}+\frac{1}{4}
\sum_{i,k}
\sum_{j,l}
(I-(-1)^{s_i}\beta)
\rho_{l_i}(r)
|\psi_i\rangle\nonumber\\
&&\times ((D^t)^{-1})_{ij}
\langle \psi_j|\bigl[\frac{1}{2}(\Sigma(\varepsilon_j)+\Sigma(\varepsilon_l))
-V^{\rm SE}_{\rm loc}\bigr]|\psi_l\rangle \nonumber\\
&& \times (D^{-1})_{lk}
\langle \psi_k|\rho_{l_k}(r)
(I-(-1)^{s_k}\beta),
\end{eqnarray}
where the summations run over $ns$ states with
the principal quantum number $n\le 3$
and over
 $np_{1/2}$,  $np_{3/2}$
$nd_{3/2}$, $nd_{5/2}$ states
 with $n\le 4$,
$\rho_{l_i}(r)=\exp{(-2\alpha Z (r/\lambdabar_C)/(1+l_i))}$,
\begin{eqnarray}
D_{ik} &=& \frac{1}{2} \langle \psi_i| (I-(-1)^{s_i}\beta)
\rho_{l_i}(r)|\psi_k \rangle\,,
\end{eqnarray}
and $V^{\rm SE}_{\rm loc}$ is defined by Eqs. (\ref{local1})-(\ref{local}) .

\section{Matrix elements of the exact self-energy operator}

To complete our construction of the model SE operator, we need
the diagonal and non-diagonal matrix elements of the exact SE operator
$\Sigma(\varepsilon)$ with the hydrogenlike wave functions. Calculations of the SE corrections
reported previously in the literature \cite{moh92a,moh92b,bei98} were performed for the diagonal matrix
elements only. In the present work, we extend these calculations to both the diagonal and
non-diagonal matrix elements of the one-loop SE operator. Our calculation was carried out into two
steps. First, we evaluated the SE matrix element for the point nucleus by using the numerical method
described in detail in Refs.~\cite{yer99a,yerokhin:05:se}. Next, we separately calculated the finite
nuclear size correction, as described in Ref.~\cite{yerokhin:11:fns}. The finite nuclear size
effect was calculated with the standard two-parameter Fermi model for the nuclear charge distribution.

The results of our calculations for the $ns$, $np_{1/2}$,
$np_{3/2}$, $nd_{3/2}$, and $nd_{5/2}$ states
with $n$ up to 5 are presented in Tables~\ref{tab:s}, \ref{tab:p1},
\ref{tab:p3}, \ref{tab:d3}, \ref{tab:d5}, respectively.
They are expressed in terms of the function $F_{ik}(\alpha Z)$
defined by
\begin{eqnarray}\label{F}
\Sigma_{ik}&\equiv&
\langle \psi_i|\frac{1}{2}(\Sigma(\varepsilon_i)+\Sigma(\varepsilon_k))|\psi_k\rangle
 \nonumber \\
&=&\frac{\alpha}{\pi}\frac{(\alpha Z)^4}{(n_i n_k)^{3/2}}
F_{ik}(\alpha Z)mc^2\,,
\end{eqnarray}
where $n_i$ and $n_k$ are the principal quantum numbers
of the $i$ and $k$ states, respectively. In the tables, the results are presented separately for
the point nucleus and for the extended nucleus (except for the cases when both results coincide).
If no error is specified, the results are supposed to be accurate to all digits quoted. In the case
of diagonal matrix elements, excellent agreement with previous results \cite{moh92a,moh92b,bei98}
is observed. In this paper, to define the model SE operator we use
 the values $F_{ik}(\alpha Z)$ with the principal quantum numbers
$n_i, n_k \le 3$  for the $s$ states and $n_i, n_k \le 4$  for the $p$ and $d$ states.
As to the other data presented in the tables, they can be used
for test calculations with H-like ions
(see the next section) as well as for extending the active space
of the model SE operator.

To obtain  the function $F_{ik}(\alpha Z)$ for values of $Z$ not listed in the tables, one may use
a polynomial interpolation, applied to the function
\begin{eqnarray}
G_{ik}(\alpha Z) = F_{ik}(\alpha Z)-\delta_{l0}(4/3){\rm ln}(\alpha Z)^{-2}\,.
\end{eqnarray}
Here, following to Ref. \cite{moh83},
we have subtracted the log term which represents the small-$\alpha Z$ behaviour
of  $F_{ik}(\alpha Z)$ for $s$ states ($l=0$).
The interpolation function is thus given by \cite{moh83}
\begin{eqnarray}
F'_{ik}(\alpha Z)&=&
\delta_{l0}(4/3){\rm ln}(\alpha Z)^{-2}
+\sum_{n=1}^{N}\Bigl[\prod_{m\ne n}\frac{Z-Z_m}{Z_n-Z_m}\Bigr]
 \nonumber \\ && \times
[F_{ik}(\alpha Z_n)-\delta_{l0}(4/3){\rm ln}(\alpha Z_n)^{-2}]\,.
\end{eqnarray}

\section{Calculations with the model self-energy operator}

Since the model SE operator is constructed
using the SE matrix elements with H-like wave functions
of the $ns$ states at $n\le 3$ and
the $np$ and $nd$ states at $n\le 4$, first of all, we
should consider how it works for H-like states
 with higher values of $n$. In  Table~\ref{tab:H-like}
we present the SE shifts
for
 the $4s$, $5s$, $5p_{1/2}$, $5p_{3/2}$, $5d_{3/2}$,
and $5d_{5/2}$ states in
H-like ions,
 obtained using the model SE operator, $\langle v|h^{\rm SE}|v\rangle $,
and compare them with the corresponding exact
results.
To demonstrate the importance of the nonlocal
part of $h^{\rm SE}$,
we present also the local  $\langle v|V_{\rm loc}^{\rm SE}|v\rangle$
contribution. As one can see from the table, for the  $s$ states
the difference between the exact and model SE operator results
does not exceed 1\%. As to the $p$ and $d$ states, despite the relative
value of the difference is significantly bigger than for the $s$ states,
its absolute value,
expressed in terms of the functions $F(\alpha Z)$, does not exceed
0.01.
We stress also the importance of the nonlocal part of the
model SE operator: for the $s$ states the difference between
the local part and the total result can amount to about 30\%.

To demonstrate the efficiency of the method, we also applied it
to calculations of the Lamb shifts in
neutral alkali metals,
Cu-like ions, superheavy atoms, and Li-like ions.

Calculations of the Lamb shift
in alkali metals were considered in Refs. \cite{sap02,lab99}.
In Ref. \cite{sap02}, it was calculated in
the potential $V_{\rm eff}(r)$, which is defined in terms of an effective
charge $Z_{\rm eff}(r)$ through
\begin{eqnarray}\label{U}
V_{\rm eff}(r) =-\frac{\alpha Z_{\rm eff}(r)}{r}\,,
\end{eqnarray}
where
\begin{eqnarray}
 Z_{\rm eff}(r)=Z_{\rm nuc}(r) -r\int_0^{\infty}dr'\frac{1}{r_{>}}
\rho_{\rm t}(r')+x_{\alpha}\Bigl[\frac{81}{32\pi^2}r\rho_{\rm t}\Bigr]^{1/3}
\end{eqnarray}
and $\rho_{\rm t}=\rho_{\rm v}+\rho_{\rm c}$ is total (valence plus core)
electron charge density,
which is determined self-consistently solving the Dirac equation with the
potential $V_{\rm eff}(r)$ (see Ref. \cite{sap02} for details).
The choice $x_{\alpha}=0$ corresponds to the Dirac-Hartree potential,
 $x_{\alpha}= 2/3$  gives the Kohn-Sham potential, and $x_{\alpha}= 1$
is the Dirac-Slater potential.
In Table~\ref{tab:alkali}, we present the results of our calculations
of the SE contribution to the Lamb
shift performed for $x_{\alpha}=0,\, 1/3,\, 2/3,\, 1$,
and related exact
data by Sapirstein and Cheng  \cite{sap02}.
Our data were obtained by averaging the model SE operator
$h^{\rm SE}$, given by Eq. (\ref{hse_fin}),
with the wave function of the valence state $v$ determined from the Dirac
equation with the potential $V_{\rm eff}(r)$. To demonstrate the importance of the nonlocal
part of $h^{\rm SE}$, in addition to the total $\langle v|h^{\rm SE}|v\rangle $ contribution,
we present also the local  $\langle v|V_{\rm loc}^{\rm SE}|v\rangle$
part. As one can see from the table,
 for all atoms the $\langle v|h^{\rm SE}|v\rangle $ values are in a good agreement
with the exact results, while the local potential
approximation,
 $\langle v|V_{\rm loc}^{\rm SE}|v\rangle$, works reasonably well only
for low-$Z$ systems.

In Ref. \cite{che06}, the QED corrections  to the transition energies
in Cu-like ions have been calculated in the Kohn-Sham potential. In Table~\ref{tab:cu},
we present the SE corrections to the $4s-4p_{1/2}$, $4s-4p_{3/2}$,
  $4p_{1/2}-4d_{3/2}$, $4p_{3/2}-4d_{3/2}$, and $4p_{3/2}-4d_{5/2}$
transition energies
 obtained by averaging the
model SE operator with the wave function of the valence electron.
This wave function was calculated by solving the KS equation
with the KS potential constructed self-consistently with the 4s state.
The comparison with the related exact results from Ref. \cite{che06}
is also given. It can be seen that the model SE operator results are
in a very good agreement with the exact results.

Experiments with superheavy elements have triggered a great interest
to calculations of the QED effects in superheavy atoms
\cite{pyy12,lab99,ind07,goi09,thi10}. In Ref. \cite{goi09},
the QED contributions to the binding energy
of the valence $7s$ electrons in Rg ($Z=111$) and Cn ($Z=112$)
have been evaluated in a local Dirac-Fock potential.
In Table~\ref{tab:comp}, we compare the related SE contributions
obtained using the model SE operator in the
Dirac-Fock method
with those
by Goidenko   \cite{goi09}. Despite the calculations
in the local DF potential
are not fully equivalent to the
calculations based on the DF equations,
 the SE contributions to the one-electron
binding energies obtained by averaging the model SE operator
with the DF wave function of the valence electron as well as
by including
this operator into the DF equations
are in a good agreement with the corresponding
Goidenko's results.
For comparison, we present also the total DF values which are
obtained as the difference between the SE contributions to
the total DF energies of the atom and the ion.
It is known \cite{eli94,eli95}
that for Rg and Cn the ionization
occurs out of the $6d_{5/2}$ shell instead of the $7s$ shell.
However, for comparison purposes, here we consider
 the  $^2S_{1/2}\rightarrow ^1S_0$ transition for Rg and
the $^1S_{0}\rightarrow ^2S_{1/2}$ transition for Cn.
The presented results are also in a reasonable agreement
 with those
based on the Welton method \cite{ind07}
and with the results obtained using a local SE
potential \cite{thi10}.
 In Table~\ref{tab:comp}, we give also the results of our
calculations of the SE corrections to the  binding energy
of the valence $8s$ electrons
in E119 and E120 and compare them
with the values obtained in Ref. \cite{thi10}.

