as a σ* or π* orbital, causes a decrease in the strength of a bond between
two atoms. The symmetry of bonding and antibonding orbitals reflects the
symmetry of the atomic orbitals and may possess an inversion symmetry.
An orbital has inversion symmetry when after inversion through the
center of the molecule, which formally would be the center of inversion,
the orbital appears to be unchanged. For example, a σgorbital has such
symmetry, as denoted by a subscript gto represent geradesymmetry (or even
symmetry; geradeis the German word for even). When the same operation
results in matching orbital values but with opposite signs for the wave-
function, the orbital, such as a σuorbital, is denoted as having ungeradesym-
metry (or uneven symmetry; ungeradeis German for uneven) as denoted
by a usubscript. The symmetry of these orbitals has a significant impact
on their involvement in optical transitions as discussed in Chapter 14.
The Hückel model
The concept of molecular orbital theory can be extended to large mole-
cules of atoms. In 1931, Erich Hückel developed an approach that could
be used for conjugated systems, for which the electron is considered free
to travel throughout the length of a conjugated system. The molecular σ
and πorbitals were treated as linear combinations of the atomic s and
p orbitals, respectively. Several approximations are made to simplify the
calculations. All carbon atoms are treated as being identical so that
the contributing atomic orbitals are identical. The interactions between
non-neighbors are neglected and the interactions between neighbors are
reduced to a single parameter, β. The calculated energies can then be shown
to split into two states that are separated in energy by the coupling para-
meter β. The resulting molecular orbitals are filled by the electrons of the
conjugated system, starting from the lowest-energy state. The two critical
states are the highest occupied molecular orbital, or HOMO, and the
lowest unoccupied molecular orbital, or LUMO. These states are sometimes
referred to as frontier orbitals of the molecule and are largely responsible
for many of the chemical properties.
Interactions in proteins
The calculation of the electronic wavefunctions for molecules in proteins
and other biological systems is difficult because proteins have heterogene-
ous surroundings. Also, the interactions with the surrounding amino acid
side chains require an analysis of the protonation state of each group, which
can be highly shifted from ideal solutions (Chapter 5). For this reason, the
factors that give rise to a well-folded protein structure are complex and still
impossible to predict based solely upon the protein structure alone.
276 PART 2 QUANTUM MECHANICS AND SPECTROSCOPY
