▁that ▁in ▁closed - shell ▁Hart ree – F ock ▁theory ▁( H F ), ▁the ▁first ▁ion ization ▁energy ▁of ▁a ▁mole cular ▁system ▁is ▁equal ▁to ▁the ▁negative ▁of ▁the ▁orb ital ▁energy ▁of ▁the ▁highest ▁occupied ▁mole cular ▁orb ital ▁( H OM O ). ▁This ▁theorem ▁is ▁named ▁after ▁T j alling ▁Ko op mans , ▁who ▁published ▁this ▁result ▁in ▁ 1 9 3 4 . ▁ ▁Ko op mans ' ▁theorem ▁is ▁exact ▁in ▁the ▁context ▁of ▁restricted ▁Hart ree – F ock ▁theory ▁if ▁it ▁is ▁assumed ▁that ▁the ▁orbit als ▁of ▁the ▁ion ▁are ▁identical ▁to ▁those ▁of ▁the ▁neutral ▁mole c ule ▁( the ▁fro zen ▁orb ital ▁approximation ). ▁I on ization ▁energ ies ▁calculated ▁this ▁way ▁are ▁in ▁qual itative ▁agreement ▁with ▁experiment   – ▁the ▁first ▁ion ization ▁energy ▁of ▁small ▁mole cules ▁is ▁often ▁calculated ▁with ▁an ▁error ▁of ▁less ▁than ▁two   elect ron ▁vol ts . ▁Therefore , ▁the ▁valid ity ▁of ▁Ko op mans ' ▁theorem ▁is ▁int imately ▁tied ▁to ▁the ▁accuracy ▁of ▁the ▁underlying ▁Hart ree – F ock ▁wave function . ▁The ▁two ▁main ▁sources ▁of ▁error ▁are ▁orb ital ▁relax ation , ▁which ▁refers ▁to ▁the ▁changes ▁in ▁the ▁F ock ▁operator ▁and ▁Hart ree – F ock ▁orbit als ▁when ▁changing ▁the ▁number ▁of ▁electrons ▁in ▁the ▁system , ▁and ▁electron ▁correlation , ▁referring ▁to ▁the ▁valid ity ▁of ▁representing ▁the ▁entire ▁many - body ▁wave function ▁using ▁the ▁Hart ree –