Finally, we applied our model approach to the Li-like ions,
for which rigorous QED calculations have been performed.
The self-energy screening diagrams
for Li-like ions to first order in $1/Z$
were first evaluated in Ref. \cite{yer99}.
In that paper, the calculations were performed in the Coulomb potential.
 Later, the same diagrams have been calculated in the Kohn-Sham
and core-Hartree potentials
\cite{koz10,sap11}. In Table~\ref{tab:li}, we present the results of our calculations of
the screened SE corrections in Li-like ions,
based on the model SE operator approach,
and compare them with the related results from Refs. \cite{koz10,sap11}.
In our approach, the screened SE corrections  were obtained
by calculating the total
ion energy with the model SE operator included into the Dirac-Fock (DF)
or the  Kohn-Sham (KS)
equation and subtracting both the related energy evaluated
without the model SE operator and the
SE contribution evaluated with the H-like wave functions.
In case of the KS method,  the KS potential
was constructed self-consistently with the valence state
under consideration.
As one can see from the table, the model SE operator
results obtained employing the KS and DF equations
 are  in a fair agreement with
the results obtained by the perturbation theory
 \cite{koz10,sap11}.
Therefore,  to a good accuracy,
the total SE corrections
can be obtained by
summing  the H-like
SE contributions and
the screened SE corrections evaluated by solving
either DF or KS equations with the model SE operator
included, as described above.
\section{Conclusion}
In this paper we have developed the model QED operator approach
to calculations of the Lamb shifts in relativistic atomic systems.
With this method, we proposed the model self-energy operator
which is given by Eq. (\ref{hse_fin}). This operator can be easily
incorporated into any calculations employing the Dirac-Coulomb-Breit
Hamiltonian. This was demonstrated by calculating the Lamb shifts
in atoms and ions with the use of the model SE operator and comparing
the obtained results with corresponding exact QED calculations.

\section{Acknowledgments}
We thank Andrey Volotka
for providing
us with details of his calculations of the self-energy screening diagrams
performed in Ref. \cite{koz10}. Valuable discussions with Jan Derezinski,
Bogumil Jeziorski,
and Jonathan Sapirstein are gratefully acknowledged.
This work was supported by RFBR (Grants No. 13-02-00630   and  No.11-02-00943-a)
and by the Ministry of Education and Science of Russian Federation
(Grant No. 8420).
\begingroup
\squeezetable
\begin{table*}

\caption{Self-energy correction for $ns$ states. Labelling $(n,n')$ denotes the
  $F_{nn'}$ function defined by Eq.~(\ref{F}). $R$ is the root-mean-square
  charge radius of the nucleus (in fermi). For each $Z$, the upper line
  shows the point-nucleus result, whereas the lower line displays the
  extended-nucleus result (if different).
\label{tab:s}}
\begin{ruledtabular}
\begin{tabular}{lclllllllllllllll}
 $Z$ & $R$     &(1,1)                           &(1,2)                           &(1,3)                           &(1,4)                           &(1,5)                           &(2,2)                           &(2,3)                           &(2,4)                           &(2,5)                           &(3,3)                           &(3,4)                           &(3,5)                           &(4,4)                           &(4,5)                           &(5,5)                         \\ \colrule\\[-5pt]
 10 & 3.005  &   4.6542    &   4.7961    &   4.8145    &   4.8193    &   4.8210(1) &   4.8944    &   4.9325    &   4.9437    &   4.9480    &   4.9524    &   4.9677    &   4.9740    &   4.9749    &   4.9824    &   4.9858     \\[2pt]
 15 & 3.189  &   3.8014    &   3.9463    &   3.9639    &   3.9681    &   3.9693    &   4.0509    &   4.0885    &   4.0992    &   4.1030    &   4.1082    &   4.1229    &   4.1288    &   4.1296    &   4.1367    &   4.1396     \\
    &        &   3.8013    &   3.9462    &   3.9639    &   3.9680    &   3.9693    &   4.0508    &   4.0885    &   4.0991    &   4.1030    &   4.1082    &   4.1229    &   4.1287    &   4.1296    &   4.1366    &   4.1396     \\[2pt]
 20 & 3.476  &   3.2463    &   3.3946    &   3.4114    &   3.4148    &   3.4155    &   3.5066    &   3.5438    &   3.5538    &   3.5572    &   3.5633    &   3.5773    &   3.5826    &   3.5834    &   3.5899    &   3.5923     \\
    &        &   3.2462    &   3.3945    &   3.4113    &   3.4147    &   3.4154    &   3.5065    &   3.5437    &   3.5537    &   3.5571    &   3.5632    &   3.5772    &   3.5825    &   3.5833    &   3.5898    &   3.5922     \\[2pt]
 25 & 3.706  &   2.8501    &   3.0023    &   3.0183    &   3.0208    &   3.0209    &   3.1230    &   3.1597    &   3.1689    &   3.1716    &   3.1789    &   3.1922    &   3.1969    &   3.1975    &   3.2034    &   3.2052     \\
    &        &   2.8499    &   3.0022    &   3.0182    &   3.0207    &   3.0207    &   3.1228    &   3.1595    &   3.1688    &   3.1715    &   3.1787    &   3.1920    &   3.1967    &   3.1974    &   3.2032    &   3.2050     \\[2pt]
 30 & 3.929  &   2.5520    &   2.7087    &   2.7238    &   2.7253    &   2.7246    &   2.8388    &   2.8750    &   2.8835    &   2.8855    &   2.8940    &   2.9064    &   2.9105    &   2.9110    &   2.9162    &   2.9173     \\
    &        &   2.5518    &   2.7084    &   2.7235    &   2.7251    &   2.7244    &   2.8386    &   2.8748    &   2.8832    &   2.8853    &   2.8937    &   2.9062    &   2.9102    &   2.9108    &   2.9159    &   2.9171     \\[2pt]
 35 & 4.163  &   2.3200    &   2.4816    &   2.4958    &   2.4963    &   2.4949    &   2.6223    &   2.6580    &   2.6656    &   2.6669    &   2.6767    &   2.6882    &   2.6915    &   2.6919    &   2.6963    &   2.6967     \\
    &        &   2.3196    &   2.4812    &   2.4954    &   2.4960    &   2.4945    &   2.6220    &   2.6576    &   2.6652    &   2.6665    &   2.6763    &   2.6878    &   2.6911    &   2.6915    &   2.6959    &   2.6963     \\[2pt]
 40 & 4.270  &   2.1352    &   2.3024    &   2.3156    &   2.3151    &   2.3129    &   2.4548    &   2.4899    &   2.4965    &   2.4970    &   2.5083    &   2.5187    &   2.5213    &   2.5215    &   2.5250    &   2.5246     \\
    &        &   2.1347    &   2.3019    &   2.3151    &   2.3146    &   2.3124    &   2.4543    &   2.4893    &   2.4960    &   2.4965    &   2.5078    &   2.5182    &   2.5208    &   2.5210    &   2.5245    &   2.5241     \\[2pt]
 45 & 4.494  &   1.9859    &   2.1594    &   2.1716    &   2.1700    &   2.1668    &   2.3247    &   2.3591    &   2.3647    &   2.3643    &   2.3772    &   2.3865    &   2.3882    &   2.3882    &   2.3907    &   2.3894     \\
    &        &   1.9853    &   2.1587    &   2.1709    &   2.1692    &   2.1661    &   2.3239    &   2.3583    &   2.3639    &   2.3635    &   2.3764    &   2.3857    &   2.3874    &   2.3874    &   2.3900    &   2.3887     \\[2pt]
 50 & 4.654  &   1.8643    &   2.0448    &   2.0560    &   2.0531    &   2.0490    &   2.2243    &   2.2580    &   2.2625    &   2.2611    &   2.2757    &   2.2837    &   2.2844    &   2.2842    &   2.2857    &   2.2833     \\
    &        &   1.8633    &   2.0438    &   2.0550    &   2.0521    &   2.0480    &   2.2233    &   2.2569    &   2.2614    &   2.2600    &   2.2747    &   2.2827    &   2.2834    &   2.2832    &   2.2846    &   2.2823     \\[2pt]
 55 & 4.804  &   1.7648    &   1.9533    &   1.9635    &   1.9593    &   1.9542    &   2.1487    &   2.1816    &   2.1847    &   2.1822    &   2.1988    &   2.2054    &   2.2050    &   2.2045    &   2.2047    &   2.2012     \\
    &        &   1.7635    &   1.9519    &   1.9621    &   1.9579    &   1.9527    &   2.1472    &   2.1800    &   2.1832    &   2.1807    &   2.1973    &   2.2039    &   2.2035    &   2.2030    &   2.2032    &   2.1997     \\[2pt]
 60 & 4.912  &   1.6838    &   1.8815    &   1.8906    &   1.8848    &   1.8786    &   2.0945    &   2.1264    &   2.1281    &   2.1243    &   2.1431    &   2.1479    &   2.1463    &   2.1455    &   2.1443    &   2.1395     \\
    &        &   1.6820    &   1.8795    &   1.8886    &   1.8829    &   1.8766    &   2.0923    &   2.1242    &   2.1259    &   2.1222    &   2.1410    &   2.1458    &   2.1442    &   2.1434    &   2.1422    &   2.1374     \\[2pt]
 65 & 5.060  &   1.6186    &   1.8267    &   1.8346    &   1.8273    &   1.8197    &   2.0596    &   2.0904    &   2.0904    &   2.0852    &   2.1064    &   2.1092    &   2.1061    &   2.1049    &   2.1021    &   2.0958     \\
    &        &   1.6161    &   1.8239    &   1.8318    &   1.8245    &   1.8169    &   2.0564    &   2.0872    &   2.0872    &   2.0821    &   2.1033    &   2.1061    &   2.1031    &   2.1019    &   2.0991    &   2.0928     \\[2pt]
 70 & 5.311  &   1.5674    &   1.7876    &   1.7942    &   1.7850    &   1.7760    &   2.0429    &   2.0723    &   2.0703    &   2.0634    &   2.0874    &   2.0878    &   2.0831    &   2.0814    &   2.0767    &   2.0686     \\
    &        &   1.5637    &   1.7833    &   1.7900    &   1.7808    &   1.7719    &   2.0381    &   2.0675    &   2.0655    &   2.0587    &   2.0826    &   2.0832    &   2.0785    &   2.0768    &   2.0721    &   2.0641     \\[2pt]
 75 & 5.339  &   1.5290    &   1.7632    &   1.7683    &   1.7570    &   1.7464    &   2.0441    &   2.0718    &   2.0674    &   2.0586    &   2.0857    &   2.0834    &   2.0767    &   2.0744    &   2.0674    &   2.0573     \\
    &        &   1.5239    &   1.7572    &   1.7624    &   1.7512    &   1.7406    &   2.0373    &   2.0650    &   2.0607    &   2.0519    &   2.0790    &   2.0767    &   2.0701    &   2.0678    &   2.0610    &   2.0509     \\[2pt]
 80 & 5.463  &   1.5028    &   1.7533    &   1.7568    &   1.7431    &   1.7305    &   2.0639    &   2.0894    &   2.0822    &   2.0710    &   2.1018    &   2.0961    &   2.0870    &   2.0840    &   2.0744    &   2.0619     \\
    &        &   1.4955    &   1.7447    &   1.7482    &   1.7346    &   1.7221    &   2.0537    &   2.0793    &   2.0722    &   2.0611    &   2.0917    &   2.0862    &   2.0772    &   2.0743    &   2.0648    &   2.0524     \\[2pt]
 85 & 5.539  &   1.4888    &   1.7587    &   1.7602    &   1.7435    &   1.7287    &   2.1039    &   2.1266    &   2.1158    &   2.1018    &   2.1369    &   2.1271    &   2.1152    &   2.1113    &   2.0985    &   2.0831     \\
    &        &   1.4784    &   1.7461    &   1.7477    &   1.7312    &   1.7165    &   2.0886    &   2.1115    &   2.1009    &   2.0871    &   2.1219    &   2.1124    &   2.1006    &   2.0968    &   2.0842    &   2.0689     \\[2pt]
 90 & 5.710  &   1.4875    &   1.7806    &   1.7797    &   1.7595    &   1.7420    &   2.1669    &   2.1860    &   2.1708    &   2.1533    &   2.1936    &   2.1786    &   2.1631    &   2.1582    &   2.1415    &   2.1225     \\
    &        &   1.4721    &   1.7615    &   1.7607    &   1.7408    &   1.7236    &   2.1431    &   2.1625    &   2.1477    &   2.1305    &   2.1702    &   2.1557    &   2.1406    &   2.1357    &   2.1194    &   2.1007     \\[2pt]
 95 & 5.905  &   1.5005    &   1.8218    &   1.8176    &   1.7931    &   1.7723    &   2.2575    &   2.2719    &   2.2510    &   2.2291    &   2.2757    &   2.2543    &   2.2344    &   2.2280    &   2.2066    &   2.1831     \\
    &        &   1.4772    &   1.7921    &   1.7882    &   1.7642    &   1.7439    &   2.2197    &   2.2345    &   2.2143    &   2.1929    &   2.2386    &   2.2180    &   2.1987    &   2.1925(1) &   2.1717    &   2.1488     \\[2pt]
100 & 5.857  &   1.5302    &   1.8863    &   1.8779    &   1.8479    &   1.8230    &   2.3831    &   2.3910    &   2.3627    &   2.3351    &   2.3897    &   2.3600    &   2.3343    &   2.3262    &   2.2987    &   2.2697     \\
    &        &   1.4961    &   1.8417    &   1.8338    &   1.8047    &   1.7806    &   2.3247    &   2.3332    &   2.3061    &   2.2795    &   2.3325    &   2.3042    &   2.2795    &   2.2717(1) &   2.2453(1) &   2.2172     \\[2pt]
105 & 5.919  &   1.5809    &   1.9811    &   1.9670    &   1.9298    &   1.8996    &   2.5553    &   2.5543    &   2.5158    &   2.4807    &   2.5457    &   2.5051    &   2.4718    &   2.4615    &   2.4261    &   2.3899     \\
    &        &   1.5286    &   1.9107    &   1.8974    &   1.8619    &   1.8331    &   2.4602    &   2.4604    &   2.4243    &   2.3911    &   2.4531    &   2.4150    &   2.3836    &   2.3737    &   2.3403(1) &   2.3058     \\[2pt]
110 & 5.993  &   1.6601    &   2.1179    &   2.0956    &   2.0487    &   2.0114    &   2.7936    &   2.7797    &   2.7272    &   2.6819    &   2.7609    &   2.7055    &   2.6617    &   2.6484    &   2.6025    &   2.5566     \\
    &        &   1.5771    &   2.0031    &   1.9824    &   1.9388    &   1.9040    &   2.6334    &   2.6222    &   2.5743    &   2.5325    &   2.6058    &   2.5552    &   2.5151    &   2.5026(1) &   2.4604(2) &   2.4179     \\[2pt]
115 & 6.088  &   1.7811    &   2.3180    &   2.2832    &   2.2224    &   2.1751    &   3.1331    &   3.1001    &   3.0270    &   2.9668    &   3.0659    &   2.9893    &   2.9307    &   2.9131    &   2.8527    &   2.7933     \\
    &        &   1.6441    &   2.1228    &   2.0917    &   2.0375    &   1.9952    &   2.8519    &   2.8248    &   2.7610    &   2.7081    &   2.7958    &   2.7288    &   2.6775    &   2.6617(1) &   2.6084(1) &   2.5556     \\[2pt]
120 & 6.175  &   1.9709    &   2.6233    &   2.5685    &   2.4866(1) &   2.4243(1) &   3.6437    &   3.5804(1) &   3.4750(1) &   3.3921(1) &   3.5219(1) &   3.4131    &   3.3319    &   3.3080    &   3.2259    &   3.1464     \\
    &        &   1.7335(1) &   2.2753(1) &   2.2294(1) &   2.1613(1) &   2.1093(1) &   3.1256(1) &   3.0760(1) &   2.9909(1) &   2.9231(1) &   3.0295(1) &   2.9411(1) &   2.8750(1) &   2.8549(2) &   2.7875(1) &   2.7216     \\[2pt]
\end{tabular}
\end{ruledtabular}
\end{table*}
\endgroup

