3. Discussion of the Hartree-Fock Equations

where the notation indicates than an electron in an occupied orbital, ψ ( )ir is removed fromthe system, becoming a “free” (unbound) electron. Neglecting the subsequent relaxation ofthe remaining occupied orbitals in the cation, we can approximate the ionization of theneutral system to form the excited state of the cation byfreeIi ≈ Ei −EHFN N N⎛N N N⎞1 1≈ ∑hk +2∑∑( Jkl −Kkl ) − ⎜∑hk +2∑∑( Jkl −Kkl)⎜⎝⎠⎟k= 1 k= 1 l= 1 k= 1 k= 1 l=1k≠i k≠i l≠i⎛⎞ NN1 1= hi 2 ( Jki Kki) 2 ( Jil Kil) ( Jii Kii)− − ∑ − − ∑ − − − ⎜k= 1 l=1⎜⎝k≠i l≠i⎠⎟⎛⎞ N=− hi ( Jki Kki)+ ∑ − −0⎜k=1⎜⎝ k≠i⎠⎟=−ε i(3.15)The fact that the orbital energies in Hartree-Fock theory approximate the negative of theionization potentials is called Koopmans’ theorem. We can approximate the “way” theelectron is removed from the molecule by noting that the electron has been removed fromthe orbital ψ ( )iz .Particularly important for chemical purposes is the highest occupied molecular orbital(HOMO), because I HOMO=− ε HOMOis the ground state ionization potential: the minimumamount of energy it takes to remove an electron from a molecule. Molecules with largehighest-occupied orbital energies (small ionization potentials) are good electrondonors/nucleophiles/Lewis bases/reducing agents. The shape of the highest-occupied orbitalcontrols the regioselectivity of the nucleophile: if ψ ( ) 2 HOMOr ≈ 0 at a point, then weobserve that the it is difficult to remove an electron from the molecule at this point becausemost of the electron density at this point is associated with electrons in orbitals that aremore tightly bound, ε i< ε HOMO. For example, the electrophilic attack of borane in thehydroboration reaction of 1,1-diphenyl-2-ethyl-1-butene occurs not on the negatively chargedaromatic rings, but at the double bond.6