▁theory ▁ ▁K ohn – Sh am ▁( KS ) ▁density ▁functional ▁theory ▁( KS - DF T ) ▁adm its ▁its ▁own ▁version ▁of ▁Ko op mans ' ▁theorem ▁( s ometimes ▁called ▁the ▁D FT - K oop mans ' ▁theorem ) ▁very ▁similar ▁in ▁spirit ▁to ▁that ▁of ▁Hart ree - F ock ▁theory . ▁The ▁theorem ▁equ ates ▁the ▁first ▁( vertical ) ▁ion ization ▁energy ▁ ▁of ▁a ▁system ▁of ▁▁ ▁electrons ▁to ▁the ▁negative ▁of ▁the ▁corresponding ▁K S ▁H OM O ▁energy ▁. ▁More ▁generally , ▁this ▁relation ▁is ▁true ▁even ▁when ▁the ▁K S ▁systems ▁describes ▁a ▁zero - temper ature ▁ensemble ▁with ▁non - integer ▁number ▁of ▁electrons ▁ ▁for ▁integer ▁▁ ▁and ▁. ▁When ▁considering ▁ ▁electrons ▁the ▁inf init es imal ▁excess ▁charge ▁enters ▁the ▁K S ▁L UM O ▁of ▁the ▁N ▁electron ▁system ▁but ▁then ▁the ▁exact ▁K S ▁potential ▁j umps ▁by ▁a ▁constant ▁known ▁as ▁the ▁" der iv ative ▁dis continu ity ". ▁It ▁can ▁be ▁argued ▁that ▁the ▁vertical ▁electron ▁aff inity ▁is ▁equal ▁exactly ▁to ▁the ▁negative ▁of ▁the ▁sum ▁of ▁the ▁L UM O ▁energy ▁and ▁the ▁derivative ▁dis continu ity . ▁ ▁Un like ▁the ▁approximate ▁status ▁of ▁Ko op mans ' ▁theorem ▁in ▁Hart ree ▁F ock ▁theory ▁( because ▁of ▁the ▁neglect ▁of ▁orb ital ▁relax ation ), ▁in ▁the ▁exact ▁K S ▁mapping ▁the ▁theorem ▁is ▁exact , ▁including ▁the ▁effect ▁of ▁orb ital ▁relax ation . ▁A ▁sketch y