Page 199 - Academic Press Encyclopedia of Physical Science and Technology 3rd Organic Chemistry

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446 Organic Chemical Systems, Theory

calculation is necessary. Here, Dewar’s perturbational MO and it underlies all of the current thinking about molec-
(PMO) method is particularly useful. ular structure. It is justified by the much larger mass of
The interaction of fully or partly localized orbitals is nuclei compared with that of electrons, which causes the
frequentlyreferredtoasconjugation.Interactionsbetween nuclei to move much more slowly than electrons. Thus,
several AOs that belong to a π system can be referred to electrons adjust their motions essentially instantaneously
as π conjugation, or conjugation for short. Interactions of to any change in the location of the nuclei, as if they had
an orbital of the π type with one or more suitably aligned no inertia.
bond orbitals of the σ type are referred to as hyperconjuga- From the viewpoint of quantum theory a molecule is
tion. Interactions between adjacent bond orbitals of the σ a quantum mechanical system composed of atomic nu-
type or one such orbital and a lone-pair orbital are referred clei and electrons. It has an infinite number of stationary
to as σ conjugation. As noted above, they occur through states, that is, states whose measurable properties do not
the interaction between two hybrid orbitals located on the change with time. Each of these states is characterized by
same atom. an energy and a wave function. A wave function is a pre-
scription for assigning a numerical value (“amplitude”)to
every possible choice of coordinates for all particles in the
3. Hypervalent Bonding
system. A square of the number assigned to any choice of
The procedures described so far will not work for hyperva- these coordinates represents the probability density that a
lent atoms that do not have enough valence AOs to provide measurement will find the system at that particular collec-
a sufficient number of localized bond orbitals. Without tion of coordinates.
going into detail, we note that in this case three-center or- In order to find the stationary states of a quantum me-
bitals can be constructed instead, and a satisfactory simple chanical system, their energies, and wave functions, one
ˆ
description results. must solve the Schr¨odinger equation Hψ = Eψ, where
ˆ
The expansion of the electron count in the valence shell H is the Hamiltonian operator of the system, ψ the wave
of an atom of a main-group element beyond eight is a function, and E the energy of the stationary state.
reflection of the inadequacy of the way in which the clas- In order to obtain a quantum mechanical description
sical rules of valence assign electrons into a valence shell. of a molecule within the Born–Oppenheimer approxima-
In reality, the true total electron density in the region of tion, at least in principle, one proceeds as follows. Fixed
space corresponding to an atomic valence shell does not molecular geometry is assumed. Mathematically, this cor-
reach these high formally assigned numbers. Fundamental responds to choosing a point in the nuclear configuration
limitations on the latter are given by the Pauli principle, space. As soon as this is done, the Hamiltonian operator
which demands that any electron density in excess of an for electronic motion is fully defined so that the corre-
octet has to occupy orbitals of the next higher principal sponding Schr¨odinger equation can be solved for ψ and
quantum number (Rydberg AOs). It is a gross oversimpli- E.Aninfinite number of solutions exist, differing in their
fication, however, to pretend that the two electrons of a energies, E, and wave functions, ψ.
bond contribute full occupancy of two to the valence shell One of the characteristics of an electronic wave func-
of each of the atoms connected by the bond, particularly tion is the number of unpaired electrons it contains. Those
when the two atoms differ greatly in electronegativity. The wave functions in which this number is zero describe sin-
degree to which the valence shell of the atom of the more glet states. Those that contain two unpaired electrons de-
electropositive element is actually filled is overestimated scribe triplet states, and so on. Wave functions of radicals
by the simple rules, and this accounts for the ease with can have one unpaired electron (a doublet state), three (a
which it enters into hypervalency. quartet state), and so on.
Of the infinite number of stationary wave functions that
are solutions to the Schr¨odinger equation for a chosen nu-
III. QUANTITATIVE ASPECTS OF clear geometry, one is of lowest energy. Almost always
MOLECULAR STRUCTURE this is a singlet wave function if the organic molecule has
an even number of electrons and a doublet wave function
A. Potential Energy Surfaces if it has an odd number of electrons. In the following, we
assume an even number of electrons. We label the lowest
1. Construction of Potential Energy Surfaces
energy singlet wave function ψ(S 0 ) and its energy E(S 0 ).
Before attempting a quantitative quantum mechanical The next higher energy singlet wave function is then iden-
treatment of molecules it is customary to separate the tified and labeled ψ(S 1 ), and its energy is identified and
motions of nuclei from those of electrons. This separa- labeled E(S 1 ). This is the wave function of the first excited
tion is known as the Born–Oppenheimer approximation, singlet state. Similarly, the wave functions of the second