&$-0.124$ &$-0.200$\
MCDHF & $-241$ & $-5061$ & $-6 \times 10^{-4}$ & $2.35\times{10^{-4}}$ & & &\
\
CI+MBPT, Ref. [@Dzuba2016a] & $-240$ & $-4762$ & & & &&\
[MCDHF]{}, Ref [@Petrasiunas2012] & $-304$ &$-3680$ & & & & &\

Electron correlation effects {#Sec:electron-correlation-eff}
----------------------------

We employ the multi-configuration Dirac-Hartree-Fock (MCDHF) method [@Grant2007; @FroeseFischer2016a] to capture the main electron correlations in the Yb$^+$ ion. In this approach, an atomic state wave-function (ASF) is represented as a linear combination of configuration state functions (CSFs) with the same parity, total angular momentum, and its component along the quantization axis. The CSFs are generated by single (S) and double (D) substitutions of orbitals occupied in the reference configurations with virtual orbitals. The reference configurations constitute the dominant CSFs of the ASF concerned. The MCDHF calculation starts from the optimization on occupied orbitals in the reference configurations. By contrast to Sec \[Sec:electronic-reduced-ml\], all of these orbitals are generated in the self-consistent field procedure. Virtual orbitals are augmented layer by layer in order to monitor the convergence of level energies and other atomic properties. Each layer includes orbitals with different angular symmetries. In addition, only the virtual orbitals in the latest added layer are variable. The 