\begingroup
\squeezetable
\begin{table*}

\caption{Self-energy correction for $np_{1/2}$ states.
\label{tab:p1}}
\begin{ruledtabular}
\begin{tabular}{ll..........}
 $Z$ & $R$     &\multicolumn{1}{c}{(2,2)}       &\multicolumn{1}{c}{(2,3)}       &\multicolumn{1}{c}{(2,4)}       &\multicolumn{1}{c}{(2,5)}       &\multicolumn{1}{c}{(3,3)}       &\multicolumn{1}{c}{(3,4)}       &\multicolumn{1}{c}{(3,5)}       &\multicolumn{1}{c}{(4,4)}       &\multicolumn{1}{c}{(4,5)}       &\multicolumn{1}{c}{(5,5)}     \\ \colrule\\[-5pt]
 10 & 3.005  &  -0.11x48   &  -0.09x62   &  -0.09x46   &  -0.09x46   &  -0.10x20   &  -0.09x41   &  -0.09x23   &  -0.09x64(1)&  -0.09x21   &  -0.09x32    \\[2pt]
 15 & 3.189  &  -0.10x45   &  -0.08x51   &  -0.08x33   &  -0.08x31   &  -0.09x01   &  -0.08x18   &  -0.07x99   &  -0.08x37   &  -0.07x94   &  -0.08x03    \\[2pt]
 20 & 3.476  &  -0.09x25   &  -0.07x21   &  -0.06x99   &  -0.06x96   &  -0.07x60   &  -0.06x74   &  -0.06x53   &  -0.06x90   &  -0.06x44   &  -0.06x53    \\[2pt]
 25 & 3.706  &  -0.07x91   &  -0.05x76   &  -0.05x50   &  -0.05x45   &  -0.06x03   &  -0.05x12   &  -0.04x89   &  -0.05x25   &  -0.04x77   &  -0.04x84    \\[2pt]
 30 & 3.929  &  -0.06x43   &  -0.04x16   &  -0.03x86   &  -0.03x79   &  -0.04x31   &  -0.03x36   &  -0.03x10   &  -0.03x44   &  -0.02x95   &  -0.02x99    \\[2pt]
 35 & 4.163  &  -0.04x83   &  -0.02x42   &  -0.02x07   &  -0.01x99   &  -0.02x44   &  -0.01x44   &  -0.01x16   &  -0.01x48   &  -0.00x97   &  -0.00x99    \\[2pt]
 40 & 4.270  &  -0.03x10   &  -0.00x54   &  -0.00x14   &  -0.00x03   &  -0.00x42   &   0.00x63   &   0.00x94   &   0.00x63   &   0.01x17   &   0.01x16    \\[2pt]
 45 & 4.494  &  -0.01x23   &   0.01x50   &   0.01x96   &   0.02x08   &   0.01x77   &   0.02x87   &   0.03x21   &   0.02x91   &   0.03x47   &   0.03x47    \\[2pt]
 50 & 4.654  &   0.00x80   &   0.03x71   &   0.04x23   &   0.04x38   &   0.04x14   &   0.05x29   &   0.05x66   &   0.05x38   &   0.05x95   &   0.05x97    \\[2pt]
 55 & 4.804  &   0.03x03   &   0.06x14   &   0.06x72   &   0.06x89   &   0.06x72   &   0.07x93   &   0.08x33   &   0.08x05   &   0.08x64   &   0.08x67    \\[2pt]
 60 & 4.912  &   0.05x48   &   0.08x81   &   0.09x46   &   0.09x65   &   0.09x56   &   0.10x83   &   0.11x25   &   0.10x98   &   0.11x58   &   0.11x61    \\
    &        &   0.05x47   &   0.08x81   &   0.09x46   &   0.09x65   &   0.09x55   &   0.10x82   &   0.11x25   &   0.10x98   &   0.11x57   &   0.11x61    \\[2pt]
 65 & 5.060  &   0.08x20   &   0.11x78   &   0.12x50   &   0.12x71   &   0.12x71   &   0.14x02   &   0.14x48   &   0.14x21   &   0.14x81   &   0.14x85    \\
    &        &   0.08x19   &   0.11x77   &   0.12x49   &   0.12x70   &   0.12x70   &   0.14x01   &   0.14x47   &   0.14x20   &   0.14x80   &   0.14x84    \\[2pt]
 70 & 5.311  &   0.11x27   &   0.15x12   &   0.15x91   &   0.16x14   &   0.16x23   &   0.17x60   &   0.18x07   &   0.17x81   &   0.18x40   &   0.18x44    \\
    &        &   0.11x26   &   0.15x10   &   0.15x90   &   0.16x12   &   0.16x21   &   0.17x58   &   0.18x05   &   0.17x78   &   0.18x38   &   0.18x41    \\[2pt]
 75 & 5.339  &   0.14x77   &   0.18x91   &   0.19x79   &   0.20x02   &   0.20x23   &   0.21x64   &   0.22x13   &   0.21x86   &   0.22x44   &   0.22x46    \\
    &        &   0.14x74   &   0.18x88   &   0.19x75   &   0.19x99   &   0.20x20   &   0.21x60   &   0.22x09   &   0.21x82   &   0.22x40   &   0.22x42    \\[2pt]
 80 & 5.463  &   0.18x82   &   0.23x29   &   0.24x24   &   0.24x48   &   0.24x83   &   0.26x26   &   0.26x77   &   0.26x49   &   0.27x04   &   0.27x03    \\
    &        &   0.18x77   &   0.23x23   &   0.24x18   &   0.24x41   &   0.24x76   &   0.26x19   &   0.26x70   &   0.26x42   &   0.26x97   &   0.26x96    \\[2pt]
 85 & 5.539  &   0.23x59   &   0.28x43   &   0.29x44   &   0.29x67   &   0.30x20   &   0.31x65   &   0.32x15   &   0.31x85   &   0.32x36   &   0.32x30    \\
    &        &   0.23x49   &   0.28x32   &   0.29x33   &   0.29x55   &   0.30x08   &   0.31x52   &   0.32x02   &   0.31x73   &   0.32x23   &   0.32x17    \\[2pt]
 90 & 5.710  &   0.29x31   &   0.34x57   &   0.35x63   &   0.35x83   &   0.36x60   &   0.38x03   &   0.38x50   &   0.38x19   &   0.38x62   &   0.38x49    \\
    &        &   0.29x12   &   0.34x36   &   0.35x42   &   0.35x62   &   0.36x37   &   0.37x80   &   0.38x27   &   0.37x95   &   0.38x38   &   0.38x25    \\[2pt]
 95 & 5.905  &   0.36x33   &   0.42x05   &   0.43x15   &   0.43x29   &   0.44x38   &   0.45x75   &   0.46x16   &   0.45x83   &   0.46x13   &   0.45x89    \\
    &        &   0.35x97   &   0.41x66   &   0.42x75   &   0.42x88   &   0.43x94   &   0.45x31   &   0.45x72   &   0.45x38   &   0.45x69   &   0.45x44    \\[2pt]
100 & 5.857  &   0.45x17   &   0.51x43   &   0.52x51   &   0.52x54   &   0.54x08   &   0.55x33   &   0.55x63   &   0.55x25   &   0.55x38   &   0.54x98    \\
    &        &   0.44x50   &   0.50x70   &   0.51x77   &   0.51x80   &   0.53x27   &   0.54x52   &   0.54x82   &   0.54x43   &   0.54x56   &   0.54x16    \\[2pt]
105 & 5.919  &   0.56x69   &   0.63x55   &   0.64x54   &   0.64x38   &   0.66x55   &   0.67x58   &   0.67x68   &   0.67x25   &   0.67x11   &   0.66x45    \\
    &        &   0.55x37   &   0.62x11   &   0.63x08   &   0.62x94   &   0.64x98   &   0.66x00   &   0.66x10   &   0.65x66   &   0.65x52   &   0.64x87    \\[2pt]
110 & 5.993  &   0.72x31   &   0.79x84   &   0.80x59   &   0.80x12   &   0.83x22   &   0.83x85   &   0.83x61   &   0.83x11   &   0.82x54   &   0.81x50    \\
    &        &   0.69x61   &   0.76x89   &   0.77x63   &   0.77x19   &   0.80x02   &   0.80x64   &   0.80x43   &   0.79x89(1)&   0.79x36   &   0.78x34    \\[2pt]
115 & 6.088  &   0.94x68   &   1.02x92   &   1.03x14   &   1.02x12   &   1.06x68   &   1.06x59   &   1.05x75   &   1.05x13   &   1.03x88   &   1.02x22    \\
    &        &   0.88x81   &   0.96x56   &   0.96x81   &   0.95x87   &   0.99x83   &   0.99x77   &   0.99x02   &   0.98x34(1)&   0.97x19(1)&   0.95x61    \\[2pt]
120 & 6.175  &   1.29x26   &   1.38x16   &   1.37x26   &   1.35x22   &   1.42x19   &   1.40x74   &   1.38x81   &   1.37x98   &   1.35x57   &   1.32x86    \\
    &        &   1.15x59   &   1.23x52   &   1.22x80   &   1.21x02   &   1.26x58   &   1.25x32(1)&   1.23x69(1)&   1.22x75(2)&   1.20x64(2)&   1.18x19    \\[2pt]
\end{tabular}
\end{ruledtabular}
\end{table*}
\endgroup

