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 TABLE III: Various components of the total energy of the boron atom. Results are reported for single-particle eigenfunctions and eigenenergies coming from self-consistent LSDA or HF calcula- tions. All energies are in Hartree. Except for the (*) values, all calculations use a radial grid of size r$_{max}$ = 40 Bohr radii and N$_{max}$ = 400 single-particle states for each angular momentum channel. The starred (*) values are difficult to converge, and the values in the table are actually uncertain to within ± 0.01: see text for details. For reference, the exact (CI) non-relativistic correlation energy is also shown as the last entry. Energy component Equation LSDA HF Difference Kinetic 24.173 24.530 1.5% Electron-ion -56.520 -56.900 0.67% Hartree 11.534 11.590 0.49% Fock exchange (22) -3.712 -3.749 1.0% Φ$_{c}$: W (0) instead of W$_{t}$ (0) (50) -0.869 -0.717 21% Φ$_{c}$: square root approx. (49) -0.820 -0.680 21% Φ$_{c}$: static approx. (46) -0.835 -0.692 20% Φ$_{c}$: main approx. (45) -0.30* -0.11* 270% Φ$_{c}$: RPA (38) -0.217 -0.171 27% Φ$_{c}$: CI (exact) Ref. [68] -0.125 33