Page 62 - Mechanism and Theory in Organic Chemistry

Molecular Orbital Theory 51

THE ORBITAL APPROXIMATION
The first approximation made in simplifying the task of solving for Y is the
orbital approximation. Each of the M electrons of the molecule is assumed to be
described by a molecular orbital function +; the total wave function for the elec-
tronic state is the product of +'s for the individual electrons (Equation A2.2).C

The problem of finding the orbital functions is commonly solved in one
of two ways. Both use the variation principle. The variation principle states that
whenever an approximate function is substituted for + in the expression

wherc Yt' now refers to any Hamiltonian operator, the value of the energy E
obtained will be greater than the true energy of the correct lowest energy wave
function. Hence, the best result possible with an approximate wave function of a
particular type will be obtained when it is chosen so as to minimize E. The more
rigorous approach to finding the orbital functions is the Hartree-Fock-Roothaan
method. It applies the variation principle to Equation A2.1, with Y expressed
as in A2.2, and yields a new result of the form A2.3, where X becomes the
Hartree-Fock or self-consistent field operator, Xs,,, and the molecular orbitals
y5i represent the best solution possible within the orbital appr~ximation.~
A
simpler procedure is to assume an approximate Hamiltonian, X,,,,,,, which
can be put directly in place of Xin A2.3. Hence, both approaches lead to the
form A2.3, which must now be solved for t,h and E.

THE LCAO APPROXIMATION
The next approximation is the expression of each molecular orbital + as a linear
combination of atomic orbitals (LCAO) (Equation A2.4), where yj are atomic orbital
functions and cj are coefficients that give the contribution of each atomic orbital
to the molecular orbital. The y, are the basis functions discussed in Section 1.2.
Valence atomic orbitals are ordinarily chosen for the basis.

We always require that orbital functions be normalized. Becfuse the proba-
bility of finding the electron in orbital + near a particular point is given'by the
value of +2 at that point and because the total probability of finding the electron

When electron spin is properly taken into account, the total wave function for the state must be an
antisymmetrized product of the orbital functions. Antisymmetrization automatically incorporates the
Pauli exclusion principle. In Hiickel theory, where orbitals are not properly antisymmetrized, it is
necessary to add the extra restriction that electrons be assigned no more than two to an orbital and
that spins of electrons occupying the same orbital be paired.
* Because Z,,, depends on its own solutions 4, an iterative procedure of successive approximations
is required. For derivations, see the sources cited in notes 2 (f), and 2 (g), p. 10.