Page 196 - Academic Press Encyclopedia of Physical Science and Technology 3rd Organic Chemistry

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Organic Chemical Systems, Theory 443

tetrahedron (Fig. 1B). Inequivalent hybrids can be used The usual approach is based on the recognition of the
for other desired valence angles. fact that the formation of bonds in molecules represents
only a small perturbation of electronic structure of the
constituent atoms, bonding energies being of the order of
B. Molecular Orbitals
1% of total atomic energies. This is because the electric
The independent-particle model can be extended to field in the vicinity of any one atom, where an electron
molecules by assuming that each electron again resides spends most of its time, is dominated by the nucleus of
in an orbital of well-defined energy. Now, however, ex- this atom, since the force of attraction of an electron by
cept for the inner-shell AOs, the orbital is spread over the a positive charge grows with the second inverse power of
space spanned by the molecular framework and is referred the distance from that positive charge.
to as a molecular orbital. To a very good approximation, This situation is acknowledged by assuming that each
the inner-shell electrons behave in exactly the same way MO can be built by mixing AOs or hybrid orbitals.
in the molecule as they would in an isolated atom, and
we will not be concerned with them further. Instead, we
C. Orbital Interactions
concentrate on the valence MOs.
The delocalized form of MOs that we have just de- A brief consideration of the ways in which orbitals interact
scribed is known as their canonical form. It can be shown will be useful not only for a description of the procedure
that the wave function describing the ground electronic in which AOs are mixed to produce MOs, but also for
configuration of a molecule, containing a certain num- subsequent reference to the mutual mixing of MOs, for
ber of doubly occupied orbitals, does not change at all instance,thoseoftwomoleculesreactingwithoneanother.
when these occupied MOs are mixed with one another In the mixing of two orbitals, the important factors are
in an arbitrary way. This degree of freedom in the wave their energies and the interaction matrix element (this is
function permits the construction of MOs that have been frequently referred to as the resonance integral β between
mixed in such a way that each one is localized in the the two orbitals). As a result of the mixing of n orbitals, n
smallest amount of space possible, using one of several new orbitals result.
possible criteria. The price one pays is that these localized It is by far easiest to consider the case of only two mutu-
MOs no longer have well-defined individual energies, al- ally interacting orbitals φ 1 and φ 2 (Fig. 2). Let the energy
though the total energy of the system is just as well defined of φ 1 be below that of φ 2 . After the interaction, the energy
as before.
It turns out that each of the localized MOs tends to be
fairly well contained within the region of one bond or one
lone pair; only in molecules with several important classi-
cal structures is this localization poor. On close inspection,
the localization is actually never perfectly complete, and
each localized orbital possesses weak “tails” extending to
other parts of the molecule. Except for the exact nature of
these tails, such a bond orbital often looks very much the
same in all molecules that contain that particular bond,
say, C C. It is along these lines that one can begin to
understand bond additivity properties and their failure in
the case of molecules in which several classical resonance
structures play an important role.
The standard qualitative model of molecular electronic
structure requires the construction of a number of valence
MOs sufficient to hold all the valence electrons. This is
normally done separately for the electrons responsible for
those bonds that are present in all important classical struc- FIGURE 2 Orbital energies. (A) Interaction of orbitals φ 1 and φ 2 to
tures of the molecule (“localized” bonds, hence “local- produce linear combinations ψ 1 and ψ 2 . (B) Interaction of atomic
ized” electrons, although strictly speaking, they are not orbitals (outside levels) to produce a bonding and an antibonding
really localized) and separately for the electrons responsi- combination (inside levels, orbital shape indicated). The effect on
the net energy of the system is indicated for occupation with one
ble for partial bonds that are present in some and absent in
to four electrons. Arrows represent electrons with spin up and
other important classical structures (“delocalized” bonds, spin down. At the bottom, the classical structural representation
hence “delocalized” electrons). is shown for comparison.