deletions | additions       diff --git a/Casida, Tamm-Dancoff, and excited-state forces.tex b/Casida, Tamm-Dancoff, and excited-state forces.tex  index e14bb4d..793118a 100644  --- a/Casida, Tamm-Dancoff, and excited-state forces.tex  +++ b/Casida, Tamm-Dancoff, and excited-state forces.tex 
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\section{Linear response in the electron-hole basis}  An alternate approach to linear response is not to solve for the response function but rather for its poles (the poles, the  excitation energies $\omega_k$) $\omega_k$,  and electric dipole matrix elements $\vec{d}_k$. The polarizability is given by \begin{align}  \alpha_{ij} \left( \omega \right) = \sum_k \left[ \frac{\left( \hat{i} \cdot \vec{d}_k \right)^{*} \left( \hat{j} \cdot \vec{d}_k \right)}{\omega_k - \omega - i \delta}  + \frac{\left( \hat{i} \cdot \vec{d}_k \right)^{*} \left( \hat{j} \cdot \vec{d}_k \right)}{\omega_k + \omega + i \delta} \right] 
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% cite Petersilka  in which only the diagonal elements of the matrix are considered.  The eigenvectors are simply the KS transitions,  as like  in the  RPA (so case, as are the  the dipole matrix elements are the same as in RPA), elements,  and the positive eigenvalues are $\omega_{cv} = \epsilon_c - \epsilon_v + A_{cv}$.  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. 
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\left< \varphi_{c'} \varphi_{v'} \right| v \left| \varphi_c \varphi_v \right>  = \int \varphi_{c'} \left( r \right) \varphi_{v'} \left( r \right) P \left[ \varphi_c \varphi_v \right] dr  \end{align}  Our basic parallelization strategy for computation of the matrix elements is by domains, as for Octopus discussed  in general, section~\ref{sec:parallelization},  but wecan  add an additional level of parallelization here over occupied-unoccupied pairs. We distribute the columns of the matrix, and do not distribute  the rows, to avoid duplication of Poisson solves. We can reduce the number of matrix elements to be computed by  almost half using the Hermitian nature of the matrix, \textit{i.e.} $M_{cv,c'v'} = M_{c'v',cv}^{*}$. If there are $N$