7.3.5.3 Ionization Energies and Electron Affinities: The Kohn–Sham
Orbitals
Ionization energies (ionization potentials) and electron affinities were discussed in
Section 5.5.5. We saw that IEs and EAs can be calculated in a straightforward way
as the energy difference between a molecule and the species derived from it by loss
or gain, respectively, of an electron. Using the energy of the optimized geometry of
the radical cation or radical anion (in the case where the species whose IE or EA we
seek is a neutral closed-shell molecule) gives the adiabatic IE or EA, while using
the energy of the ionized species at the geometry of the neutral gives the vertical IE
or EA. Muchall et al. have reported adiabatic and vertical ionization energies and
electron affinities of eight carbenes, calculated in this way by semiempirical,
ab initio, and DFT methods [ 124 ]. They recommend B3LYP/6-31þG//B3LYP/
3-21G()as the method of choice for predicting first ionization energies; the use of
the small 3-21G()basis with B3LYP for the geometry optimization is unusual – see
Section 5.4.2 – usually the smallest basis used with a correlated method is 6-31G.
This combination is relatively undemanding and gave accurate (largest absolute
error 0.14 eV) adiabatic and vertical ionization energies for the carbenes studied.
Table7.11shows the results of applying this method to some other (non-carbene)
molecules. The B3LYP/6-31+G ionization energies are essentially the same with
B3LYP/3-21G()geometries and AM1 geometries; they are good estimates of the
experimental IE [ 125 , 126 ], are somewhat better than the ab initio MP2 ionization
energies, and are considerably better than the MP2 Koopmans’ theorem (below)
IEs. Of course, for unusual molecules (like the carbenes studied by Muchall et al.
[ 124 ]) AM1 may not give good geometries, and for such species it would be safer to
use B3LYP/3-21G()or B3LYP/6-31G geometries for the single-point B3LYP/6-
31+G* calculations. Golas et al. obtained fairly good IEs (-0.2 eV for IEs of ca.
8–9 eV) and useful EAs (-0.4 eV for EAs of ca. 1–2 eV) with B3LYP/6-311+G*
energies on B3LYP/6-31G geometries [ 127 ].
In wavefunction theory an alternative way to find IEs for removal of an electron
from a molecular orbital (usually the highest), is to invoke Koopmans’ theorem:
Table 7.11 Some ionization energies (eV). TheDE ionization energy values (cation energy minus
neutral energy) correspond to adiabatic and (in parentheses) vertical IEs; the Koopmans theorem
values are vertical IEs. Experimental IEs are adiabatic (CH 3 OH and CH 3 COCH 3 [ 125 ], CH 3 SH
[ 126 ]). The use of B3LYP/3-21G()geometries is based on [ 124 ]. That the vertical IE is smaller
than the adiabatic for the B3LYP/6-31þG//AM1 calculation on CH 3 SH is presumably due to a
somewhat inaccurate geometry, probably for the cation (experimental vertical IEs are always
bigger than adiabatic since it takes energy to distort the relaxed-geometry cation to the geometry of
the neutral)
DE¼IE Koopmans’
(MP2(fc)/6-31G*)
Exp
B3LYP/6-31þG*//
B3LYP/3-21G(*)B3LYP/6-
31 þG*//AM1MP2(fc)/
6-31G*
CH 3 OH 10.77 (10.92) 10.76 (10.85) 10.6 12.1 10.9
CH 3 SH 9.40 (9.43) 9.53 (9.36) 9 9.2 9.4
CH 3 COCH 3 9.60 (9.70) 9.67 (9.68) 9.6 11.2 9.7
7.3 Applications of Density Functional Theory 495