\begingroup
\squeezetable
\begin{table*}

\caption{Self-energy correction for $np_{3/2}$ states.
\label{tab:p3}}
\begin{ruledtabular}
\begin{tabular}{ll..........}
 $Z$ & $R$     &\multicolumn{1}{c}{(2,2)}       &\multicolumn{1}{c}{(2,3)}       &\multicolumn{1}{c}{(2,4)}       &\multicolumn{1}{c}{(2,5)}       &\multicolumn{1}{c}{(3,3)}       &\multicolumn{1}{c}{(3,4)}       &\multicolumn{1}{c}{(3,5)}       &\multicolumn{1}{c}{(4,4)}       &\multicolumn{1}{c}{(4,5)}       &\multicolumn{1}{c}{(5,5)}     \\ \colrule\\[-5pt]
 10 & 3.005  &   0.13x04   &   0.13x35   &   0.13x16   &   0.13x01   &   0.14x21   &   0.14x54   &   0.14x57   &   0.14x74   &   0.14x96   &   0.15x03    \\[2pt]
 15 & 3.189  &   0.13x66   &   0.13x99   &   0.13x80   &   0.13x65   &   0.14x90   &   0.15x25   &   0.15x27   &   0.15x46   &   0.15x69   &   0.15x76    \\[2pt]
 20 & 3.476  &   0.14x38   &   0.14x73   &   0.14x55   &   0.14x40   &   0.15x72   &   0.16x07   &   0.16x10   &   0.16x30   &   0.16x54   &   0.16x62    \\[2pt]
 25 & 3.706  &   0.15x19   &   0.15x56   &   0.15x37   &   0.15x23   &   0.16x63   &   0.16x99   &   0.17x02   &   0.17x25   &   0.17x49   &   0.17x58    \\[2pt]
 30 & 3.929  &   0.16x06   &   0.16x46   &   0.16x27   &   0.16x12   &   0.17x61   &   0.17x99   &   0.18x03   &   0.18x27   &   0.18x52   &   0.18x61    \\[2pt]
 35 & 4.163  &   0.16x99   &   0.17x41   &   0.17x22   &   0.17x06   &   0.18x66   &   0.19x06   &   0.19x09   &   0.19x36   &   0.19x62   &   0.19x72    \\[2pt]
 40 & 4.270  &   0.17x96   &   0.18x41   &   0.18x21   &   0.18x05   &   0.19x77   &   0.20x19   &   0.20x22   &   0.20x52   &   0.20x78   &   0.20x90    \\[2pt]
 45 & 4.494  &   0.18x97   &   0.19x45   &   0.19x25   &   0.19x08   &   0.20x93   &   0.21x37   &   0.21x40   &   0.21x73   &   0.22x00   &   0.22x13    \\[2pt]
 50 & 4.654  &   0.20x01   &   0.20x53   &   0.20x32   &   0.20x15   &   0.22x14   &   0.22x60   &   0.22x64   &   0.22x99   &   0.23x28   &   0.23x41    \\[2pt]
 55 & 4.804  &   0.21x07   &   0.21x65   &   0.21x43   &   0.21x24   &   0.23x40   &   0.23x88   &   0.23x92   &   0.24x31   &   0.24x61   &   0.24x75    \\[2pt]
 60 & 4.912  &   0.22x16   &   0.22x79   &   0.22x56   &   0.22x37   &   0.24x71   &   0.25x21   &   0.25x25   &   0.25x68   &   0.25x99   &   0.26x15    \\[2pt]
 65 & 5.060  &   0.23x28   &   0.23x97   &   0.23x73   &   0.23x53   &   0.26x05   &   0.26x60   &   0.26x63   &   0.27x10   &   0.27x43   &   0.27x60    \\[2pt]
 70 & 5.311  &   0.24x41   &   0.25x17   &   0.24x93   &   0.24x72   &   0.27x45   &   0.28x03   &   0.28x07   &   0.28x58   &   0.28x92   &   0.29x10    \\[2pt]
 75 & 5.339  &   0.25x56   &   0.26x41   &   0.26x16   &   0.25x94   &   0.28x89   &   0.29x51   &   0.29x55   &   0.30x11   &   0.30x47   &   0.30x67    \\
    &        &   0.25x55   &   0.26x40   &   0.26x15   &   0.25x93   &   0.28x88   &   0.29x50   &   0.29x54   &   0.30x10   &   0.30x46   &   0.30x66    \\[2pt]
 80 & 5.463  &   0.26x72   &   0.27x67   &   0.27x42   &   0.27x18   &   0.30x38   &   0.31x05   &   0.31x09   &   0.31x71   &   0.32x08   &   0.32x30    \\
    &        &   0.26x71   &   0.27x66   &   0.27x41   &   0.27x17   &   0.30x36   &   0.31x03   &   0.31x07   &   0.31x69   &   0.32x07   &   0.32x28    \\[2pt]
 85 & 5.539  &   0.27x89   &   0.28x96   &   0.28x71   &   0.28x46   &   0.31x91   &   0.32x64   &   0.32x69   &   0.33x36   &   0.33x76   &   0.33x99    \\
    &        &   0.27x87   &   0.28x94   &   0.28x69   &   0.28x44   &   0.31x89   &   0.32x61   &   0.32x66   &   0.33x33   &   0.33x73   &   0.33x96    \\[2pt]
 90 & 5.710  &   0.29x07   &   0.30x28   &   0.30x03   &   0.29x77   &   0.33x50   &   0.34x28   &   0.34x34   &   0.35x07   &   0.35x50   &   0.35x75    \\
    &        &   0.29x04   &   0.30x24   &   0.30x00   &   0.29x73   &   0.33x46   &   0.34x24   &   0.34x30   &   0.35x03   &   0.35x45   &   0.35x70    \\[2pt]
 95 & 5.905  &   0.30x24   &   0.31x61   &   0.31x38   &   0.31x11   &   0.35x12   &   0.35x98   &   0.36x05   &   0.36x85   &   0.37x30   &   0.37x57    \\
    &        &   0.30x20   &   0.31x56   &   0.31x32   &   0.31x05   &   0.35x06   &   0.35x92   &   0.35x98   &   0.36x78   &   0.37x23   &   0.37x50    \\[2pt]
100 & 5.857  &   0.31x41   &   0.32x97   &   0.32x75   &   0.32x47   &   0.36x79   &   0.37x73   &   0.37x81   &   0.38x68   &   0.39x17   &   0.39x46    \\
    &        &   0.31x35   &   0.32x89   &   0.32x67   &   0.32x39   &   0.36x70   &   0.37x64   &   0.37x71   &   0.38x58   &   0.39x06   &   0.39x35    \\[2pt]
105 & 5.919  &   0.32x56   &   0.34x33   &   0.34x14   &   0.33x85   &   0.38x49   &   0.39x53   &   0.39x63   &   0.40x56   &   0.41x09   &   0.41x40    \\
    &        &   0.32x48(1)&   0.34x21   &   0.34x02   &   0.33x73   &   0.38x36   &   0.39x38   &   0.39x48   &   0.40x41   &   0.40x93   &   0.41x24    \\[2pt]
110 & 5.993  &   0.33x68   &   0.35x67   &   0.35x52   &   0.35x23   &   0.40x20   &   0.41x34   &   0.41x46   &   0.42x47   &   0.43x04   &   0.43x38    \\
    &        &   0.33x56(1)&   0.35x52   &   0.35x35   &   0.35x06   &   0.40x02   &   0.41x13   &   0.41x25   &   0.42x25   &   0.42x81   &   0.43x15    \\[2pt]
115 & 6.088  &   0.34x73   &   0.36x98   &   0.36x88   &   0.36x60   &   0.41x89   &   0.43x14   &   0.43x29   &   0.44x37   &   0.44x98   &   0.45x35(1) \\
    &        &   0.34x57(1)   &   0.36x76   &   0.36x65   &   0.36x35   &   0.41x63   &   0.42x85   &   0.43x00   &   0.44x07   &   0.44x67   &   0.45x02(1) \\[2pt]
120 & 6.175  &   0.35x67   &   0.38x19   &   0.38x16   &   0.37x89   &   0.43x48   &   0.44x84   &   0.45x04   &   0.46x17   &   0.46x83   &   0.47x22(1) \\
    &        &   0.35x48(1)   &   0.37x92   &   0.37x86   &   0.37x57   &   0.43x16(1)&   0.44x48   &   0.44x66   &   0.45x80   &   0.46x43   &   0.46x81(1) \\[2pt]
\end{tabular}
\end{ruledtabular}
\end{table*}
\endgroup

