Page 48 - Advanced Organic Chemistry Part A - Structure and Mechanisms, 5th ed (2007) - Carey _ Sundberg

minimization of total energy is reached. It is not possible to carry out these calculations 27
exactly, so various approximations are made and/or parameters introduced. Particular
sets of approximations and parameters characterize the various MO methods. We SECTION 1.2
discuss some of these methods shortly. Molecular Orbital
Theory and Methods
The output of an MO calculation includes atomic positions and the fractional
contribution of each basis set orbital to each MO, that is, the values of c . The energy

of each MO is calculated, and the total binding energy of the molecule is the sum of
the binding energies of the filled MOs:

E = f +h ii (1.13)
ii
The electronic charge at any particular atom can be calculated by the equation

q = n c 2 (1.14)
r j jr
where c is the contribution of the atom to each occupied orbital. Thus, MO calcula-
jr
tions give us information about molecular structure (from the nuclear positions), energy
(from the total binding energy), and electron density (from the atomic populations).
The extent of approximation and parameterization varies with the different MO
methods. As computer power has expanded, it has become possible to do MO calcu-
lations on larger molecules and with larger basis sets and fewer approximations and
parameters. The accuracy with which calculations can predict structure, energy, and
electron density has improved as better means of dealing with the various approxima-
tions have been developed. In the succeeding sections, we discuss three kinds of MO
calculations: (1) the Hückel MO method, (2) semiempirical methods, and (3) ab initio
methods, and give examples of the application of each of these approaches.

1.2.1. The Hückel MO Method

The Hückel MO (HMO) method was very important in introducing the concepts
of MO theory into organic chemistry. The range of molecules that the method can
treat is quite limited and the approximations are severe, but it does provide insight
into a number of issues concerning structure and reactivity. Furthermore, the mathe-
matical formulation is simple enough that it can be used to illustrate the nature of
the calculations. The HMO method is restricted to planar conjugated systems such as
polyenes and aromatic compounds. The primary simplification is that only the 2p
z
orbitals are included in the construction of the HMOs. The justification is that many of
the properties of conjugated molecules are governed by the orbitals that arise from
the p atomic orbitals. A further approximation of the HMO calculations is that only
z
adjacent p orbitals interact. This allows construction of mathematical formulations for
z
the MOs for such systems as linear and branched-chain polyenes, cyclic polyenes,
and fused-ring polyenes. For conjugated linear polyenes such as 1,3,5-hexatriene, the
energy levels are given by the equation

E = +m (1.15)
j
where m = 2cos j /n+1 for j = 1 2 3 n, with n being the number of carbon
j
atoms in the conjugated polyene.
The quantity is called the Coulomb integral; it represents the binding of an
2
electron in a 2p orbital and is considered to be constant for all sp carbon atoms. The
z