 angular momentum projection $M$. The DCB Hamiltonian is conventionally
defined by
\begin{eqnarray}
    H_{\rm DCB} = \sum_i h_{\rm D}(i) + \sum_{i<j} \left[ V_{C}(i,j)+
    V_{B}(i,j)\right]\,,
\end{eqnarray}
where the indices $i,j = 1,\ldots,N$ numerate the electrons, $h_D$ is the one-particle
Dirac-Coulomb Hamiltonian, and $V_{C}$ and $V_B$ are the Coulomb and the frequency-independent
Breit parts of the electron-electron interaction, see, e.g., \cite{johnson:book} for details. It is
assumed that $H_{\rm DCB}$ acts in the space of the wave functions constructed from the
positive-energy eigenfunctions of some one-particle Dirac Hamiltonian (the so-called no-pair
approximation).

In the CI method, the eigenfunctions $\Psi(PJM)$ of Eq.~(\ref{eql1}) are represented by a (finite)
sum of the configuration-state functions (CSFs) with the same $P$, $J$, and $M$,
\begin{equation}\label{eq4}
  \Psi(PJM) = \sum_r c_r \Phi(\gamma_r PJM)\,,
\end{equation}
where $\gamma_r$ denotes the set of additional quantum numbers that determine the CSF. The CSFs are
constructed as linear combinations of antisymmetrized products of one-electron orbitals $\psi_n$.
In the present work, the one-electron orbitals $\psi_n$ are solutions of the frozen-core Dirac-Fock
equation.

The energy of the system is given by one of the roots of the secular equation
\begin{eqnarray}
    {\rm det} \bigl\{\langle \gamma_r PJM|H_{\rm DCB}|\gamma_s PJM\rangle -E_r\,\delta_{rs}\bigr\} =
    0\,,
\end{eqnarray}
where ``det'' denotes the determinant of the matrix. The elements of the Hamiltonian matrix are
represented as linear combinations of one- and two-particle radial integrals (see, e.g.,
\cite{johnson:book}),
\begin{align}\label{eq7}
\langle \gamma_r PJM| &\,  H_{\rm DCB}|\gamma_s PJM\rangle = \sum_{ab}
 d_{rs}(ab)\,I(ab)
  \nonumber \\
 + &\, \alpha \sum_k \sum_{abcd} v_{rs}^{(k)}(abcd)\,
  R_{k}(abcd)\,.
\end{align}
Here, $a$, $b$, $c$, and $d$ numerate the one-electron orbitals, $d_{rs}$ and $v^{(k)}_{rs}$ are
the angular coefficients \cite{grant:73:cpc,grant:76:jpb,gaigalas:01:cpc,gaigalas:02:cpc}, $I(ab)$
are the one-electron radial integrals, and $R_{k}(abcd)$ are the relativistic generalization of
Slater radial integrals \cite{johnson:88:b}.

Our implementation of the CI method has been described in the previous papers
\cite{yerokhin:08:pra:ra,yerokhin:08:pra,yerokhin:12:lilike}. We shall therefore discuss here only
those issues that are new to the present calculation. The first difficulty arises in the
calculation of the core-excited states, whose energies are well above the autoionization threshold,
i.e., above the continuum of the valence-excited states. For a large basis set we are using here,
the atomic states of interest turn out to be very far away from the lowest eigenvalue of the
Hamiltonian matrix. The numerical approach we were using previously for determining the eigenvalues
of a large matrix (the implementation of the Davidson algorithm by Stathopoulos and Froese Fischer
\cite{stathopoulos:94:cpc}) was suitable for the computation of just the lowest (highest) matrix
eigenvalues. In the present work, we use the Jacobi-Davidson algorithm as implemented within the
JDQZ package \cite{JDQZ,fokkema:98}. The JDQZ package, although significantly slower than the one
by Stathopoulos and Froese Fischer, was able to provide us with eigenvalues and the corresponding
eigenvectors around an arbitrary energy target far from the lowest eigenvalue.

