Contrary to our intuition, level-shifting has significantly worse performance than CPCM, with only 4 out of 25 peptides (1uao, 2ceh, 2jof, 3fva) showing non-trivial reduction (more than 50%) of DIIS iterations (Figure ). Moreover, 3vfa and 2jof yield unphysical negative HOMO-LUMO gaps with level-shifting, and their energies are also 0.11 eV and 2.80 eV higher than their gasphase result, indicating that level-shifting may cause the electronic structure converges to a different state (Figure ). Therefore, CPCM performs better in some tricky molecules that cannot be handled by level-shifting. CPCM's acceleration mechanism cannot be explained in the same way as level-shifting by merely opening the HOMO-LUMO gap. This motivates us to investigate the different mechanisms in CPCM vs. level shifting in opening the gap in Section III C.