▁proof ▁of ▁this ▁exact ▁relation ▁goes ▁in ▁three ▁stages . ▁First , ▁for ▁any ▁finite ▁system ▁ ▁determ ines ▁the ▁ ▁asympt otic ▁form ▁of ▁the ▁density , ▁which ▁dec ays ▁as ▁. ▁Next , ▁as ▁a ▁cor ollary ▁( since ▁the ▁phys ically ▁interact ing ▁system ▁has ▁the ▁same ▁density ▁as ▁the ▁K S ▁system ), ▁both ▁must ▁have ▁the ▁same ▁ion ization ▁energy . ▁Finally , ▁since ▁the ▁K S ▁potential ▁is ▁zero ▁at ▁infinity , ▁the ▁ion ization ▁energy ▁of ▁the ▁K S ▁system ▁is , ▁by ▁definition , ▁the ▁negative ▁of ▁its ▁H OM O ▁energy ▁and ▁thus ▁finally : ▁. ▁ ▁While ▁these ▁are ▁exact ▁statements ▁in ▁the ▁formal ism ▁of ▁D FT , ▁the ▁use ▁of ▁approximate ▁exchange - cor relation ▁potential s ▁makes ▁the ▁calculated ▁energ ies ▁approximate ▁and ▁often ▁the ▁orb ital ▁energ ies ▁are ▁very ▁different ▁from ▁the ▁corresponding ▁ion ization ▁energ ies ▁( even ▁by ▁several ▁e V ! ). ▁ ▁A ▁tun ing ▁procedure ▁is ▁able ▁to ▁" im pose " ▁Ko op mans ' ▁theorem ▁on ▁D FT ▁approxim ations ▁thereby ▁impro ving ▁many ▁of ▁its ▁related ▁predictions ▁in ▁actual ▁applications . ▁ ▁In ▁approximate ▁D FT s ▁one ▁can ▁estimate ▁to ▁high ▁degree ▁of ▁accuracy ▁the ▁dev iance ▁from ▁Ko op mans ' ▁theorem ▁using ▁the ▁concept ▁of ▁energy ▁curv ature . ▁ ▁It ▁provides ▁exc itation ▁energ ies ▁to ▁z er oth - order ▁and ▁. ▁ ▁References ▁ ▁External ▁links ▁▁▁▁▁▁ ▁Category : Quant um ▁chem istry ▁Category