\begingroup
\squeezetable
\begin{table*}

\caption{Self-energy correction for $nd_{3/2}$ states.
\label{tab:d3}}
\begin{ruledtabular}
\begin{tabular}{ll..........}
 $Z$ & $R$     &\multicolumn{1}{c}{(3,3)}       &\multicolumn{1}{c}{(3,4)}       &\multicolumn{1}{c}{(3,5)}       &\multicolumn{1}{c}{(4,4)}       &\multicolumn{1}{c}{(4,5)}       &\multicolumn{1}{c}{(5,5)}     \\ \colrule\\[-5pt]
 10 & 3.005  &  -0.04x27(1)&  -0.03x52   &  -0.03x44(1)&  -0.04x07   &  -0.03x71   &  -0.03x95(1) \\[2pt]
 15 & 3.189  &  -0.04x24   &  -0.03x49   &  -0.03x41   &  -0.04x03   &  -0.03x68   &  -0.03x91    \\[2pt]
 20 & 3.476  &  -0.04x20   &  -0.03x45   &  -0.03x37   &  -0.03x99   &  -0.03x63   &  -0.03x87    \\[2pt]
 25 & 3.706  &  -0.04x15   &  -0.03x40   &  -0.03x32   &  -0.03x93   &  -0.03x58   &  -0.03x81    \\[2pt]
 30 & 3.929  &  -0.04x10   &  -0.03x34   &  -0.03x27   &  -0.03x87   &  -0.03x51   &  -0.03x74    \\[2pt]
 35 & 4.163  &  -0.04x04   &  -0.03x28   &  -0.03x20   &  -0.03x79   &  -0.03x43   &  -0.03x66    \\[2pt]
 40 & 4.270  &  -0.03x96   &  -0.03x20   &  -0.03x13   &  -0.03x71   &  -0.03x34   &  -0.03x56    \\[2pt]
 45 & 4.494  &  -0.03x88   &  -0.03x10   &  -0.03x04   &  -0.03x60   &  -0.03x23   &  -0.03x45    \\[2pt]
 50 & 4.654  &  -0.03x78   &  -0.03x00   &  -0.02x93   &  -0.03x48   &  -0.03x10   &  -0.03x31    \\[2pt]
 55 & 4.804  &  -0.03x66   &  -0.02x87   &  -0.02x80   &  -0.03x34   &  -0.02x95   &  -0.03x16    \\[2pt]
 60 & 4.912  &  -0.03x53   &  -0.02x72   &  -0.02x66   &  -0.03x17   &  -0.02x77   &  -0.02x98    \\[2pt]
 65 & 5.060  &  -0.03x38   &  -0.02x55   &  -0.02x49   &  -0.02x98   &  -0.02x57   &  -0.02x76    \\[2pt]
 70 & 5.311  &  -0.03x21   &  -0.02x36   &  -0.02x29   &  -0.02x76   &  -0.02x33   &  -0.02x52    \\[2pt]
 75 & 5.339  &  -0.03x02   &  -0.02x13   &  -0.02x06   &  -0.02x51   &  -0.02x06   &  -0.02x23    \\[2pt]
 80 & 5.463  &  -0.02x79   &  -0.01x87   &  -0.01x80   &  -0.02x21   &  -0.01x74   &  -0.01x90    \\[2pt]
 85 & 5.539  &  -0.02x54   &  -0.01x57   &  -0.01x49   &  -0.01x88   &  -0.01x37   &  -0.01x52    \\[2pt]
 90 & 5.710  &  -0.02x25   &  -0.01x23   &  -0.01x14   &  -0.01x49   &  -0.00x95   &  -0.01x08    \\[2pt]
 95 & 5.905  &  -0.01x92   &  -0.00x83   &  -0.00x73   &  -0.01x04   &  -0.00x46   &  -0.00x58    \\[2pt]
100 & 5.857  &  -0.01x54   &  -0.00x37   &  -0.00x26   &  -0.00x53   &   0.00x10   &   0.00x01    \\[2pt]
105 & 5.919  &  -0.01x11   &   0.00x15   &   0.00x29   &   0.00x06   &   0.00x74   &   0.00x68    \\
    &        &  -0.01x12   &   0.00x14   &   0.00x28   &   0.00x05   &   0.00x73   &   0.00x67    \\[2pt]
110 & 5.993  &  -0.00x62   &   0.00x74   &   0.00x91   &   0.00x74   &   0.01x48   &   0.01x45    \\
    &        &  -0.00x63   &   0.00x73   &   0.00x90   &   0.00x72   &   0.01x46   &   0.01x43    \\[2pt]
115 & 6.088  &  -0.00x07   &   0.01x42   &   0.01x63   &   0.01x51   &   0.02x32   &   0.02x32    \\
    &        &  -0.00x09   &   0.01x39   &   0.01x60   &   0.01x48   &   0.02x29   &   0.02x29    \\[2pt]
120 & 6.175  &   0.00x55   &   0.02x19   &   0.02x45   &   0.02x38   &   0.03x28   &   0.03x32(1) \\
    &        &   0.00x51(1)&   0.02x14   &   0.02x39   &   0.02x32   &   0.03x21   &   0.03x24(1) \\[2pt]
\end{tabular}
\end{ruledtabular}
\end{table*}
\endgroup

\begingroup
\squeezetable
\begin{table*}

\caption{Self-energy correction for $nd_{5/2}$ states.
\label{tab:d5}}
\begin{ruledtabular}
\begin{tabular}{ll..........}
 $Z$ & $R$     &\multicolumn{1}{c}{(3,3)}       &\multicolumn{1}{c}{(3,4)}       &\multicolumn{1}{c}{(3,5)}       &\multicolumn{1}{c}{(4,4)}       &\multicolumn{1}{c}{(4,5)}       &\multicolumn{1}{c}{(5,5)}     \\ \colrule\\[-5pt]
 10 & 3.005  &   0.04x08   &   0.03x77   &   0.03x54   &   0.04x28   &   0.04x24   &   0.04x40    \\[2pt]
 15 & 3.189  &   0.04x12   &   0.03x81   &   0.03x58   &   0.04x33   &   0.04x29   &   0.04x45    \\[2pt]
 20 & 3.476  &   0.04x17   &   0.03x86   &   0.03x63   &   0.04x40   &   0.04x35   &   0.04x52    \\[2pt]
 25 & 3.706  &   0.04x24   &   0.03x93   &   0.03x70   &   0.04x48   &   0.04x43   &   0.04x60    \\[2pt]
 30 & 3.929  &   0.04x32   &   0.04x01   &   0.03x77   &   0.04x57   &   0.04x53   &   0.04x70    \\[2pt]
 35 & 4.163  &   0.04x41   &   0.04x10   &   0.03x86   &   0.04x67   &   0.04x63   &   0.04x81    \\[2pt]
 40 & 4.270  &   0.04x52   &   0.04x20   &   0.03x96   &   0.04x79   &   0.04x75   &   0.04x94    \\[2pt]
 45 & 4.494  &   0.04x63   &   0.04x31   &   0.04x07   &   0.04x93   &   0.04x89   &   0.05x08    \\[2pt]
 50 & 4.654  &   0.04x75   &   0.04x43   &   0.04x19   &   0.05x07   &   0.05x04   &   0.05x24    \\[2pt]
 55 & 4.804  &   0.04x89   &   0.04x57   &   0.04x32   &   0.05x23   &   0.05x20   &   0.05x42    \\[2pt]
 60 & 4.912  &   0.05x03   &   0.04x72   &   0.04x46   &   0.05x41   &   0.05x38   &   0.05x60    \\[2pt]
 65 & 5.060  &   0.05x19   &   0.04x87   &   0.04x62   &   0.05x60   &   0.05x57   &   0.05x81    \\[2pt]
 70 & 5.311  &   0.05x36   &   0.05x04   &   0.04x78   &   0.05x80   &   0.05x78   &   0.06x02    \\[2pt]
 75 & 5.339  &   0.05x53   &   0.05x22   &   0.04x96   &   0.06x01   &   0.06x00   &   0.06x26    \\[2pt]
 80 & 5.463  &   0.05x72   &   0.05x41   &   0.05x14   &   0.06x24   &   0.06x23   &   0.06x50    \\[2pt]
 85 & 5.539  &   0.05x91   &   0.05x61   &   0.05x34   &   0.06x48   &   0.06x47   &   0.06x76    \\[2pt]
 90 & 5.710  &   0.06x12   &   0.05x82   &   0.05x54   &   0.06x73   &   0.06x73   &   0.07x04    \\[2pt]
 95 & 5.905  &   0.06x33   &   0.06x03   &   0.05x75   &   0.07x00   &   0.07x00   &   0.07x32    \\[2pt]
100 & 5.857  &   0.06x54   &   0.06x26   &   0.05x97   &   0.07x27   &   0.07x28   &   0.07x62    \\[2pt]
105 & 5.919  &   0.06x77   &   0.06x48   &   0.06x19   &   0.07x55   &   0.07x57   &   0.07x92    \\[2pt]
110 & 5.993  &   0.06x99   &   0.06x72   &   0.06x42   &   0.07x83   &   0.07x86   &   0.08x24    \\[2pt]
115 & 6.088  &   0.07x22   &   0.06x95   &   0.06x65   &   0.08x12   &   0.08x16   &   0.08x55(1) \\[2pt]
120 & 6.175  &   0.07x45   &   0.07x19   &   0.06x88   &   0.08x41   &   0.08x46   &   0.08x87(1) \\[2pt]
\end{tabular}
\end{ruledtabular}
\end{table*}
\endgroup

\begin{table}[ht]
\begin{center}
\vspace{1mm}
\caption{
The self energy function $F(\alpha Z)$, defined
 by $\Delta E^{\rm SE}=\frac{\alpha}{\pi}\frac{(\alpha Z)^4}{n^3}F(\alpha Z)mc^2$,
 for H-like ions. $\langle v|h^{\rm SE}|v\rangle$ denotes the results of the
 model SE operator approach, $\langle v|V_{\rm loc}^{\rm SE}|v\rangle$ is the contribution of
 the local part of the model SE operator,
and "Exact" labels the results of the exact calculation.
}
\label{tab:H-like}
\vspace{2mm}

