the IE for an orbital is the negative of the orbital energy; Section 5.5.5. By the
“ionization energy” we usually mean the lowest one, corresponding to removing an
electron from the HOMO. In Chapters5 and 6both the energy difference and the
Koopmans’ theorem methods were used to calculate some IEs (Tables 5.17 and 6.6).
The problem with applying Koopmans’ theorem to DFT is that in “strict” DFT there
are no molecular orbitals, only electron density, while in Kohn–Sham DFT (practical
DFT) the MOs, the orbitalscKSthat make up the Slater determinant of Eq.7.19,
were, as explained in Section7.2.3.2, introduced only to provide a way to calculate
the energy (note Eqs.7.21, 7.22, and 7.26). The problem is to see if these Kohn–
Sham MOs are, as there was a tendency to view them, mere mathematical artifices or
if they arein themselvesuseful. There was at one time a fair amount of argument over
the physical meaning, if any, of the Kohn–Sham orbitals. Baerends and coworkers
compared DFT with Hartree–Fock theory and concluded that “The Kohn–Sham
orbitals are physically sound and may be expected to be more suitable for use
in qualitative molecular orbital theory than either Hartree–Fock or semiempirical
orbitals” [ 128 ]. Cramer echoes this in pointing out that there are reasons to even
preferthe Kohn–Sham MOs: they all feel the same external potential, while HF MOs
feel varying potentials, the virtual MOs carrying this to an extreme [ 129 ]. Stowasser
and Hoffmann showed that the KS orbitals resemble those of conventional wavefunc-
tion theory (extended H€uckel and Hartree–Fock ab initio, Chapters4 and 5)inshape,
symmetry, and, usually, energy ordering [ 130 ]. They conclude that these orbitals can
indeed be treated much like the more familiar orbitals of wavefunction theory.
Furthermore, they showed that although the KS orbital energy values (the eigenva-
luesefrom diagonalization of the DFT Fock matrix – Section7.2.3.3) are not good
approximations to the ionization energies of molecular orbitals (as revealed by
photoelectron spectroscopy), there is a linear relation between |ei(KS)"ei(HF)| and
ei(HF). Salzner et al., too, showed that in DFT, unlike ab initio theory calculations,
negative HOMO energies are not good approximations to the IE (with anexact
functional Koopmans’ theorem would be exact), but, surprisingly, HOMO-LUMO
gaps from hybrid functionals agreed well with thep!p* UV transitions of unsatu-
rated molecules [ 131 ]. Vargas et al. introduced a “Koopmans-like approximation” to
obtain a relation between the Kohn–Sham orbital energies and vertical IEs and EAs,
and assert that their method improves the calculation of electron density indexes
(below) of hardness, electronegativity and electrophilicity [ 132 ]. The utility of the
Kohn–Sham orbital energies to predict IE, EA and the hardness index was studied by
Zhan et al. [ 133 ], and Zhang et al. explored the ability of various functionals to use
these orbitals to predict IE, EA and the lowest-energy UV transition [ 134 ].
Concerning electron affinities, in Hartree–Fock calculations the negative LUMO
energy of a species M corresponds to the electron affinity not of M but rather of the
anion M"[ 135 ]. However, Salzner et al. reported that the negative LUMOs from
LSDA functionals gave rough estimates of EA (ca. 0.3–1.4 eV too low; gradient-
corrected functionals were much worse, ca. 6 eV too low) [ 131 ]. Brown et al. found
that for eight medium-sized organic molecules the energy difference method using
gradient-corrected functionals predicted electron affinities fairly well (average
mean error less than 0.2 eV) [ 103 ].
496 7 Density Functional Calculations