deletions | additions       diff --git a/Casida, Tamm-Dancoff, and excited-state forces.tex b/Casida, Tamm-Dancoff, and excited-state forces.tex  index c576eae..83da779 100644  --- a/Casida, Tamm-Dancoff, and excited-state forces.tex  +++ b/Casida, Tamm-Dancoff, and excited-state forces.tex 
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Using the result of a calculation of excited states by one of these methods, and a previous calculation  of vibrational modes with the Sternheimer equation, we can compute forces in each excited state, which can be used   for excited-state structural relaxation or molecular dynamics \cite{Strubbe_forces}. Our formulation allows us to do this without introducing any extra summations over empty states as in states, unlike  previous force  implementations \cite{Sitt2007,Tsukagoshi2012,Hutter2003}. The energy of a given excited state $k$ is a sum of the ground-state energy and the excitation energy: $E_k = E_0 + \omega_k$.  The force is then given by the ground-state force, minus the derivative of the excitation energy:  \begin{align}