\begin{tabular}{ccccc}
\hline \hline\\[-2mm]
$Z$ & State &   $\langle v|V_{\rm loc}^{\rm SE}|v\rangle$ &  $\langle v|h^{\rm SE}|v\rangle$  & Exact \\
\hline\\[-2mm]
10  & $4s$ &4.60 & 4.96 & 4.97  \\
    & $5s$ &4.59  & 4.96 & 4.99 \\
    & $5p_{1/2}$ &-0.15  & -0.10 & -0.09 \\
    & $5p_{3/2}$ &0.17  & 0.15 & 0.15 \\
    & $5d_{3/2}$ &-0.07  & -0.05 & -0.04 \\
    & $5d_{5/2}$ &0.07  & 0.05 & 0.04 \\
\hline\\[-2mm]
20  & $4s$ &3.11 & 3.57 & 3.58  \\
    & $5s$ &3.11  & 3.57 & 3.59 \\
    & $5p_{1/2}$ &-0.12  & -0.08 & -0.07 \\
    & $5p_{3/2}$ &0.18  & 0.16 & 0.17 \\
    & $5d_{3/2}$ &-0.07  & -0.05 & -0.04 \\
    & $5d_{5/2}$ &0.07  & 0.05 & 0.05 \\
\hline\\[-2mm]
40  & $4s$ &1.91 & 2.51 & 2.52  \\
    & $5s$ &1.90  & 2.50 & 2.52 \\
    & $5p_{1/2}$ &-0.04  & 0.00 & 0.01 \\
    & $5p_{3/2}$ &0.22  & 0.21 & 0.21 \\
    & $5d_{3/2}$ &-0.06  & -0.04 & -0.04 \\
    & $5d_{5/2}$ &0.07  & 0.06 & 0.05 \\
\hline\\[-2mm]
60  & $4s$ &1.46 &2.13 & 2.14  \\
    & $5s$ &1.44  & 2.12 & 2.14 \\
    & $5p_{1/2}$ &0.06  & 0.11 & 0.12 \\
    & $5p_{3/2}$ &0.26  & 0.26 & 0.26 \\
    & $5d_{3/2}$ &-0.05  & -0.04 & -0.03 \\
    & $5d_{5/2}$ &0.08  & 0.06 & 0.06 \\
\hline\\[-2mm]
83  & $4s$ &1.37 & 2.08 & 2.09  \\
    & $5s$ &1.34  & 2.05 & 2.06 \\
    & $5p_{1/2}$ &0.21  & 0.29 & 0.30 \\
    & $5p_{3/2}$ &0.30  & 0.33 & 0.33 \\
    & $5d_{3/2}$ &-0.04  & -0.03 & -0.02 \\
    & $5d_{5/2}$ &0.09  & 0.07 & 0.07 \\
\hline\\[-2mm]
92  & $4s$ &1.44 & 2.15 & 2.16  \\
    & $5s$ &1.40  & 2.10 & 2.12 \\
    & $5p_{1/2}$ &0.29  & 0.40 & 0.41 \\
    & $5p_{3/2}$ &0.32  & 0.36 & 0.36 \\
    & $5d_{3/2}$ &-0.03  & -0.02 & -0.01 \\
    & $5d_{5/2}$ &0.09  & 0.08 & 0.07 \\
\hline\\[-2mm]
\hline
\end{tabular}
\end{center}
\end{table}
\begin{table}[ht]
\begin{center}
\vspace{1mm}
\caption{ The self energy function $F(\alpha Z)$, defined
 by $\Delta E^{\rm SE}=\frac{\alpha}{\pi}\frac{(\alpha Z)^4}{n^3}F(\alpha Z)mc^2$,
 for neutral alkali metals
in different potentials.
$\langle v|h^{\rm SE}|v\rangle$ denotes the results of the
 model operator approach, calculated by averaging the model
 SE operator with the valence electron wave
 function in the corresponding potential.
$\langle v|V_{\rm loc}^{\rm SE}|v\rangle$ is the contribution of
 the local part of the model SE operator,
and "Exact" labels the results of the
 exact calculation of Ref. \cite{sap02}.
}
\label{tab:alkali}
\vspace{2mm}

\begin{tabular}{cccccc}
\hline \hline
&&&&&\\[-2mm]
Atom & Method  & \hspace{1mm}  $x_{\alpha}=0$   \hspace{1mm}
&   \hspace{1mm}  $x_{\alpha}=1/3$ \hspace{1mm}
&   \hspace{1mm}  $x_{\alpha}=2/3$ \hspace{1mm}
&   \hspace{1mm}  $x_{\alpha}=1$ \hspace{1mm}
   \\[0mm]
\hline
&&&&& \\[-1mm]
Na $3s_{1/2}$ & $\langle v|V_{\rm loc}^{\rm SE}|v\rangle$ &   0.166~ &   0.163~ &  0.176~  & 0.214~  \\[0mm]
              &  $\langle v|h^{\rm SE}|v\rangle$    &   0.170~ &   0.168~ &  0.183~  & 0.224~ \\[0mm]
              & Exact \cite{sap02}   & 0.169 & 0.167 & 0.181  & 0.223
               \\[2mm]
K  $4s_{1/2}$ &  $\langle v|V_{\rm loc}^{\rm SE}|v\rangle$  & 0.067~ & 0.067~  & 0.076~  & 0.100~  \\[0mm]
              &  $\langle v|h^{\rm SE}|v\rangle$  & 0.072~ & 0.072~  & 0.083~  & 0.110~  \\[0mm]
              &    Exact  \cite{sap02}    & 0.072 & 0.072 & 0.083 & 0.110
               \\[2mm]
Rb $5s_{1/2}$ &  $\langle v|V_{\rm loc}^{\rm SE}|v\rangle$ & 0.0187~ &  0.0193~  &  0.0230~  & 0.0320~
              \\[0mm]
              &  $\langle v|h^{\rm SE}|v\rangle$  & 0.0229~ &  0.0237~  &  0.0284~  & 0.0397~
               \\[0mm]
              &  Exact \cite{sap02}    & 0.0228 & 0.0236 & 0.0283 & 0.0396
               \\[2mm]
Cs $6s_{1/2}$ &  $\langle v|V_{\rm loc}^{\rm SE}|v\rangle$ &  0.0093~   & 0.0097~    & 0.0118~    & 0.0171~
               \\[0mm]
              &  $\langle v|h^{\rm SE}|v\rangle$  &  0.0127~   & 0.0132~    & 0.0163~    & 0.0236~
               \\[0mm]
              &  Exact \cite{sap02}    & 0.0126 & 0.0132 & 0.0162 & 0.0235
              \\[2mm]
Fr $7s_{1/2}$ &  $\langle v|V_{\rm loc}^{\rm SE}|v\rangle$ & 0.0047~ &   0.0052~  & 0.0067~ & 0.0102~ \\[0mm]
              &  $\langle v|h^{\rm SE}|v\rangle$  & 0.0069~ &   0.0076~  & 0.0099~ & 0.0151~  \\[0mm]
              &  Exact \cite{sap02}    & 0.0068 & 0.0075 & 0.0098 & 0.0150
              \\[2mm]
\hline
\hline
\end{tabular}
\end{center}
\end{table}
\begin{table}[ht]
\begin{center}
\vspace{1mm}
\caption{The self energy contribution to the $4s-4p_{1/2}$, $4s-4p_{3/2}$,
 $4p_{1/2}-4d_{3/2}$, $4p_{3/2}-4d_{3/2}$, and $4p_{3/2}-4d_{5/2}$
transition energies
in Cu-like ions, in eV.
$\langle v|h^{\rm SE}|v\rangle$ denotes the results of the
 model operator approach, calculated by averaging the
 model SE operator with the Dirac-Kohn-Sham wave
function of the valence electron.
"Exact" labels the results of the exact calculation of Ref. \cite{che06}.
}
\label{tab:cu}
\vspace{2mm}

\begin{tabular}{cccc}
\hline \hline\\[-2mm]
Ion & Transition &  $\langle v|h^{\rm SE}|v\rangle$  & Exact \cite{che06} \\
\hline\\[-2mm]
Yb$^{41+}$  & $4s-4p_{1/2}$  & -1.29 & -1.28  \\
            & $4s-4p_{3/2}$  & -1.21 & -1.21 \\
            & $4p_{1/2}-4d_{3/2}$  & -0.10 & -0.11 \\
            & $4p_{3/2}-4d_{3/2}$  & -0.18 & -0.18 \\
            & $4p_{3/2}-4d_{5/2}$  & -0.14 & -0.14 \\
\hline\\[-1.5mm]
W$^{45+}$  & $4s-4p_{1/2}$  & -1.64 &-1.64  \\
           & $4s-4p_{3/2}$  & -1.55 & -1.56 \\
            & $4p_{1/2}-4d_{3/2}$  & -0.16 & -0.17 \\
            & $4p_{3/2}-4d_{3/2}$  & -0.25 & -0.25 \\
            & $4p_{3/2}-4d_{5/2}$  & -0.19 & -0.19 \\

\hline\\[-1.5mm]
Os$^{47+}$  & $4s-4p_{1/2}$  & -1.85 &-1.84  \\
            & $4s-4p_{3/2}$  & -1.75 & -1.76 \\
            & $4p_{1/2}-4d_{3/2}$  & -0.19 & -0.20 \\
            & $4p_{3/2}-4d_{3/2}$  & -0.28 & -0.28 \\
            & $4p_{3/2}-4d_{5/2}$  & -0.22 & -0.22 \\

\hline\\[-1.5mm]
Au$^{50+}$  & $4s-4p_{1/2}$  & -2.18 &-2.18  \\
            & $4s-4p_{3/2}$  & -2.10 & -2.10 \\
            & $4p_{1/2}-4d_{3/2}$  & -0.26 & -0.28 \\
            & $4p_{3/2}-4d_{3/2}$  & -0.35 & -0.35 \\
            & $4p_{3/2}-4d_{5/2}$  & -0.27 & -0.28 \\
\hline\\[-1.5mm]
Pb$^{53+}$  & $4s-4p_{1/2}$  & -2.57 &-2.56  \\
            & $4s-4p_{3/2}$  & -2.49 & -2.50 \\
            & $4p_{1/2}-4d_{3/2}$  & -0.35 & -0.37 \\
            & $4p_{3/2}-4d_{3/2}$  & -0.43 & -0.43 \\
            & $4p_{3/2}-4d_{5/2}$  & -0.34 & -0.34 \\

\hline\\[-1.5mm]
Bi$^{54+}$  & $4s-4p_{1/2}$  & -2.71 &-2.70  \\
            & $4s-4p_{3/2}$  & -2.64 & -2.64 \\
            & $4p_{1/2}-4d_{3/2}$  & -0.39 & -0.40 \\
            & $4p_{3/2}-4d_{3/2}$  & -0.46 & -0.46 \\
            & $4p_{3/2}-4d_{5/2}$  & -0.36 & -0.37 \\

\hline\\[-1.5mm]
Th$^{61+}$  & $4s-4p_{1/2}$  & -3.85 &-3.85  \\
            & $4s-4p_{3/2}$  & -3.88 & -3.89 \\
            & $4p_{1/2}-4d_{3/2}$  & -0.73 & -0.74 \\
            & $4p_{3/2}-4d_{3/2}$  & -0.70 & -0.71 \\
            & $4p_{3/2}-4d_{5/2}$  & -0.56 & -0.57 \\

