2.3 Electronic Structures
29
HOMO
-5.802 eV
LUMO
-0.980 eV
LUMO+1
-0.164 eV
HOMO-1
-6.549 eV
Fig. 2.25 Selected MO patterns of naphthalene (D 2h symmetry) obtained by DFT/B3LYP/6-31G**
with each MO energy
α -HOMO
-5.262 eV
α -LUMO
1.325 eV
β-HOMO
-7.757 eV
β-LUMO
-1.872 eV
Fig. 2.26 Selected α-and and β-spin MO patterns of allyl radical (C 2v symmetry) obtained by
unrestricted DFT/B3LYP/6-31G** with each MO energy. Note in this molecule that the patterns of
α-HOMO and β-LUMO are the same in this molecule
The MO energies can be utilized to simulate the spectra obtained by the ultraviolet
photoelectron spectroscopy (UPS) and/or X-ray photoelectron spectroscopy (XPS)
measurements of the molecule indicating the energy levels of the valence and inner
electrons in molecules. Although the MO energy is given as only a numerical value
from theoretical calculations, the experimental data have normally a certain energy
width whose size depends on the resolution of each spectrometer. Hence the shape
of an expansion of the energy peaks from the occupied MO energy levels by the
Gaussian curves, called density of valence states (DOVS), is usually compared with
that of the UPS/XPS spectra. The DOVS N(E) can be calculated by
N (E) =
i
exp
−(E − ε i )
2
a
(2.11)