deletions | additions       diff --git a/The_kink_we_found_at__.tex b/The_kink_we_found_at__.tex  index e3abcdd..cceb282 100644  --- a/The_kink_we_found_at__.tex  +++ b/The_kink_we_found_at__.tex 
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that electron (band) effective mass close to $1$ and that $k_\text{F} = 0.72/a_0$ for the effective homogeneous electron gas of $\mathrm{Cu}$ $\mathrm{s}$-electrons \cite{Ashcroft_2003}, we can derive an approximate value of $v_\text{kink}$ caused by the participation of $\mathrm{d}$-electrons.  Based in this DFT ground state density of states plus conservation laws, we obtain an estimate of $v_\text{kink} = \Delta/(2\hbar k_\text{F}) = 0.41~\mathrm{a.u.}$ in near agreement with our TDDFT prediction.  In reality, the $\mathrm{d}$-band is about $\Delta_\text{exp} = 2~\mathrm{eV}$ below the Fermi energy as indicated by ARPES ~\cite{Knapp_1979}, which means that both the DFT-based estimate and the full TDDFT result should be giving an underestimation of 25\% of the kink location.  The second (negative) kink at $v = 0.3~\mathrm{a.u.}$ is more difficult to explain precisely as the qualitative description in terms of $k_\text{F}$ (as in the homogeneous electron gas) becomes more ambiguous, but it is related to the point at which the whole conduction band ($11$ $\mathrm{s} `$\mathrm{s}  + \mathrm{d}$ \mathrm{d}$'  electrons) starts participating in the process.