using non-equilibrium ionization of at least hydrogen, and ideally also helium . Computing the non-equilibrium ionization balance in post processing is not possible. Therefore, we chose to compute the electron density from non-LTE statistical equilibrium for the contribution from hydrogen, and in LTE from all other elements [since they add a minimal number of electrons to the total in the upper chromosphere]{}. The details for this procedure are given in Section \[sec:linerized\_eqs\].

Radiative transfer computations {#sec:radtrans}
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We numerically solve the non-LTE radiative transfer problem with the Multi3D code [@2009ASPC..415...87L] in a model atmosphere discretized on a Cartesian three-dimensional (3D) grid. The code treats one atom in non-LTE by simultaneously integrating the transfer equation and solving the statistical equilibrium equations for the atomic level populations. This is done iteratively using the multi-level accelerated $\Lambda$-iteration (M-ALI) method with preconditioned radiative rates following . The transfer equation is integrated at spectral points covered by radiative transitions of the atom using a domain-decomposed and parallelized method of short characteristics [e.g., @1987JQSRT..38..325O]. For angle integrations we use a quadrature with 24 rays [set A4 from @carlson1963]. The opacities and the source function are approximated along the characteristics using linear interpolation, which is the most 