deletions | additions       diff --git a/Casida, Tamm-Dancoff, and excited-state forces.tex b/Casida, Tamm-Dancoff, and excited-state forces.tex  index 2568993..d7f6034 100644  --- a/Casida, Tamm-Dancoff, and excited-state forces.tex  +++ b/Casida, Tamm-Dancoff, and excited-state forces.tex 
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%\begin{align}  %f_k = \frac{2 m \omega_k}{\hbar^2} \left| d_k \right|^2  %\end{align}  The simplest approximation to use is the random-phase approximation (RPA), in which the excitation energies are given by the differences of unoccupied and occupied Kohn-Sham KS  eigenvalues, $\omega_{c v \sigma} = \epsilon_c - \epsilon_v$. The corresponding dipole matrix elements are $\vec{d}_{cv} = \left< \varphi_c \left| \vec{r} \right| \varphi_v \right>$ \cite{Onida2002}. (As implemented in the code, this section will refer only to the case of a system without partially occupied levels.) The RPA is not a very satisfactory approximation however. The full solution within TDDFT is given by a non-Hermitian matrix eigenvalue equation, with a basis consisting of both occupied-unoccupied ($v \rightarrow c$) and unoccupied-occupied ($c \rightarrow v$) Kohn-Sham KS  transitions. \begin{align}  \left[ \begin{array}{c|c}  A & B \\ \hline