deletions | additions       diff --git a/Casida, Tamm-Dancoff, and excited-state forces.tex b/Casida, Tamm-Dancoff, and excited-state forces.tex  index 5ca9839..0a56749 100644  --- a/Casida, Tamm-Dancoff, and excited-state forces.tex  +++ b/Casida, Tamm-Dancoff, and excited-state forces.tex 
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in which the lowest triplet state is lower in energy than the ground state \cite{Casida2009}.  The dipole matrix elements are now a superposition of the KS ones:  \begin{align}  \vec{d}_k = \sum_{cv} \vec{d}_{cv} \vec{d}_{cv}\,  x_{cv} \end{align}  When the wavefunctions are real, the full problem can be collapsed into a Hermitian one of the same 
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\end{align}  The dipole matrix elements are  \begin{align}  \vec{d}_k = \sum_{cv} x_{cv} x_{cv}\,  d_{cv} / \sqrt{\epsilon_c - \epsilon_c} \end{align}  An alternate approach for finding excitation energies is to look for many-body eigenstates