deletions | additions       diff --git a/Casida, Tamm-Dancoff, and excited-state forces.tex b/Casida, Tamm-Dancoff, and excited-state forces.tex  index 83da779..209ebfe 100644  --- a/Casida, Tamm-Dancoff, and excited-state forces.tex  +++ b/Casida, Tamm-Dancoff, and excited-state forces.tex 
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where $\hat{v}_{\rm c}$ is the Coulomb kernel, and $\hat{f}_{\rm xc}$ is the exchange-correlation kernel (currently only supported for LDA-type functionals in Octopus).  We do not solve the full equation in Octopus, but provide a hierarchy of approximations. An example calculation for the N$_2$ molecule with each theory level is shown in Table \ref{tab:nitrogen_casida}.  The lowest approximation we use is RPA. The next is the single-pole approximation of Petersilka \textit{et al.} \cite{Petersilka1996}, in which only the diagonal elements of the matrix are considered.  The eigenvectors are simply the KS transitions,  like Like  in the RPA case,as are the  the eigenvectors and  dipole matrix elements,  and elements are simply  the KS transitions. The  positive eigenvalues are $\omega_{cv} = \epsilon_c - \epsilon_v + A_{cvcv}$. This can be a reasonable approximation when there is little mixing between KS transitions,  but generally fails when there are degenerate or nearly degenerate transitions.