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1 s –2 p at 6.4 Ha. As the transition energy sweeps through each atomic transition, we see large contributions to the correlation energies at first, but then the transition is “exhausted” and the contributions become small again. 

The obvious difference between the three curves is that for the static case all the con- tributions are always negative, tend to be large in magnitude, and keep adding up to yield a large negative value. On the other hand, for the dynamic approximation and the exact result, the initial low-energy contributions are large and negative, may then become slightly positive, but rapidly become small in magnitude. To understand this difference, we write the modified screened interaction as W$_{t}$ ( ω ) = ε$_{t}$ ( ω ) − $^{1}$V in terms of a modified dielectric function ε$_{t}$ ( ω ). Solving the RPA/Casida Eq. (36) with transition t missing gives us a set modified plasma modes with energies ˜ ω$_{p}$ and mode functions | ˜ p 〉 . Based on Eq. (35), we 

