deletions | additions       diff --git a/The_kink_we_found_at__.tex b/The_kink_we_found_at__.tex  index 4fd45f4..783b01e 100644  --- a/The_kink_we_found_at__.tex  +++ b/The_kink_we_found_at__.tex 
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Taking into account that DFT band structure predicts that the $\mathrm{d}$-band edge is $\Delta_\text{DFT} = 1.6~\mathrm{eV}$ below the Fermi energy (see for example, Fig.~3(a) in Ref.~\cite{Lin_2008}),   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 a value of th $v_\text{kink}$ 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 qualitative agreement with the 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}, ~\cite{Knapp_1979},  that means that both the DFT-based estimate and the 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 $\mathrm{s} + \mathrm{d}$ electrons) starts participating in the process.