Page 197 - Academic Press Encyclopedia of Physical Science and Technology 3rd Organic Chemistry

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444 Organic Chemical Systems, Theory

of the more stable of the two new resulting orbitals ψ 1 will
lie below that of φ 1 and that of the less stable one ψ 2 will
lie above that of φ 2 , as if the two original energies repelled
one another. The amount by which each orbital energy is
shifted will depend on the strength of the interaction ele-
ment β and on the difference
ε in the original energies. If

ε |β|, the shifts are approximately inversely propor-
tional to
ε (this approximation is known as first-order
perturbation theory). When the two orbitals are originally
degenerate (
ε = 0), the shifts are the largest.
An important characteristic of the interaction be-
tween two orbitals φ 1 and φ 2 is their overlap integral

S 12 = φ 1 φ 2 dτ, where the integration is over all space.
For normalized orbitals (S 11 = S 22 = 1) the value of S 12
can vary between −1 and +1. Usually if S 12 is positive,
the interaction element β is negative, and if S 12 is negative,
β is positive. If S 12 = 0, orbitals φ 1 and φ 2 are said to be
orthogonal; they can still have a nonvanishing resonance
FIGURE 3 Overlap between orbitals.
integral. The energy shifts caused by the interaction of
two orthogonal orbitals are opposite in direction but equal
in magnitude. When two nonorthogonal orbitals interact, π type (Fig. 3). Intermediate situations are also possible.
the less stable new orbital ψ 2 is destabilized more than the The interaction matrix element β between two AOs or
more stable new orbital ψ 1 is stabilized. hybrid orbitals located at different atoms is approximately
The more stable of the two new orbitals is referred to as proportional to the negative of their overlap integral S 12 .
bonding and the less stable as antibonding with respect to The two new orbitals ψ 1 and ψ 2 that result from the
the interaction considered. An orbital with the same en- interaction are linear combinations of the two old orbitals
ergy as before the interaction is called nonbonding; such (Fig.2).Inthebondingorbital ψ 1 ,theoverlappingportions
orbitals often result when more than two AOs are mixed. of the entering old orbitals ψ 1 and ψ 2 have the same sign.
If two electrons are available to fill each bonding orbital, In the antibonding orbital ψ 2 , the overlapping lobes of the
maximum stabilization will result relative to the situation original partners are of opposite signs, and its sign changes
before orbital interaction. If too many electrons are avail- as one goes across the interaction region from the center
able and some must be placed into antibonding orbitals, of φ 1 to the center of φ 2 . The surface where the sign of ψ 2
some or all of the stabilization is lost and net destabi- changes is called a nodal surface (plane).
lization may result. For this reason, the interaction of a In summary, the bonding combination ψ 1 lacks a nodal
doubly occupied with an unoccupied orbital leads to a net plane and has an increased electron density between the
stabilization and the interaction of two doubly occupied atoms (in-phase mixing if S 12 > 0), while the antibonding
orbitals to a net destabilization. combination ψ 2 has a nodal plane and a reduced elec-
These general concepts can now be specialized to the tron density between the atoms (out-of-phase mixing if
case of mixing of AOs or hybrid orbitals to produce bond S 12 < 0). The bonding orbital ψ 1 will accommodate two
orbitals. Two valence-shell AOs or properly constructed electrons and thus produce a net stabilization (a chemical
hybrid orbitals located at the same atom are always orthog- bond). This is the origin of the classical “shared electron
onal. For two orbitals located at the same center, the inter- pair.” For maximum bonding stabilization, it is desirable
action element β is zero if one or both are pure AOs. How- to maximize the absolute value of the overlap integral S 12 .
ever, two hybrid orbitals at the same center have a nonvan- A third electron would have to enter the antibonding
ishing mutual interaction element. Its magnitude is related orbital φ 2 , causing a loss of much and possibly all of the
to the promotion energy (i.e., the energy difference be- overall stabilization. A fourth electron would also have to
tween the AOs from which the hybrids were constructed). enter φ 2 , and now interaction would definitely lead to a
Two AOs or hybrid orbitals located at different atoms net destabilization. For this reason, filled AOs (lone pairs)
can be, but do not need to be, orthogonal. A nonvanishing repel and avoid one another (“lone-pair repulsion”). For
overlap integral between two p AOs can be produced in minimum destabilization, it is desirable to minimize |S 12 |.
two geometrically distinct ways. When the axes of the two For instance, if the lone pairs are of the p type and are on
◦
orbitals are aimed at one another, the overlap is said to be adjacent atoms, a dihedral angle of close to 90 will be
of the σ type; when they are parallel, it is said to be of the preferred (e.g., in H 2 O 2 ).