\hline\\[-1.5mm]
U$^{63+}$  & $4s-4p_{1/2}$  & -4.24 &-4.24  \\
           & $4s-4p_{3/2}$  & -4.32 & -4.33 \\
           & $4p_{1/2}-4d_{3/2}$  & -0.87 & -0.88 \\
            & $4p_{3/2}-4d_{3/2}$  & -0.79 & -0.79 \\
            & $4p_{3/2}-4d_{5/2}$  & -0.63 & -0.65 \\
\hline
\end{tabular}
\end{center}
\end{table}

\begin{table*}[ht]
\begin{center}
\vspace{1mm}
\caption{The self energy contribution to the binding energy of the valence
electrons in Rg, Cn, E119 and E120, in eV.
In this work, the perturbation theory (PT) value is obtained by
averaging the model SE operator with the Dirac-Fock wave
function of the valence electron, while the DF and total DF
values are obtained by including
this operator into the DF equations.
The DF value is given by the SE contribution to
the one-electron binding energy
  whereas the total DF value is obtained as
the difference between the SE contributions to
the total DF energies of the atom and the ion
 (the  $^2S_{1/2}\rightarrow ^1S_0$
and the $^1S_{0}\rightarrow ^2S_{1/2}$
transitions are considered for Rg and Cn, respectively;
see the text of the paper).
 In Refs. \cite{lab99,goi09}, the calculations
were performed with a local DF potential.
}
\label{tab:comp}
\vspace{2mm}

\begin{tabular}{cccccc}
\hline \hline
Atom & Valence electron & Method & This work & Ref. \cite{goi09} & Other works \\
\hline
Rg   & $7s$  & PT & -0.088 & -0.089 & -0.087$^a$\\
     &        & DF & -0.105 & -0.102 &  \\
     &        & Total DF & -0.096 &       \\
     &        & Welton method &  &  &  -0.084$^b$   \\
     &        & Local SE pot. &  & &  -0.089$^c$ \\
\hline
Cn   & $7s$  & PT & -0.101 & -0.103 &   \\
     &       & DF & -0.105 & -0.110 & \\
     &       & Total DF & -0.098 &       \\
     &        & Local SE pot. &  & &  -0.091$^c$ \\
\hline
E119   & $8s$ & PT  & -0.0233 & & -0.0274$^a$  \\
       &      & DF  & -0.0250 & &   \\
       &      & Total DF & -0.0232 & &     \\
       &      & Local SE pot. &  & &  -0.0210$^c$ \\
\hline
E120   & $8s$  & PT & -0.0331 &  \\
       &       & DF & -0.0265 &  \\
       &       & Total DF & -0.0250 & &    \\
       &       & Local SE pot. &  & & -0.0226$^c$ \\
\hline
\end{tabular}
\end{center}
$^a$Taken from Ref. \cite{lab99};
$^b$Taken from Ref. \cite{ind07};
$^c$Taken from Ref. \cite{thi10}.\\
\end{table*}

\begin{table}[ht]
\begin{center}
\vspace{1mm}
\caption{The screened self energy for the $2s$, $2p_{1/2}$, and $2p_{3/2}$ states
of Li-like ions, in eV. The Kohn-Sham (KS) and Dirac-Fock (DF) results
 are obtained by calculating the total
ion energy with the model SE operator included into the KS and DF
equations, respectively, and substructing the related energies
evaluated without the
model SE operator and the self energy contributions
 calculated with the hydrogenlike
wave functions.
 Comparison with the calculations performed
by perturbation theory (PT) in the  Kohn-Sham
potential \cite{koz10,sap11}
is given.
}
\label{tab:li}
\vspace{2mm}

\begin{tabular}{cccccc}
\hline \hline
$Z$ & State & KS  &  DF
&   PT \cite{koz10} &PT \cite{sap11}
   \\[0mm]
\hline
20  & \hspace{1mm} $2s$ & -0.047  & -0.045   &   -0.044  & -0.046    \\
    & \hspace{1mm} $2p_{1/2}$ & -0.009 &-0.008           &   -0.008 & -0.008   \\
    &  \hspace{1mm}  $2p_{3/2}$   & -0.012    & -0.011          & -0.013 &  -0.013    \\
\hline
40  & \hspace{1mm} $2s$ & -0.277  & -0.269   &   & -0.260    \\
    & \hspace{1mm} $2p_{1/2}$ & -0.063 &-0.059           &   & -0.059   \\
    &  \hspace{1mm}  $2p_{3/2}$   & -0.077    & -0.073        &  &  -0.085    \\
\hline
50  & \hspace{1mm} $2s$  & -0.50 & -0.49   &-0.48  \\
    & \hspace{1mm} $2p_{1/2}$    & -0.13 & -0.12           &   -0.12      \\
    &  \hspace{1mm}  $2p_{3/2}$   & -0.14       & -0.14          &   -0.16    \\
\hline
60  & \hspace{1mm} $2s$  & -0.84 & -0.82   &  & -0.80 \\
    & \hspace{1mm} $2p_{1/2}$  & -0.24 &    -0.24       &  & -0.25     \\
    &  \hspace{1mm}  $2p_{3/2}$   & -0.24       &-0.23    &      &  -0.27   \\
\hline
74  & \hspace{1mm} $2s$  & -1.58 &-1.55 &   &  -1.55  \\
    & \hspace{1mm} $2p_{1/2}$  & -0.57 &-0.56    &       &  -0.62      \\
    &  \hspace{1mm}  $2p_{3/2}$   & -0.45       & -0.43   &       & -0.53     \\
\hline
83  & \hspace{1mm} $2s$ & -2.35  & -2.25   &   -2.32  &   -2.26   \\
    & \hspace{1mm} $2p_{1/2}$       & -0.97   &-0.98           &     -1.07  &  -1.07    \\
    &  \hspace{1mm}  $2p_{3/2}$      &    -0.65   & -0.61          &  -0.75  &  -0.76 \\
\hline
92  & \hspace{1mm} $2s$ & -3.47  & -3.19   &     &   -3.81   \\
    & \hspace{1mm} $2p_{1/2}$       & -1.67   &-1.69           &    &  -1.58    \\
    &  \hspace{1mm}  $2p_{3/2}$      &    -0.91   & -0.86          &   &  -1.04 \\

\hline
\hline
\end{tabular}
\end{center}
\end{table}

redpajama_set_name: RedPajamaArXiv


Boys' Latin uses several software applications and web-based services which are hosted in the cloud provided by third-party vendors. These applications and services offer many educational advantages including flexibility for teachers to select the best resources to complement their curriculum, a wide range of Internet-based solutions and tools that can be quickly accessed anytime and anywhere, and increased opportunities for sharing, collaboration, and interactive learning.
In accordance with federal regulations included in the Children's Online Privacy Protection Act (COPPA), parental approval is required for Boys' Latin to continue to provide your son with the most effective web-based tools and applications for learning.
For our students to use these programs and services, certain personally identifying information, including the student's name and email address, must be provided to the website operator. Students will be able to maintain individual and personalized online access and files by providing this information to these operators. Importantly, Boys' Latin only chooses to partner with service providers that do not share personal identifying information. Under federal law, these websites must provide parental notification and obtain parental consent before collecting personal information from children under the age of 13. The law permits schools such as Boys' Latin to consent to the collection of personal information on behalf of all of its students, thereby eliminating the need for individual parental consent given directly to the website operator.


In 2017 Lucy Spoors is part of the Women's Eight to compete at World Cup II, World Cup III, and Henley Royal Regatta. Lucy has been part of the RNZ Elite team since 2014 and has recently made a successful transition from sculling to sweeping.
In 2016 she competed in the women's quad at the Final Qualification regatta in Lucerne. The crew finished third, and missed out on qualifying for the Rio 2016 Olympic Games by just three seconds. In 2015 Lucy raced the quad at the World Championships finishing 6th with the same crew make up as in 2014 where they finished 5th.
Lucy Spoors first represented New Zealand at the Junior World Rowing Championships in 2007 in the women's quad. In 2008 she again went to the Junior World Championship in the four winning gold.
In 2009 she made the transition to under-23s where she raced in the quad, and was again a member of this crew in 2010. In 2010 she also raced at the elite World Rowing Championships at Lake Karapiro in the women's four finishing 4th.
In 2011 Lucy returned to the quad at the under-23 level, and in 2012 had success winning bronze in the quad.
In 2014 Lucy gained selection in the elite women's quadruple scull alongside Erin-Monique Shelton, Georgia Perry and Sarah Gray, this crew then went onto win bronze at the second and third world cups held in Aiguebelette, France and Lucerne, Switzerland.


The Tibetan Yungdrung Bon Library was inaugurated by His Holiness the 14th Dalai Lama on the 21st April 2007. The Tibetan Yungdrung Bon Library is the culmination of His Holiness the 33rd Menri Trizin's vision for a religious study center at Menri Monastery. The library's aim is to have a comprahensive collection of important books and rare manuscripts on the Yungdrung Bon and Buddhism, history and culture of Tibet and the Zhang Zhung. Since 1969, His Holiness the 33rd Menri Trizin and members of the Bonpo community have collected Bonpo texts from around the world to be preserved in the Library. The Library now has an extensive collection of Bonpo Canonical texts and books and the Buddist Kangyur, Tengyur and commentaries about each of the four Buddhist schools. So far approximately thirty thousand Tibetan and Zhang Zhung texts and ten thousand books in other languag4s have been collected. The Library itself is a four-floor building in traditional Tibetan styles.


Potters Bar war memorial is located in St John's Churchyard in High Street, Potters Bar, England. The memorial was designed by the Arts and Crafts architect and designer C.F.A. Voysey. It has been Grade II listed on the National Heritage List for England since December 1973. Voysey's only other free standing war memorial, the Malvern Wells War Memorial, was erected in 1920 in Malvern Wells in Worcestershire.
The memorial is on the site of the former St. John's Church which was replaced by St. Mary's Church, The Walk, in 1915 after St. John's became dilapidated.
^ Historic England. "The War Memorial, St John's Churchyard (1174621)". National Heritage List for England. Retrieved 22 August 2016.
^ The story of Potters Bar and its church. St Mary the Virgin and All Saints. Retrieved 28 August 2016.
Wikimedia Commons has media related to Potters Bar war memorial.
This article about a Hertfordshire building or structure is a stub. You can help Wikipedia by expanding it.
This page was last edited on 13 April 2018, at 10:16 (UTC).


Roblox Free Download For Computer – Tag : roblox free download for computer, . Roblox Toys are modest collectible mini-figures and playsets predicated on many well known Roblox characters, developers and users. They're generated by the business Jazwares. Each toy includes a distinctive code which may be redeemed for a distinctive exclusive thing, based upon the toy you purchased, you might secure a great deal such as for instance a ring master, Shedletsky.
Players can get the toys from other stores and places. Such cases include Toys"R"Us, Walmart, CVS Pharmacy, Target and Amazon. Roblox also comes with a website on Jazwares which lists the different figurines and collections readily available, residences the code salvation platform, and lists where players could get the toys.
Every single Roblox Toy has the capabilities of having their roles blended with additional Roblox Toys. The toy arms are all fashioned in R 6 shape, while the legs have been organised in to R 15 shape. Users attempting to mimic the movement of their Roblox toys on their own avatar often make use of the toy cartoon bunch.
Though the toys were received with all entire positive reception, most customers criticized it saying they wore too much like Lego collectible figurines. Some accessories and figurines also have Roblox's old"R" emblem to them instead of the more modern square variant.