Another new feature of the present calculations is the identification of computed levels in terms
of the nonrelativistic $LS$ coupling scheme. In our relativistic CI calculations we use the $jj$
coupling scheme, which is natural in the relativistic case. In order to compare the computed levels
with experiments and previous nonrelativistic calculations, we had to identify our calculated
levels within the $LS$ coupling scheme. In the case of the core-exited states of beryllium-like
ions such identification is not straightforward (as it was in the case of lithium-like ions),
because of the large number and high density of levels. In order to identify the levels, we
calculated the expectation values of the squares of the orbital momentum operator $L^2$ and the
spin operator $S^2$ with the eigenfunctions of the DCB Hamiltonian. The matrix elements of the
$L^2$ and $S^2$ operators are obtained as
\begin{align}
\langle \gamma_r PJM| &\,  L^2 |\gamma_s PJM\rangle = \sum_{ab}
 d_{rs}(ab)\,I^{(L)}(ab)
  \nonumber \\
 + &\, \sum_{abcd} v_{rs}^{(1)}(abcd)\,
  R^{(L)}(abcd)\,,
\end{align}
and
\begin{align}
\langle \gamma_r PJM| &\,  S^2 |\gamma_s PJM\rangle = \frac34\, \sum_{a}
 d_{rs}(aa)
  \nonumber \\
 + &\, \sum_{abcd} v_{rs}^{(1)}(abcd)\,
  R^{(S)}(abcd)\,,
\end{align}
where the angular coefficients $d_{rs}(ab)$ and $v_{rs}^{(k)}(abcd)$ are the same as in
Eq.~(\ref{eq7}) and the radial integrals $I^{(L)}(ab)$ and $R^{(L,S)}(abcd)$ are presented in
Appendix.

An essential part of the present calculation is the systematic estimation of the uncertainties of
the obtained theoretical predictions. For each atomic state of interest, we perform our CI
calculations with many (typically, about 20) different sets of configuration-state functions. From
these computations, we then deduce an estimate of how well our CI results were converged, by
analyzing the successive increments of the results obtained with the basis set being increased in
various directions.

\subsection{QED effects}

The QED effects are calculated in the present work by means of two different approaches. By
comparing the results from these approaches, we estimate the uncertainty of our treatment. The
first method is based on summing up the self-energy and vacuum-polarization QED corrections
calculated for each one-electron orbital in an effective screening potential. The total QED
correction for a given many-electron state is obtained by adding the QED contributions from all
one-electron orbitals, weighted by their fractional occupation numbers as obtained from the
eigenvectors of the CI calculation,
\begin{align}
\delta E_{\rm QED} = \sum_a q_a\,\bigl[\langle a| \Sigma_{\rm SE}(\varepsilon_a)|a\rangle
+  \langle a| V_{\rm VP} |a\rangle  \bigr]\,.
\end{align}
Here, the index $a$ runs over all one-electron orbitals contributing to the given many-electron
state, $q_a$ is the occupation number of the one-electron orbital, $\Sigma_{\rm SE}$ is the
self-energy operator, $\varepsilon_a$ is the Dirac energy of the one-electron state $a$, and $V_{\rm VP}$
is the vacuum polarization potential. This approximate treatment of QED corrections was used in our
previous work on lithium-like ions \cite{yerokhin:12:lilike} and similarly also by other authors,
in particular, for beryllium-like ions by Chen and Cheng \cite{chen:97}.

Our second method of evaluating the QED effects is based on the model QED operator $h^{\rm QED}$
formulated recently by Shabaev et al.~\cite{shabaev:13:qedmod} and implemented in the QEDMOD
Fortran package \cite{shabaev:14:qedmod}. The QEDMOD package is a tool that efficiently calculates
matrix elements of $h^{\rm QED}$ with the (bound-state) one-electron wave functions. In the present
work, we add the model Q