Question

Koopman's theorem states that the ionization energy of a Hartree-Fock orbital is equal to its eigenvalue when acted on by the Fock operator. In this problem we will show Koopman's theorem by comparing the molecular energy before and after an electron is removed from the system, assuming the orbitals themselves remain unchanged. We will make use of the fact that the Hartree-Fock energy Exp(N) of a state with N electrons in spinorbitals {i} is given by Eur(N) = $(dilfulds) + ŞI 601 5160) - 01), where the sums are performed over all N occupied spinorbitals and hy is the score Hamiltonian" that does not include electron-electron interactions. (a) Using the Fock equations F 6.) = € :), show that « = (blubov) + Žito) 160) - Coub 16), where is the energy of the ith spinorbital (6). Keep in mind that F61%) = + ŠIS 630) 1053)61 (5)diz- / 6; (31 0.)6565)dr.). (b) Calculate the ionization energy EHF(N-1) - Enr(N), where E F(N) is the energy of a Hartree-Fock state with N electrons occupying spinorbitals {i}, and EHF(N-1) is the energy of a Hartree-Fock state occupying those same spinorbitals except that the electron in the futh orbital has been removed. Use your answer from part a to show that E F(N-1)- EHF(N) is equal to -Ek where ek is the energy of the spinorbital Ók-