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text:
Sprankle named the 2019 Paula Milone-Nuzzo Nursing Alumni Volunteer of the Year
UNIVERSITY PARK, Pa. — Jennifer Sprankle, 1992 nursing, was selected as this year's Paula Milone-Nuzzo Nursing Alumni Volunteer of the Year. This award, established in 2018, is presented annually by the Penn State Nursing Alumni Society to an alumnus/alumna who has provided outstanding volunteer service to the College of Nursing. Volunteer service can be in the areas of mentoring students, board and council service, serving as a preceptor for students enrolled in any education program in the college, etc.
"True love and gratitude are what keeps Jennifer coming back for more when it comes to volunteering at Penn State in the College of Health and Human Development and College of Nursing. Your volunteerism is inspiring, and the College of Nursing Alumni Society would not have been the same without your wisdom and leadership to pave the way for future volunteers," said one of her nominators.
Sprankle has been a dedicated volunteer for the nursing program for 14 years, serving as president of the alumni society when the nursing program was housed in the College of Health and Human Development. Her years of experience on the HHD Board, serving as a member, committee chair, and society president were invaluable to the College of Nursing. Jennifer served on the College of Nursing Alumni Society Board during the probationary period and served as the first At Large Member of the College of Nursing Alumni Society's Executive Board.
"Jennifer's insight and experience as a volunteer leader were integral to me as I served as the co-chair of the Probationary Board and then as president of the College of Nursing Alumni Society Board," Sprankle's nominator added.
In addition to her bachelor's degree in nursing, Sprankle holds a master's degree in nursing from Villanova University.
Inaugural conference to address substance misuse held at Penn State
New research project to analyze cognitive training programs for older adults
Penn State promotions in academic rank, effective July 1, 2017
Health and Human Development, Nursing

redpajama_set_name: RedPajamaCommonCrawl


Role Based Access Control (RBAC) is the new permissions model in Microsoft Exchange Server 2010. With RBAC, you don't need to modify and manage access control lists (ACLs), which was done in Exchange Server 2007. ACLs created several challenges in Exchange 2007, such as modifying ACLs without causing unintended consequences, maintaining ACL modifications through upgrades, and troubleshooting problems that occurred due to using ACLs in a nonstandard way.
RBAC enables you to control, at both broad and granular levels, what administrators and end-users can do. RBAC also enables you to more closely align the roles you assign users and administrators to the actual roles they hold within your organization. In Exchange 2007, the server permissions model applied only to the administrators who managed the Exchange 2007 infrastructure. In Exchange 2010, RBAC now controls both the administrative tasks that can be performed and the extent to which users can now administer their own mailbox and distribution groups.
If you want to change what roles are assigned to role groups, you need to change the role assignments that link the role groups to roles. Unless the assignments built into Exchange 2010 don't suit your needs, you won't have to change these assignments. For more information, see Understanding Management Role Assignments.
Management role assignment policies associate end-user management roles to users. Role assignment policies consist of roles that control what a user can do with his or her mailbox or distribution groups. These roles don't allow management of features that aren't directly associated with the user. When you create a role assignment policy, you define everything a user can do with his or her mailbox. For example, a role assignment policy may allow a user to set the display name, set up voice mail, and configure Inbox rules. Another role assignment policy might allow a user to change the address, use text messaging, and set up distribution groups. Every user with an Exchange 2010 mailbox, including administrators, is given a role assignment policy by default. You can decide which role assignment policy should be assigned by default, choose what the default role assignment policy should include, override the default for certain mailboxes, or not assign role assignment policies by default at all.
Management role assignment policy The management role assignment policy is a special object in Exchange 2010. Users are associated with the role assignment policy when their mailboxes are created or if you change the role assignment policy on a mailbox. This is also what you assign end-user management roles to. The combination of all the roles on a role assignment policy defines everything that the user can manage on his or her mailbox or distribution groups.
If you want to change what roles are assigned to role assignment policies, you need to change the role assignments that link the role assignment policies to roles. Unless the assignments built into Exchange 2010 don't suit your needs, you won't have to change these assignments. For more information, see Understanding Management Role Assignments.
In addition to the Recipient Management - Vancouver custom role group, Jane also needs a role assignment policy to manage her own mailbox's configuration settings. The organization administrators have decided that all users, except for senior management, receive the same permissions when they manage their own mailboxes. They can configure their voice mail, set up retention policies and change their address information. The default role assignment policy provided with Exchange 2010 now reflects these requirements.
To give Jane the ability to manage her own mailbox settings, a role assignment policy needs to be configured with the required permissions.
All end-user roles are removed from the default role assignment policy, except for: MyBaseOptions, MyContactInformation, MyVoicemail and MyRetentionPolicies. MyBaseOptions is included because this management role provides the basic user functionality in Microsoft Office Outlook Web App, such as Inbox rules, calendar configuration, and other tasks.
For more information about customizing the default role assignment policy, see Change the Default Assignment Policy.
As with Jane, Joe also needs permissions to manage his own mailbox. He is also given the same permissions as Jane: He can set up his voice mail and retention policies, and change his address information.
For more information about adding a user to a role group, see Add Members to a Role Group.


The intense saga "Mere Apne" was a directorial debut of ever versatile Gulzar Sahab. The movie revolved around an old lady (Meena Kumari) who tries to be a peacemaker amongst two rival groups led by dashing Vinod Khanna (late) & Shatrughan Sinha.
The audience loved the street rivalry between these two lead actors.Vinod Khanna was portrayed more of a sophasticated amongst the two, whereas ShotGun was raw,brute and more violent.
The movie was packed by young guns such as Danny Denzongpa, Asrani, Paintal, Sudhir Thakkar and Dinesh Thakur(late) – all fresh from the FTII (Pune),working in tandem with lead characters - Shatrughan Sinha & Vinod Khanna.
Music was always the hallmark of this movie and it was Gulzar Sahab's masterstroke to have Salil Choudhury at the realm of things and the legend,himself, penning down the lyrics.
WorthITT (www.worthitt.com) is a portal for an artist (mainly those into performing arts) across all genres to showcase one's talent to the world.
1.	Is a "One Stop Point" for artists as well as talent scouts to collaborate and engage amongst like-minded associates.
2.	Act as a "CATALYST" between talent scouts and the artists.
3.	Is an "All in One" content GEO-INDEPENDENT & GENRE-INDEPENDENT portal – Portal to digitally create an artist's profile.
4.	A tool for all the upcoming artists to digitally share their content to the world – free of cost.
5.	Regular online contest - wherein artist can win prizes/accessories from esteemed jury.
6.	Opportunity to collaborate in the in-house content generation like albums, web-series.


text:
Apple AirPods 3 could get an all-new design — goodbye toothbrush stems
By Darragh Murphy
A potential look at the third-generation AirPods and case
(Image credit: Regan Coule/Laptop Magazine)
Apple's next-generation AirPods may be getting a design overhaul to look more like the AirPods Pro (one of the best wireless earbuds), according to leaked images showing off a shorter stem and smaller case of the third-gen wireless earbuds. You know what they say, the third generation's the charm.
As shared by Chinese site 52audio, the new design for Apple's rumoured AirPods 3, which are set to come out in 2021, is looking to lose the long stem of the previous generation, along with adopting silicone ear tips of the AirPods Pro. Well, that seems to fit the description of recent rumours.
Apple AirPods Pro 2 expected to drop in first half of 2021
Apple AirPods Pro review
Samsung Galaxy Buds Pro vs AirPods Pro
(Image credit: 52audio)
The third-gen AirPods are expected to look like the Pro in every way, and to even boast a longer battery life. However, the upcoming earbuds won't have certain advanced features, such as the Pro's popular noise cancellation. That makes sense, otherwise, it would just be a supposedly cheaper AirPods Pro.
That said, 52audio also states the new Airpods will have the AirPods Pro's pressure-relieving system, which equalizes pressure in the user's ear to make it more comfortable over long periods. As for spatial audio, the article points out the special Pro feature is "highly" likely for the AirPods 3, although that's just speculation.
As for the charging case, it also shares similarities to the AirPods Pro's design, with a more compact look along with an LED charging indicator.
While 52audio has been known to have false reports, including an image of the AirPods 3 shared back in November which was proven to be fake (as pointed out by MacRumors), these designs share many similarities with what the Apple rumour mill is churning out these days. However, like always, take it all with a pinch of salt.
The AirPods Pro 2 are also set to release alongside the Apple AirPods 3 in the first half of 2021, which will be cutting out the stems altogether, much like the Samsung Galaxy Buds. If these rumours prove to be true, then it might be time to lay off getting your hands on the 2019 AirPods Pro (even though they are pretty great).
Today's best Apple AirPods Pro deals
Darragh Murphy
Darragh Murphy is fascinated by all things bizarre, which usually leads to assorted coverage varying from washing machines designed for AirPods to the mischievous world of cyberattacks. Whether it's connecting Scar from The Lion King to two-factor authentication or turning his love for gadgets into a fabricated rap battle from 8 Mile, he believes there's always a quirky spin to be made. With a Master's degree in Magazine Journalism from The University of Sheffield, along with short stints at Kerrang! and Exposed Magazine, Darragh started his career writing about the tech industry at Time Out Dubai and ShortList Dubai, covering everything from the latest iPhone models and Huawei laptops to massive Esports events in the Middle East. Now, he can be found proudly diving into gaming, gadgets, and letting readers know the joys of docking stations for Laptop Mag.
The Witcher Remake: Everything we know so far
Xbox Game Pass just won me back — Hi-Fi Rush showcases why Xbox is special
The M2 Pro MacBook Pro is a beast — but let's talk about its mediocre aspects, too
10 secret Windows features you should be using


Exclusively at Casino Royale. The automated roulette table Karma™ from Spintech. Karma™ is an innovative electronic roulette game, designed around player comfort.
The energy inside Casino Royale was high as the final table of the inaugural Card Player Poker Tour St. Maarten main event played out to the sounds of the Mayweather-Pacquiao fight in the background.
[responsive][/responsive] Here are the lucky winners of our Valentine's Day Promo together with x104.3, your extreme party radio station in St Maarten. Winners received an all-inclusive weekend at Sonesta Maho Beach Resort.
Super Bowl 2015 at Casino Royale!
[responsive][/responsive] On Friday December 5, we had a happy winner at Casino Royale's Poker Room with a Royal Flush Jackpot that paid out $5000. Julien Levrat from France with a Royal Flush in Hearts was the winner of $3000 bonus and the rest of the table players shared $2000.
$5000 Jackpot in one of our slot machines!
[responsive][/responsive] Although she didn't win in our $100,000 Tournament, Suzanne Mikel was happy to take home a $5000 jackpot on one of our slots… 10 x 10 Pay!


Jennie Boden currently serves as Managing Director of Strategic Relationships and Senior Consultant at Quantum Governance. She brings to the position more than 25 years of leadership and management experience in the nonprofit sector. During her tenure at Quantum Governance, she has worked with clients as varied as Camphill Village, the Center for Arms Control and Non-Proliferation, Children of Faith Missions, the Credit Union Executives Society, the Friends of the National Arboretum and the Tipping Point Community.
Previously Jennie served as Executive Director of the Severna Park Community Center, a community-based nonprofit that enhances family life and individual development. During her tenure, SPCC grew its revenues from $1.1 million to more than $1.6 million in 16+ months. Prior, she served as the ED of the Maryland Coalition Against Sexual Assault, realigning its programs, developing its first strategic plan; re-building its Board; and managing nearly $1 million in state and federal grants.
At First Candle, Jennie served as the VP for the National Campaign for Cribs funded by the Bill & Melinda Gates Foundation and VP of Development – responsible for raising $23+ million in revenue. She negotiated corporate partnerships valued at $1+ million; oversaw special events with $500,000+ in annual net revenue; and launched a CSR program with 30 participating companies. Prior to joining First Candle, Jennie served as a Senior Director at the National Mental Health Association overseeing $3.5 million in corporate funds. She also served as the Director of External Relations for the Ethics Resource Center (ERC), where she worked with Fortune 500 companies and other national NGOs. She is the author of the 2001 Baltimore City Sexual Assault Response Team Annual Report and the ERC's Insights 2000. She has served as a member of the governor's grant review team for the Maryland Board of Victim Services and the Office on Crime Control & Prevention; as well as on the Boards of ASPIRE and the Patrick Allison House and the Advisory Board of the Indiana SIDS Alliance.
Ms. Boden received a BA from the University of California at Berkeley.
For more information on Jennie and the Quantum Governance team, contact her at: jennie@quantumgovernance.net.